Deformation quantization modules

Deformation quan tization mo dules Masaki Kashiwara and Pier re Schapira 2 Con t en ts 1 Mo dules ov er formal deformations 11 1.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2 F ormal deformations of a sheaf of rings . . . . . . . . . . . . . 16 1.3 A v aria n t of the preceding results . . . . . . . . . . . . . . . . 28 1.4 ~ -graduation and ~ -lo calization . . . . . . . . . . . . . . . . . 32 1.5 Cohomologically complete mo dules . . . . . . . . . . . . . . . 36 1.6 Cohomologically complete A -mo dules . . . . . . . . . . . . . 42 2 DQ -algebroids 49 2.1 Algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.2 DQ-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.3 DQ-algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.4 DQ-mo dules supp orted by the diagonal . . . . . . . . . . . . . 75 2.5 Dualizing complex for DQ-a lgebroids . . . . . . . . . . . . . . 79 2.6 Almost free resolutions . . . . . . . . . . . . . . . . . . . . . . 8 4 2.7 DQ-algebroids in the algebraic case . . . . . . . . . . . . . . . 86 3 Kernels 91 3.1 Con v olution of k ernels: deﬁnition . . . . . . . . . . . . . . . . 91 3.2 Con v olution of k ernels: ﬁniteness . . . . . . . . . . . . . . . . 93 3.3 Con v olution of k ernels: duality . . . . . . . . . . . . . . . . . 95 3.4 Action of k ernels on Grothendiec k groups . . . . . . . . . . . . 99 4 Ho ch schild classes 103 4.1 Ho c hsc hild homology and Ho c hsc hild class es . . . . . . . . . . 103 4.2 Comp osition of Ho c hsc hild classes . . . . . . . . . . . . . . . . 106 4.3 Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.4 Graded and lo calized Ho ch sc hild classes . . . . . . . . . . . . 114 5 The commu tat ive case 117 5.1 Ho c hsc hild class of O -mo dules . . . . . . . . . . . . . . . . . . 117 3 4 CONTENTS 5.2 co-Ho c hsc hild class . . . . . . . . . . . . . . . . . . . . . . . . 122 5.3 Chern and Euler classes of O - mo dules . . . . . . . . . . . . . 124 5.4 Proo f of Theorem 5.3.2 . . . . . . . . . . . . . . . . . . . . . . 127 6 Symplectic case and D -mo dules 133 6.1 Deformation quan tization on c ot a ngen t bundles . . . . . . . . 133 6.2 Ho chsc hild homology o f A . . . . . . . . . . . . . . . . . . . . 1 36 6.3 Euler classes of A lo c -mo dules . . . . . . . . . . . . . . . . . . 142 6.4 Ho chsc hild classes of D -mo dules . . . . . . . . . . . . . . . . . 144 6.5 Euler classes of D - mo dules . . . . . . . . . . . . . . . . . . . . 146 7 Holonomic D Q -modules 149 7.1 A -mo dules a lo ng a Lagrangian submanifold . . . . . . . . . . 149 7.2 Holonomic DQ-mo dules . . . . . . . . . . . . . . . . . . . . . 1 5 7 7.3 Lagra ngian cycles . . . . . . . . . . . . . . . . . . . . . . . . . 159 7.4 Simple ho lonomic mo dules . . . . . . . . . . . . . . . . . . . . 161 7.5 Inv ariance by deformation . . . . . . . . . . . . . . . . . . . . 163 In tro duction In a few w ords these Not es could b e considered b oth a s an intro duction to non comm utativ e complex analytic geometry and to the study of micro diﬀeren tial systems . Indeed, on a complex manifold X , w e replace the structure sheaf O X with a formal deformation of it, that is, a D Q-algebra, or b etter, a D Q- algebroid, and study mo dules ov er this ring, extending to this framew ork classical r esults of Cartan-Serre and Grauert, and a lso classical results on Ho c hsc hild classes and the index theorem. Here, DQ stands for “ deformation quan tization”. But the theory of mo dules o ver DQ-algebroids is also a natural generalization o f that of D -mo dules. I ndeed, when the P oisson structure underlying t he deformation is symplec tic, the study of DQ-mo dules naturally generalizes that of micro diﬀeren tial mo dules, and some times mak es it easier (see Theorem 7.2.3). The notion of a star pro duct is now a classical sub ject studied b y man y authors and natura lly app earing in v arious con texts. Tw o cornerstones of its his to r y are the pap er [1] (see also [2, 3]) who deﬁnes ⋆ -pro ducts and the fundamen tal result of [46] which, roughly sp eaking, asserts that any real P oisson manifold ma y b e “quan tized”, that is, endo w ed with a star algebra to whic h the P oisson structure is asso ciated. It is now a w ell-kno wn fact ( see [36, 45 ]) that, in order to quantize complex P oisson manifolds, shea ves of algebras are no t well-suite d and ha v e to b e replaced b y algebroid stac ks. W e refer to [13, 65] for further dev elopmen ts. In this pap er, w e consider complex manifolds endo w ed with DQ-a lg ebroids, that is, algebroid stack s lo cally asso ciated to sheav es of star-algebras, and study mo dules o ve r suc h algebroids. The main results of this pap er are: • a ﬁniteness theorem, whic h asserts that the con v olution of tw o coheren t k ernels, satisfying a suitable pro p erness assumption, is coheren t (a kind of Grauert’s theorem), • the construction of the dualizing complex and a dualit y theorem, whic h asserts that dualit y comm utes with con v olution, • the construction of t he Ho c hsc hild class o f coheren t DQ-mo dules and 5 6 CONTENTS the theorem whic h asserts that Ho c hsc hild class comm utes with con vo- lution, • the link b etw een Ho c hsc hild classes and Chern classes and also with Euler classes, in the commutativ e case, • the constructibilit y of the complex of solutions of an holonomic mo dule in to another one in the the symplectic case. Let us describ e this pap er with some details. In Chapter 1 , w e systematically study rings ( i . e ., shea v es of rings) whic h are forma l deformations of rings, and mo dules ov er suc h deformed rings. More precisely , consider a top ological space X , a comm utative unital ring K and a sheaf A of K [[ ~ ] ]-a lgebras on X whic h is ~ - complete and without ~ -torsion. W e also assume tha t there exists a base of op en subsets of X , acyclic for coheren t modules ov er A 0 := A / ~ A . W e ﬁrst show ho w to deduce v arious prop erties of the ring A from the corresp onding prop erties on A 0 . F or example, A is a No etherian ring as so on as A 0 is a No etherian ring, and an A -mo dule M is coheren t as so on as it is lo cally ﬁnitely generated and ~ n M / ~ n +1 M is A 0 -coheren t for a ll n ≥ 0. Then, w e introduce the prop erty of b eing cohomologically complete for an ob ject of the deriv ed category D( A ). W e prov e that this notio n is lo cal, stable b y direct images and a n ob ject M with b ounded coheren t cohomology is cohomologically complete. Conv erse ly , if M is cohomologically complete, it has coheren t cohomology ob j ects as so on a s its gra ded mo dule A 0 L ⊗ A M has coheren t cohomology ov er A 0 (see Theorem 1.6.4). W e also giv e a similar criterion whic h ensures that an A - mo dule is ﬂat. In C hapter 2 w e consider the case where X is a complex manifold, K = C , A 0 = O X and A is lo cally isomorphic to an alg ebra ( O X [[ ~ ] ] , ⋆ ) where ⋆ is a star-pro duct. It is an algebra ov er C ~ := C [[ ~ ] ]. W e call suc h an algebra A a DQ-algebra. W e also consider DQ-algebroids, that is, C ~ -algebroids (in the sens e of stac ks) lo cally equiv alen t to the algebroid asso ciated with a DQ-algebra. Remark that a DQ-algebroid on a manifo ld X deﬁnes a P oisson structure on it. Con v ersely , a fa mous theorem of Kon tsevic h [46 ] asserts that on a real P oisson manifo ld there exists a DQ-algebra to whic h this Poiss on structure is asso ciated. In the complex case, there is a similar result using DQ-algebroids. This is a theorem of [45] after a related result of [36] in the con tact case. If ( X, A X ) is a complex manifold X endo w ed with a DQ-algebroid A X , w e denote b y X a the manifold X endow ed with the DQ-algebroid A op X opp osite to A X . CONTENTS 7 W e deﬁne t he external pro duct A X 1 × X 2 of tw o DQ-algebroids A X 1 and A X 2 on manifolds X 1 and X 2 . There exists a canonical A X × X a -mo dule C X on X × X a supp orted b y the diago nal, whic h corresp onds to the A X -bimo dule A X . On a complex manifold X endo we d with a DQ-alg ebroid, w e construct the C ~ -algebroid D A X , a deforma t io n quan tization of the ring D X of diﬀeren tial op erators. It is a C ~ -subalgebroid of E nd C ~ ( A X ). It turns out t ha t D A X is equiv alent to D X [[ ~ ] ]. This new algebroid allows us to construct the dualizing complex ω A X asso ciated to a D Q-algebroid A X . This complex is the dual ov er D A X of A X , similarly to the case of O X -mo dules. Note that the dualizing complex for DQ-alg ebras has already b een considered in a mo r e particular situation b y [20, 21]. W e a lso adapt to algebroids a results o f [40] whic h allows us to replace a coheren t A X -mo dule by a complex of “ almost free” mo dules, suc h an ob ject b eing a lo cally ﬁnite sum ⊕ i ∈ I ( L i ) U i , the L i ’s b eing free A X -mo dules of ﬁnite rank deﬁned on a neighborho o d of U i . W e giv e a similar result for algebraic manifolds. Chapter 3 . Consider three complex manifolds X i endo w ed with DQ- algebroids A X i ( i = 1 , 2 , 3). Let K i ∈ D b coh ( A X i × X a i +1 ) ( i = 1 , 2) b e tw o coheren t kerne ls and deﬁne t heir conv olution by setting K 1 ◦ K 2 := R p 14 !  ( K 1 ⊠ K 2 ) L ⊗ A X 2 × X a 2 C X 2  . Here p 14 denotes the pro jection of the pro duct X 1 × X a 2 × X 2 × X a 3 to X 1 × X a 3 . W e pro v e in The orem 3.2.1 tha t , under a nat ura l prop erness hypothesis, the conv olution K 1 ◦ K 2 b elongs t o D b coh ( A X 1 × X a 3 ) and in Theorem 3 .3.3 that the con v olution of kerne ls comm utes with duality . F or further applications, it is a lso in teresting to consider the lo calized algebroid A lo c X = C ~ , lo c ⊗ C ~ A X , where C ~ , lo c = C (( ~ ) ). An A lo c X -mo dule M is go o d if for an y relativ ely compact op en subset U of X , there exists a coheren t A U -mo dule whic h generates M | U . Then w e pro ve that there is a natural map of the Grothendiec k gr oups K gd ( A lo c X ) − → K coh (gr ~ A U ) and t ha t this map is compatible to the comp o sition of k ernels. Note t hat these t heorems extend classical results o f Cartan, Serre and Grauert on ﬁniteness a nd duality for coheren t O -mo dules on complex mani- folds to DQ-algebroids. F or pap ers related to DQ-algebras and D Q-algebroids on complex P oisson manifolds, and particularly to t heir classiﬁcation, w e refer to [5, 6 , 8, 13, 50 , 51, 61, 64]. Chapter 4. W e in tro duce the Ho c hsc hild homology H H ( A X ) o f the 8 CONTENTS algebroid A X : HH ( A X ) := C X a L ⊗ A X × X a C X , an ob j ect o f D b ( C ~ X ) , and, using the dualizing complex, we construct a natura l con v olutio n mor- phism ◦ X 2 : R p 13 ! ( p − 1 12 HH ( A X 1 × X a 2 ) L ⊗ p − 1 23 HH ( A X 2 × X a 3 )) − → HH ( A X 1 × X a 3 ) . T o an ob ject M of D b coh ( A X ), w e naturally asso ciate its Ho chs child class hh X ( M ), an elemen t of H 0 Supp( M ) ( X ; H H ( A X )). The main r esult of this c hapter is Theorem 4.3.5 whic h asserts tha t taking the Ho ch sc hild class com- m utes with the conv olution: hh X 1 × X a 3 ( K 1 ◦ K 2 ) = hh X 1 × X a 2 ( K 1 ) ◦ X 2 hh X 2 × X a 3 ( K 2 ) . (0.0.1) In Chapter 5 , w e consider the case whe re the deformation is trivial. In this case, there is no need of the parameter ~ and w e are in the we ll-known ﬁeld of complex analytic geometry . Although the results of t his chapter are considered as w ell-kno wn (see in particular [33]), a t least from the sp ecialists, w e ha v e decid ed to include this c hapter. Indeed, to our o pinion, there is no satisfactory pro of in the literature of the fa ct tha t the Ho c hsc hild class of coheren t O X -mo dules is functorial with resp ect to con v olution. W e recall in particular the formu la, in whic h the T o dd class app ears, whic h mak es the link b et w een Ho chs child class and Chern classes. This f orm ula w as conjecturally stated b y the ﬁrst named author around 1 991 and has only b een pro v ed v ery recen tly b y Ramadoss [53] in the algebraic setting and b y Griv aux [3 0] in the general case. F or other pap ers closely related to this c hapter, see [14, 15, 33, 48, 58]. In C hapter 6 w e study Ho c hsc hild homology and Ho c hsc hild classe s in the case w here the Poiss on structure asso ciated to the deformation is symplectic. W e pro v e then that the dualizing complex ω A X is isomor phic to C X shifted by d X , the complex dimension o f X , and w e construct canonical morphisms ~ d X / 2 C ~ X [ d X ] − → H H ( A X ) − → ~ − d X / 2 C ~ X [ d X ] (0.0.2) whose composition is the canonical inclusion. The morphisms in (0.0.2) in- duce a n isomorphism C ~ , lo c X [ d X ] ≃ H H ( A lo c X ) . (0.0.3) CONTENTS 9 The ﬁrst morphism in (0.0.2) giv es an intrins ic construction of the canonical class in H − d X ( X ; H H ( A X )) studied and used b y sev eral autho rs (see [12, 11, 25]). The isomorphism (0 .0.3) allow s us to asso ciate an Euler class eu X ( M ) ∈ H d X Λ ( X ; C ~ , lo c X ) to an y coheren t A X -mo dule M supp orted b y a closed se t Λ. Then w e show ho w our results apply to D - mo dules. W e recov er in par- ticular the Riemann-Ro ch theorem for D - mo dules of [47] a s w ell as the func- toriality of the Euler class of D -mo dules of [57]. Finally , in Chapter 7 , w e study holonomic A lo c X -mo dules on complex symplectic manifolds. W e pro v e that if L and M are tw o holono mic A lo c X - mo dules, then the complex R H om A loc X ( M , L ) is p ervers e (hence, in partic- ular, C -constructible) ov er the ﬁeld C ~ , lo c . If the in tersection o f the supp orts of the holonomic mo dules L and M is compact, form ula (0.0.1) give s in particular χ (RHom A loc X ( M , L )) = Z X (eu X ( M ) · eu X ( L )) . The Euler class of a holonomic module ma y b e in terpreted as a Lagra ng ia n cycle, whic h make s its calculation quite easy . If the mo dules L and M are simple along smo ot h Lag r a ngian submani- folds, then one can estimate the microsupp ort of this complex . This partic- ular case had b een already treated in [42] in the analytic framew or k, that is, using analytic deformations ( in the sense of [54]), not forma l deforma t io ns, and the pro of giv en here is muc h simpler. W e also pro v e (Theorem 7.5.2) that if L a is family of holonomic mo dules indexed b y a holomorphic parameter a , then, under suitable geometrical h y- p otheses, the complex of global sections R Hom A loc X ( M , L a ), whic h b elongs to D b f ( C ~ , lo c ), do es not dep end o n a . This is a kind of in v ariance b y Hamiltonian symplectomorphism of this complex. W e ha v e dev elop ed the t heory in the framew ork o f complex analytic mani- folds. Ho we ver, all along the manus cript, w e explain how t he r esults extend (and sometimes simplify) in the algebraic setting, tha t is on quasi-compact and separated smo oth v arieties ov er C . The main results of this paper, with the exce ption of Chapter 7, hav e been announced in [43, 44]. Ac knowled gmen t s W e w ould like to thank Andrea D’Agnolo, Pietro P ole- sello, St ´ ephane Guillermou, Jean-Pierre Sc hneiders and Boris Tsygan fo r useful commen ts a nd remarks . 10 CONTENTS Chapter 1 Mo dules o v er formal deformations 1.1 Preliminary Some notations Throughout this pap er, K denotes a comm utative unital r ing . W e shall mainly follow the notations of [41 ]. In particular, if C is a category , w e denote by C op the opp o site category . If C is an additive cat- egory , w e denote by C( C ) the category of complexes of C and b y C ∗ ( C ) ( ∗ = + , − , b) the full sub category consisting of complexes b ounded from b e- lo w (resp. b ounded from ab o v e, resp. b ounded). If C is an ab elian category , w e denote b y D ( C ) the derive d categor y of C and by D ∗ ( C ) ( ∗ = + , − , b) the full tria ngulated sub category consisting of ob jects with b ounded fro m b elo w ( r esp. b ounded fro m ab o ve , resp. b ounded) cohomology . W e denote as usual b y τ ≥ n , τ ≤ n etc. the truncation f unctors in D( C ). If A is a ring (or a sheaf of rings on a top ological space X ), an A -mo dule means a left A -mo dule. W e denote b y A op the opp osite ring of A . Hence an A op -mo dule is nothing but a righ t A -mo dule. W e denote b y Mo d( A ) the category o f A -mo dules. W e set for short D( A ) := D(Mod( A )) and w e write similarly D ∗ ( A ) instead of D ∗ (Mo d( A )). W e denote b y D b coh ( A ) the full triangulated sub category of D b ( A ) consisting of ob jects with coheren t coho- mology . If K is No etherian, one denotes simply b y D b f ( K ) the full sub category of D b ( K ) consisting of ob jects with ﬁnitely generated cohomolog y . W e denote b y D ′ X the dualit y functor f or K X -mo dules: D ′ X ( • ) := R H om K X ( • , K X ) . (1.1.1) 11 12 CHAPTER 1. MODULES OVER FORMAL DEFORMA TIONS and we simply denote by ( • ) ⋆ the dualit y functor o n D b ( K ): ( • ) ⋆ = RHom K ( • , K ) . (1.1.2) If K is No etherian and with ﬁnite global dimension, ( • ) ⋆ sends (D b f ( K )) op to D b f ( K ). W e denote by { pt } t he set with a single elemen t. Finiteness conditions Let X b e a top ological space and let A b e a K X -algebra ( i.e., a sheaf of K -algebras) on X . Let us recall a few classical deﬁnitions. • An A - mo dule M is lo cally ﬁnitely g enerated if there lo cally exists an exact sequence L 0 − → M − → 0 (1.1.3) suc h that L 0 is lo cally free of ﬁnite rank ov er A . • An A -mo dule M is lo cally of ﬁnite presen tation if there lo cally exists an exact sequence L 1 − → L 0 − → M − → 0 (1.1.4) suc h that L 1 and L 0 are lo cally free of ﬁnite rank ov er A . This is equiv alent t o sayin g that there lo cally exists an exact sequence 0 − → K u − → N − → M − → 0 (1.1.5) where N is lo cally free of ﬁnite rank and K is lo cally ﬁnitely g ener- ated. This is also equiv alen t to sa ying that there lo cally exis ts an exact sequence K − → N − → M − → 0 (1.1.6) where N is lo cally of ﬁnite presen tation and K is lo cally ﬁnitely gen- erated. • An A -mo dule M is pseudo-coheren t if for a n y lo cally deﬁned mor- phism u : N − → M with N of ﬁnite presen tation, Ker u is lo cally ﬁnitely generated. This is also equiv alent to sa ying that any lo cally deﬁned A - submodule of M is lo cally of ﬁnite presen tation a s so o n as it is lo cally ﬁnitely generated. 1.1. PRELIMINAR Y 13 • An A - mo dule M is coheren t if it is lo cally ﬁnitely generated and pseudo-coheren t. A ring is a coheren t ring if it is so as a mo dule o v er itself. One denotes by Mo d coh ( A ) the f ull additive sub category of Mo d( A ) consisting of coherent mo dules. Note that Mo d coh ( A ) is a full ab elian subcategory of Mo d( A ), stable b y exten sion, and the natural functor Mo d coh ( A ) − → Mo d( A ) is exact (see [41, Exe. 8.23 ]) . • An A -mo dule M is No etherian (see [37, D ef. A.7]) if it is coheren t, M x is a No etherian A x -mo dule for an y x ∈ X , and for an y op en subse t U ⊂ X , an y ﬁltran t family o f coheren t submo dules of M | U is lo cally stationary . (This means that given a family { M i } i ∈ I of coheren t sub- mo dules of M | U indexed b y a ﬁltrant ordered set I , with M i ⊂ M j for i ≤ j , there lo cally exists i 0 ∈ I suc h that M i 0 ∼ − → M j for any j ≥ i 0 .) A ring is a No etherian ring if it is so as a left mo dule ov er itself. Mittag-Leﬄer c ondition and pro-ob ject s W e refer to [55 ] for the notions of ind-ob ject and pro-ob ject as w ell as to [41] for an exp osition. T o an ab elian category C , one asso ciates the a b elian category Pro( C ) o f its pro-ob jects. Then C is a full abelian sub category of Pro( C ) stable b y kernel, cok ernel and extension, the natural functor C ֒ → Pro( C ) is e xact, and the functor “lim ← − ” : F ct( I op , C ) − → Pro( C ) is exact for any small ﬁltra nt category I . In the sequel, w e identify C with a full sub category of Pro( C ). If C admits small pro jectiv e limits, w e denote by π the left exact functor π : Pro( C ) − → C , “lim ← − ” i X i 7→ lim ← − i X i . If C has enough injective s, then π admits a righ t deriv ed functor (lo c. cit.): R π : D +  Pro( C )  − → D + ( C ) . Deﬁnition 1.1.1. W e sa y that an ob ject M ∈ Pro( C ) satisﬁes the Mittag- Leﬄer condition if, for any N ∈ C and an y morphism M − → N in Pro( C ), Im( M − → N ) is represen table b y an ob ject of C . By the deﬁnition, any quotien t of an ob ject whic h satisﬁes the Mittag - Leﬄer conditio n also satisﬁes the Mitta g -Leﬄer condition. Lemma 1.1.2. L et { M n } n ∈ Z ≥ 1 b e a pr oje ctive system in an ab elian c ate gory C , and set M = “ lim ← − ” n M n ∈ Pro( C ) . Then the fol lowing c onditions ar e e quivalent: (i) M satisﬁes the Mittag-L eﬄer c ondition , 14 CHAPTER 1. MODULES OVER FORMAL DEFORMA TIONS (ii) { M n } n ∈ Z ≥ 1 satisﬁes the Mittag-L eﬄer c ondition ( that is, for an y p ≥ 1 , the se quenc e { Im( M n − → M p ) } n ≥ p is stationary ) , (iii) ther e exis ts a pr oje c tive system { M ′ n } n ∈ Z ≥ 1 in C such that the mor- phism M ′ n +1 − → M ′ n is an epimorph i s m for an y n ≥ 1 and we have a n isomorphism M ≃ “lim ← − ” n M ′ n in Pro( C ) . Pr o of. ( i) ⇒ (ii). F or an y p ≥ 1, Im( M − → M p ) ≃ “lim ← − ” n ≥ p Im( M n − → M p ) is represen table b y a n ob ject of C . Hence, the sequenc e { Im( M n − → M p ) } n ≥ p is stationary . (ii) ⇒ (iii). Set M ′ n = Im ( M k − → M n ) for k ≫ n . Then the morphisms M ′ n − → M n induce a morphism f : “lim ← − ” n M ′ n − → “ lim ← − ” n M n . On the other ha nd, for eac h n , M − → M n decomp oses as M − → M ′ n ֌ M n , since taking k ≫ n suc h that M ′ n = Im ( M k − → M n ), w e ha v e a morphism M − → M k − → M ′ n . These morphisms induce a morphism g : “lim ← − ” n M n = M − → “lim ← − ” n M ′ n . It is easy to see that f and g are inv erse to eac h o ther. (iii) ⇒ (i). F or an y N ∈ C a nd an y morphism f : M − → N in Pro( C ), there exists p suc h that f decomp oses in to M − → M ′ p − → N . Then Im ( M − → N ) ≃ “lim ← − ” n ≥ p Im( M ′ n − → N ) ≃ Im( M ′ p − → N ). Q.E.D. Note that the followin g lemma is w ell kno wn. Lemma 1.1.3. L et { M n } n ≥ 1 b e a p r oje ctive system of Z -mo dules. Then R i π (“lim ← − ” n M n ) ≃ 0 for i 6 = 0 , 1 . I f { M n } n ≥ 1 satisﬁes the Mittag-L eﬄer c ondition, then H 1  R π “lim ← − ” n M n  ≃ 0 . Here a nd in the sequel, w e make t he f ollo wing conv en tion. Con v ention 1.1.4. When w e ha v e a left exac t functor C F − → C ′ of a b elian categories and X ∈ D ( C ), the notation R i F ( X ) stands for H i  R F ( X )  . F or example, R i π RΓ( U ; M ) means H i  R π RΓ( U ; M )  . Lemma 1.1.5. L et R b e an algebr a ov e r a top olo gic al sp ac e X , and let { M n } n ≥ 0 b e a pr oje ctive system of R -mo dules. Set M = “lim ← − ” n M n ∈ Pro(Mo d( R )) . L et U b e an op en subset of X and let i ∈ Z . Then we have an exact se quenc e 0 − → R 1 π  “lim ← − ” n H i − 1 ( U ; M n )  − → H i ( U ; R π M ) − → lim ← − n H i ( U ; M n ) − → 0 . 1.1. PRELIMINAR Y 15 Pr o of. W e ha v e RΓ( U ; R π M ) ≃ R π RΓ( U ; M ) and w e also hav e H i ( U ; M ) ≃ “lim ← − ” n H i ( U ; M n ). Consider the distinguished tria ngle R π τ 0 a pr oje ctive system of ab elian she aves on X and F := lim ← − n F n . Assume the fol lowing c onditions: (a) for any x ∈ X and any inte ger i , we have lim − → x ∈ U R 1 π “lim ← − ” n H i ( U ; F n ) ≃ 0 , wher e U r anges over an op en neighb orho o d system of x , (b) for any x ∈ X and i > 0 , lim − → x ∈ U  lim ← − n H i ( U ; F n )  = 0 , wher e U r anges ove r an op en neighb orho o d system of x , 16 CHAPTER 1. MODULES OVER FORMAL DEFORMA TIONS Then for any i , the morphism h i : H i ( X ; F ) − → lim ← − n H i ( X ; F n ) is surje ctive. I f mor e over { H i − 1 ( X ; F n ) } n satisﬁes the Mittag-L eﬄer c ondi- tion, then h i is an isomorph i sm. Pr o of. Set M = “lim ← − ” n F n . By the preceding lem ma, we hav e a n exact sequence 0 − → R 1 π  “lim ← − ” n H i − 1 ( U ; F n )  − → H i ( U ; R π M ) − → lim ← − n H i ( U ; F n ) − → 0 . F or an y x , taking the inductive limit with resp ect to U in an op en neigh b o r- ho o d system of x , w e obtain (R i π M ) x = 0 for i 6 = 0. Hence w e conclude R π M ≃ F . Then the exact sequence ab ov e reads as 0 − → R 1 π  “lim ← − ” n H i − 1 ( X ; F n )  − → H i ( X ; F ) − → lim ← − n H i ( X ; F n ) − → 0 . Hence w e hav e the desired res ult. Q.E.D. 1.2 F ormal deformation s of a sh eaf of rin g s No w we consider t he follow ing situation: X is a top ological space, A is a K -algebra on X and ~ is a section of A con tained in the cen ter o f A . W e set A 0 := A / ~ A Let M b e an A -mo dule. W e set c M := lim ← − n M / ~ n M , (1.2.1) and call it the ~ - completion o f M . W e sa y tha t • M has no ~ - torsion if ~ : M − → M is injectiv e, • M is ~ - separated if M − → c M is a monomorphism, i.e. , T n ≥ 0 ~ n M = 0, • M is ~ - complete if M − → c M is an isomorphism. Lemma 1.2.1. L et M ∈ Mo d( A ) and assume that M has no ~ -torsion. Then 1.2. FORMAL DEFORMA TIONS O F A SHEAF OF RINGS 17 (i) c M has no ~ -torsion, (ii) M / ~ n M ∼ − → c M / ~ n c M , (iii) c M ∼ − → M b b , i.e., c M is ~ -c omplete. Pr o of. ( i) Consider the exact sequence 0 − → M / ~ n M ~ a − − → M / ~ n + a M − → M / ~ a M − → 0 . Applying the left exact functor lim ← − n w e get the exact sequence 0 − → c M ~ a − − → c M − → M / ~ a M , whic h give s the result. (ii) Consider the commutativ e diagram with exact rows : 0 / / M   ~ n / / M   / / M / ~ n M ≀   / / 0 0 / / c M ~ n / / c M / / M / ~ n M . (iii) Apply the functor lim ← − to the isomorphism in (ii). Q.E.D. In this paper, with the exce ption of § 1.3, w e assume the follo wing con- ditions:                (i) A ha s no ~ -torsion, (ii) A is ~ -complete, (iii) A 0 is a left Noetherian ring, (1.2.2) and          (iv) there exists a base B of op en subsets of X suc h that for any U ∈ B and any coheren t ( A 0 | U )-mo dule F , w e ha v e H n ( U ; F ) = 0 for any n > 0. (1.2.3) It follows from (1 .2.2) that, for an o p en set U and a n ∈ A ( U ) ( n ≥ 0), P n ≥ 0 ~ n a n is a w ell-deﬁned eleme nt of A ( U ). 18 CHAPTER 1. MODULES OVER FORMAL DEFORMA TIONS By (1.2.2) (ii), ~ A x is contained in the Jacobson r a dical of A x for a n y x ∈ X . Indeed, for an y a ∈ ~ A x , 1 − a is in ve rtible in A x since a is deﬁned on a n open neigh b o r ho o d U of x , and 1 − a is in ve rtible on U . Hence Nak a y ama’s lemma implies the following lemma that w e frequen tly use. Lemma 1.2.2. L et M b e a lo c al ly ﬁnitely gener ate d A -mo dule. (i) If M satisﬁes M = ~ M , then M = 0 . (ii) L et f : N − → M b e a morph i s m of A -mo dules. If the c omp osition N − → M − → M / ~ M is an epimorphism , then f is an ep i m orphism. F or n ∈ Z ≥ 0 , set A n = A / ~ n +1 A . Note that there is an equiv alence of categories b etw een the category Mo d( A n ) and the full sub category of Mo d( A ) consisting o f mo dules M satisfying ~ n +1 M ≃ 0. Lemma 1.2.3. L et n ∈ Z ≥ 0 . (i) An A n -mo dule N is lo c al ly ﬁnitely gener ate d as an A n -mo dule if and only if it is so as an A -mo dule. (ii) An A n -mo dule N is lo c al ly of ﬁn ite pr esentation as an A n -mo dule if and o nly i f it is so as an A -mo dule. (iii) An A n -mo dule N is c oher ent as an A n -mo dule if and only if it is so as an A -mo dule. (iv) A n is a left No e theri a n ring. Pr o of. No te that since w e ha v e A n ≃ A / A ~ n +1 , A n is an A - mo dule lo cally of ﬁnite presen tation. (i) is o b vious. (ii)-(a) Let M b e an A n -mo dule lo cally of ﬁnite presen tation a nd consider an exact sequence of A n -mo dules as in (1.1.5). Then K is lo cally ﬁnitely generated as an A -mo dule, N is lo cally of ﬁnite presen tation as an A - mo dule and u is A -linear. Hence, M is lo cally of ﬁnite presen tation as an A -mo dule. (ii)-(b) Con v ersely assume that M is an A n -mo dule whic h is lo cally of ﬁnite presen tation as an A -mo dule. Consider an exact sequence of A -mo dules as in ( 1 .1.4). Applying the functor A n ⊗ A • , w e ﬁnd and exact sequence of A n - mo dules as in (1.1.4), whic h prov es that M is lo cally of ﬁnite presen tation as an A n -mo dule. 1.2. FORMAL DEFORMA TIONS O F A SHEAF OF RINGS 19 (iii) follo ws from (i) and (ii) since a mo dule is coheren t if it is lo cally ﬁnitely generated and an y submo dule lo cally ﬁnitely generated is lo cally of ﬁnite presen tation. (iv) Let us prov e that A n is a coheren t ring . Since A 0 is a coheren t ring b y the assumption, A 0 is a coheren t A -mo dule. Using the exact sequence s of A -mo dules 0 − → A n − 1 ~ − → A n − → A 0 − → 0 , w e get b y induction on n that A n is a coheren t A - mo dule. Hence (iii) implies that A n is a coherent r ing . One pro ves similarly by induction on n that ( A n ) x is a No etherian ring for all x ∈ X and that a ny ﬁltrant fa mily o f coheren t A n -submo dules of a coheren t A n -mo dule is lo cally stationa ry . Q.E.D. Lemma 1.2.4. L et U ∈ B , and n ≥ 0 . (i) F or any c oher ent A n -mo dule N , we ha v e H k ( U ; N ) = 0 for k 6 = 0 . (ii) F or any epi m orphism N − → N ′ of c oher ent A n -mo dules, N ( U ) − → N ′ ( U ) is surje ctive, (iii) A ( U ) − → A n ( U ) is surje ctive. Pr o of. ( i) is pro ve d b y induction on n , using the exact sequence 0 − → ~ N − → N − → N / ~ N − → 0 . (1.2.4) (ii) follows immediately from (i) and t he f act that A n is a coherent r ing . (iii) By (ii), A n +1 ( U ) − → A n ( U ) is surjectiv e for an y n ≥ 0. Hence, the morphism lim ← − m A m ( U ) − → A n ( U ) is surjectiv e. Sin ce the functor lim ← − com- m utes with the functor Γ( U ; • ), A ( U ) ∼ − → lim ← − m A m ( U ) and the result follo ws. Q.E.D. Prop erties of A Recall that A satisﬁes (1.2.2) and (1 .2.3). Theorem 1.2.5. (i) A is a left No etherian ring. (ii) L et M b e a lo c a l l y ﬁnitely gen e r ate d A -mo dule. Then M is c oher ent if and only if ~ n M / ~ n +1 M is a c oh er ent A 0 -mo dule for any n ≥ 0 . 20 CHAPTER 1. MODULES OVER FORMAL DEFORMA TIONS (iii) Any c oher ent A -mo dule M is ~ -c omplete, i.e. , M ∼ − → c M . (iv) Converse ly, an A -mo dule M is c oher ent if and only if it is ~ -c om plete and ~ n M / ~ n +1 M is a c oher ent A 0 -mo dule for any n ≥ 0 . (v) F o r an y c oher ent A -m o dule M and any U ∈ B , we have H j ( U ; M ) = 0 for any j > 0 . The pro of of Theorem 1.2.5 decomp oses in to sev eral lemmas. Lemma 1.2.6. L et L b e a lo c al ly fr e e A - m o dule of ﬁnite r ank a n d let N b e an A -submo dule of L . Assume that (a) ( N + ~ L ) / ~ L is a c oher ent A 0 -mo dule, (b) N ∩ ~ n L ⊂ ~ N + ~ 1+ n L fo r any n ≥ 1 . Then we have (i) N is a lo c al ly ﬁ nitely gener ate d A -mo dule, (ii) N ∩ ~ n L = ~ n N fo r a ny n ≥ 0 , (iii) T n ≥ 0 ( N + ~ n L ) = N . Pr o of. F irst, let us sho w tha t N ∩ ~ L ⊂ ~ N + ~ n L for an y n ≥ 0. (1.2.5) Indeed, (1.2.5) is trivial for n ≤ 1. L et us argue by induction, and let n ≥ 2, assuming the assertion for n − 1. W e hav e N ∩ ~ L ⊂ N ∩ ( ~ N + ~ n − 1 L ) = ~ N + ( N ∩ ~ n − 1 L ) ⊂ ~ N + ( ~ N + ~ n L ) b y the a ssumption (b). This pro v es (1.2.5). Set f N = \ n ≥ 0 ( N + ~ n L ) . Then N ⊂ f N and f N ∩ ~ L ⊂ ~ f N . (1.2.6) Indeed w e hav e f N ∩ ~ L ⊂ ( N + ~ n +1 L ) ∩ ~ L ⊂ N ∩ ~ L + ~ n +1 L ⊂ ~ N + ~ n +1 L = ~ ( N + ~ n L ) fo r any n . 1.2. FORMAL DEFORMA TIONS O F A SHEAF OF RINGS 21 Set ¯ N = ( N + ~ L ) / ~ L = ( f N + ~ L ) / ~ L . By the hy p o t hesis (a), ¯ N is A 0 -coheren t. Hence w e ma y assume that there exist ﬁnitely man y sections s i of N suc h that ¯ N = P i A 0 s i , where s i is the image of s i in L / ~ L . By hy p o t hesis (a) and Lemma 1.2.4 (ii), we hav e for any U ∈ B , ¯ N ( U ) = P i A 0 ( U ) s i . Since A ( U ) − → A 0 ( U ) is surjectiv e b y Lemma 1.2.4 ( iii) , we ha v e f N ( U ) ⊂ P i A ( U ) s i + ~ L ( U ). Since f N ∩ ~ L = ~ f N , w e hav e f N ( U ) ⊂ X i A ( U ) s i + ~ f N ( U ) . F or v ∈ f N ( U ), we shall deﬁne a seque nce { v n } n ≥ 0 in f N ( U ) and seque nces { a i,n } n ≥ 0 in A ( U ), induc tively on n : set v 0 = v , a nd write v n = X i a i,n s i + ~ v n +1 . Hence w e hav e ~ n v n = P i ~ n a i,n s i + ~ n +1 v n +1 and w e obtain v = v 0 = X i ( X n ≥ 0 ~ n a i,n ) s i . Th us w e ha v e f N = P i A s i . Hence N = f N whic h pro v es (i) and (iii). Since f N ∩ ~ L = ~ f N b y (1.2.6), we obta in (ii) f or n = 1 . F or n ≥ 1 w e ha v e by induction N ∩ ~ n L ⊂ ~ N ∩ ~ n L = ~ ( N ∩ ~ n − 1 L ) ⊂ ~ · ~ n − 1 N . Q.E.D. Lemma 1.2.7. L et L b e a lo c al ly fr e e A -mo dule of ﬁnite r ank, an d let N b e an A -submo dule of L . Assume that ( N + ~ n +1 L ) / ~ n +1 L is a c oher ent A -mo dule for any n ≥ 0 . Then we have (i) N is a lo c al ly ﬁnitely gener a te d A -mo dule, (ii) T n ≥ 0 ( N + ~ n L ) = N , (iii) lo c al ly, ~ n L ∩ N ⊂ ~ ( ~ n − 1 L ∩ N ) for n ≫ 0 , (iv) N / ~ n N is a c oher ent A -mo dule fo r any n ≥ 0 . 22 CHAPTER 1. MODULES OVER FORMAL DEFORMA TIONS Pr o of. W e em b ed L into the A [ ~ − 1 ]-mo dule K [ ~ , ~ − 1 ] ⊗ K [ ~ ] L = S n ∈ Z ~ n L . Note that ~ n induces an isomorphism ~ n : ( L ∩ ~ − n N + ~ L ) / ~ L ∼ − → ( N ∩ ~ n L + ~ n +1 L ) / ~ n +1 L . Since ( N ∩ ~ n L + ~ n +1 L ) / ~ n +1 L ≃  ( N + ~ n +1 L ) / ~ n +1 L  \  ~ n L / ~ n +1 L  is A -coheren t, { ( L ∩ ~ − n N + ~ L ) / ~ L } n ≥ 0 is a n increasing sequenc e of coheren t A 0 -submo dules of L / ~ L . Hence it is lo cally stationa r y: lo cally there exists n 0 ≥ 0 suc h that L ∩ ~ − n N + ~ L = L ∩ ~ − n 0 N + ~ L for an y n ≥ n 0 . Set N 0 := L ∩ ~ − n 0 N . (1.2.7) Then ( N 0 + ~ L ) / ~ L is a coheren t A 0 -mo dule and N 0 ∩ ~ n L ⊂ ~ n ( ~ − n − n 0 N ∩ L ) ⊂ ~ n ( N 0 + ~ L ) ⊂ ~ N 0 + ~ n +1 L for an y n > 0. Hence b y Lemma 1.2.6 : • N 0 is lo cally ﬁnitely generated o v er A , • T n ≥ 0 ( N 0 + ~ n L ) = N 0 , • N 0 ∩ ~ n L = ~ n N 0 for an y n ≥ 0. (i) Since N ∩ ~ n 0 L = ~ n 0 N 0 b y ( 1 .2.7), the mo dule N / ~ n 0 N 0 ≃ N / ( N ∩ ~ n 0 L ) ≃ ( N + ~ n 0 L ) / ~ n 0 L is A -coheren t. Sinc e ~ n 0 N 0 is lo cally ﬁnitely generated o v er A , N is also lo cally ﬁnitely g enerated o ve r A . (ii) W e hav e \ n ≥ n 0 ( N + ~ n L ) ⊂ ( N + ~ n 0 L ) \ ∩ n ≥ n 0 ( N + ~ n L ) ⊂ N + ~ n 0 L \ ∩ n ≥ n 0 ( N + ~ n L ) ⊂ N + ∩ n ≥ n 0 ( ~ n 0 L ∩ N + ~ n L ) ⊂ N + ∩ n ≥ n 0 ( ~ n 0 N 0 + ~ n L ) ⊂ N + ~ n 0 N 0 = N . 1.2. FORMAL DEFORMA TIONS O F A SHEAF OF RINGS 23 (iii) F or n > n 0 , we hav e ~ n L ∩ N ⊂ ~ n 0 ( L ∩ ~ − n 0 N ) ∩ ~ n L ⊂ ~ n 0 ( N 0 ∩ ~ n − n 0 L ) ⊂ ~ n 0 ~ n − n 0 N 0 = ~ n N 0 ⊂ ~ ( N ∩ ~ n − 1 L ) . (iv) Since N has no ~ -torsion, w e ha v e the exact sequence 0 − → N / ~ n N ~ − → N / ~ n +1 N − → N / ~ N − → 0 . Hence, it is enough to show that N / ~ N is coheren t. By (i), the images o f N and ~ N in L / ~ n L are coheren t. Since N ∩ ~ n L ⊂ ~ N for some n , b y (ii) , we hav e the exact sequence ~ N ~ N ∩ ~ n L − → N N ∩ ~ n L − → N ~ N − → 0 , whic h implies tha t N / ~ N is coheren t. Q.E.D. Corollary 1.2.8. Assume that M is a lo c al ly ﬁnitely gener ate d A -mo dule. If M / ~ n M is a c oher ent A -mo dule for al l n > 0 , then M is an A -mo dule lo c a l ly of ﬁnite pr es e ntation and T n ≥ 0 ~ n M = 0 . Pr o of. W e ma y a ssume that M = L / N for a lo cally free A -mo dule L of ﬁnite rank and N ⊂ L . F rom the exact sequence 0 − → ( N + ~ n L ) / ~ n L − → L / ~ n L − → M / ~ n M − → 0 , w e deduce that ( N + ~ n L ) / ~ n L is coheren t f o r a ny n . Hence N is lo cally ﬁnitely generated b y Lemma 1.2.7, which implies tha t M is lo cally o f ﬁnite presen tation. Since T n ≥ 0 ( N + ~ n L ) = N by Lemma 1.2 .7, \ n ≥ 0 ~ n M ≃  \ n ≥ 0 ( N + ~ n L )  / N v anishe s. Q.E.D. Prop osition 1.2.9. A is c oher ent. Pr o of. L et I be a lo cally ﬁnitely generated A -submo dule of A . Since ( I + ~ n +1 A ) / ~ n +1 A ≃ I / ( I ∩ ~ n +1 A ) ⊂ A / ~ n +1 A , the A -mo dule I / ~ n I is c oherent b y Lemm a 1.2.7 (iv). Hence Corol- lary 1.2.8 implies that I is lo cally o f ﬁnite presen tatio n. Q.E.D. 24 CHAPTER 1. MODULES OVER FORMAL DEFORMA TIONS Lemma 1.2.10. A ny ﬁltr ant family of c oher en t A -submo dules of A is lo c al ly stationary. Pr o of. L et { I i } i ∈ I b e a family of coherent A - submo dules of A indexed b y a ﬁltran t ordered set I , with I i ⊂ I j for an y i ≤ j . Then { ( ~ − k I i ∩ A + ~ A ) / ~ A } i ∈ I , k ≥ 0 is increasing with resp ect to k and i ∈ I . Hence lo cally there exist i 0 and k 0 suc h that ~ − k I i ∩ A + ~ A = ~ − k 0 I i 0 ∩ A + ~ A for an y i ≥ i 0 and k ≥ k 0 . Then, for i ≥ i 0 , the ideal J i := A ∩ ~ − k 0 I i satisﬁes J i ∩ ~ m A ⊂ ~ m ( ~ − m − k 0 I i ∩ A ) ⊂ ~ m ( ~ − k 0 I i ∩ A + ~ A ) ⊂ ~ J i + ~ m +1 A for any m > 0. Hence Lemma 1.2 .6 implies that J i ∩ ~ A = ~ J i . Since w e hav e J i ⊂ J i 0 + ~ A , w e ha v e J i ⊂ J i 0 + ( J i ∩ ~ A ) ⊂ J i 0 + ~ J i . Then Nak a y ama’s lemma implies J i = J i 0 , or equiv alently , ~ − k 0 I i ∩ A = ~ − k 0 I i 0 ∩ A for i ≥ i 0 . Th us { I i ∩ ~ k 0 A } i is lo cally statio na ry . Since { I i / ( I i ∩ ~ k 0 A ) } i is a ﬁltrant family of coheren t submo dules of A k 0 − 1 , it is also lo cally stationa ry and it follows that { I i } i is lo cally stationary . Q.E.D . Lemma 1.2.11. F or any x ∈ X , A x is a c oher ent ring. Pr o of. Any morphism f : A ⊕ n x − → A x extends to a morphism ˜ f : A ⊕ n | U − → A | U for some op en neigh b orho o d U of x . Since N := Ker ˜ f is coheren t, N x ≃ Ker f is a ﬁnitely generated A x -mo dule. Q.E.D. Lemma 1.2.12. F or any x ∈ X and a ﬁnitely gener ate d left ide al I of A x , I ∩ ~ n +1 A x = ~ ( I ∩ ~ n A x ) for n ≫ 0 . Pr o of. L et us ta k e a coheren t ideal I of A deﬁned on a neigh b orho o d of x suc h that I = I x . Then Lemma 1.2.7 implies that I ∩ ~ n +1 A = ~ ( I ∩ ~ n A ) for n ≫ 0. Q.E.D. Lemma 1.2.13. F or any x ∈ X , A x is a No etherian ring. Pr o of. Set A = A x . Let us sho w tha t an increasing seq uence { I n } n of ﬁnitely generated left ideals of A is stationary . Since { ( ~ − k I n ∩ A + ~ A ) / ~ A } n,k is increasing with resp ect n, k , there exist n 0 and k 0 suc h that ~ − k I n ∩ A + ~ A = ~ − k 0 I n 0 ∩ A + ~ A fo r n ≥ n 0 and k ≥ k 0 . F o r an y n ≥ n 0 there exists k ≥ k 0 suc h that ~ − k I n ∩ ~ A = ~ ( ~ − k I n ∩ A ) b y Lemma 1.2.12. Hence w e hav e ~ − k I n ∩ A ⊂ ~ − k I n ∩ ( ~ − k 0 I n 0 ∩ A + ~ A ) ⊂ ~ − k 0 I n 0 ∩ A + ( ~ − k I n ∩ ~ A ) ⊂ ~ − k 0 I n 0 ∩ A + ~ ( ~ − k I n ∩ A ). Since ~ − k I n ∩ A is ﬁnitely generated b y Lemma 1.2.11, Nak a y ama’s lemma implies that ~ − k I n ∩ A = ~ − k 0 I n 0 ∩ A . Hence ~ − k 0 I n ∩ A = ~ − k 0 I n 0 ∩ A for any n ≥ n 0 . Therefore I n ∩ ~ k 0 A = ~ k 0 ( ~ − k 0 I n ∩ A ) is stationa ry . Since { I n / ( I n ∩ ~ k 0 A ) } n is stationary , { I n } n is stationary . Q.E.D. 1.2. FORMAL DEFORMA TIONS O F A SHEAF OF RINGS 25 Th us, w e hav e pro v ed that A is a No etherian ring. Lemma 1.2.14. L et { M n } n ≥ 0 b e a pr oje ctive system of c oher ent A -mo dules. Assume that ~ n +1 M n = 0 and the induc e d morphism M n +1 / ~ n +1 M n +1 − → M n is an isomorp h ism for any n ≥ 0 . Th e n M := lim ← − n M n is a c oher ent A -mo dule and M / ~ n +1 M − → M n is an isomorph i sm for any n ≥ 0 . Pr o of. Since the question is lo cal, w e may assume that X ∈ B and there exist a free K -mo dule V of ﬁnite rank and a morphism V − → M 0 ( X ) whic h induces an epimorphism L := A ⊗ K V ։ M 0 . Since M n +1 ( X ) − → M n ( X ) is surjectiv e and V is pro jectiv e, w e ha v e a pro jectiv e syste m of mor phisms { V − → M n ( X ) } n : V   & & , , - - Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z · · · / / / / M n ( X ) / / / / M n − 1 ( X ) / / / / · · · / / / / M 1 ( X ) / / / / M 0 ( X ) , whic h induces a pro jectiv e system of mo r phisms { L − → M n } n . Hence we ma y assume that there exists a morphism L − → M suc h that the comp osition L − → M − → M 0 is an epimorphism. Since L − → M n / ~ M n ∼ − → M 0 is an epimorphism, L − → M n is an epimorphism b y Lemma 1.2.2. Set L n = L / ~ n +1 L , and let N n b e the k ernel of L n − → M n . Set N = lim ← − n N n . Then w e hav e a comm utativ e diagram with exact rows : 0 / / N / /   L / /   M   0 / / N n / / L n / / M n / / 0 . In the comm utativ e diagram 0   0   ~ n +1 L n +1 / /   ~ n +1 M n +1 / /   0 0 / / N n +1 / /   L n +1 / /   M n +1 / /   0 0 / / N n / / L n / /   M n   / / 0 0 0 26 CHAPTER 1. MODULES OVER FORMAL DEFORMA TIONS the ro ws and the columns are exact. Hence the left v ertical arrow N n +1 − → N n is an epimorphism. Therefore, N n +1 ( U ) − → N n ( U ) is surjectiv e for an y U ∈ B , and N ( U ) ∼ − → lim ← − m N m ( U ) − → N n ( U ) is surjectiv e. Hence N − → N n is an epimorphism for an y n ≥ 0, and { N n ( U ) } n satisﬁes the Mittag-Leﬄer condition. Th us in the fo llo wing comm utative diag ram 0 / / N ( U ) / / ∼   L ( U ) / / ∼   M ( U ) ∼   / / 0 0 / / lim ← − n N n ( U ) / / lim ← − n L n ( U ) / / lim ← − n M n ( U ) / / 0 , the b o ttom row is exact. Hence 0 − → N − → L − → M − → 0 is exact. Since N − → N n is an epimorphism, w e ha v e M / ~ n +1 M ≃ Coker( N − → L n ) ≃ Cok er( N n − → L n ) ≃ M n . Since M is lo cally ﬁnitely g enerated and M / ~ n +1 M is coherent for an y n ≥ 0, M is coherent b y Corolla r y 1.2.8 and Prop osition 1.2.9. Q.E.D. Prop osition 1.2.15. L et M b e a c oher ent A -m o dule. Then w e have the fol lowing pr op erties. (i) M is ~ -c om plete, i.e., M ∼ − → c M , (ii) for any U ∈ B , H k ( U ; M ) = 0 for any k > 0 . Pr o of. ( i) Since the ke rnel o f M − → c M is T n ≥ 0 ~ n M , the morphism M − → c M is a mono morphism b y Corollary 1.2.8. Let us show that M − → c M is a n epimorphism. By the preceding lemma, c M is a coherent A -mo dule, and c M / ~ c M ≃ M / ~ M . Hence Nak a y ama’s lemma implies that M − → c M is an epimorphism. (ii) F or an y U ∈ B , the map Γ( U ; M / ~ n +1 M ) − → Γ( U ; M / ~ n M ) is surjec- tiv e, and H k ( U ; M / ~ n M ) = 0 for any k > 0. Hence Lemma 1.1 .6 implies (ii). Q.E.D. Corollary 1.2.16. L et M b e an A -mo dule. If M satisﬁes the fol lowing c onditions (i) and (ii) , then M is a c oher e n t A - m o dule. (i) M is ~ -c om plete, (ii) ~ n M / ~ n +1 M is a c oher ent A 0 -mo dule for al l n ≥ 0 . 1.2. FORMAL DEFORMA TIONS O F A SHEAF OF RINGS 27 Pr o of. Set M n = M / ~ n +1 M . Then it is a coheren t A -mo dule b y (ii), and lim ← − n M n is a coheren t A - mo dule b y Lemma 1 .2.14. Q.E.D. This completes the pro of of Theorem 1 .2.5. Lemma 1.2.17. L et M b e a c oher ent A -mo dule without ~ - torsio n. If M / ~ M is a lo c al ly f r e e A 0 -mo dule of r ank r ∈ Z ≥ 0 , then M i s a lo c al ly fr e e A -mo dule of r ank r . Pr o of. W e ma y assume that t here exists a morphism of A - mo dules f : L := A ⊕ r − → M suc h that L / ~ L − → M / ~ M is an isomorphism. Then, Nak ay ama’s lemma implies that f is an epimorphism. Let N b e the k ernel of f . Since M has no ~ - torsion, w e ha v e an exact seque nce 0 − → N / ~ N − → L / ~ L − → M / ~ M − → 0. Hence N / ~ N = 0 and Nak ay ama’s lemma implies N = 0. Q.E.D. The following pro p osition giv es a criterion for the coherence of the pro- jectiv e limit of coheren t mo dules, generalizing Lemma 1 .2.14. Prop osition 1.2.18. L et { N n } n ≥ 1 b e a pr oje ctive system of c oher ent A - mo dules. Assume (a) the pr o-obje c t “lim ← − ” n N n / ~ N n is r epr esentable by a c oher e n t A 0 -mo dule, (b) the pr o-obje ct “lim ← − ” n Ker( N n ~ − → N n ) is r epr esentable by a c oher ent A 0 - mo dule. Then (i) N := lim ← − n N n is a c oher ent A -mo dule, (ii) N / ~ k +1 N ∼ − → “lim ← − ” n N n / ~ k +1 N n for any k ≥ 0 , (iii) Ker( N ~ − → N ) ∼ − → “lim ← − ” n Ker( N n ~ − → N n ) . (iv) Assume mor e over that for e ach n ≥ 1 ther e exists k ≥ 0 such that ~ k N n = 0 . Then the pr oje ctive system { N n } n satisﬁes the Mittag- L eﬄer c ondition. 28 CHAPTER 1. MODULES OVER FORMAL DEFORMA TIONS Pr o of. F o r any k ≥ 0, set S k := “lim ← − ” n N n / ~ k +1 N n . Then S 0 is represen table b y a coheren t A -mo dule b y hy p o thesis (a ). W e shall sho w that S k is represen table b y a coheren t A -mo dule for all k ≥ 0 b y induction on k . Consider the exact seq uences 0 − → ~ N n / ~ k +1 N n − → N n / ~ k +1 N n − → N n / ~ N n − → 0 , (1.2.8) Ker( N n ~ − → N n ) − → N n / ~ k N n ~ − → ~ N n / ~ k +1 N n − → 0 . (1.2.9) Assume that S k − 1 is represen table b y a coheren t A - mo dule. Applying the functor “lim ← − ” n to the exact sequence ( 1 .2.9), w e deduce that the ob ject “lim ← − ” n ~ N n / ~ k +1 N n is represen table b y a coheren t A -mo dule. Then apply- ing the functor “lim ← − ” n to the exact sequence (1.2.8), w e deduce that S k is represen table by a coheren t A -mo dule. Since N n ≃ lim ← − k N n / ~ k +1 N n b y Theorem 1.2.5 (iii ), we hav e N ≃ lim ← − k ,n N n / ~ k +1 N n ≃ lim ← − k S k . Since S k +1 / ~ k +1 S k +1 ≃ S k , Lemma 1.2.14 implies (i), (ii). The prop erty (iii) is o b vious. Let us pro ve (iv). By t he a ssumption, N n ≃ “lim ← − ” k N n / ~ k N n . Hence “lim ← − ” n N n ≃ “lim ← − ” k ,n N n / ~ k N n ≃ “lim ← − ” k S k . Since { S k } k satisﬁes the Mittag-L eﬄer condition, { N n } n satisﬁes the Mitta g - Leﬄer conditio n b y Lemma 1 .1.2. Q.E.D. Remark 1.2.19. In Prop osition 1.2.18 (iv), the conditio n ~ k N n = 0 ( k ≫ 0) is necessary as seen b y considering the pro jectiv e system N n = ~ n A , ( n ∈ N ). 1.3 A v ari ant of the prec eding res ults Here, w e consider rings whic h satisfy hypotheses (1.2.2), but in whic h (1.2.3) is replaced with another h yp othesis. Indeed, as w e shall see, the ring D X [[ ~ ] ] 1.3. A V ARIANT OF THE PRECEDING RESUL TS 29 of diﬀerential op era t o rs on a complex manifold X has nice prop erties, al- though D X do es not satisfy (1 .2 .3). The study of mo dules o ver D X [[ ~ ] ] is p erformed in [17]. W e assume that X is a Ha usdorﬀ lo cally compact space. By a basis B of compact subsets of X , we mean a family of compact subsets suc h that fo r an y x ∈ X and an y o p en neighborho o d U of x , there exists K ∈ B suc h that x ∈ Int( K ) ⊂ U . W e consider a K -algebra A on X and a section ~ of A contained in the cen ter of A . Set A 0 = A / ~ A . W e assume the conditions (1.2 .2) and                                                                                  (iv’) there exist a base B of compact subsets o f X and a prestac k U 7→ Mo d gd ( A 0 | U ) ( U op en in X ) suc h that (a) for any K ∈ B a nd an op en subset U suc h that K ⊂ U , t here exists K ′ ∈ B suc h that K ⊂ Int( K ′ ) ⊂ K ′ ⊂ U , (b) U 7→ Mo d gd ( A 0 | U ) is a full subprestac k of U 7→ Mo d coh ( A 0 | U ), (c) for an op en subset U and M ∈ Mo d coh ( A 0 | U ), if M | V b elongs to Mo d gd ( A 0 | V ) for an y relativ ely compact op en subset V of U , then M b elongs to Mo d gd ( A 0 | U ), (d) for any op en subset U of X , Mo d gd ( A 0 | U ) is stable by sub ob jects, quotien ts and extension in Mo d coh ( A 0 | U ), (e) for a n y K ∈ B , any o p en set U containing K , an y M ∈ Mo d gd ( A 0 | U ) and an y j > 0, one has H j ( K ; M ) = 0, (f ) for any M ∈ Mo d coh ( A 0 | U ), there exists an op en co v- ering U = S i U i suc h that M | U i ∈ Mo d gd ( A 0 | U i ), (g) A 0 ∈ Mo d gd ( A 0 ). (1.3.1) Note that Lemmas 1.2.2 and 1.2.3 still hold. The prestac k U 7→ Mo d gd ( A 0 | U ) b eing giv en, a coheren t mo dule whic h b elongs to Mo d gd ( A 0 | U ) will be called a go o d mo dule. Note that in view of h yp othesis (iv’) (f), hypothesis ( iv’) (g) could b e deleted since all the results of this subsection will b e of lo cal nature. How ev er, w e kee p it for simplicit y . Example 1.3.1. Let X b e a complex manifold, O X the structure sheaf and let D X denote the C -algebra of diﬀeren tial o p erators. One c hec ks easily that, 30 CHAPTER 1. MODULES OVER FORMAL DEFORMA TIONS taking for B t he set of Stein compact subsets and for A 0 the C -algebra D X , the prestack of go o d D X -mo dules in the sense of [37] satisﬁes the h yp otheses (1.3.1). Deﬁnition 1.3.2. A cohere nt A -mo dule M is go o d if b oth the k ernel and the cok ernel o f ~ : M − → M are go o d A 0 -mo dules. One denotes by Mo d gd ( A ) the category of go o d A - mo dules. Note that an A 0 -mo dule is go o d if and only if it is go o d as an A -mo dule. This allows us t o state: Deﬁnition 1.3.3. An A n -mo dule M is go o d if it is go o d as an A - mo dule. Lemma 1.3.4. The c ate go ry Mo d gd ( A ) is a sub c ate g o ry of Mo d coh ( A ) stable by sub obje cts, quotients and extension. Pr o of. F irst not e that ~ n M / ~ n +1 M is a go o d A 0 -mo dule for any M ∈ Mo d gd ( A ) and an y in teger n ≥ 0. Indeed, it is a quotien t of M / ~ M . F or an A -mo dule N , set N ~ := Ker( ~ : N − → N ). W e shall sho w that a n y coheren t A -submo dule N o f a goo d A -mo dule M is a go o d A -mo dule. It is ob vious that N ~ is a go o d A 0 -mo dule, because it is a coheren t submo dule of M ~ . W e shall show that N / ( ~ N + N ∩ ~ k +1 M ) is a go o d A 0 -mo dule for an y k ≥ 0 . W e argue by induction on k . F or k = 0 , it is a go o d A 0 -mo dule since it is a coheren t submo dule o f M / ~ M . F or k > 0, w e ha v e an exact sequence 0 − → ~ N + N ∩ ~ k M ~ N + N ∩ ~ k +1 M − → N ~ N + N ∩ ~ k +1 M − → N ~ N + N ∩ ~ k M − → 0 . (1.3.2) Since ( N ∩ ~ k M ) / ( N ∩ ~ k +1 M ) is a coheren t submo dule of ~ k M / ~ k +1 M , it is a go o d A 0 -mo dule. Since ( ~ N + N ∩ ~ k M ) / ( ~ N + N ∩ ~ k +1 M ) is a quotien t of ( N ∩ ~ k M ) / ( N ∩ ~ k +1 M ), the left term in (1.3.2) is a go o d A 0 -mo dule. Hence the induction pro ceeds and w e conclude that N / ( ~ N + N ∩ ~ k +1 M ) is a go o d A 0 -mo dule. On an y compact set, we hav e N ∩ ~ k +1 M ⊂ ~ N for k ≫ 0. Hence, ( N / ~ N ) | V is a go o d ( A 0 | V )-mo dule for an y relativ ely compact subset V . Hence N b elongs to Mo d gd ( A ) by (iv’) (c). Consider an exact sequence 0 − → M ′ − → M − → M ′′ − → 0 of coheren t A -mo dules. It gives rise t o an exact sequence of coheren t A 0 -mo dules 0 − → M ′ ~ − → M ~ − → M ′′ ~ − → M ′ / ~ M ′ − → M / ~ M − → M ′′ / ~ M ′′ − → 0 . 1.3. A V ARIANT OF THE PRECEDING RESUL TS 31 If M is a go o d A -mo dule, then so is M ′ . Hence the exact seq uence a b o v e implies that M ′′ ~ and M ′′ / ~ M ′′ are go o d A 0 -mo dules. This sho ws tha t Mo d gd ( A ) is stable by quotien ts. Finally , let us show that Mo d gd ( A ) is stable by extens ion. If M ′ ~ , M ′′ ~ , M ′ / ~ M ′ and M ′′ / ~ M ′′ are go o d A 0 -mo dules, then so a r e M ~ and M / ~ M b y the exact sequence a b o v e. Q.E.D. Lemma 1.3.5. L et K ∈ B , and n ≥ 0 . (i) F or any g o o d A n -mo dule N , we have H j ( K ; N ) = 0 f o r j 6 = 0 . (ii) F or any epim o rphism N − → N ′ of g o o d A n -mo dules, N ( K ) − → N ′ ( K ) is surje ctive. (iii) A ( K ) − → A n ( K ) is surje ctive. Pr o of. ( i) is pro ve d b y induction on n , using the exact sequence (1.2.4). (ii) follows immediately fro m (i) a nd t he fact that the ke rnel of a mo r phism of go o d mo dules is go o d. (iii) By (ii), A n +1 ( K ) − → A n ( K ) is surjectiv e for any n ≥ 0. Hence lim ← − m  A m ( K )  − → A n ( K ) is surjectiv e. F or s ∈ A n ( K ), there exist K ′ ∈ B and s ′ ∈ A n ( K ′ ) suc h tha t K ⊂ In t( K ′ ) a nd s ′ | K = s . Then s ′ is in the image of lim ← − m  A m ( K ′ )  − → A n ( K ′ ). Hence s is in the image of A ( K ) − → A n ( K ), b ecause lim ← − m  A m ( K ′ )  − → A n ( K ′ ) − → A n ( K ) decomp oses in to lim ← − m  A m ( K ′ )  − → lim ← − m  A m (In t( K ′ ))  ≃ A ( Int( K ′ )) − → A ( K ) − → A n ( K ) . Q.E.D. The pro o f of the following theorem is almost the same as the pro of of Theorem 1.2.5, and we do not rep eat it. Theorem 1.3.6. Assume (1.2.2) and (1.3.1) . (i) A is a left No etherian ring. (ii) L et M b e a lo c a l l y ﬁnitely gen e r ate d A -mo dule. Then M is c oher ent if and only if ~ n M / ~ n +1 M is a c oh er ent A 0 -mo dule for any n ≥ 0 . (iii) F or any c o h er ent A -mo dule M , M is ~ -c omplete, i.e., M ∼ − → c M . 32 CHAPTER 1. MODULES OVER FORMAL DEFORMA TIONS (iv) Converse ly, an A -mo dule M i s c ohe r ent if and only if M is ~ -c omplete and ~ n M / ~ n +1 M is a c oher ent A 0 -mo dule for any n ≥ 0 . (v) F o r any go o d A -mo dule M and any K ∈ B , we have H j ( K ; M ) = 0 for any j > 0 . 1.4 ~ -graduation and ~ - lo caliz ation In this section, A is a sheaf of algebras satisfying h yp otheses (1 .2.2) and either (1.2.3) or (1.3.1). Graded mo dules Let R b e a Z [ ~ ]-algebra on a top ological space X . W e a ssume that R has no ~ - torsion. W e set R 0 := R / R ~ . Deﬁnition 1.4.1. W e denote by gr ~ : D ( R ) − → D( R 0 ) the left deriv ed func- tor of the righ t exact functor Mo d( R ) − → Mo d( R 0 ) giv en by M 7→ M / ~ M . F or M ∈ D( R ) w e call gr ~ ( M ) t he g raded mo dule associated to M . W e hav e gr ~ ( M ) ≃ R 0 L ⊗ R M ≃ Z X L ⊗ Z X [ ~ ] M . Lemma 1.4.2. L et M ∈ D ( R ) and let a ∈ Z . Then we have an exact se quenc e of R 0 -mo dules 0 − → R 0 ⊗ R H a ( M ) − → H a (gr ~ ( M )) − → T or R 1 ( R 0 , H a +1 ( M )) − → 0 . Although this kind of results is w ell-know n, w e giv e a pro of for the reader’s con v enience. Pr o of. The exact sequenc e 0 − → R ~ − − → R − → R 0 − → 0 gives rise t o the distinguished triangle M ~ − − → M − → gr ~ ( M ) +1 − − − → . It induces a long exact sequence H a ( M ) ~ − − → H a ( M ) − → H a (gr ~ ( M )) − → H a +1 ( M ) ~ − − → H a +1 ( M ) . 1.4. ~ - GRADUA TION AND ~ -LOCALIZA TION 33 The result then follows from R 0 ⊗ R H a ( M ) ≃ Coke r ( H a ( M ) ~ − − → H a ( M )) , T or R 1 ( R 0 , H a +1 ( M )) ≃ Ker( H a +1 ( M ) ~ − − → H a +1 ( M )) . Q.E.D. Prop osition 1.4.3. (i) L et K 1 ∈ D( R op ) and K 2 ∈ D( R ) . Then gr ~ ( K 1 L ⊗ R K 2 ) ≃ gr ~ ( K 1 ) L ⊗ R 0 gr ~ ( K 2 ) . (1.4.1) (ii) L et K i ∈ D( R ) ( i = 1 , 2) . Then gr ~ (R H om R ( K 1 , K 2 )) ≃ R H om R 0 (gr ~ ( K 1 ) , gr ~ ( K 2 )) . (1.4.2) Pr o of. ( i) W e ha v e gr ~ ( K 1 L ⊗ R K 2 ) ≃ K 1 L ⊗ R K 2 L ⊗ Z X [ ~ ] Z X ≃ K 1 L ⊗ R gr ~ ( K 2 ) ≃ K 1 L ⊗ R R 0 L ⊗ R 0 gr ~ ( K 2 )) ≃ ( K 1 L ⊗ R R 0 ) L ⊗ R 0 gr ~ ( K 2 ) ≃ gr ~ ( K 1 ) L ⊗ R 0 gr ~ ( K 2 ) . (ii) The pro of is similar. Q.E.D. Prop osition 1.4.4. L et f : X − → Y b e a morphism of top olo gic a l sp ac es. L et M ∈ D( Z X [ ~ ]) an d N ∈ D ( Z Y [ ~ ]) . Then gr ~ R f ∗ M ≃ R f ∗ gr ~ M , gr ~ f − 1 N ≃ f − 1 gr ~ N . Pr o of. This fo llo ws immediately from the fact that for a complex of Z X [ ~ ]- mo dules M , gr ~ ( M ) is represen ted by the mapping cone of M ~ − → M and similarly for Z Y [ ~ ]-mo dules. Q.E.D. Recall that A is a sheaf of algebras satisfying h yp otheses (1.2.2) and either (1.2.3) or (1.3.1). The functor gr ~ induces a functor (w e k eep the same notation): gr ~ : D b coh ( A ) − → D b coh ( A 0 ) . (1.4.3) The f o llo wing prop osition is an immediate consequence o f Lemma 1 .4.2 and Nak a y ama’s lemma. 34 CHAPTER 1. MODULES OVER FORMAL DEFORMA TIONS Prop osition 1.4.5. L et M ∈ D b coh ( A ) and let a ∈ Z . The c onditions b elow ar e e quivalent: (i) H a (gr ~ ( M )) ≃ 0 , (ii) H a ( M ) ≃ 0 and H a +1 ( M ) ha s no ~ -torsion. Corollary 1 .4.6. The functor gr ~ in (1 .4.3) is c on s e rvative (i.e., a m orphism in D b coh ( A ) is an isomo rphism as so on as its image by gr ~ is an isomorphism in D b coh ( A 0 ) ) . Pr o of. Consider a morphism ϕ : M − → N in D b coh ( A ) and a ssume that it induces an isomorphism gr ~ ( ϕ ) : g r ~ ( M ) − → gr ~ ( N ) in D b coh ( A 0 ). Let M − → N − → L +1 − − − → b e a distinguished triangle. Then g r ~ L ≃ 0 , and hence all the cohomologies of L v anishes b y the prop osition ab ov e, whic h means that L ≃ 0 . Q.E.D. Homological dimension In the sequel, for a left No etherian K -a lgebra R , w e shall sa y that a coherent R -mo dule P is lo cally pro jectiv e if, for a n y o p en subset U ⊂ X , the functor H om R ( P , • ) : Mo d coh ( R | U ) − → Mo d( K U ) is exact. This is equiv alent to o ne o f the fo llo wing conditions: (i) for eac h x ∈ X , the stalk P x is pro jectiv e as an R x -mo dule, (ii) fo r each x ∈ X , the stalk P x is ﬂat as a n R x -mo dule, (iii) P is lo cally a direct summand of a free R -mo dule of ﬁnite rank. Lemma 1.4.7. A c oher ent A -mo dule P is lo c al ly pr oje ctive if and only if P has no ~ -torsion and gr ~ P is a lo c al ly pr oje ctive A 0 -mo dule. Pr o of. W e set for short A := A x and A 0 := ( A 0 ) x . Note that A 0 ≃ gr ~ A . Let P b e a ﬁnitely generated A - mo dule. (i) Assume that P is pro jective . Then P is a direct summand of a free A - mo dule. It follows tha t P has no ~ -to rsion and gr ~ P is also a direct summand of a f r ee A 0 -mo dule. (ii) Assume that P has no ~ -to rsion a nd gr ~ P is pro jectiv e. Consider an exact sequence 0 − → N u − → L − → P − → 0 in whic h L is free of ﬁnite rank. Applying the functor gr ~ w e ﬁnd the exact sequence 0 − → gr ~ N gr ~ u − − → gr ~ L − → gr ~ P − → 0 and gr ~ P b eing pro jectiv e, t here exists a map v : gr ~ L − → gr ~ N suc h that 1.4. ~ - GRADUA TION AND ~ -LOCALIZA TION 35 v ◦ gr ~ u = id gr ~ N . Let us choose a map v : L − → N suc h that gr ~ ( v ) = v . Since gr ~ ( v ◦ u ) = id gr ~ N , w e may write v ◦ u = id N − ~ ϕ where ϕ : N − → N is an A - linear map. The map id N − ~ ϕ is inv ertible and w e denote by ψ its in v erse. Then ψ ◦ v ◦ u = id N , whic h pro v es that P is a direct summand of a free A -mo dule. Q.E.D. Theorem 1.4.8. L et d ∈ N . Assume that an y c oher ent A 0 -mo dule lo c a l l y admits a r esolution of length ≤ d by fr e e A 0 -mo dules of ﬁnite r ank. Then (a) for any c oher ent lo c al ly pr oje ctive A -mo dule P , ther e lo c al ly exis ts a fr e e A - m o dule of ﬁnite r ank F such that P ⊕ F is fr e e of ﬁ n ite r ank, (b) any c oher ent A -mo dule lo c al ly admits a r esolution of length ≤ d + 1 by fr e e A - m o dules of ﬁnite r ank. Pr o of. ( a) It is w ell-known (see e.g., [56, Lem. B.2 .2]) that the result in (a) is true when replacing A with A 0 . Now, let P b e as in the statemen t. Then gr ~ P is pro jectiv e and coheren t. Therefore, there exists a lo cally free A -mo dule F suc h that gr ~ P ⊕ gr ~ F is free of ﬁnite rank o v er A 0 . This implies that P ⊕ F is f ree of ﬁnite ra nk ov er A by Lemma 1.2.17. (b)-(i) Let M ∈ Mo d coh ( A ) and let us ﬁrst assume that M has no ~ - torsion. Since A is coheren t, there exists lo cally a n exact sequence 0 − → K − → L d − 1 − → · · · − → L 0 − → M − → 0 , the A -mo dules L i (0 ≤ i ≤ d − 1) b eing free o f ﬁnite ra nk. Applying the functor g r ~ , w e ﬁnd an exact sequence of A 0 -mo dules and it f o llo ws that gr ~ ( K ) is pro j ectiv e and ﬁnitely generated. Therefore K is pro jectiv e a nd ﬁnitely g enerated. Let F b e as in the statemen t (a). Replacing K and L d − 1 with K ⊕ F a nd L d − 1 ⊕ F resp ectiv ely , the result fo llows in this case. (b)-(ii) In general, any coheren t A - mo dule M lo cally admits a resolution 0 − → N − → L − → M − → 0 , where L is a free A - mo dule of ﬁnite rank. Since N has no ~ -torsion, N admits a free resolution with length d , and the result follo ws. Q.E.D. Corollary 1.4.9. We make the hyp otheses of Th e or em 1.4.8. L et M • b e a c omplex of A - mo dules c onc e ntr a te d in de gr e es [ a, b ] and ass ume that H i ( M ) is c oher ent for al l i . Then, in a neighb orho o d of e ach x ∈ X , ther e exists a quasi-isomorphism L • − → M • wher e L • is a c omplex of fr e e A -mo dules of ﬁnite r ank c on c entr ate d in de gr e es [ a − d − 1 , b ] . Pr o of. The pro of uses [41, Lem. 13.2.1] (or rather the dual statemen t). Since w e do not use this result here, details a re left to the reader. Q.E.D. 36 CHAPTER 1. MODULES OVER FORMAL DEFORMA TIONS Lo calization F or a Z X [ ~ ]-algebra R with no ~ -t o rsion, w e set R lo c := Z X [ ~ , ~ − 1 ] ⊗ Z X [ ~ ] R , (1.4.4) and we call R lo c the ~ -lo c alization of R . F or an R - mo dule M , we a lso set M lo c := R lo c ⊗ R M ≃ Z X [ ~ , ~ − 1 ] ⊗ Z X [ ~ ] M . Lemma 1.4.10. The a l g e br a A lo c is No etherian . Pr o of. L et T b e a n indeterminate. One know s b y [37, Th. A.30] that A [ T ] is No etherian. Since A lo c ≃ A [ T ] / A [ T ]( T ~ − 1 ), the result fo llo ws. Q.E.D. 1.5 Cohomologicall y c o mplete mo dules In order to give a criterion for the coherency of the cohomolog ies of a complex of mo dules ov er an algebra A satisfying (1.2.2) and either (1.2.3) or (1.3.1). w e in tro duce the notion of cohomologically complete complexes. In this section, R is a Z [ ~ ]-a lg ebra satisfying R has no ~ -torsion. (1.5.1) Recall that M lo c := Z [ ~ , ~ − 1 ] ⊗ Z [ ~ ] M for an R -mo dule M . Lemma 1.5.1. F or M , M ′ ∈ D b ( R lo c ) , we have R H om R loc ( M , M ′ ) ∼ − → R H om R ( M , M ′ ) . Pr o of. W e hav e R lo c L ⊗ R M ≃ M . Hence, R H om R loc ( M , M ′ ) ≃ R H om R loc ( R lo c L ⊗ R M , M ′ ) ≃ R H o m R ( M , M ′ ) . Q.E.D. The next result is obv ious. Lemma 1.5.2. The triangulate d c ate gory D( R lo c ) is e quivalent to the ful l sub c ate gory of D( R ) c onsisting of obje cts M s atisfying one of the fo l lowing e quivalent c onditions: 1.5. COHOMOLOGICALL Y COMPLETE MOD ULES 37 (i) gr ~ ( M ) = 0 , (ii) ~ : H i ( M ) − → H i ( M ) is an isomorphism for any inte g e r i , (iii) M − → R lo c L ⊗ R M is an isomorphism, (iv) R H om R ( R lo c , M ) − → M is an isomorphism , (v) R H om R ( R lo c / R , M ) ≃ 0 . Lemma 1.5.3. L et K b e a Z [ ~ ] -mo dule with pr oje ctive dimension ≤ 1 . T hen for any M ∈ D( R ) , any op en subset U an d any inte ger i , we have an exact se quenc e 0 − → Ext 1 Z [ ~ ]  K, H i − 1 ( U ; M )  − → H i  U ; R H om Z [ ~ ] ( K , M )  − → Hom Z [ ~ ]  K, H i ( U ; M )  − → 0 . Pr o of. W e hav e a distinguished triang le RHom Z [ ~ ]  K, τ 0. Therefore, R i π ( L L ⊗ R M ) ≃ 0 for i > 1. O n the other hand, since { Γ( U ; M n ) } n satisﬁes the Mittag-Leﬄer condition, we get that R 1 π ( L L ⊗ R M ) ≃ 0. Q.E.D. 42 CHAPTER 1. MODULES OVER FORMAL DEFORMA TIONS Hence, M is cohomologically complete if and only if the mor phism M − → R π  “lim ← − ” n ( R / R ~ n ) L ⊗ R M  is an isomorphism. Prop osition 1.5.12. L et f : X − → Y b e a c ontinuous m ap, and M ∈ D( Z X [ ~ ]) . If M is c ohomolo gic al ly c omplete, then so is R f ∗ M . Pr o of. It immediately follows f r o m R H om Z Y [ ~ ] ( Z Y [ ~ , ~ − 1 ] , R f ∗ M ) ≃ R f ∗ R H om Z X [ ~ ] ( Z X [ ~ , ~ − 1 ] , M ) . Q.E.D. 1.6 Cohomologicall y c o mplete A -mo dules In this section, A is a K -alg ebra satisfying h yp otheses (1.2.2) and either (1.2.3) or (1.3.1). Theorem 1.6.1. L et M ∈ D b coh ( A ) . Then M is c ohomolo gic al ly c omplete. Pr o of. Since any coherent mo dule is an extension of a mo dule without ~ - torsion by an ~ - torsion mo dule, it is enough to tr eat each case. Assume ﬁrst that M is an ~ -torsion coheren t A -mo dule. Since t he ques - tion is lo cal, w e ma y assume tha t there exists n suc h that ~ n M = 0. The n the action of ~ on the cohomology groups of R H om A ( A lo c , M ) is nilp oten t and inv ertible, and hence t he coho mo lo gy groups v anishes . No w a ssume tha t M is a coherent A -mo dule without ~ -to r sion. Then Corollary 1.5.7 sho ws that M is cohomologically complete. Q.E.D. Corollary 1.6.2. If M ∈ D b coh ( A ) and N ∈ D( A ) , then R H om A ( N , M ) is c ohomolo gic al ly c omplete. Pr o of. It is a n immediate conseq uence of Prop osition 1.5.10 and the theorem ab o ve . Q.E.D. In the course of the pro of of Theorem 1.6.4 b elow, w e shall use the fol- lo wing elemen tary lemma t hat we state here without pro of. Lemma 1.6.3 (Cross Lemma) . L et C b e an ab elian c ate gory and c onsider an ex act d iagr am in C X 2   X 1 / / Y   / / Z 1 Z 2 . 1.6. COHOMOLOGICALL Y COMPLETE A -MODULES 43 Then the c onditions b elo w ar e e quivalent: (a) Im( X 2 − → Z 1 ) ∼ − → Im( Y − → Z 1 ) , (b) Im( X 1 − → Z 2 ) ∼ − → Im( Y − → Z 2 ) , (c) X 1 ⊕ X 2 − → Y is an epimorphism. Theorem 1.6.4. L et M ∈ D + ( A ) and assume: (a) M is c ohomolo gic al ly c omplete, (b) gr ~ ( M ) ∈ D + coh ( A 0 ) . Then, M ∈ D + coh ( A ) , and we have the iso m orphism H i ( M ) ∼ − → lim ← − n H i ( A n L ⊗ A M ) for al l i ∈ Z . Pr o of. W e s hall assume (1.2.3). The case of hypothesis ( 1 .3.1) could be treated with sligh t mo diﬁcatio ns. Recall that A n := A / ~ n +1 A and set M n = A n L ⊗ A M , N j n := H j ( M n ). (1) F or each n ∈ N , the distinguished triangle A / ~ n A ~ − − → A / ~ n +1 A − → A / ~ A +1 − − − → induces the dis tinguished triangle M n − 1 ~ − − → M n − → M 0 +1 − − − → . (1.6.1) This triangle giv es rise to the long exact sequence N j − 1 0 − → N j n − 1 ~ − − → N j n − → N j 0 − → N j +1 n − 1 (1.6.2) from whic h w e deduce b y induction on n that N j n is a coheren t A - mo dule for an y j and n ≥ 0 by using the h yp othesis ( b). (2) Let us sho w tha t “lim ← − ” n Cok er( N j n ~ − → N j n ) and “lim ← − ” n Ker( N j n ~ − → N j n ) are lo cally represen table for all j ∈ Z . (1.6.3) Consider the distinguished tr ia ngle: M 0 ~ n +1 − − − − → M n +1 − → M n +1 − − − → . (1.6.4) 44 CHAPTER 1. MODULES OVER FORMAL DEFORMA TIONS It gives rise to the long exact sequence · · · − → N j 0 ~ n +1 − − − − → N j n +1 − → N j n ϕ j n − − − → N j +1 0 − → · · · . (1.6.5) No w consider the exact diag r a m, deduced from (1.6.2) and (1.6.5): N j n +1   N j n − 1 ~ · / / ϕ j n − 1 $ $ H H H H H H H H N j n ϕ j n   / / N j 0 N j +1 0 . (1.6.6) Here the commutativit y of t he triangle f ollo ws from the comm utative diagram M 0 ~ n / / id   M n / / ~   M n − 1 +1 / / ~   M 0 ~ n +1 / / M n +1 / / M n +1 / / Hence Im( ϕ j n − 1 ) ⊂ Im( ϕ j n ) ⊂ N j +1 0 . Therefore, the sequence { Im ϕ j n } n of coheren t A - submo dules of N j +1 0 is increasing and th us lo cally stationary . It fo llows from ( 1 .6.6) and Lemma 1.6.3 that the decreasing sequence { Im( N j n − → N j 0 ) } n is lo cally station- ary for an y j ∈ Z . (1.6.7) Since Coke r ( N j n − 1 ~ − → N j n ) ≃ Im( N j n − → N j 0 ) b y (1.6.2), we deduce that “lim ← − ” n Cok er( N j n ~ − → N j n ) ≃ “lim ← − ” n Cok er( N j n − 1 ~ − → N j n ) is lo cally represen table. Since Ker( N j n − 1 ~ − → N j n ) ≃ N j − 1 0 / Im( N j − 1 n − → N j − 1 0 ) by (1 .6.2), w e get that “lim ← − ” n Ker( N j n ~ − → N j n ) ≃ “lim ← − ” n Ker( N j n − 1 ~ − → N j n ) is lo cally repre- sen table. Therefore, w e ha v e pro ved (1.6.3). Then b y Prop osition 1.2.1 8 , lim ← − n N j n is a coherent A -mo dule and { N j n } n satisﬁes the Mittag-Leﬄer condition. 1.6. COHOMOLOGICALL Y COMPLETE A -MODULES 45 (3) Hence it remains t o prov e that H j ( M ) ∼ − → lim ← − n N j n for a n y j . Set M ′ = (“lim ← − ” n A n ) L ⊗ A M ∈ D + (Pro(Mo d( A ) ) ) and N j = H j ( M ′ ) ≃ “lim ← − ” n N j n ∈ Pro(Mo d( A )) . Lemma 1 .5.11 implies that M ∼ − → R π M ′ . Since the N j n ’s are coheren t A -mo dules, for an y an y U ∈ B , H i ( U ; N j n ) = 0 ( i > 0) and { N j n ( U ) } n satisﬁes the Mittag- Leﬄer condition. Hence in the exact sequence 0 − → R 1 π  “lim ← − ” n H i − 1 ( U ; N j n )  − → H i ( U ; R π N j ) − → lim ← − n H i ( U ; N j n ) − → 0 , the ﬁrst and the last term v anish, and we obtain R i π N j = 0 for an y i > 0. Let us sho w that H j ( M ) ∼ − → lim ← − n N j n b y induction on j . Assuming H j ( M ) ∼ − → lim ← − n N j n for j < c , let us sho w that H c ( M ) ∼ − → lim ← − n N c n . By the a ssumption, H i ( M ) ∼ − → R π ( N i ) for an y i < c . Hence τ a . F or an y n ∈ Z w e can lo cally ﬁnd a ﬁnite complex L of free A op -mo dules of ﬁnite rank suc h that the re exists a dis tinguished tria ng le L L ⊗ A M − → N L ⊗ A M − → G where G ∈ D n + a . Hence N L ⊗ A M is cohomolog ically complete. Q.E.D. 46 CHAPTER 1. MODULES OVER FORMAL DEFORMA TIONS Flatness Theorem 1.6.6. Assume that A op / ~ A op is a No e therian ring and the ﬂabby dimension of X is ﬁn i te. L et M b e an A -mo dule. Assume the fol lowing c onditions: (a) M has n o ~ -torsion, (b) M is c ohom olo gic al ly c omplete, (c) M / ~ M is a ﬂat A 0 -mo dule. Then M is a ﬂat A -mo dule. Pr o of. L et N b e a coheren t A op -mo dule. It is enough to show that w e hav e H i ( N L ⊗ A M ) = 0 for a n y i < 0. W e kno w b y Prop osition 1.6.5 that N L ⊗ A M is cohomologically complete. Since gr ~ ( N L ⊗ A M ) ≃ (gr ~ N ) L ⊗ A 0 (gr ~ M ) b e- longs to D ≥ 0 ( Z X ), we ha v e N L ⊗ A M ∈ D ≥ 0 ( Z [ ~ ] X ) b y Prop osition 1 .5 .8. Q.E.D. Corollary 1.6.7. In the si tuation of The or em 1.6.6, assume mor e over that M / ~ M is a faithful l y ﬂat A 0 -mo dule. Then M is a faithful ly ﬂat A -mo dule. Pr o of. L et N b e a coheren t A op -mo dule suc h that N ⊗ A M ≃ 0. W e hav e to show that N ≃ 0. By Theorem 1.6.6, w e know that M is ﬂat, so that N ⊗ A M ≃ N L ⊗ A M . Therefore (gr ~ N ) L ⊗ A 0 (gr ~ M ) ≃ gr ~ ( N ⊗ A M ) ≃ 0 and the h yp othesis tha t M / ~ M is fa it hf ully ﬂat implies t hat gr ~ N ≃ 0. Since N is coheren t, Corollary 1.4.6 implies that N ≃ 0 . Q.E.D. Prop osition 1.6.8. Assume ( 1 .2.2) and (1.2.3) . L et U b e an op en subset of X satisfying: U ∩ V ∈ B for any V ∈ B . (1.6.8) Then for any c o her ent A -mo dule M , we ha v e (i) R n Γ U ( M ) = 0 for any n 6 = 0 , (ii) Γ U ( A ) ⊗ A M − → Γ U ( M ) is an isomorphism, (iii) Γ U ( A ) is a ﬂat A op -mo dule. 1.6. COHOMOLOGICALL Y COMPLETE A -MODULES 47 Pr o of. ( i) Since R n Γ U ( M ) is the sheaf asso ciated with the presheaf V 7→ H n ( U ∩ V ; M ), (i) follo ws from Theorem 1.2.5 (v). (ii) The question b eing lo cal, w e ma y assume that w e hav e an exact se- quence 0 − → N − → L − → M − → 0 , where L is a free A -mo dule of ﬁnite rank. Then, w e hav e a comm utativ e diagram with exact rows by (i): Γ U ( A ) ⊗ A N / /   Γ U ( A ) ⊗ A L / / ≀   Γ U ( A ) ⊗ A M / /   0 0 / / Γ U ( N ) / / Γ U ( L ) / / Γ U ( M ) / / 0 . Since the middle ve rtical arrow is an isomorphism, Γ U ( A ) ⊗ A M − → Γ U ( M ) is an epimorphism. Applying this to N , Γ U ( A ) ⊗ A N − → Γ U ( N ) is an epimorphism. Hence, Γ U ( A ) ⊗ A M − → Γ U ( M ) is a n isomorphism. (iii) By (i) and (ii), M 7→ Γ U ( A ) ⊗ A M is an exact functor on the category of coheren t A -mo dules. It follo ws that f or all x ∈ X , the functor M 7→ (Γ U ( A )) x ⊗ A x M x is exact on the category Mo d coh ( A ). Therefore, (Γ U ( A )) x is a ﬂa t A op x -mo dule. Q.E.D. Remark 1.6.9. The results of this c hapter can b e generalized in the fo llo wing situation. Let A b e a she af of rings on a topolo gical space X and let I b e a b oth-sided sheaf of ideals of A . W e assume that: there exists lo cally a section s of I s uch that A ∋ a 7→ as and A ∋ a 7→ sa giv e isomorphisms A ∼ − → I . W e set A 0 = A / I , A ( − n ) = I n ⊂ A and A ( n ) = R H om A ( A ( − n ) , A ) for n ≥ 0. Then w e hav e A ( n ) ⊂ A ( n + 1), and A ( n ) ⊗ A A ( m ) ≃ A ( n + m ). W e set A lo c = lim − → n A ( n ) and for an A -mo dule M , w e set M ( n ) = A ( n ) ⊗ A M . W e sa y tha t M is I -torsion free if M ( − 1) − → M is a monomorphism. Of course, A is I -torsion free. Finally , for an A -mo dule M w e set c M := lim ← − n Cok er( M ( − n ) − → M ). Instead of (1.2 .2), we assume          (i) A ∼ − → c A , (ii) A 0 is a left Noetherian ring. (1.6.9) Under the assumptions (1.6.9) and (1.2.3), all the results of this c hapter ho ld with suitable mo diﬁcations. 48 CHAPTER 1. MODULES OVER FORMAL DEFORMA TIONS In par t icular, our theory can b e a pplied when X = T ∗ M is the cotangent bundle to a complex manifold M and A = c E X (0) is the ring of f o rmal micro diﬀeren tial op erators o f o r der 0 (see Section 6.1 for more details on the ring o f for ma l microdiﬀerential op erato rs). Chapter 2 DQ -algebroids 2.1 Algebroi ds In this section, X denotes a t o p ological space and recall that K is a com- m utativ e unital ring. A K -linear category means a category C suc h that Hom C ( X , Y ) is endow ed with a K -mo dule structure for an y X , Y ∈ C , and the composition map Hom C ( X , Y ) × Hom C ( Y , Z ) − → Hom C ( X , Z ) is K - bilinear for a ny X , Y , Z ∈ C . One deﬁnes similarly the notion of a K -linear stac k. The notion of an algebroid has b een in tro duced in [45]. W e refer to [18] fo r a more systematic study and to [41] for an intro duction to stac ks. Recall tha t a K -algebroid A on X is a K -linear stack lo cally non empt y and suc h that for any op en subset U of X , an y t w o ob jects o f A ( U ) are lo cally isomorphic. If A is a K -algebra (an algebra, not a sheaf of algebras), w e denote b y A + the K -linear category with one ob ject and hav ing A a s the endomorphism ring of this ob ject. Let A b e a sheaf of K -algebras on X and consider the prestac k U 7→ A ( U ) + ( U op en in X ). W e denote by A + the a ssociat ed stack . Then A + is a K -algebroid and is called the K -algebroid associated with A . The category A + ( X ) is equiv alen t to the full sub category o f Mo d( A op ) consisting of ob jects lo cally isomorphic to A op . Con v ersely , if A is an algebroid on X and σ ∈ A ( X ), then A is equiv a- len t to the algebroid E nd A ( σ ) + . F or an algebroid A and σ , τ ∈ A ( U ), the K -algebras E nd A ( σ ) and E n d A ( τ ) are lo cally isomorphic. Hence, any deﬁnition of lo cal na t ure con- cerning sheav es of K -algebras, suc h as b eing coheren t or No etherian, extends to K -algebroids. Recall that fo r a n a lg ebroid A , the algebroid A op is deﬁned b y A op ( U ) = 49 50 CHAPTER 2. DQ -ALGEBROIDS ( A ( U )) op ( U o p en in X ). Then, if A is a sheaf of K -a lg ebras, ( A op ) + ≃ ( A + ) op . Con v ention 2.1.1. If A is a sheaf of algebras and if there is no risk of confusion, we shall k eep t he same notatio n A to denote the a sso ciated alge- broid. Note that tw o algebras ma y no t b e isomorphic even if the asso ciated algebroids are equiv alent. Example 2.1.2. Let X b e a complex manifold, L a line bundle on X a nd denote as usual by D X the ring of diﬀerential op erators on X . The ring of L -t wisted diﬀerential op erato rs is g iv en by D L X := L ⊗ O X D X ⊗ O X L ⊗ − 1 . In general the tw o algebras D X and D L X are not isomorphic although the asso ciated algebroids are equiv alen t. The equiv alence is obtained by using the bi-in v ertible mo dule D X ⊗ O X L ⊗ − 1 (see Deﬁnition 2.1 .10 and Lemma 2 .1.11 b elo w). Let U = { U i } i ∈ I b e an op en co v ering of X . In the sequel w e set U ij := U i ∩ U j , U ij k := U i ∩ U j ∩ U k , etc. Consider the data of  a K -a lgebroid A on X , σ i ∈ A ( U i ) and isomorphisms ϕ ij : σ j | U ij ∼ − → σ i | U ij . (2.1.1) T o these data , we asso ciate: • A i = E nd A ( σ i ), • f ij : A j | U ij ∼ − → A i | U ij , the K -algebra isomorphism a 7→ ϕ ij ◦ a ◦ ϕ − 1 ij , • a ij k , the inv ertible elemen t of A i ( U ij k ) giv en by ϕ ij ◦ ϕ j k ◦ ϕ − 1 ik . Then:  f ij ◦ f j k = Ad( a ij k ) ◦ f ik on U ij k , a ij k a ik l = f ij ( a j k l ) a ij l on U ij k l . (2.1.2) (Recall that Ad( a )( b ) = aba − 1 .) Con v ersely , let A i b e K -algebras on U i ( i ∈ I ) , let f ij : A j | U ij ∼ − → A i | U ij ( i, j ∈ I ) b e K -a lgebra isomorphisms, and let a ij k ( i, j, k ∈ I ) b e inv ertible sections of A i ( U ij k ) satisfying (2.1.2). One calls ( { A i } i ∈ I , { f ij } i,j ∈ I , { a ij k } i,j,k ∈ I ) (2.1.3) 2.1. ALGEBROIDS 51 a gluing datum for K -algebroids on U . The following result, whic h easily follo ws fro m [27, L em 3.8.1], is stated (in a diﬀeren t form) in [36] and go es bac k to [28 ]. Prop osition 2.1.3. Assume that X is p ar ac omp a ct. Consider a gluing da - tum (2.1.3) on U . T hen ther e exist an algebr oid A on X and { σ i , ϕ ij } i,j ∈ I as in (2.1.1) to whic h this gluing datum is asso ciate d. Mor e over, the data ( A , σ i , ϕ ij ) ar e unique up to an e quivalenc e of stacks, this e quivale nc e b eing unique up to a unique isomorphism . W e will giv e another construction in Prop osition 2.1.13, whic h may b e applied to non paracompact spaces suc h as alg ebraic v arieties. F or a n algebroid A , o ne deﬁnes the K -linear ab elian catego ry Mo d( A ), whose ob jects are called A -mo dules, by setting Mo d( A ) := Fct K ( A , M od ( K X )) . (2.1.4) Here M od ( K X ) is the K -linear stac k of shea v es of K -mo dules o n X and, for t w o K -linear stac ks A 1 and A 2 , Fct K ( A 1 , A 2 ) is the catego r y o f K -linear functors of stacks from A 1 to A 2 . If A is the algebroid asso ciated with a K -algebra A on X , then Mo d( A ) is equiv alen t to Mo d( A ). The category Mo d( A ) is a Grothendiec k category and w e denote b y D( A ) its deriv ed cat- egory and by D b ( A ) its b ounded deriv ed category . F or a K -algebroid A , the K -linear prestac k U 7→ Mo d( A | U ) is a stac k and w e denote it b y Mod ( A ). In the sequel, w e shall write for short “ σ ∈ A ” instead of “ σ ∈ A ( U ) fo r some o p en set U ”. Deﬁnition 2.1.4. An A -mo dule L is in ve rtible if it is lo cally isomor phic to A , na mely fo r an y σ ∈ A , t he E nd A ( σ )-mo dule L ( σ ) is lo cally isomorphic to E n d A ( σ ). This terminology is motiv ated b y the fact that for an inv ertible mo dule L , if w e set B := ( E nd A ( L )) op , then H om A ( L , A ) ⊗ A L ≃ B and L ⊗ B H om A ( L , A ) ≃ A . W e denote b y Inv( A ) the full sub category of Mo d( A ) consisting of in v ertible A - mo dules and b y Inv ( A ) the corresp onding full substac k of Mod ( A ). Then w e hav e equiv alences of K -linear stacks A ∼ − → Inv ( A op ) ∼ − → Inv ( A ) op . Recall that for t wo K -linear categories C and C ′ , one deﬁnes their tensor pro duct C ⊗ K C ′ b y setting O b( C ⊗ K C ′ ) = Ob( C ) × Ob( C ′ ) a nd Hom C ⊗ K C ′ (( M , M ′ ) , ( N , N ′ )) = Hom C ( M , N ) ⊗ K Hom C ′ ( M ′ , N ′ ) 52 CHAPTER 2. DQ -ALGEBROIDS for M , N ∈ C and N , N ′ ∈ C ′ . Then C ⊗ K C ′ is a K - linear category . F or a pair of K -alg ebroids A and A ′ , the K -alg ebroid A ⊗ K A ′ is the K -linear stac k asso ciated with the prestack U 7→ A ( U ) ⊗ K A ′ ( U ) ( U o p en in X ). W e ha ve Mo d( A ⊗ K A ′ ) ≃ Fct K ( A , M od ( A ′ )) . F or a K -a lgebroid A , Mo d( A ⊗ K A op ) has a canonical ob j ect g iv en by A ⊗ K A op ∋ ( σ, σ ′ op ) 7→ H om A ( σ ′ , σ ) ∈ Mod ( K X ) . W e denote this ob ject by the same letter A . If A is asso ciated with a K - algebra A , this ob ject corr esponds to A , regarded as an ( A ⊗ K A op )-mo dule. F or K -algebroids A i ( i = 1 , 2 , 3), we hav e the tensor pro duct functor • ⊗ A 2 • : Mo d( A 1 ⊗ K A op 2 ) × Mo d( A 2 ⊗ K A op 3 ) (2.1.5) − → Mo d( A 1 ⊗ K A op 3 ) , and the H om functor H om A 1 ( • , • ) : Mo d( A 1 ⊗ K A op 2 ) op × Mo d( A 1 ⊗ K A op 3 ) (2.1.6) − → Mo d( A 2 ⊗ K A op 3 ) . In particular, w e hav e • ⊗ A • : Mo d( A op ) × Mo d( A ) − → Mo d( K X ) , H om A ( • , • ) : Mod( A ) op × Mo d( A ) − → Mo d( K X ) , H om A ( • , A ) : Mo d( A ) op − → Mo d( A op ) . Since Mo d( A ) is a Grothendiec k category , any left exact functor f r o m Mo d( A ) to an ab elian category admits a rig ht deriv ed functor. No w consider the tensor pro duct in (2.1.5). It admits a left deriv ed func- tor as so on as A 3 is K -ﬂat. Indeed, an y M ∈ Mo d( A 2 ⊗ ( A 3 ) op ) is a quotien t of an A 2 -ﬂat mo dule since there is an exact sequence M s ∈ Hom ( L , M | U ) L − → M − → 0 , where U ranges ov er the family of op en subsets of X and L ∈ ( A 2 ⊗ ( A 3 ) op ) op ( U ). (Recall that for a K -algebroid A , A op ( U ) is equiv alen t to Inv ( A )( U ).) Not e tha t L is A 2 -ﬂat since ( A 3 ) op is K -ﬂat. The followin g lemma is obv ious. 2.1. ALGEBROIDS 53 Lemma 2.1.5. L et A and A ′ b e K -algebr oids. T o giv e a functor of alge- br oids ϕ : A ′ − → A is e quivalent to giv ing an ( A ′ ⊗ A op ) -mo dule L which is lo c a l ly is o morphic to A ( i.e. for σ ∈ A an d σ ′ ∈ A ′ , L ( σ ′ ⊗ σ op ) is lo c al ly isomorphic to E nd A ( σ ) as an E nd A ( σ ) op -mo dule ) . The A ′ ⊗ A op -mo dule L corresp onding to ϕ is the mo dule induced from the A ⊗ A op -mo dule A by ϕ ⊗ A op : A ′ ⊗ A op − → A ⊗ A op . The forgetful functor Mo d( A ) − → Mo d( A ′ ) is isomorphic to M 7→ L ⊗ A M . Let f : X − → Y b e a con tin uous map and let A b e a K -algebroid on Y . W e denote b y f − 1 A the K -linear stac k asso ciated with the prestac k S giv en b y: S ( U ) = { ( σ, V ) ; V is a n op en subset o f Y suc h that f ( U ) ⊂ V and σ ∈ A ( V ) } for a n y op en su bset U of X , Hom S ( U )  ( σ , V ) , ( σ ′ , V ′ )  ) = Γ( U ; f − 1 H om A ( σ , σ ′ )) . Then f − 1 A is a K -algebroid. W e hav e functors f ∗ , f ! : Mo d( f − 1 A ) − → Mo d( A ) , f − 1 : Mo d( A ) − → Mod( f − 1 A ) . F or t w o to p ological spaces X 1 and X 2 , let p i : X 1 × X 2 − → X i b e the pro jection. Let A i b e a K -algebroid o n X i ( i = 1 , 2). W e deﬁne a K - algebroid on X 1 × X 2 , called the external tensor pro duct of A 1 and A 2 , b y setting: A 1 ⊠ A 2 := p − 1 1 A 1 ⊗ p − 1 2 A 2 . W e hav e a canonical bi-functor • ⊠ • : Mo d( A 1 ) × Mo d( A 2 ) − → Mo d( A 1 ⊠ A 2 ) . Bi-in v ertible mo dules The following no t ion of bi- in v ertible mo dules will app ear a ll along these No t es since it describ es equiv alenc es of algebroids. 54 CHAPTER 2. DQ -ALGEBROIDS Deﬁnition 2.1.6. Let A and A ′ b e t w o shea v es of K -a lg ebras. An A ⊗ A ′ - mo dule L is called bi-invertible if there exists lo cally a section w of L suc h that A ∋ a 7→ ( a ⊗ 1) w ∈ L and A ′ ∋ a ′ 7→ (1 ⊗ a ′ ) w ∈ L g iv e isomorphisms of A -mo dules and A ′ -mo dules, resp ectiv ely . Lemma 2.1.7. L et L b e a bi-invertible A ⊗ A ′ -mo dule and let u b e a se ction of L . If A ∋ a 7→ ( a ⊗ 1) u ∈ L is an isomorphism of A -mo dules, then A ′ ∋ a ′ 7→ (1 ⊗ a ′ ) u ∈ L is also an isomorphism of A ′ -mo dules. Pr o of. L et w b e as ab ov e. There exist a ∈ A and b ∈ A suc h that u = ( a ⊗ 1) w and w = ( b ⊗ 1) u . Then w e ha v e u = ( ab ⊗ 1) u and hence ab = 1. Similarly w = ( ba ⊗ 1 ) w implies ba = 1. Hence we ha ve a comm utative diagram A ′ w ∼ / / u & & M M M M M M M M M M M M L ≀ a ⊗ 1   L and we obta in the des ired result. Q.E.D. Remark 2.1.8. Let A a nd B b e t wo K -algebras and let L b e an ( A ⊗ B op )- mo dule. Ev en if L is isomorphic to A as a n A -mo dule and isomorphic to B op as a B op -mo dule, L is not neces sarily bi- in v ertible, as shown by the follow ing example. Let I b e an inﬁnite set and tak e o ∈ I . Set I ∗ = I \ { o } . Then there exists a bijection v : I ∗ − → I . Set X = { a ∈ Hom Set ( I , I ); a (o) = o } , Y = { b ∈ Hom Set ( I , I ); b (o) = o and b ( I ∗ ) ⊂ I ∗ } . Set Z = X . Then X a nd Y are se mi-g r oups and X acts on Z from the left and Y acts on Z from the righ t. Let v ′ ∈ Z b e the unique elemen t extending v . Then id I ∈ Z g iv es an isomorphism X ∼ − → Z ( X ∋ a 7→ a ∈ Z ) and v ′ ∈ Z induces an isomorphism Y ∼ − → Z ( Y ∋ b 7→ v ′ ◦ b ∈ Z ). Let A = K [ X ] and B = K [ Y ] b e the semigroup algebras corresp onding t o X and Y . Set L = K [ Z ]. Then L is an ( A ⊗ B op )-mo dule and L is isomorphic to A as an A -mo dule and isomorphic to B op as an B op -mo dule. Let u b e the elemen t of L corresp onding to id I . Then u giv es an isomorphism A ∋ a 7→ ( a ⊗ 1) u ∈ L . Since the imag e of B op ∋ b 7→ (1 ⊗ b ) u ∈ L is K [ Y ] 6 = L , L is not bi- inv ertible in view o f Lemma 2.1.7. Ho w ev er the follow ing partial result holds. 2.1. ALGEBROIDS 55 Lemma 2.1.9. L et A and A ′ b e K -algeb r as and let L b e an A ⊗ A ′ -mo dule. Assume that L is isomorphic to A a s an A -mo dule and isomorphic to A ′ as an A ′ -mo dule. If w e assume mor e over that A x is a left no etherian ring for any x ∈ X , then L is bi-invertible. Pr o of. Assume that A ∋ a 7→ ( a ⊗ 1) u ∈ L and A ′ ∋ a ′ 7→ (1 ⊗ a ′ ) v ∈ L are isomorphisms for some u, v ∈ L . Set v = ( a ⊗ 1) u and u = (1 ⊗ a ′ ) v . There exists a ′′ ∈ A suc h that (1 ⊗ a ′ ) u = ( a ′′ ⊗ 1) u . Then w e hav e u = (1 ⊗ a ′ ) v = (1 ⊗ a ′ )( a ⊗ 1) u = ( a ⊗ 1)(1 ⊗ a ′ ) u = ( aa ′′ ⊗ 1 ) u . Hence w e obtain aa ′′ = 1. Therefore the A -linear endomorphism f : A ∋ z 7→ z a ′′ is a n epimorphism ( f ( z a ) = z ). Since A x is a left no etherian ring, f is a n isomorphism. Hence, a ′′ , as we ll as a , is an in v ertible elemen t. Then the follow ing comm utativ e diagram implies the desired res ult: A ′ v ∼ / / u & & M M M M M M M M M M M M L ≀ a ⊗ 1   L. Q.E.D. Deﬁnition 2.1.10. F or t w o K -algebroids A and A ′ , we sa y t ha t an ( A ⊗ A ′ )-mo dule L is bi-in v ertible if fo r an y σ ∈ A and σ ′ ∈ A ′ , L ( σ ⊗ σ ′ ) is a bi-in v ertible E nd A ( σ ) ⊗ E nd A ′ ( σ ′ )-mo dule. Lemma 2.1.11. T o give an e q uivalenc e A ′ ∼ − → A is e quivalen t to giv- ing a bi-inve rtible ( A ′ ⊗ A op ) -mo dule. Mor e pr e cisely, the for getful func- tor Mod ( A ) − → M od ( A ′ ) is given by M 7→ L ⊗ A M for a bi-invertible ( A ′ ⊗ A op ) -mo dule L . Let M ∈ Mo d( A ). W e shall denote by E nd K ( M ) the stac k asso ciated with the pres tack S whose ob jects are those of A and H o m S ( σ , σ ′ ) = H om K ( M ( σ ) , M ( σ ′ )) for σ , σ ′ ∈ A ( U ). Then E nd K ( M ) is a K -algebroid and there exists a natural functor o f K -algebroids A − → E nd K ( M ). Note that M ma y be rega r ded a s an E nd K ( M )-mo dule. In particular, E nd K ( A ) is a K -algebroid, there is a functor of K -a lgebroids A ⊗ A op − → E nd K ( A ), and A ma y b e regarded as a n E n d K ( A )- mo dule. Lemma 2.1.12. L et A and A ′ b e K -algebr oids and let M ∈ Mo d( A ) , M ′ ∈ Mo d( A ′ ) . Assume that M and M ′ ar e lo c al ly isomorph i c as K - mo dules, that is, for any σ ∈ A and σ ′ ∈ A ′ , M ( σ ) and M ′ ( σ ′ ) ar e lo c al ly isomorphic as K X -mo dules. Th en E nd K ( M ) a nd E nd K ( M ′ ) a r e e quivalent as K -algebr oids. 56 CHAPTER 2. DQ -ALGEBROIDS Pr o of. F o r σ ∈ A and σ ′ ∈ A ′ , set L ( σ ′ ⊗ σ op ) = H om K ( M ( σ ) , M ′ ( σ ′ )). Then L is a n ( E nd K ( M ′ ) ⊗ E nd K ( M ) op )-mo dule. By the assumption, L is a bi-inv ertible  E n d K ( M ′ ) ⊗ E nd K ( M ) op  -mo dule. Hence w e obtain the desired result. Q.E.D. Since Prop osition 2.1 .3 do es not apply to algebraic v arieties, w e need an alternativ e lo cal des cription of algebroids. Let U = { U i } i ∈ I b e an op en co ve ring of X . Consider the data of  a K -a lgebroid A on X , σ i ∈ A ( U i ). (2.1.7) T o these data , we asso ciate • A i := E nd A ( σ i ), • L ij := H om A i | U ij ( σ j | U ij , σ i | U ij ), (hence L ij is a bi-inv ertible A i ⊗ A op j - mo dule on U ij ), • the natura l isomorphisms a ij k : L ij ⊗ A j L j k ∼ − → L ik in Mo d( A i ⊗ A op k | U ij k ). (2.1.8) Then the dia g ram b elo w in Mo d( A i ⊗ A op l | U ij kl ) commutes : L ij ⊗ L j k ⊗ L k l a ij k / / a j kl   L ik ⊗ L k l a ikl   L ij ⊗ L j l a ij l / / L il . (2.1.9) Con v ersely , let A i b e shea v es of K -algebras on U i ( i ∈ I ), let L ij b e a bi- in v ertible A i ⊗ A op j -mo dule on U ij , a nd let a ij k b e isomorphisms as in (2.1.8) suc h that the diagr a m (2.1.9 ) commute s. One calls ( { A i } i ∈ I , { L ij } i,j ∈ I , { a ij k } i,j,k ∈ I ) (2.1.10) an a lg ebraic g luing datum for K -algebroids on U . Prop osition 2.1.13. Consider an algebr aic gluing datum (2.1.10) o n U . Then ther e exist an alge br oid A on X and { σ i , ϕ ij } i,j ∈ I as in (2.1.1) to which this gluing datum is as s o ciate d. Mor e over, the data ( A , σ i , ϕ ij ) ar e unique up to an e quivalenc e of stacks, this e quivalenc e b eing unique up to a unique isomorphism. 2.1. ALGEBROIDS 57 Sketch o f pr o of. W e deﬁne a category Mod( A X ) as follows . An ob ject M ∈ Mo d( A X ) is deﬁned as a family { M i , q ij } i,j ∈ I with M i ∈ Mod( A i ) and the q ij ’s are isomorphisms q ij : L ij ⊗ A j M j ∼ − → M i making the diagram b elow comm utative: L ij ⊗ L j k ⊗ M k q j k / / a ij k   L ij ⊗ M j q ij   L ik ⊗ M k q ik / / M i . A morphism { M i , q j i } i,j ∈ I − → { M ′ i , q ′ j i } i,j ∈ I in Mod( A X ) is a family of mor- phisms u i : M i − → M ′ i satisfying the natural compatibilit y conditions. Re- placing X with U op en in X , we deﬁne a prestack U 7→ Mo d( A U ) and one easily c hec ks that this prestack is a stack and moreo v er that Mo d( A U i ) is equiv alen t to Mo d( A i ). W e denote it b y Mod ( A ). Then w e deﬁne the algebroid A X as the substac k of ( Mod ( A )) op consisting of ob jects lo cally isomorphic to A i on U i . Q.E.D. In v ert ible algebroids In this subsection, ( X, R ) denotes a top ological space endo we d with a sheaf of commutativ e K -algebras. Recall (see [41, Chap.19 § 5 ]) that an R -linear stac k S is a K -linear stac k S t o gether with a morphism of K -algebras R − → E n d (id S ). Here, E nd (id S ) is the sheaf of endomorphisms of t he identit y functor id S from S to it self. Deﬁnition 2.1.14. (i) An R -algebroid P is a K -algebroid P o n X en- do w ed with a morphism of K -algebras R − → E nd (id P ). (ii) An R -algebroid P on X is called an inv ertible R -algebroid if R U − → E n d P ( σ ) is an isomorphism for an y op en subset U o f X and an y σ ∈ P ( U ). W e shall state some prop erties of in v ertible R -algebroids. Since the pro ofs are more or less obv ious, w e o mit them. F or t w o R -a lgebroids P 1 and P 2 , the R -algebroid P 1 ⊗ R P 2 is deﬁned as the R - linear stack asso ciated with the prestac k S giv en b y S ( U ) = P 1 ( U ) × P 2 ( U ) , H om S  ( σ 1 , σ 2 ) , ( σ ′ 1 , σ ′ 2 )  ) = H om P 1 ( σ 1 , σ ′ 1 ) ⊗ R H om P 2 ( σ 2 , σ ′ 2 ) . 58 CHAPTER 2. DQ -ALGEBROIDS If P 1 and P 2 are inv ertible, then so is P 1 ⊗ R P 2 . W e hav e a functor of K -linear stacks P 1 ⊗ K X P 2 − → P 1 ⊗ R P 2 . Note that If P 1 and P 2 are t w o inv ertible R -algebroids and F : P 1 − → P 2 is a functor o f R - linear stac ks, then F is an equiv alence. (2.1.11) F or a n y in ve rtible R -alg ebroid P , P ⊗ R P op is equiv alen t to R a s a n R -algebroid. (2.1.12) The set of equiv alence classes of in v ertible R -algebroids has a structure o f an additive group by the op era t io n • ⊗ R • deﬁned a b o v e, and this g roup is isomorphic to H 2 ( X ; R × ) (see [7, 41]). Here R × denotes the ab elian sheaf of inv ertible sections of R . (2.1.13) F or t w o in v ertible R -algebroids P 1 and P 2 , there is a natural functor • ⊗ R • : Mo d( P 1 ) × Mo d( P 2 ) − → Mo d( P 1 ⊗ R P 2 ) , and its deriv ed vers ion. (2.1.14) In v ert ible O X -algebroids In this subsection, ( X , O X ) denotes a complex manifold. As a particular case of Deﬁnition 2 .1.14, taking K = C and R = O X , we get the notions o f an O X -algebroid as w ell as that of an in ve rtible O X -algebroid. Lemma 2.1.15. A ny C -algebr a endomorphism of O X is e qual to the identity. Although this result is elemen tary and well-kno wn, w e g ive a pro of. Pr o of. L et ϕ b e a C - a lgebra endomorphism of O X . F or x ∈ X , denote by ϕ x the C -algebra endomorphism of O X,x induced b y ϕ and b y m x the unique maximal ideal of the ring O X,x . Then ϕ x sends m x to m x , ϕ x induces an C -algebra homomorphism u x : O X,x / m x − → O X,x / m x . Since the comp osition C ∼ − → O X,x / m x u x − − − → O X,x / m x ∼ − → C is the iden tity , w e obtain that u x is the iden tit y . Hence , for any f ∈ O X , ϕ ( f )( x ) = f ( x ). Therefore ϕ ( f ) = f . Q.E.D. Lemma 2.1.16. L et P b e a C -algebr oid on a c omplex manifold X . Assume that, for any σ ∈ P , E nd P ( σ ) is lo c al ly isom orphic to O X as a C -al g ebr a. Then P is uniquely endowe d with a structur e of O X -algebr oid, and P is invertible. 2.1. ALGEBROIDS 59 Pr o of. By Lemma 2.1 .1 5, for an op en subset U and σ ∈ P ( U ), there exists a unique C -algebra isomorphism O X | U ∼ − → E nd P ( σ ). It giv es a structure of O X -algebroid on P . The r emaining statemen ts a re ob vious. Q.E.D. Let P b e an in ve rtible O X -algebroid. F or σ , σ ′ ∈ P ( U ), t he tw o O X - mo dule structures on H om P ( σ , σ ′ ) ind uced b y E n d P ( σ ) ≃ O X and b y E n d P ( σ ′ ) ≃ O X coincide, and H om P ( σ , σ ′ ) is a n in v ertible O X -mo dule. Let f : X − → Y b e a morphism of complex manifolds. F o r an in v ertible O Y -algebroid P Y , w e set f ∗ P Y := O X ⊗ f − 1 O Y f − 1 P Y , where the tensor pr o duct ⊗ f − 1 O Y is deﬁned similarly as fo r K -algebroids. Then f ∗ P Y is an in ve rtible O X -algebroid. W e ha v e functors f ∗ : Mo d( P Y ) − → Mo d( f ∗ P Y ) , L f ∗ : D b ( P Y ) − → D b ( f ∗ P Y ) , (2.1.15) and f ! , f ∗ : Mo d( f ∗ P Y ) − → Mo d( P Y ) , R f ! , R f ∗ : D b ( f ∗ P Y ) − → D b ( P Y ) . (2.1.16) Let f : X − → Y b e a morphism of complex manifo lds, and let P X (resp. P Y ) b e an in ve rtible O X -algebroid (resp. an inv ertible O Y -algebroid ). If f − 1 P Y − → P X is a functor of C - linear stac ks, then it deﬁnes a functor of C -linear stac ks f ∗ P Y − → P X and this la st functor is an equiv alence b y the preceding results. Remark 2.1.17. Inv ertible O X -algebroids are trivial in the algebraic case. Indeed, f o r a smo oth algebraic v ariet y X , the g r o up H 2 ( X ; O × X ) is zero. Here the cohomolo gy is calculated with r esp ect to the Zariski top ology . (With the ´ etale top ology , it do es not v anish in g eneral.) This result and its pro of below ha v e b een comm unicated to us by Prof. Josep h Oesterl ´ e, and we tha nk him here. Let K b e the ﬁeld of rational functions on X , K × X , the constan t sheaf with the ab elian group K × as stalks, and denote b y X 1 the set o f closed irreducible h yp ersurfaces of X . O ne ha s a n exact se quence 0 − → O × X − → K × X − → M S ∈ X 1 Z S − → 0 . Since K × X is constan t, it is a ﬂabb y sheaf for the Z ariski t o p ology . On the other hand the sheaf L S ∈ X 1 Z S is also ﬂabb y . It follo ws that H j ( X ; O × X ) is zero for j > 1. 60 CHAPTER 2. DQ -ALGEBROIDS 2.2 DQ -algebras F rom now on, X will b e a complex manifold. W e denote by δ X : X ֒ → X × X the diagonal embedding and we set ∆ X = δ X ( X ). W e denote by O X the structure sheaf on X , by d X the complex dimension, by Ω X the sheaf of holomorphic forms of maximal degree and b y Θ X the sheaf o f ho lomorphic v ector ﬁelds. As usual, w e denote b y D X the sheaf of rings of (ﬁnite order) diﬀeren tial op erators on X a nd by F n ( D X ) the sheaf of diﬀeren tial op erators of order ≤ n . Recall that a bi- diﬀeren tial op erator P on X is a C - bilinear morphism O X × O X − → O X whic h is obtained as the comp osition δ − 1 X ◦ e P where e P is a diﬀ erential op erator on X × X deﬁned on a neigh b or ho o d of the diagonal and δ − 1 is the restriction to the diago nal: P ( f , g )( x ) = ( e P ( x 1 , x 2 ; ∂ x 1 , ∂ x 2 )( f ( x 1 ) g ( x 2 )) | x 1 = x 2 = x . (2.2.1) Hence the sheaf of bi-diﬀeren tial op erators is isomorphic to D X ⊗ O X D X , where b oth D X are rega r ded as O X -mo dules b y the left multiplic at io ns. Star-pro ducts Notation 2.2.1. W e den ot e b y C ~ the ring C [[ ~ ] ] of formal p ow er series in an indeterminate ~ and by C ~ , lo c the ﬁeld C (( ~ ) ) o f Lauren t series in ~ . Then C ~ , lo c is the fraction ﬁeld of C ~ . W e set O X [[ ~ ] ] := lim ← − n O X ⊗ ( C ~ / ~ n C ~ ) ≃ Q n ≥ 0 O X ~ n . Let us recall a classical deﬁnition (see [1, 46]). Deﬁnition 2.2.2. An asso ciative m ultiplication law ⋆ on O X [[ ~ ] ] is a star- pro duct if it is C ~ -bilinear and satisﬁes f ⋆ g = X i ≥ 0 P i ( f , g ) ~ i for f , g ∈ O X , (2.2.2) where t he P i ’s are bi-diﬀeren tial op erators s uch that P 0 ( f , g ) = f g a nd P i ( f , 1) = P i (1 , f ) = 0 for all f ∈ O X and i > 0. W e call ( O X [[ ~ ] ] , ⋆ ) a star-algebra. Note that 1 ∈ O X ⊂ O X [[ ~ ] ] is a unit with resp ect to ⋆ . Note also tha t w e ha ve ( P i ≥ 0 f i ~ i ) ⋆ ( P i ≥ 0 g i ~ i ) = P n ≥ 0  P i + j + k = n P k ( f i , g j )  ~ n . 2.2. DQ -ALG EBRAS 61 Recall that a star-pro duct deﬁnes a Poisson structure on ( X , O X ) b y setting for f , g ∈ O X : { f , g } = P 1 ( f , g ) − P 1 ( g , f ) = ~ − 1 ( f ⋆ g − g ⋆ f ) mo d ~ O X [[ ~ ] ] , (2.2.3) and that lo cally , (globally in the real case), an y P oisson manifo ld ( X , O X ) ma y b e endo w ed with a star- pro duct to whic h the P oisson structure is asso- ciated. This is a famous theorem of Konts evic h [46]. Prop osition 2.2.3. L et ⋆ and ⋆ ′ b e star-pr o ducts and let ϕ : ( O X [[ ~ ] ] , ⋆ ) − → ( O X [[ ~ ] ] , ⋆ ′ ) b e a morph ism of C ~ -algebr as. Then ther e ex ists a unique se- quenc e of diﬀer ential op er ators { R i } i ≥ 0 on X s uch that R 0 = 1 a n d ϕ ( f ) = P i ≥ 0 R i ( f ) ~ i for any f ∈ O X . I n p articular, ϕ is an is o morphism. First, w e nee d a lemma. In this lemma, we set F ∞ ( D X ) = D X . Lemma 2.2.4. L et l ∈ Z ≥− 1 ⊔ { ∞} , and ϕ ∈ End C X ( O X ) . If [ ϕ , g ] ∈ F l ( D X ) for al l g ∈ O X , then ϕ ∈ F l +1 ( D X ) . Pr o of. W e may assume that X is an op en subset of C n and we denote by ( x 1 , . . . , x n ) the co ordinates. Set P i = [ ϕ, x i ]. Then [ P i , x j ] = [ [ ϕ, x i ] , x j ] = [ [ ϕ, x j ] , x i ] = [ P j , x i ] . This implies the existence of P ∈ F l +1 ( D X ) suc h that [ P , x i ] = P i for all i . Setting ψ := ϕ − P , w e hav e [ ψ , x i ] = 0 f or all i = 1 , . . . , n. Let us show t ha t ψ ∈ O X . Replacing ψ with θ := ψ − ψ (1), w e get by induction on the order of the polynomials t ha t θ ( Q ) = 0 and [ θ , Q ] = 0 for all Q ∈ C [ x 1 , . . . , x n ]. Let f ∈ O X . W e shall pro v e that θ ( f )( x ) = 0 for all x ∈ X . It is enough t o prov e it for x = 0. Then, writing f = f (0) + P i x i f i , w e get θ ( f ) = θ ( f (0)) + X i θ ( x i f i ) = θ ( f (0)) + X i  x i θ ( f i ) + [ θ , x i ] f i  = X i x i θ ( f i ) , whic h v anishes at x = 0 . Q.E.D. Pr o of of Pr op osition 2.2.3. L et us write ϕ ( f ) = X i ≥ 0 ~ i ϕ i ( f ) , f ∈ O X . (2.2.4) 62 CHAPTER 2. DQ -ALGEBROIDS By Lemma 2.1.15, ϕ 0 = 1. W e shall pro ve b y induction t ha t the ϕ i ’s in (2.2.4) are diﬀeren tial op erators and w e assume that this is so for all i < n for n ∈ Z > 0 . Let { P i } and { P ′ i } b e the sequence of bi-diﬀeren tial op erato rs asso ciated with the star-pro ducts ⋆ and ⋆ ′ , resp ectiv ely . W e hav e ϕ ( f ⋆ g ) = ϕ ( X j ≥ 0 ~ j P j ( f , g )) = X i,j ≥ 0 ~ i + j ϕ i ( P j ( f , g )) , ϕ ( f ) ⋆ ′ ϕ ( g ) = X i ≥ 0 ~ i ϕ i ( f ) ⋆ ′ X j ∈ N ~ j ϕ j ( g ) = X i,j,k ≥ 0 ~ i + j + k P ′ k ( ϕ i ( f ) , ϕ j ( g )) . Since ϕ ( f ⋆ g ) = ϕ ( f ) ⋆ ′ ϕ ( g ) , we get: X n = i + j ϕ i ( P j ( f , g )) = X n = i + j + k P ′ k ( ϕ i ( f ) , ϕ j ( g )) . (2.2.5) By the induction hypothesis, the left hand side of (2.2.5) ma y b e written as ϕ n ( f g ) + Q n ( f , g ) where Q n is a bi-diﬀerential o p erator. Similarly , the righ t hand side of (2.2.5) may b e written as ϕ n ( f ) g + f ϕ n ( g ) + R n ( f , g ) where R n is a bi-diﬀeren tia l op erator. F o r any g ∈ O X , considering g as an endomorphism of O X , w e get [ ϕ n , g ]( f ) := ϕ n ( f g ) − g ϕ n ( f ) = f ϕ n ( g ) + S n ( f ) , where S n is a diﬀeren tial op erator. Then, the result fo llo ws from Lemma 2.2.4. Q.E.D. DQ -algebras Deﬁnition 2.2.5. A DQ-alg ebra A on X is a C ~ -algebra lo cally isomorphic to a star- algebra ( O X [[ ~ ] ] , ⋆ ) as a C ~ -algebra. Clearly a DQ-algebra A satisﬁes the conditio ns:                  (i) ~ : A − → A is inj ectiv e, (ii) A − → lim ← − n A / ~ n A is an isomorphism, (iii) A / ~ A is isomorphic to O X as a C -algebra. (2.2.6) F or a C ~ -algebra A satisfying (2.2.6), the C - algebra isomorphism A / ~ A ∼ − → O X in (2.2.6) (iii) is unique b y Lemma 2.1 .15. W e denote by σ 0 : A − → O X (2.2.7) 2.2. DQ -ALG EBRAS 63 the C ~ -algebra morphism A − → A / ~ A ∼ − → O X . If ϕ is a C -linear sec- tion of σ 0 : A − → O X , then ϕ extends t o an isomorphism of C ~ -mo dules e ϕ : O X [[ ~ ] ] ∼ − → A , giv en b y e ϕ ( P i f i ~ i ) = P i ϕ ( f i ) ~ i . Deﬁnition 2.2.6. W e sa y that a C -linear se ction ϕ : O X − → A of A − → O X is standard if there exists a sequence of bi-diﬀerential op erators P i suc h that ϕ ( f ) ϕ ( g ) = X i ≥ 0 ϕ ( P i ( f , g )) ~ i for an y f , g ∈ O X . (2.2.8) Consider a standard section ϕ : O X − → A of A − → O X . Deﬁne a star- pro duct ⋆ on O X [[ ~ ] ] by setting f ⋆ g = X i ≥ 0 P i ( f , g ) ~ i for an y f , g ∈ O X . Then w e get an isomorphism of C ~ -algebras e ϕ : ( O X [[ ~ ] ] , ⋆ ) ∼ − → A . (2.2.9) W e call e ϕ in (2.2.9) a standar d isomo rphism . Hence, a DQ-algebra is nothing but a C ~ -algebra satisfying (2.2.6) and admitting lo cally a standard section. Remark 2.2.7. W e conjecture that a C ~ -algebra satisfying (2.2.6) lo cally admits a standard section. Let A b e a DQ- algebra. F or f , g ∈ O X , taking a , b ∈ A suc h that σ 0 ( a ) = f and σ 0 ( b ) = g , w e set { f , g } = σ 0 ( ~ − 1 ( ab − ba )) ∈ O X . (2.2.10) Then this deﬁnition do es not dep end on the choice of a , b and it deﬁnes a P oisson structure on X . In particular, t w o DQ-algebras induce the same P oisson structure on X as so on as they are lo cally isomorphic. By Prop osition 2.2.3, if ϕ, ϕ ′ : O X − → A are t w o standard sections, then there exists a unique sequence of diﬀeren tial op erators { R i } i ≥ 0 suc h that ϕ ′ ( f ) = P i ≥ 0 ~ i ϕ ( R i ( f )) for any f ∈ O X . Clearly , a D Q-algebra satisﬁe s the hy p o t heses (1.2.2) and (1.2.3). Hence, a D Q-algebra is a righ t and left No etherian ring (in particular, coheren t). Lemma 2.2.8. L et A b e a DQ -a lgebr a. T hen the o p p osite algebr a A op is also a DQ -algebr a. Pr o of. This follo ws from (2.2.2). Q.E.D. 64 CHAPTER 2. DQ -ALGEBROIDS Let X and Y b e complex manifolds endo we d with tw o star-pro ducts ⋆ X and ⋆ Y . Denote b y { P i } i and { Q j } j the bi-diﬀeren tial op erators asso ciated to these star-pro ducts as in (2.2.2). L et P i ⊠ Q j b e a bi- diﬀeren tial op erator o n X × Y deﬁned as follows . Let us tak e diﬀeren tial op erators e P i ( x 1 , x 2 , ∂ x 1 , ∂ x 2 ) and e Q j ( y 1 , y 2 , ∂ y 1 , ∂ y 2 ) corresponding to P i and Q j as in (2.2 .1 ). Then w e set ( P i ⊠ Q j )( f , g )( x, y ) =  e P i ( x 1 , x 2 , ∂ x 1 , ∂ x 2 ) e Q j ( y 1 , y 2 , ∂ y 1 , ∂ y 2 )( f ( x 1 , y 1 ) g ( x 2 , y 2 ))  | x 1 = x 2 = x y 1 = y 2 = y . Hence, P i ⊠ Q j is the unique bi-diﬀeren tial op era t or on X × Y suc h that ( P i ⊠ Q j )( f 1 ( x ) g 1 ( y ) , f 2 ( x ) g 2 ( y ) ) = P i ( f 1 ( x ) , f 2 ( x )) · Q j ( g 1 ( y ) , g 2 ( y ) ) fo r an y f ν ( x ) ∈ O X and g ν ( y ) ∈ O Y ( ν = 1 , 2). One deﬁnes the external pro duct of the star- pro ducts ⋆ X and ⋆ Y on O X × Y [[ ~ ] ] b y se tting f ⋆ g = X n ≥ 0 ~ n X i + j = n ( P i ⊠ Q j )( f , g ) . Hence: Lemma 2.2.9. L et X and Y b e c omplex m anifolds, and let A X b e a DQ - algebr a on X and A Y a DQ -algebr a o n Y . Th e n ther e exists a DQ - a lgebr a A on X × Y whic h c ontains A X ⊠ C ~ A Y as a C ~ -sub algebr a. Mor e over such an A is unique up to a unique isomorphism. W e call A the external pro duct of the DQ-algebra A X on X a nd the DQ-algebra A Y on Y , and denote it by A X ⊠ A Y . Remark 2.2.10. (i) An y commutativ e DQ-algebra is lo cally isomorphic to ( O X [[ ~ ] ] , ⋆ ) where ⋆ is the t r ivial star-pro duct f ⋆ g = f g . (ii) F or the trivial DQ-algebra O X [[ ~ ] ], w e hav e A ut C ~ -alg ( O X [[ ~ ] ]) ≃ ~ Θ X [[ ~ ] ] := Y n ≥ 1 ~ n Θ X , (recall that Θ X is t he sheaf of vec tor ﬁelds on X ) and w e a ssociat e to v := P n ≥ 1 ~ n v n the automorphism f 7→ exp( v ) f . The ring D A X and another construction for DQ -algebras W e deﬁne the C ~ -algebra D X [[ ~ ] ] := lim ← − n D X ⊗ ( C ~ / ~ n C ~ ) ≃ Y n ≥ 0 D X ~ n . 2.2. DQ -ALG EBRAS 65 Then O X [[ ~ ] ] has a D X [[ ~ ] ]-mo dule structure, and D X [[ ~ ] ] ⊂ E nd C ~ ( O X [[ ~ ] ]). Let A X b e a DQ-alg ebra. Cho ose (lo cally) a standard section ϕ giving rise to a standard isomorphism of C ~ -mo dules e ϕ : O X [[ ~ ] ] ∼ − → A X . This last isomorphism induces an isomorphism Φ : E nd C ~ ( O X [[ ~ ] ]) ∼ − → E nd C ~ ( A X ) . (2.2.11) Deﬁnition 2.2.11. Let A X b e a DQ-algebra and let ϕ b e a standard section. The sheaf of rings D A X is the C ~ -subalgebra of E nd C ~ ( A X ), the image of D X [[ ~ ] ] ⊂ E nd C ~ ( O X [[ ~ ] ]) by the isomorphism Φ in (2.2.11). It is easy to see that D A X ⊂ E nd C ~ ( A X ) do es not dep end o n the choice of the standard se ction ϕ in virtue of Prop o sition 2.2.3. Hence D A X is w ell- deﬁned on X although standard sections only lo cally exist. By its construction, w e hav e D A X ∼ − → lim ← − n D A X / ~ n D A X . Moreo v er, the image of the algebra morphism A X ⊗ A op X − → E nd C ~ ( A X ), as w ell as the one of δ − 1 X A X × X a − → E nd C ~ ( A X ) is con tained in D A X . Hence w e hav e algebra morphisms A X ⊗ A X a − → δ − 1 X A X × X a − → D A X . W e shall sho w how to construct a star-algebra from the dat a of se ctions of D X [[ ~ ] ] satisfying suitable commu ta t ion prop erties. Let A X := ( O X [[ ~ ] ] , ⋆ ) b e a star-algebra. There are t wo C ~ -linear mor- phisms from O X [[ ~ ] ] to D X [[ ~ ] ] giv en b y Φ l : f 7→ f ⋆, Φ r : f 7→ ⋆ f . (2.2.12) Hence, for f ∈ O X , w e hav e: Φ l ( f ) = X i ≥ 0 P i ( f , • ) ~ i , Φ r ( f ) = X i ≥ 0 P i ( • , f ) ~ i . Then Φ l : A X − → D X [[ ~ ] ] and Φ r : A op X − → D X [[ ~ ] ] are t wo C ~ -algebra morphisms, and induce a C ~ -algebra morphism A X ⊗ A op X − → D X [[ ~ ] ]. Assume to b e giv en a lo cal co ordinate system x = ( x 1 , . . . , x n ) on X and for i = 1 , . . . , n , set Φ l ( x i ) = A i and Φ r ( x i ) = B i . Then { A i , B j } i,j =1 ,...,n are sections of D X [[ ~ ] ] whic h satisfy    A i (1) = B i (1) = x i , A i ≡ x i mo d ~ D X [[ ~ ] ], B i ≡ x i mo d ~ D X [[ ~ ] ], [ A i , B j ] = 0 ( i, j = 1 , . . . , n ). (2.2.13) Con v ersely , we ha v e the following result. 66 CHAPTER 2. DQ -ALGEBROIDS Prop osition 2.2.12. L et { A i , B j } i,j =1 ,...,n b e se ctions of D X [[ ~ ] ] which satisfy (2.2.13) . Deﬁne the sub algebr a A X ⊂ D X [[ ~ ] ] by A X = { a ∈ D X [[ ~ ] ] ; [ a, B i ] = 0 , i = 1 , . . . , n } ( 2 .2.14) and deﬁne the C ~ -line ar map ψ : A X − → O X [[ ~ ] ] by setting ψ ( a ) = a (1) . Then (a) ψ is a C ~ -line ar isomo rp h ism, (b) the pr o duct on O X [[ ~ ] ] given by ψ ( a ) ⋆ ψ ( b ) := ψ ( a · b ) is a star-pr o duct, A X is a D Q -algebr a and ψ − 1 is a standa r d isomorphism, (c) the alge b r a A op X is obtaine d by r eplacing A i with B i ( i = 1 , . . . , n ) in the ab ove c onstruction. Pr o of. ( a)-(i) A X ∩ ~ D X [[ ~ ] ] = ~ A X , since [ ~ a, B i ] = 0 implies [ a, B i ] = 0. Hence w e hav e A X / ~ j A X ⊂ D X [[ ~ ] ] / ~ j D X [[ ~ ] ] for an y j . (a)-(ii) A X ∼ − → lim ← − j A X / ~ j A X . Indeed, let a = P ∞ i =0 ~ i a i and assume that [ k X i =0 ~ i a i , B l ] = 0 mo d ~ k +1 ( l = 1 , . . . , n ) for all k ∈ N . Then [ a, B l ] = 0 for l = 1 , . . . , n . (a)-(iii) Let ψ j : ~ j A X / ~ j +1 A X − → ~ j O X / ~ j +1 O X b e the morphisms induce d b y ψ . By (a)-(ii) it is enough to che ck that all ψ j ’s are isomorphisms. Since all ~ j A X / ~ j +1 A X are isomorphic and all ~ j O X / ~ j +1 O X are isomorphic, we are reduced to prov e that ψ 0 : A X / ~ A X − → O X is an isomorphism. (a)-(iv) ψ 0 is injectiv e. Let a 0 ∈ A X / ~ A X ⊂ D X . Since [ a 0 , x i ] ∈ ~ D X [[ ~ ] ] implies [ a 0 , x i ] = 0, we get a 0 ∈ O X . Therefore, a 0 (1) = 0 implies a 0 = 0. (a)-(v) ψ 0 is surjectiv e. Let y = ( y 1 , . . . , y n ) b e a lo cal co ordinate system o n a cop y of X . Notice ﬁr st that the sections y i − A i of D X × Y [[ ~ ] ] are in v ertible on the op en sets { y i 6 = x i } . Let f ( x 1 , . . . , x n ) ∈ O X . Deﬁne the section G ( f ) of D X [[ ~ ] ] by G ( f ) = 1 (2 π i ) n I f ( y )( y 1 − A 1 ) − 1 · · · ( y n − A n ) − 1 dy 1 · · · dy n . (2.2.15) Then [ G ( f ) , B i ] = 0 for all i . It is ob vious that G ( f ) − f ∈ ~ D X [[ ~ ] ] and ψ 0 ( G ( f )) = f . (b) Clearly , the algebra ( O X [[ ~ ] ] , ⋆ ) satisﬁes (2.2.6). Moreo v er, f 7→ G ( f ) is a standard section since there exist P i ( f ) ∈ D X [[ ~ ] ] ( i ∈ N ) suc h that G ( f ) = P i P i ( f ) ~ i and P i ( f ) is obtained as the action of a bidiﬀeren tial op erator P i on f . (c) fo llows f rom A op = { b ∈ E nd C ~ ( A X ); [ b, A X ] = 0 } . Q.E.D. 2.3. DQ -ALG EBR OIDS 67 Example 2.2.13. Let M := { a ij } i,j =1 ,...,n b e an n × n sk ew-symmetric matrix with en tries in C . Let X = C n and consider the sections of D X [[ ~ ] ]: A i = x i + ~ 2 X j a ij ∂ j , B i = x i − ~ 2 X j a ij ∂ j . Then { A i , B j } i,j =1 ,...,n satisfy (2 .2 .13), thus deﬁne a DQ-algebra A X . Note that the Poiss on structure asso ciated with the D Q-alg ebra A X is symplectic if a nd only if the ma t r ix M is non-degenerate. 2.3 DQ -algebroids Let us in tro duce the notion of a deformation quantization algebroid, a DQ- algebroid for short. Deﬁnition 2.3.1. A DQ-algebroid A o n X is a C ~ -algebroid suc h that for eac h o p en set U ⊂ X and each σ ∈ A ( U ), the C ~ -algebra E nd A ( σ ) is a DQ-algebra on U . Note that a DQ-alg ebroid is called a twis ted asso ciativ e deformation of O X in [64]. By (2.2 .10), a DQ-algebroid A on the complex manifold X deﬁnes a P oisson structure on X . It is pro v ed in [45] that, conv ersely , an y complex P oisson manifold X ma y b e endo we d w ith a DQ-algebroid to whic h this P oisson structure is asso ciated. According to Con v ention 2.1.1, if A is a DQ-algebra, w e shall often use the same notation A for the asso ciated DQ-algebroid. Note that an y DQ-algebroid A on X may b e obtained as the stac k associ- ated with a gluing datum as in (2.1.3), where the sheav es A i are DQ-algebras. Let A b e a DQ-algebroid on X . F or an A - mo dule M , the lo cal notions of b eing coheren t or lo cally free, etc. make sense. The category Mo d( A ) is a G rothendiec k category . W e denote b y D( A ) its deriv ed category and b y D b ( A ) its b ounded deriv ed category . W e still call an ob ject o f this deriv ed category an A -mo dule. W e denote by D b coh ( A ) the full triangulated sub category of D b ( A ) consisting of ob jects with coheren t cohomologies. Opp osite str ucture If X is endo w ed with a DQ-algebroid A X , then w e denote b y X a the manifold X endow ed with the algebroid A op X , that is: A X a = A op X . (2.3.1) 68 CHAPTER 2. DQ -ALGEBROIDS This is a DQ-algebroid by Lemma 2.2 .8 . External pro duct Assume that complex manifo lds X and Y are endo w ed with DQ-algebroids A X and A Y resp ectiv ely . By Lemma 2.2.9, there is a canonical D Q-algebroid A X ⊠ A Y on X × Y lo cally equiv alen t to the stack a sso ciated with the external pro duct A X ⊠ A Y of the DQ-algebras and there is a fait hf ul functor of C ~ - algebroids A X ⊠ A Y − → A X ⊠ A Y , (2.3.2) whic h induces a functor for : Mo d( A X ⊠ A Y ) − → Mo d( A X ⊠ A Y ) . (2.3.3) When there is no risk o f confusion, we set A X × Y := A X ⊠ A Y . Then A X × Y b elongs to Mo d( A X × Y ⊗ ( A X a ⊠ A Y a )) and the functor fo r ad- mits a left adjoin t functor K 7→ A X × Y ⊗ A X ⊠ A Y K : Mo d( A X × Y ) for / / Mo d( A X ⊠ A Y ) . o o (2.3.4) W e denote by • ⊠ • the bi-functor A X × Y ⊗ A X ⊠ A Y ( • ⊠ • ): • ⊠ • : Mo d( A X ) × Mo d( A Y ) − → Mo d( A X × Y ) . (2.3.5) Lemma 2.3.2. If M is an A X -mo dule without ~ -torsion, then the functor M ⊠ • : Mo d( A Y ) − → Mo d( A X × Y ) is an exact functor. Pr o of. W e may a ssume that A X and A Y are DQ-algebras. Hence it is enough to show that for any ( x, y ) ∈ X × Y , setting N := A X × Y ⊗ A X M , N ( x,y ) is a ﬂat mo dule ov er A op Y ,y . W e ma y assume further that M is a cohere nt A X - mo dule without ~ -to rsion. F or an y Stein op en subset U , let p U : U × Y − → Y b e t he pro jection. Set N U := ( p U ) ∗  ( A X × Y ⊗ A X M ) | U × Y  . Then it is easy t o c hec k the conditions (a)–(c) in Theorem 1.6.6 are satisﬁed (( c) follo ws fr om the O -mo dule v ersion of this lemma), and w e conclude that N U is a ﬂat A op Y -mo dule. Hence, N ( x,y ) ≃ lim − → x ∈ U ( N U ) y is a ﬂat ( A op Y ) y -mo dule. Q.E.D. 2.3. DQ -ALG EBR OIDS 69 Hence the left deriv ed functor • L ⊠ • : D ( A X ) × D( A Y ) − → D( A X × Y ) satisﬁes M • L ⊠ N • ∼ − → M • ⊠ N • as so on as M • or N • is a complex b ounded from ab ov e o f mo dules without ~ -torsion. Graded mo dules F or a C ~ -algebroid B on X , one denotes b y gr ~ ( B ) the C -algebroid asso ciated with the prestac k S giv en by Ob( S ( U )) = Ob( B ( U )) for an op en subset U of X , Hom S ( U ) ( σ , σ ′ ) = Hom B ( σ , σ ′ ) / ~ Hom B ( σ , σ ′ ) f or σ , σ ′ ∈ B ( U ) . Let no w A X b e a D Q-algebroid on X . Then it is easy to see t ha t gr ~ ( A X ) is an in v ertible O X -algebroid and that w e ha ve a natural functor A X − → gr ~ ( A X ) of C -algebroids. This functor induces a functor for : Mo d(gr ~ ( A X )) − → Mo d( A X ) . (2.3.6) The functor for ab ov e is fully fait hf ul a nd Mo d(gr ~ ( A X )) is equiv alen t to the full sub category o f Mod( A X ) consisting of ob jects M suc h that ~ : M − → M v anishe s. The functor for : Mo d(gr ~ ( A X )) − → Mo d( A X ) admits a left adjoin t functor M 7→ M / ~ M ≃ C ⊗ C ~ M . The functor for is exact and it induces a functor for : D(gr ~ ( A X )) − → D( A X ) . (2.3.7) Remark 2.3.3. The f unctor in (2 .3 .7) is not full in general. Indeed, choose X = pt, A X = C ~ and L = C ~ / ~C ~ view ed as a gr ~ ( A )-mo dule. Then Hom D b ( C ~ )  for ( L ) , for ( L [1])  ≃ C ~ / ~C ~ , Hom D b ( C ) ( L, L [1]) ≃ 0 . It could b e also sho wn that this functor is not faithful in general. One extends Deﬁnition 1.4.1 to the algebroid A X . As an ( A X ⊗ A X a )- mo dule, gr ~ ( A X ) is isomorphic t o C ⊗ C ~ A X ≃ A X / ~ A X . W e get the functor gr ~ : D( A X ) − → D(gr ~ ( A X )) , M 7→ gr ~ ( A X ) L ⊗ A X M ≃ C L ⊗ C ~ M . (2.3.8) Note that Lemma 1.4.2, Prop ositions 1.4.3 and 1.4.5 a s w ell a s Corollary 1.4.6 still hold. Moreov er 70 CHAPTER 2. DQ -ALGEBROIDS Corollary 2.3.4. L et M ∈ D b coh ( A X ) . Then its supp ort, Supp( M ) , is a close d c omplex an alytic s ubse t of X . Pr o of. By Corolla ry 1.4.6, Supp( M ) = Supp ( g r ~ ( M )). Since gr ~ ( M ) ∈ D b coh (gr ~ ( A X )) and gr ~ ( A X ) is lo cally isomorphic to O X , the result follows. Q.E.D. Let d X denote the complex dimension of X . Applying Theorem 1.4.8, we get Corollary 2.3.5. L et A X b e a DQ -algebr a and let M ∈ Mo d coh ( A X ) . Then, lo c a l ly, M admits a r esolution by fr e e mo dules of ﬁnite r ank of len g th ≤ d X + 1 . Prop osition 2.3.6. The functors gr ~ in (2.3.8) and for i n (2.3.7) deﬁne p airs of adjoint functors (gr ~ , for ) and ( for , gr ~ [ − 1]) . Pr o of. Consider a pair ( B , C ) in whic h either B = A X and C = g r ~ ( A X ) or B = gr ~ ( A X ) and C = A X , and let K b e a ( B , C )-bimo dule. W e ha v e the adjunction formula, for M ∈ D b ( B ) a nd N ∈ D( C ): Hom D( B ) ( K L ⊗ C N , M ) ≃ Hom D( C ) ( N , R H om B ( K , M )) . (2.3.9) (i) Let us apply form ula (2.3.9) with B = gr ~ ( A X ), C = A X and K = gr ~ ( A X ) considered as a (gr ~ ( A X ) , A X )-bimo dule. W e get Hom D(gr ~ ( A X )) (gr ~ ( A X ) L ⊗ A X M , N ) ≃ Hom D( A X ) ( M , R H om gr ~ ( A X ) (gr ~ ( A X ) , N )) , and when remarking that R H om gr ~ ( A X ) (gr ~ ( A X ) , N ) ≃ for ( N ), we get the ﬁrst adjunction pairing. (ii) Let us apply f orm ula (2.3 .9) with C = gr ~ ( A X ), B = A a nd K = gr ~ ( A X ) considered as an ( A X , gr ~ ( A X ))-bimo dule. W e get Hom D( A X ) (gr ~ ( A X ) L ⊗ gr ~ ( A X ) N , M ) ≃ Hom D(gr ~ ( A X )) ( N , R H om A X (gr ~ ( A X ) , M )) . W e ha v e gr ~ ( A X ) L ⊗ gr ~ ( A X ) N ≃ for ( N ) and to get the second adjunction pairing, notice that R H om A X (gr ~ ( A X ) , M ) ≃ R H om A X (gr ~ ( A X ) , A X ) L ⊗ A X M , and R H om A X (gr ~ ( A X ) , A X ) ≃ g r ~ ( A X ) [ − 1]. Q.E.D. 2.3. DQ -ALG EBR OIDS 71 Dualit y Let A X b e a DQ-algebroid on X . Deﬁnition 2.3.7. Let M ∈ D( A X ). Its dua l D ′ A X M ∈ D( A X a ) is giv en by D ′ A X M := R H om A X ( M , A X ) . ( 2 .3.10) When there is no risk of confusion, w e write D ′ A instead of D ′ A X . By Corollary 2.3.5, D ′ A sends D b coh ( A X ) to D b coh ( A X a ): D ′ A : D b coh ( A X ) − → D b coh ( A X a ) . Assume that M ∈ D b coh ( A X ). Then there is a canonical isomorphism: M ∼ − → D ′ A D ′ A M . (2.3.11) F or a g r ~ ( A X )-mo dule M , denote b y D ′ O M its dual, D ′ O M := R H om gr ~ ( A X ) ( M , gr ~ ( A X )) . (2.3.12) Prop osition 2.3.8. L et M ∈ D b coh ( A X ) . T hen gr ~ (D ′ A M ) ≃ D ′ O (gr ~ ( M )) . Pr o of. This follo ws from Prop osition 1 .4.3. Q.E.D. Corollary 2.3.9. L et L ∈ D b coh ( A X ) and j ∈ Z . L et us assume that E xt j gr ~ ( A X ) (gr ~ ( L ) , g r ~ ( A X )) ≃ 0 . Then E xt j A X ( L , A X ) ≃ 0 . Pr o of. Applying the ab o v e propo sition, w e get E xt j gr ~ ( A X ) (gr ~ ( L ) , gr ~ ( A X )) = H j (D ′ O (gr ~ ( L ))) ≃ H j (gr ~ (D ′ A ( L ))) . Then the result follows from Prop osition 1 .4.5. Q.E.D. Simple mo dules Deﬁnition 2.3.10. Let Λ b e a smo oth submanifold of X and let L b e a coheren t A X -mo dule supp orted b y Λ . One say s that L is simple along Λ if gr ~ ( L ) is concen trated in degree 0 and H 0 (gr ~ ( L )) is an in ve rtible O Λ ⊗ O X gr ~ ( A X )-mo dule. (In particular, L has no ~ - t o rsion.) 72 CHAPTER 2. DQ -ALGEBROIDS Prop osition 2.3.11. L et Λ b e a close d submanifold of X of c o d i m ension l and let L b e a c oher ent A X -mo dule simple along Λ . Then H j (D ′ A ( L )) = E xt j A X ( L , A X ) vanishes for j 6 = l and H l (D ′ A ( L )) i s a c oher ent A X a -mo dule simple along Λ . Pr o of. The question b eing lo cal, we ma y assume that A X is a DQ-algebra so that gr ~ ( A X ) ≃ O X and gr ~ ( L ) ≃ O Λ . Then, w e ha v e E xt j O X (gr ~ ( L ) , O X ) ≃ 0 f o r j 6 = l . Therefore, E xt j A X ( L , A X ) = 0 for j 6 = l by Corolla ry 2 .3.9 a nd gr ~ ( E xt l A X ( L , A X )) ≃ D ′ O (gr ~ L ) [ l ] ≃ E xt l O X (gr ~ ( L ) , O X ) b y Prop osition 2.3.8. If gr ~ ( L ) is lo cally isomorphic to O Λ , then so is E xt l O X (gr ~ ( L ) , O X ). Q.E.D. Homological dimension of A X -mo dules The co dimension o f the supp ort of a coheren t O X -mo dule F is related to the v anishing of the E xt j O X ( F , O X ). Similar results ho ld for A X -mo dules. Prop osition 2.3.12. L et M b e a c oher ent A X -mo dule. Then (a) E xt j A X ( M , A X ) ≃ 0 for j < co dim Supp M , (b) co dim Supp E xt j A X ( M , A X ) ≥ j . Pr o of. ( a) Fir st, note that Supp( M ) = Supp(gr ~ M ). Therefore, E xt j gr ~ ( A X ) (gr ~ M , gr ~ ( A X )) ≃ 0 fo r j < co dim Supp M and the r esult follo ws from Corollary 2.3.9. (b) By Prop osition 1.4.5, w e kno w that Supp E xt j A X ( M , A X ) ⊂ Supp E xt j gr ~ ( A X ) (gr ~ M , gr ~ ( A X )) , and co dim Supp E xt j gr ~ ( A X ) (gr ~ M , gr ~ ( A X )) ≥ j b y classical results for O X - mo dules. Q.E.D. 2.3. DQ -ALG EBR OIDS 73 Extension of the base ring Recall that C ~ , lo c := C (( ~ ) ) is the fraction ﬁeld of C ~ . T o a D Q-algebroid A X w e asso ciate the C ~ , lo c -algebroid A lo c X = C ~ , lo c ⊗ C ~ A X (2.3.13) and w e call A lo c X the ~ -lo c alization of A X . It follo ws fro m Lemma 1.4.1 0 that the algebroid A lo c X is No etherian. There naturally exists a faithful functor of C ~ -algebroid A X − → A lo c X . (2.3.14) This functor giv es rise to a pair of adjo int functors ( lo c , for ): Mo d( A lo c X ) for / / Mo d( A X ) . lo c o o (2.3.15) Both functors are exact and w e k eep the same notations for their deriv ed functors D b ( A lo c X ) for / / D b ( A X ) . lo c o o (2.3.16) F or N ∈ D b ( A X ), we set N lo c := lo c( N ) ≃ C ~ , lo c ⊗ C ~ N . (2.3.17) W e sa y that an A X -mo dule M 0 is a submodule of an A lo c X -mo dule M if there is a mono morphism M 0 − → for ( M ) in Mo d( A X ). If M is an A lo c X -mo dule, M 0 an A X -submo dule and M 0 ⊗ C ~ C ~ , lo c ∼ − → M , then w e shall say t ha t M 0 generates M . The following result is of constan t use and follo ws from [37, App endix A]. Lemma 2.3.13. Any lo c al ly ﬁnitely g e n er ate d A X -submo dule of a c oher ent A lo c X -mo dule is c oher ent, i.e., any c oher ent A lo c X -mo dule is pseudo-c oher ent as an A X -mo dule. Deﬁnition 2.3.14. A coheren t A X -submo dule M 0 of a coheren t A lo c X -mo dule M is called an A X -lattice of M if M 0 generates M . W e extend D eﬁnition 2.3.7 t o A lo c X -mo dules and, for M ∈ D b ( A lo c X ), w e set D ′ A loc M := R H om A loc X ( M , A lo c X ) . (2.3.18) 74 CHAPTER 2. DQ -ALGEBROIDS Prop osition 2.3.15. L et M b e a c oher ent A lo c X -mo dule. Then (a) E xt j A loc X ( M , A lo c X ) ≃ 0 for j < co dim Supp M , (b) co dim Supp E xt j A loc X ( M , A lo c X ) ≥ j . Pr o of. The result is lo cal and w e ma y c ho ose an A X -lattice M 0 of M . Then the result follo ws from Prop osition 2.3 .12. Q.E.D. Go o d mo dules Deﬁnition 2.3.16. ( i) A coherent A lo c X -mo dule M is go o d if, for any rel- ativ ely compact open subset U of X , there exists an ( A X | U )-lattice of M | U . (ii) One denotes b y Mod gd ( A lo c X ) the full subcategory of Mo d coh ( A lo c X ) con- sisting of go o d mo dules. (iii) One denotes b y D b gd ( A lo c X ) the full sub category of D b coh ( A lo c X ) consisting of ob jects M suc h that H j ( M ) is g o o d for all j ∈ Z . Roughly sp eaking, a coheren t A lo c X -mo dule M is go o d if it is endo wed with a g o o d ﬁltration (see [37]) on eac h op en relativ ely compact subset of X . Prop osition 2.3.17. (a) Th e c ate gory Mo d gd ( A lo c X ) is a thick sub c ate gory of Mod coh ( A lo c X ) , ( i.e., stable by ke rn e ls, c okernels and extensio n ) . (b) The ful l sub c ate gory D b gd ( A lo c X ) of D b coh ( A lo c X ) is triangulate d. (c) An obje ct M ∈ D b coh ( A lo c X ) is go o d if and only if, for any o p en r el a tively c omp act subset U of X , ther e exists an A X | U -mo dule M 0 ∈ D b coh ( A X | U ) such that M lo c 0 is iso morphic to M | U . Since the pro of is sim ilar to that o f [37, Prop. 4 .23], w e shall not repeat it. Prop osition 2.3.18. L et M ∈ D b coh ( A lo c X ) . Th e n Supp( M ) is a close d c omplex an alytic subset of X , invo l utive (i.e., c o-isotr opic ) for the Poisson br acket on X . Pr o of. Since the pro blem is lo cal, w e may assume t ha t A X is a D Q-algebra. Then the pro p osition follow s fro m Gabb er’s theorem [26]. Q.E.D. Remark 2.3.19. One shall b e aw are that the supp ort of a coheren t A X - mo dule is not inv olutiv e in general. Indeed, for a DQ- algebra A X , an y co- heren t O X -mo dule ma y b e r ega rded as a n A X -mo dule. Hence any closed analytic subset can b e the supp ort of a coheren t A X -mo dule. 2.4. DQ -MODULES SUPPOR TED BY THE DIAGONAL 75 2.4 DQ -mo dule s sup p ort e d b y the diagon al Let X b e a complex manifold endo w ed with a DQ-algebroid A X . W e denote b y A X × X a the external pro duct of A X and A X a on X × X a . W e still denote b y δ X : X ֒ → X × X a the diagonal em b edding and w e denote by Mo d ∆ X ( A X ⊠ A X a ) the category of ( A X ⊠ A X a )-mo dules supp orted by the diag onal ∆ X . Then δ X ∗ : Mo d( A X ⊗ A X a ) − → Mo d ∆ X ( A X ⊠ A X a ) giv es an equiv alence o f catego ries, with quasi-in v erse δ − 1 X . W e shall o f ten iden tify these t w o categories b y this equiv alence . Recall that we ha v e a canonical ob ject A X in Mo d( A X ⊗ A X a ) (see § 2.1). W e iden tify A X with an ( A X ⊠ A X a )-mo dule supp orted by the diagonal ∆ X of X × X a . In fact, it has a structure of A X × X a -mo dule. More generally , w e ha v e: Lemma 2.4.1. L et M b e an ( A X ⊗ A X a ) -mo dule. (a) The fol lowing c onditions ar e e quivalent: (i) M is a bi-invertible ( A X ⊗ A X a ) -mo dule ( se e Deﬁnition 2.1.1 0) , (ii) M i s in vertible as an A X -mo dule ( se e D eﬁnition 2.1.4) , that is, M is lo c al ly isomorphic to A X as an A X -mo dule, (iii) M is invertible as an A X a -mo dule. (b) Under the e quivale n t c onditions in (a) , δ X ∗ M − → A X × X a ⊗ A X ⊠ A X a δ X ∗ M is an isomorphism and δ X ∗ M has a structu r e of an A X × X a -mo dule. Mor e over, δ X ∗ M is a simple A X × X a -mo dule along the dia gonal of X × X a . (c) Convers e l y, if N is a simple A X × X a -mo dule a long the diag o nal of X × X a , then δ − 1 X N sa tisﬁes the e quivalent c ondition s (a) (i)–(iii) . Pr o of. The statemen t is lo cal and w e ma y assume that A X = ( O X [[ ~ ] ] , ⋆ ). (a) Assume (ii) and tak e a generator u ∈ M as an A X -mo dule. Then for any a ∈ A X , there exists a unique θ ( a ) ∈ A X suc h that ua = θ ( a ) u . Then θ : A X − → A X giv es a C ~ -algebra endomorphism of A X . Hence θ is an isomorphism b y Propo sition 2.2.3. Thu s we obtain (i). Similarly (iii) implies (i). (b) Let us c ho ose u ∈ M as in (a) and iden tify M with O X [[ ~ ] ] that w e regard as a sheaf supp orted b y the diagonal. The action of A X ⊗ A op X on M 76 CHAPTER 2. DQ -ALGEBROIDS can b e expressed b y diﬀeren tial op erato rs. Namely , there exist diﬀeren tial op erators { S i ( x, ∂ x 1 , ∂ x 2 , ∂ x 3 ) } i ∈ N suc h that f ⋆ a ⋆ θ ( g ) = X i  S i ( x, ∂ x 1 , ∂ x 2 , ∂ x 3 ) f ( x 1 ) g ( x 2 ) a ( x 3 )  | x 1 = x 2 = x 3 = x ~ i for f , g ∈ A X and a ∈ O X [[ ~ ] ]. Then this action extends to a n action of A X × X a b y setting f ( x, y ) ⋆ a ( x ) = X i  S i ( x, ∂ x 1 , ∂ x 2 , ∂ x 3 ) f ( x 1 , x 2 ) a ( x 3 )  | x 1 = x 2 = x 3 = x ~ i for f ∈ A X × X a and a ∈ O X [[ ~ ] ]. W e denote b y f M the A X × X a -mo dule thus obtained. Then, as an ( A X ⊗ A X a )- mo dule, it is isomorphic to M . Hence f M is a lo cally ﬁnitely generated A X × X a -mo dule. Since ~ n f M / ~ n +1 f M is isomorphic to O X , f M is a coheren t A X × X a -mo dule b y Theorem 1.2.5 (ii). Let ˜ I b e the annihilator of u ∈ M ≃ f M . Then ˜ I is a cohere nt left ideal of A X × X a . In the exact sequence T or C ~ 1 ( f M , C ) − → ˜ I / ~ ˜ I − → A X × X a / ~ A X × X a − → f M / ~ f M − → 0 , T or C ~ 1 ( f M , C ) v anishes. Therefore w e obtain an exact sequence 0 − → ˜ I / ~ ˜ I − → O X × X a − → O X − → 0 , and ˜ I / ~ ˜ I is isomorphic to the deﬁning ideal I ∆ ⊂ O X × X a of t he diagonal set ∆ ⊂ X × X a . This sho ws that f M is simple along the diago na l. Denote b y I ′ the left ideal of A X ⊗ A op X generated by the sections { a ⊗ 1 − 1 ⊗ θ ( a ) } where a ranges o v er the family of sections o f A X and b y I the left ideal of A X × X a generated b y I ′ . Set M ′ := A X × X a ⊗ A X ⊠ A X a M . W e hav e: M ≃ ( A X ⊗ A X a ) / I ′ , M ′ ≃ A X × X a / I . There exists a surjectiv e A X × X a -linear mo r phism M ′ ։ f M , and hence I ⊂ ˜ I . Since I / ~ I − → ˜ I / ~ ˜ I ≃ I ∆ is surjectiv e, w e conclude that I = ˜ I . Hence w e obtain M ′ ≃ f M . (c) By the assumption, p 1 ∗ gr ~ ( N ) ≃ gr ~ ( δ − 1 X N ) is an inv ertible O X -mo dule. where p 1 : X × X a − → X is t he pro jection. Hence Theorem 1.2.5 (iv) implies that δ − 1 X N is a coheren t A X -mo dule. It is lo cally isomorphic to A X b y Lemma 1.2.17 b ecause gr ~ ( δ − 1 X N ) is lo cally isomor phic to O X . Q.E.D. 2.4. DQ -MODULES SUPPOR TED BY THE DIAGONAL 77 Th us w e obtain: Prop osition 2.4.2. The c ate go ry of bi-invertible ( A X ⊗ A X a ) -mo dules is e quivalent to the c ate g o ry of c oh e r ent A X × X a -mo dules simp l e alo n g the d i a g- onal. Deﬁnition 2.4.3. W e regard δ X ∗ A X as an A X × X a -mo dule supp orted by the diagonal and denote it b y C X . W e call it the canonical mo dule asso ciated with the diagonal. The next corollary immediately follo ws from Lemma 2.4.1. Corollary 2.4.4. The A X × X a -mo dule C X is c oher ent and simp le along the diagonal. Mor e o ver, A X × X a ⊗ A X ⊠ A X a C X − → C X is an isomorph ism in Mo d( A X × X a ) , and A X − → δ − 1 X ( C X ) is an isomorphism in Mo d( A X ⊗ A X a ) . The next result is obvious . Lemma 2.4.5. L et Y b e another c omplex m anifold e ndowe d with a D Q - algebr oid A Y . Then, ther e is a natur al isomorphism C X L ⊠ C Y ≃ C X × Y . Her e, we id entify ( X × X a ) × ( Y × Y a ) with ( X × Y ) × ( X × Y ) a . Deﬁnition 2.4.6. W e sa y that P ∈ D b ( A X ⊗ A X a ) is bi-inv ertible if P is concen trated to some degree n and H n ( P ) is bi-in v ertible (see D eﬁni- tion 2.1.10). W e sometimes consider a bi-in v ertible ( A X ⊗ A X a )-mo dule as an ob ject of D b coh ( A X × X a ) supp orted b y the diagonal. F or a pair of bi- in v ertible ( A X ⊗ A X a )-mo dules P 1 and P 2 , P 1 L ⊗ A X P 2 is also a bi-inv ertible ( A X ⊗ A X a )-mo dule. Hence the category of bi- in v ertible ( A X ⊗ A X a )-mo dules has a structure of a tensor categor y (see e.g. [41, § 4 .2]). It is easy to see that C X is a unit ob ject. Namely , for any bi-in v ertible ( A X ⊗ A X a )-mo dule P , w e hav e: C X L ⊗ A X P ≃ P L ⊗ A X C X ≃ P . W e hav e P L ⊗ A X R H om A X ( P , A X ) ∼ − → C X , R H om A X a ( P , A X ) L ⊗ A X P ∼ − → C X . Hence w e hav e R H om A X ( P , A X ) ≃ R H om A X a ( P , A X ). 78 CHAPTER 2. DQ -ALGEBROIDS Deﬁnition 2.4.7. F or a bi- inv ertible ( A X ⊗ A X a )-mo dule P , w e set P ⊗− 1 = R H om A X ( P , A X ) ≃ R H om A X a ( P , A X ) . Hence w e hav e P ⊗− 1 L ⊗ A X P ≃ P L ⊗ A X P ⊗− 1 ≃ C X . Note that , for tw o bi-inv ertible ( A X ⊗ A X a )-mo dules P 1 and P 2 , w e hav e R H om A X ( P 1 , P 2 ) ≃ P ⊗− 1 1 L ⊗ A X P 2 , R H om A X a ( P 1 , P 2 ) ≃ P 2 L ⊗ A X P ⊗− 1 1 . F or a bi-in v ertible ( A X ⊗ A X a )-mo dule P a nd M , N ∈ D( A X × Y × Z ), w e ha v e the isomorphism R H om A X × Y ( M , N ) ≃ R H om A X × Y ( P L ⊗ A X M , P L ⊗ A X N ) (2.4.1) in D( C ~ X × Y ⊠ A Z ). Remark 2.4.8. Although it is sometimes con v enien t to iden tify ( X × Y a ) a with Y × X a , we do not tak e this p oin t view in this Note. W e iden tify ( X × Y a ) a with X a × Y . Hence, for example, w e hav e f unctors D ′ A X × Y a : D b ( A X × Y a ) − → D b ( A X a × Y ) , D ′ A X × X a : D b ( A X × X a ) − → D b ( A X a × X ) . The next result may b e useful. Lemma 2.4.9. ( i) L et X and Y b e m anifolds endowe d with DQ -algebr oids A X and A Y , let M b e an A X × Y a -mo dule and let Q b e a bi-i n vertible ( A Y ⊗ A Y a ) -mo dule. Then D ′ A X × Y a ( M ⊗ A Y Q ) ≃ Q ⊗− 1 ⊗ A Y D ′ A X × Y a ( M ) . (ii) L et P and Q b e bi-invertible ( A X ⊗ A X a ) -mo dules. Then D ′ A X × X a ( P ⊗ A X Q ) ≃ Q ⊗− 1 ⊗ A X D ′ A X × X a P ≃ D ′ A X × X a Q ⊗ A X P ⊗− 1 , D ′ A X × X a C X ⊗ A X P ≃ P ⊗ A X D ′ A X × X a C X ≃ D ′ A X × X a ( P ⊗− 1 ) , (D ′ A X × X a C X ) ⊗− 1 ⊗ A X P ≃ P ⊗ A X (D ′ A X × X a C X ) ⊗− 1 . 2.5. DUALIZ ING COMPLEX FOR DQ -ALGEBROIDS 79 Pr o of. ( i) W e ha v e the isomorphism D ′ A X × Y a ( M ⊗ A Y Q ) = H om A X × Y a ( M ⊗ A Y Q , A X × Y a ) ≃ H om A X × Y a ( M , A X × Y a ⊗ A Y Q ⊗− 1 ) ≃ H om A X × Y a ( M , Q ⊗− 1 ⊗ A Y A X × Y a ) ≃ Q ⊗− 1 ⊗ A Y D ′ A X × Y a ( M ) . (ii) The ﬁrst isomorphism follo ws from (i) and the the second is prov ed similarly . The t w o last isomorphisms follo w. Q.E.D. The next result follows immediately from Corollary 2.4.4. Lemma 2.4.10. L et M ∈ D b ( A X ) , L ∈ D b coh ( A X ) and N ∈ D b ( A X a ) . Identifying ∆ X and X , ther e ar e natur al isomorphisms M ≃ A X L ⊗ A X M ≃ R H om A X ( A X , M ) in D( A X ) , N L ⊗ A X M ≃ ( N L ⊠ M ) L ⊗ A X × X a C X in D( C ~ X ) , R H om A X ( L , M ) ≃ D ′ A L L ⊗ A X M in D( C ~ X ) , R H om A X ( M , L ) ≃ R H om A X × X a ( M L ⊠ D ′ A L , C X ) in D( C ~ X ) . 2.5 Dualizing c omplex for DQ -algeb roids The algebroid D A X W e ha v e seen that the C ~ -algebra D A X ⊂ E nd C ~ ( A X ) is well-deﬁne d for a DQ-algebra A X on X . No w let A X b e a DQ-algebroid. Then w e can regard A X as an ( A X ⊗ A op X )-mo dule. In § 2.1, w e ha v e deﬁned the C ~ -algebroid E nd C ~ ( A X ) and in tro duced a functor of C ~ -algebroids A X ⊗ A op X − → E nd C ~ ( A X ). Deﬁnition 2.5.1. The C ~ -algebroid D A X is the C ~ -substac k of E nd C ~ ( A X ) asso ciated to the prestac k S deﬁned as follo ws. The ob jects of S a re those of A X ⊗ A op X . F or σ 1 , σ 2 ∈ A X ⊗ A op X , with σ 1 = τ 1 ⊗ λ op 1 , σ 2 = τ 2 ⊗ λ op 2 , we c ho ose isomorphisms ϕ i : τ i ≃ λ i ( i = 1 , 2 ) and ϕ 3 : τ 1 ≃ τ 2 . Set B = E nd A X ( λ 1 ). It is a D Q-algebra. The isomorphisms ϕ i ( i = 1 , 2 , 3) induce an isomorphism ψ : H om C ~ ( B , B ) ∼ − → H om C ~ ( H om ( λ 1 , τ 1 ) , H om ( λ 2 , τ 2 )) ∼ − → H om C ~ ( A X ( σ 1 ) , A X ( σ 2 )) . 80 CHAPTER 2. DQ -ALGEBROIDS W e deﬁne H om S ( σ 1 , σ 2 ) ⊂ H om C ~ ( A X ( σ 1 ) , A X ( σ 2 )) a s the image of D B X b y ψ . (This do es not dep end on the choice o f the isomorphism ϕ i ( i = 1 , 2 , 3) in virtue of Prop osition 2.2.3 .) Then there are functors of C ~ -algebroids A X ⊗ A X a − → δ − 1 X A X × X a − → D A X − → E nd C ~ ( A X ) and A X ma y b e regarded as a n ob ject of Mo d( D A X ). Prop osition 2.5.2. (i) The C ~ -algebr oid E n d C ~ ( A X ) is e quivalent to the C ~ -algebr oid E nd C ~ ( O X [[ ~ ]]) ( r e gar ding the C ~ -algebr a E nd C ~ ( O X [[ ~ ]]) as a C ~ -algebr oid ) . (ii) The e quivalenc e in (i) induc es a n e quivalenc e of C ~ -algebr oids D A X ≃ D X [[ ~ ] ] . (iii) The e quivalenc e in ( ii) induc es an e quivalenc e of C ~ -line ar stacks Mod ( D A X ) ≃ M od ( D X [[ ~ ] ]) . Mor e over, the D A X -mo dule A X is sent to the D X [[ ~ ] ] -mo dule O X [[ ~ ]] by this e quival e n c e. (iv) The e quivalenc e in (ii) also induc es an e quivalenc e of C -algebr oids gr ~ ( D A X ) ≃ D X , and an e quivalenc e of C -line ar stacks Mod (gr ~ ( D A X )) ≃ M od ( D X ) . Mor e over the gr ~ ( D A X ) -mo dule gr ~ ( A X ) is sent to the D X -mo dule O X by this e quivalenc e. Pr o of. R ecall ﬁrst that for tw o C ~ -algebroids B and B ′ , to giv e an equiv a- lence of C ~ -algebroids B ≃ B ′ is equiv alen t to giving a bi- in v ertible B op ⊗ B ′ - mo dule (Lemma 2.1.11). (i) fo llo ws from Lemma 2.1.12. More precisely , w e deﬁne a n  E n d C ~ ( A X ) ⊗ ( E nd C ~ ( O X [[ ~ ]])) op  -mo dule L ′ as follow s. F or σ = ( σ 1 ⊗ σ op 2 ) ∈ A X ⊗ A op X , set L ′ ( σ ) := H o m C ~ ( O X [[ ~ ]] , H om A X ( σ 2 , σ 1 )) . Clearly , L ′ is bi-in v ertible. (ii) F or σ = ( σ 1 ⊗ σ op 2 ) ∈ A X ⊗ A op X , let us c ho o se an isomorphism ψ : σ 1 ∼ − → σ 2 and a standard isomorphism e ϕ : O X [[ ~ ] ] ∼ − → E nd A X ( σ 1 ). Then they give an isomorphism f : O X [[ ~ ]] ∼ − → H om A X ( σ 2 , σ 1 ) . 2.5. DUALIZ ING COMPLEX FOR DQ -ALGEBROIDS 81 W e deﬁne a ( D A X ⊗ D X [[ ~ ]] op )-submo dule L o f L ′ as follows: let L ( σ ) b e the D X [[ ~ ]] op -submo dule of L ′ ( σ ) generated b y f . Then L ( σ ) coin- cides with the su bmo dule generated b y f ov er the C ~ -algebra E nd D A X ( σ ) ⊂ E n d C ~ ( H om A X ( σ 2 , σ 1 )). Moreov er, L ( σ ) does not dep end on the c hoice of ψ and e ϕ . It is easy to see that L is a bi-inv ertible ( D A X ⊗ D X [[ ~ ]] op )-mo dule. (iii) The ( D A X ⊗ D X [[ ~ ]] op )-mo dule L giv es an equiv alence of categories L ⊗ D X [[ ~ ]] • : Mo d( D X [[ ~ ]]) ∼ − → Mo d( D A X ) , (2.5.1) whic h is isomorphic to the func tor induced by the algebroid equiv alenc e D A X ∼ − → D X [[ ~ ]]. Consider the ( D X [[ ~ ]] ⊗ ( D A X ) op )-mo dule L ∗ := H om D A X ( L , D A X ) . A quasi-in v erse of the equiv alence (2.5.1) is given by L ∗ ⊗ D A X • ≃ H om D A X ( L , • ) : Mod( D A X ) ∼ − → Mo d( D X [[ ~ ]]) . The results follow. Q.E.D. Dualizing c omplex Let A X b e a DQ-algebroid o n X . W e shall construct a deforma t io n of the sheaf of diﬀerential fo rms of maximal degree a nd then the dualizing complex for A X . Lemma 2.5.3. (i) A X has lo c al ly a r esolution of length d X by fr e e D A X - mo dules of ﬁnite r an k. (ii) gr ~ ( E xt d X D A X ( A X , D A X )) ≃ Ω X . ( Note that gr ~ ( E xt d X D A X ( A X , D A X )) is a mo dule o v er gr ~ ( A X ) ⊗ O X gr ~ ( A X a ) ≃ O X by (2.1.12)) . (iii) E xt i D A X ( A X , D A X ) = 0 for i 6 = d X . Pr o of. W e ha ve D A X ≃ D X [[ ~ ] ] and A X ≃ O X [[ ~ ] ] as D A X -mo dules. Then the results follow from R H om D X [[ ~ ]] ( O X [[ ~ ] ] , D X [[ ~ ] ]) ≃  Ω X [[ ~ ] ]  [ − d X ] . (ii) follows from gr ~ (R H om D A X ( A X , D A X )) ≃ R H om gr ~ ( D A X ) (gr ~ ( A X ) , gr ~ ( D A X )) ≃ R H om D X ( O X , D X ) ≃ Ω X [ − d X ] . Q.E.D. 82 CHAPTER 2. DQ -ALGEBROIDS W e set Ω A X := E xt d X D A X ( A X , D A X ) ∈ Mo d( A X ⊗ A X a ) . (2.5.2) Lemma 2.5.4. The ( A X ⊗ A op X ) -mo dule Ω A X is bi-invertible. Pr o of. Under the equiv alence D A X ≃ D X [[ ~ ] ], w e hav e Ω A X ≃ Ω X [[ ~ ] ]. Hence w e h av e an isomorphism Ω A X ∼ − → lim ← − n Ω A X / ~ n Ω A X . Since gr ~ (Ω A X ) ≃ Ω X is a coheren t gr ~ ( A X )-mo dule, Ω A X is a coheren t A X -mo dule by Theorem 1.2.5 (iv). Since gr ~ (Ω A X ) is an in v ertible O X -mo dule and Ω A X has no ~ - torsion, Ω A X is lo cally isomorphic to A X as an A X -mo dule. Hence Ω A X is a bi-in v ertible ( A op X ⊗ A X )-mo dule b y Lemma 2 .4.1 (a ). Q.E.D. Lemma 2.5.5. One h a s the isomorphisms Ω A X L ⊗ D A X A X [ − d X ] ≃ R H om D A X ( A X , A X ) ≃ C ~ X . (2.5.3) Pr o of. The ﬁrst isomorphism is obv ious b y Lemma 2 .5 .3. Hence, it is enough to prov e that the natural morphism C ~ X − → R H om D A X ( A X , A X ) is a n isomor- phism. By the equiv alence D A X ≃ D X [[ ~ ]], we ma y a ssume that A X = O X [[ ~ ] ] and D A X = D X [[ ~ ] ]. Then R H om D A X ( A X , A X ) is represen ted b y an inﬁnite pro duct of the de Rham complexes: Q n ~ n Ω • X . Then the assertion follows from a classical result: Ω • X ( U ) is quasi-isomorphic to C when U is a con- tractible Stein op en subset. Q.E.D. Note that Ω A X and Ω A X a are isomorphic as A X ⊗ A X a -mo dules. Deﬁnition 2.5.6. W e set ω A X := δ X ∗ Ω A X [ d X ] ≃ δ X ∗ R H om D A X ( A X , D A X )[2 d X ] ∈ D b ( A X × X a ) and call ω A X the A X -dualizing sheaf. Note that ω A X is bi-inv ertible (see D eﬁnition 2.4.6). Using (2.5.3) and the morphism δ X ∗ Ω A X L ⊗ A X × X a C X − → Ω A X L ⊗ D A X A X , w e g et the natural morphism ω A X a L ⊗ A X × X a C X − → δ X ∗ C ~ X [2 d X ] . (2.5.4) Applying the functor gr ~ to the ab ov e morphisms, w e get the morphism δ X ∗ (gr ~ ω A X a ) L ⊗ gr ~ A X × X a ( δ X ∗ gr ~ C X ) − → δ X ∗ ( C X [2 d X ]) , (2.5.5) 2.5. DUALIZ ING COMPLEX FOR DQ -ALGEBROIDS 83 whic h coincides with the morphism deriv ed fr o m δ − 1 X  δ X ∗ (gr ~ ω A X a ) L ⊗ gr ~ A X × X a ( δ X ∗ gr ~ C X )  − → Ω X [ d X ] − → C X [2 d X ] . (2.5.6) Here we used the functor of algebroids δ − 1 X (gr ~ A X × X a ) − → O X . Let Y b e another manifold endow ed with a DQ- algebroid A Y . W e in tro- duce the notation: ω A X × Y / Y = ω A X L ⊠ C Y ∈ D b ( A X × X a × Y × Y a ) . Then ω A X × Y / Y also b elongs to D b  ( D A X ) op ⊠ A Y × Y a  , and w e ha v e an isomor- phism ω A X × Y / Y L ⊗ D A X A X ≃ C ~ X ⊠ A Y . Hence w e hav e a canonical morphism ω A X a × Y / Y L ⊗ A X × X a C X − → ( C ~ X ⊠ C Y )[2 d X ] (2.5.7) in D b ( C ~ X ⊠ A Y × Y a ). The pro of of the follo wing fundamen tal result will b e giv en later at the end of § 3.3. Theorem 2.5.7. We have the isomorphism ω A X ≃ (D ′ A X a × X C X a ) ⊗− 1 in D b ( A X × X a ) . (2.5.8) Note that in form ula (2.5.8), D ′ A X a × X is the dual o v er A X a × X and ( • ) ⊗− 1 is the dual ov er A X . Corollary 2.5.8. F or M ∈ D b ( A X × X a × Y ) , we have C X a L ⊗ A X × X a M ≃ R H om A X × X a ( C X , ω A X L ⊗ A X M ) ≃ R H o m A X × X a ( C X , M L ⊗ A X ω A X ) . Pr o of. W e hav e C X a L ⊗ A X × X a M ≃ R H om A X × X a (D ′ A X a × X C X a , M ) ≃ R H om A X × X a ( ω A X L ⊗ A X D ′ A X × X a C X a , ω A X L ⊗ A X M ) ≃ R H om A X × X a ( C X , ω A X L ⊗ A X M ) . The other isomorphism is similarly prov ed. Q.E.D. 84 CHAPTER 2. DQ -ALGEBROIDS One shall b e aw are that, although Ω A X is lo cally isomorphic to A X as an A X -mo dule, it is not alw ay s lo cally isomorphic to A X as an A X ⊗ A X a - mo dule. Example 2.5.9. Let X = C 2 with co or dina t es ( x 1 , x 2 ) and let A X b e the DQ-algebra giv en b y the relatio n [ x 1 , x 2 ] = ~ x 1 . Let ( y 1 , y 2 ) denotes the co ordinates on X a . Hence [ y 1 , y 2 ] = − ~ y 1 . Then C X is the A X × X a -mo dule A X × X a · u where the generator u satisﬁes ( x i − y i ) · u = 0 ( i = 1 , 2). Therefore C X is quasi-isomorphic to the complex 0 − → A X × X a α − → A ⊕ 2 X × X a β − → A X × X a − → 0 , (2.5.9) where A X × X a on the r ig h t is in degree 0, α ( a ) = ( − a ( x 2 − y 2 + ~ ) , a ( x 1 − y 1 )) and β ( b, c ) = b ( x 1 − y 1 ) + c ( x 2 − y 2 ). It fo llows that D ′ A ( C X ) [2] is isomorphic to A X × X a · w where the generator w satisﬁes ( x 1 − y 1 ) · w = 0, ( y 2 − x 2 + ~ ) · w = 0. The mo dules D ′ A ( C X ) [2] and C X are isomorphic on x 1 6 = 0 by u ↔ x 1 w . How ev er, D ′ A ( C X ) [2] and C X are not isomorphic on a neighborho o d of x 1 = 0. Indeed if they w ere isomorphic b y u ↔ aw for a ∈ A X , then x 1 a = ax 1 and x 2 a = a ( x 2 − ~ ). Then { x 2 , σ 0 ( a ) } = − σ 0 ( a ). Since { x 2 , • } = − x 1 ∂ x 1 , w e hav e x 1 ∂ x 1 σ 0 ( a ) = σ 0 ( a ), whic h contradicts the f a ct that σ 0 ( a ) is in ve rtible. Remark 2.5.10. The fact tha t D ′ A C X is concen trated in a single degree and pla ys the role of a dualizing complex in the sense of [60 ] was already prov ed (in a more restrictiv e situation) in [2 0, 21 ]. 2.6 Almost free re s olutions W e recall here and adapt to the framew ork of algebroids some results of [40]. In this section, K denotes a comm utative unital ring, X a pa racompact and lo cally compact space and A a K -alg ebroid on X . Let us tak e a family S of op en subs ets of X . W e assume the follo wing t w o conditions on S :                (i) for a n y x ∈ X , { U ∈ S ; x ∈ U } is a neigh b orho o d sys - tem of x , (ii) for U , V ∈ S , U ∩ V is a ﬁnite union of open subsets b elonging to S . (2.6.1) 2.6. ALMOST FREE RESOLUTIONS 85 Recall that in v ertible mo dules are deﬁned in Deﬁnition 2 .1.4. Deﬁnition 2.6.1. (i) W e deﬁne the additiv e category Mo d af ( A ) of S - almost free A - mo dules a s f o llo ws. (a) An ob ject of Mo d af ( A ) is the data of { I , { U i , U ′ i , L i } i ∈ I } where I is an index set, U i and U ′ i are op en subsets of X , U i ∈ S , U i ⊂ U ′ i , the family { U ′ i } i ∈ I is lo cally ﬁnite and L i is an in v ertible A | U ′ i -mo dule. (b) Let N = { J, { V j , V ′ j , K j } j ∈ J } and M = { I , { U i , U ′ i , L i } i ∈ I } b e t w o ob jects of Mo d af ( A ). A morphism u : N − → M is the data of u ij ∈ Γ( V j ; H om A ( K j , L i )) f o r all ( i, j ) ∈ I × J suc h that V j ⊂ U i . (c) The comp osition of morphisms is the natural one. (d) W e denote by Φ : Mo d af ( A ) − → Mo d( A ) the functor whic h sends { I , { U i , U ′ i , L i } i ∈ I } to L i ∈ I ( L i ) U i and whic h sends an elemen t u ij of Γ( V j ; H om A ( K j , L i )) to its image in Hom A (( K j ) V j , ( L i ) U i ) if V j ⊂ U i and 0 otherwise. (ii) Similarly , w e deﬁne the additive category Mo d af ( A ) as follo ws. (a) The set of ob jects of Mo d af ( A ) is the same as the one of Mo d af ( A ). (b) Let N = { J, { V j , V ′ j , K j } j ∈ J } and M = { I , { U i , U ′ i , L i } i ∈ I } b e t w o ob jects of Mo d af ( A ). A morphism u : N − → M is the data of u ij ∈ Γ( U i ; H om A ( K j , L i )) for all ( i, j ) ∈ I × J suc h that U i ⊂ V j . (c) The comp osition of morphisms is the natural one. (d) W e denote by Ψ : Mo d af ( A ) − → Mo d( A ) t he functor whic h sends { I , { U i , U ′ i , L i } i ∈ I } to L i ∈ I Γ U i ( L i ) and whic h sends an elemen t u ij of Γ( U i ; H om A ( K j , L i )) to its image in Hom A (Γ V j ( K j ) , Γ U i ( L i )) if U i ⊂ V j and 0 otherwise. Note that Mo d af ( A ) is equiv a lent to Mo d af ( A op ) op b y the functor whic h sends { I , { U i , U ′ i , L i } i ∈ I } to { I , { U i , U ′ i , H om A ( L i , A ) } i ∈ I } . Recall that f o r an additiv e category C , we denote by C − ( C ) (resp. C + ( C )) the category of complexes of C bounded fr om ab ov e (resp. from b elow). The followin g theorem is prov ed similarly as in [40, App endix]. Theorem 2.6.2. L et A b e a left c oher ent algebr oid and let M ∈ D − coh ( A ) . Then ther e exist L • ∈ C − (Mo d af ( A )) and an isomorphism Φ( L • ) ≃ M in D − ( A ) . There is a dual v ersion of Theorem 2.6.2. Theorem 2.6.3. Assume 86 CHAPTER 2. DQ -ALGEBROIDS (a) A b eing r e gar de d as an obje ct of Mo d( A ⊗ A op ) , RΓ U ( A ) is c onc en- tr ate d in de gr e e 0 for al l U ∈ S , (b) A is a right and left c oher ent algebr oid, (c) ther e exists an inte ge r d s uch that, for any op en subset U , any c oher e nt A | U -mo dule admits lo c al ly a ﬁn ite f r e e r esolution of length d . L et M ∈ D + coh ( A ) . Then ther e exist L • ∈ C + (Mo d af ( A )) and an isomor- phism M ≃ Ψ( L • ) in D + ( A ) . Pr o of. D enote by D the dua lity functor R H om A ( • , A ) and ke ep the same notation with A op instead of A . This functor sends D + coh ( A ) t o D − coh ( A op ) b y (c). It a lso sends D − coh ( A op ) to D + coh ( A ), and the comp osition D + coh ( A ) D − − → D − coh ( A op ) D − − → D + coh ( A ) is isomorphic to the iden tity f unctor. On the o ther hand, if L is an in v ertible A op -mo dule, then D( L ) is an in v ertible A -mo dule, and by the h yp othesis (a), w e ha v e D( L U ) ≃ Γ U (D( L )) for an y U ∈ S . Then w e get t he result b y applying Theorem 2.6.2 to D( M ) ∈ D − coh ( A op ) and using M ∼ − → D( D ( M )). Q.E.D. 2.7 DQ -algebroids in th e algebr aic case In this section, X denotes a quasi-compact separated smo oth algebraic v ari- et y ov er C . Clearly , the notions of a D Q-algebra and of a DQ-algebroid mak e sense in this settings and a detailed study of DQ-alg ebroids on algebraic v ariet y is p erformed in [64]. Assume t ha t X is endo w ed with a DQ-alg ebroid A X for the Zariski top ol- ogy . Then, in view of Remark 2.1.17, gr ~ ( A X ) ≃ O X . Ho w eve r, this equiv a- lence is not unique in general. Let us denote b y X an the complex analytic manifold asso ciated with X and b y ρ : X an − → X the natura l morphism. Then we can naturally asso ciate a D Q-algebroid A X an to A X and there is a natural functor ρ − 1 A X − → A X an , whose construction is left to the reader. Then it induces functors Mo d( A X ) − → Mo d( A X an ) (2.7.1) 2.7. DQ -ALG EBR OIDS IN THE ALGEBRAIC CASE 87 and Mo d coh ( A X ) − → Mo d coh ( A X an ) . (2.7.2) When X is pro jectiv e, t he classical G A GA theorem of Serre extends to DQ- algebroids and it is prov ed in [16] that (2.7.2) is an equiv alence. Lemma 2.7.1. L et M ∈ Mo d coh ( A lo c X ) . The two c on ditions b elow ar e e quiv- alent. (a) M is the inductive limit of its c o h er ent sub- A X -mo dules, (b) ther e exi s ts an A X -lattic e of M ( s e e Deﬁnition 2.3 .14 ) . Pr o of. ( a) ⇒ (b) Let M = lim − → N where N ranges ov er the ﬁltrant family of coheren t A X -submo dules of M . Since A lo c X is No etherian, the family { C ~ , lo c ⊗ C ~ N } is lo cally stationary . Since X is quasi-compact, this family is stationary . (b) ⇒ (a) is ob vious. Q.E.D. Deﬁnition 2.7.2. Let M ∈ Mo d coh ( A lo c X ). W e sa y that M is alg ebraically go o d if it satisﬁes the equiv alen t conditions in Lemma 2.7 .1. W e still denote b y M o d gd ( A lo c X ) the full sub category of Mo d coh ( A lo c X ) consisting of algebraically go o d mo dules. The pro of of [3 7, Prop. 4 .2 3] extends to this case and Mo d gd ( A lo c X ) is a thick ab elian s ub category of Mo d coh ( A lo c X ). Hence, we still denote b y D b gd ( A lo c X ) the f ull triangula ted sub categor y o f D b coh ( A lo c X ) consisting of ob- jects M suc h that H j ( M ) is a lgebraically go o d for all j ∈ Z . Remark 2.7.3. W e do no t k now if ev ery c oherent A lo c X -mo dule is alge- braically go o d. Almost free resolutions Recall that X is endo w ed with a D Q-algebroid A X for the Zariski top olog y . W e den ot e b y B the family o f aﬃne o p en subsets U of X on whic h the algebroid A U is a sheaf of algebras. No t e t hat this family is stable b y in tersection. Moreov er, hy p o theses (1.2.2) and (1.2.3) are satisﬁed. Lemma 2.7.4. Assume that X is aﬃne and A X is a D Q -algebr a. Then, for any M ∈ Mo d coh ( A X ) , ther e exist a fr e e A X -mo dule L of ﬁnite r ank and an epimorphism u : L ։ M . 88 CHAPTER 2. DQ -ALGEBROIDS Pr o of. Set M 0 = M / ~ M . Then M 0 is a coherent O X -mo dule and there exist ﬁnitely man y sec tio ns ( v 1 , . . . , v N ) of M 0 on X whic h generate M 0 o v er O X . By Theorem 1.2.5, Γ( X ; M ) − → Γ( X ; M 0 ) is surjectiv e. Let ( u 1 , . . . , u N ) b e sections of M whose image by this morphism are ( v 1 , . . . , v N ). Let L = A N X and denote b y ( e 1 , . . . , e N ) its canonical basis. It remains to deﬁne u b y setting u ( e i ) = u i . Q.E.D. Theorem 2.7.5. L et M ∈ Mo d coh ( A X ) . T hen ther e exists an isomorphism M ≃ L • in D b ( A X ) such that L • is a b ounde d c omplex of A X -mo dules and e ach L i is a ﬁnite dir e ct sum of mo dules of the form i U ∗ L U , wh er e i U : U ֒ → X is the em b e dding of an aﬃne op en se t U such that A U is isomorphic to a D Q -algebr a and L U is a lo c al ly fr e e A U -mo dule of ﬁnite r ank. Before pro ving Theorem 2.7.5, w e need some preliminary results . Let U = { U i } i ∈ I b e a ﬁnite cov ering of X by aﬃne op en sets suc h that A X | U i is a DQ-algebra for all i . W e denote b y Σ the category of non emp ty subsets of I (the morphisms are the inclusions maps). F or σ ∈ Σ, w e denote b y | σ | its cardinal. F or σ ∈ Σ, we set U σ = \ i ∈ σ U i , ι σ : U σ ֒ → X the natura l embedding . W e in tro duce a category Mo d( A , U ) as follo ws. An o b ject M of Mo d( A , U ) is t he data of a fa mily ( { M σ } σ ∈ Σ , { q M σ ,τ } τ ⊂ σ ∈ Σ ), where M σ ∈ Mo d( A U σ ) and q M σ ,τ : M τ | U σ − → M σ are morphisms for ∅ 6 = τ ⊂ σ ∈ Σ satisfying q M σ ,σ = id a nd for an y σ 1 ⊂ σ 2 ⊂ σ 3 , the diagram b elow comm utes M σ 1 | U σ 3 q M σ 2 ,σ 1 / / q M σ 3 ,σ 1 % % K K K K K K K K K K M σ 2 | U σ 3 q M σ 3 ,σ 2   M σ 3 . (2.7.3) A morphism M − → M ′ in Mo d( A , U ) is a family of morphisms M σ − → M ′ σ satisfying the natural compatibility conditions. Clearly , Mod( A , U ) is an abelian category . T o an ob ject M ∈ Mo d( A , U ) w e shall asso ciate a Koszul complex C • ( M ) using the construc tion of [41, § 12.4]. T o M w e asso ciate a functor F : Σ − → Mo d( A X ) as follow s: F ( σ ) = ι σ ∗ M σ , and F ( τ ⊂ σ ) : F ( τ ) − → F ( σ ) is giv en by the comp osition ι τ ∗ M τ − → ι σ ∗ ( M τ | U σ ) q M σ,τ − − → ι σ ∗ M σ . 2.7. DQ -ALG EBR OIDS IN THE ALGEBRAIC CASE 89 According to lo c. cit., w e get a Koszul complex C • ( M ) C • ( M ) := · · · − → 0 − → C 1 ( M ) d 1 − − → C 2 ( M ) d 2 − − → · · · (2.7.4) where C i ( M ) = M | σ | = i ι σ ∗ M σ is in degree i . This construction being functorial, we get a f unctor C • : Mo d( A , U ) − → C b (Mo d( A X )) . (2.7.5) It is con v enien t to in tro duce some notations. W e set Mo d coh ( A , U ) = { M ∈ Mo d( A , U ) ; M σ ∈ Mo d coh ( A U σ ) fo r all σ ∈ Σ } , Mo d ﬀ ( A , U ) = { M ∈ Mo d ( A , U ); M σ is a lo cally free A U σ -mo dule of ﬁnite rank for all σ ∈ Σ } . Clearly , Mo d coh ( A , U ) is a full ab elian subcategory of Mo d( A , U ) a nd Mo d ﬀ ( A , U ) is a full additive sub category o f Mo d coh ( A , U ) . Lemma 2.7.6. Th e functor C • : Mod coh ( A , U ) − → C b (Mo d( A X )) ind uc e d by (2.7.5) is exact. Pr o of. By Prop osition 1.6.8 the functor ι σ : Mo d coh ( A U σ ) − → Mo d( A X ) is exact f or each σ ∈ Σ. The result then easily follows. Q.E.D. Let us denote b y λ : Mo d coh ( A X ) − → Mo d coh ( A , U ) (2.7.6) the functor whic h, to M ∈ Mod coh ( A X ), asso ciates the ob ject M where M σ = M | U σ and q M σ ,τ : M τ | U σ − → M σ is the restriction morphism. Lemma 2.7.7. The natur al morphi s m M − → C • ( λ ( M )) [1] is a quasi - isomorphism. Pr o of. Apply [41, Th. 1 8 .7.4 (ii)] with A = “ F ” i ∈ I U i , u : A − → X . By this result, the complex F • u := 0 − → M − → C 1 ( λ ( M )) d 1 − → C 2 ( λ ( M )) d 2 − → · · · is exact. Q.E.D. 90 CHAPTER 2. DQ -ALGEBROIDS Lemma 2.7.8. L et M ∈ Mo d coh ( A , U ) . Then ther e exists an epim o rp hism L ։ M in Mo d( A , U ) with L ∈ Mo d ﬀ ( A , U ) . Pr o of. Applying Lemma 2 .7.4, w e choo se for eac h σ ∈ Σ an epimorphism L ′ σ ։ M σ with a lo cally free A U σ -mo dule L ′ σ of ﬁnite rank. Set L σ := M ∅6 = τ ⊂ σ L ′ τ | U σ and deﬁne the morphism L σ − → M σ b y the comm utativ e diagra ms in whic h τ ⊂ σ : L σ / / M σ L ′ τ | U σ / / O O M τ | U σ . O O F or τ ⊂ σ , the morphism q L σ ,τ : L τ | U σ − → L σ is deﬁned b y the mor phisms ( λ ⊂ τ ): L τ | U σ q L σ,τ / / L σ . L ′ λ | U σ O O 9 9 r r r r r r r r r r r Clearly , the family of morphisms q L σ ,τ satisﬁes t he compatibility conditions similar to those in diagram (2.7.3). W e ha v e th us constructed an o b j ect L ∈ Mo d( A , U ), and the family of morphisms L σ − → M σ deﬁnes the epimorphism L ։ M in Mod( A , U ). Q.E.D. Pr o of of The o r em 2.7.5. By Lemma 2 .7.8, there exists an exact sequence in Mo d coh ( A , U ) 0 − → L d X +1 − → · · · − → L 1 − → L 0 − → λ ( M ) − → 0 (2.7.7) with the L i ’s in Mo d ﬀ ( A , U ) (see Corollary 2.3.5). Consider the complex L • := · · · − → L 1 − → L 0 − → 0 . (2.7.8) Hence, w e hav e a quasi-isomorphism L • q is − → λ ( M ). Using Lemma 2.7 .6 , w e ﬁnd a quasi-isomorphism C • ( L • ) q is − → C • ( λ ( M )) . (2.7.9) Then, the result follows from Lemma 2.7.7. Q.E.D. Chapter 3 Kernels 3.1 Con v oluti on of k ernels : deﬁni t ion In tegral transforms, also called “corresp ondences”, are o f constant use in al- gebraic and analytic geometry and w e refer to t he b o ok [33] for an expo sition. Here, we shall dev elop a similar formalism in the framew ork of DQ-mo dules ( i.e., mo dules o v er DQ-algebroids). Consider complex manifolds X i endo w ed with DQ-algebroids A X i ( i = 1 , 2 , . . . ). Notation 3.1.1. (i) Consider a pro duct of manifolds X × Y × Z . W e denote b y p i the i -th pro jection and b y p ij the ( i, j )-th pro jection ( e.g., p 13 is t he pro jection from X 1 × X a 1 × X 2 to X 1 × X 2 ). W e use similar notations for a pro duct of fo ur manifolds. (ii) W e write A i and A ij a instead of A X i and A X i × X a j and similarly with other pro ducts. W e use the same notat io ns fo r C X i . (iii) When there is no risk of confusion, w e do note write the sym b ols p − 1 i and similarly with i replaced with ij , etc. Let K i ∈ D b ( A X i × X a i +1 ) ( i = 1 , 2). W e set K 1 L ⊗ A 2 K 2 := p − 1 12 K 1 L ⊗ p − 1 2 A 2 p − 1 23 K 2 ≃ ( K 1 L ⊠ K 2 ) L ⊗ A 2 ⊠ A 2 a C 2 ∈ D b ( A 1 ⊠ C ~ X 2 ⊠ A 3 a ) . (3.1.1) Similarly , for K i ∈ D b ( A X i × X i +1 ) ( i = 1 , 2), w e set R H om A 2 ( K 1 , K 2 ) := R H om p − 1 2 A 2 ( p − 1 12 K 1 , p − 1 23 K 2 ) . (3.1.2) 91 92 CHAPTER 3. KERNELS Here w e iden tify X 1 × X 2 × X a 3 with the diagonal set of X 1 × X a 2 × X 2 × X a 3 . This tensor pro duct is not w ell suited to tr eat DQ - mo dules. F or example, A X × Y 6 = A X ⊠ A Y . This leads us to introduce a kind of completion of the tensor pro duct as follow s. Deﬁnition 3.1.2. Let K i ∈ D b ( A X i × X a i +1 ) ( i = 1 , 2). W e se t K 1 L ⊗ A 2 K 2 = δ − 1 2 ( K 1 L ⊠ K 2 ) L ⊗ A 22 a C 2 (3.1.3) = p − 1 12 K 1 L ⊗ p − 1 12 A 1 a 2 A 123 L ⊗ p − 1 23 A 23 a p − 1 23 K 2 . It is an ob ject of D b ( p − 1 13 A 13 a ) where p 13 : X 1 × X 2 × X 3 − → X 1 × X 3 is the pro jection. W e hav e a morphism in D b ( p − 1 1 A X 1 ⊗ p − 1 3 A X a 3 ): K 1 L ⊗ A 2 K 2 − → K 1 L ⊗ A 2 K 2 . (3.1.4) Note that (3.1.4) is an isomorphism if X 1 = pt or X 3 = pt. Deﬁnition 3.1.3. Let K i ∈ D b ( A X i × X a i +1 ) ( i = 1 , 2). W e se t K 1 ◦ X 2 K 2 = R p 13 ! ( K 1 L ⊗ A 2 K 2 ) ∈ D b ( A X 1 × X a 3 ) , (3.1.5) K 1 ∗ X 2 K 2 = R p 13 ∗ ( K 1 L ⊗ A 2 K 2 ) ∈ D b ( A X 1 × X a 3 ) . (3.1.6) W e call ◦ X 2 the conv olution of K 1 and K 2 (o v er X 2 ). If there is no risk of confusion, w e write K 1 ◦ K 2 for K 1 ◦ X 2 K 2 and similarly with ∗ . Note that in case where X 3 = pt w e get: K 1 ◦ K 2 ≃ R p 1 ! ( K 1 L ⊗ A 2 p − 1 2 K 2 ) , and in the general case, we hav e: K 1 ◦ X 2 K 2 ≃ ( K 1 L ⊠ K 2 ) ◦ X 2 × X a 2 C X 2 ≃ R p 14 !  ( K 1 L ⊠ K 2 ) L ⊗ A 22 a C 2  , (3.1.7) where p 14 is the pro jection X 1 × X 2 × X a 2 × X a 3 − → X 1 × X a 3 . There are canonical isomorphisms K 1 ◦ X 2 C X 2 ≃ K 1 and C X 1 ◦ X 1 K 1 ≃ K 1 . (3.1.8) 3.2. CONV OLUTION OF KERNELS: FINITENESS 93 One shall be a w are that ◦ and ∗ a r e not asso ciativ e in general. (See Prop o- sition 3.2.4 (ii).) Ho w ev er, if L is a bi-in v ertible A X 2 ⊗ A X a 2 -mo dule and the K i ’s ( i = 1 , 2) are as a b o v e, there are natur a l isomorphisms K 1 ◦ X 2 L ≃ K 1 L ⊗ A X 2 L , L ◦ X 2 K 2 ≃ L L ⊗ A X 2 K 2 , ( K 1 ◦ X 2 L ) ◦ X 2 K 2 ≃ K 1 ◦ X 2 ( L ◦ X 2 K 2 ) . F or a closed subs et Λ i of X i × X i +1 ( i = 1 , 2 ), we set Λ 1 ◦ Λ 2 := p 13 ( p − 1 12 Λ 1 ∩ p − 1 23 Λ 2 ) (3.1.9) = p 14 ((Λ 1 × Λ 2 ) ∩ ( X 1 × ∆ 2 × X 3 )) ⊂ X 1 × X 3 . Note that if Λ i is a closed complex analytic sub v ariet y of X i × X a i +1 ( i = 1 , 2) and p 13 is prop er on p − 1 12 Λ 1 ∩ p − 1 23 Λ 2 , then Λ 1 ◦ Λ 2 is a closed complex analytic sub v ariet y of X 1 × X a 3 . Let us still denote by ◦ the con v olution of gr ~ ( A )- mo dules. More precisely for L i ∈ D b (gr ~ ( A X i × X a i +1 )) ( i = 1 , 2), w e set L 1 ◦ L 2 = R p 14 !  ( L 1 L ⊠ L 2 ) L ⊗ gr ~ ( A 22 a ) gr ~ ( C 2 )  . Prop osition 3.1.4. F or K i ∈ D b ( A X i × X a i +1 ) ( i = 1 , 2) , we have gr ~ ( K 1 ◦ K 2 ) ≃ g r ~ ( K 1 ) ◦ gr ~ ( K 2 ) . (3.1.10) Pr o of. Applying Prop osition 1.4.3, it remains to remark that the functor gr ~ comm utes with the functors o f in v erse images and prop er direct images as w ell as with t he f unctor ⊠ . Q.E.D. 3.2 Con v oluti on of k ernels : ﬁnite n ess In this section, w e use Notation 3.1.1 Consider complex manifolds X i endo w ed with DQ-algebroids A X i ( i = 1 , 2 , . . . ). W e denote by d X the complex dimension of X and we write for short d i instead of d X i . W e shall prov e the follo wing coherency theorem for D Q-mo dules by re- ducing it to the corresp onding result for O -mo dules due to Grauert ([29]). In the seq uel, for a closed subs et Λ of X , w e denote b y D b coh , Λ ( A X ) the full triangulated su b catego ry of D b coh ( A X ) consisting of o b j ects suppo rted b y Λ. W e deﬁne similarly D b gd , Λ ( A lo c X ). 94 CHAPTER 3. KERNELS Theorem 3.2.1. F o r i = 1 , 2 , let Λ i b e a clos e d subset of X i × X i +1 and K i ∈ D b coh , Λ i ( A X i × X a i +1 ) . Assume that Λ 1 × X 2 Λ 2 is pr op er over X 1 × X 3 , an d set Λ = Λ 1 ◦ Λ 2 . T h e n the obje ct K 1 ◦ K 2 b elongs to D b coh , Λ ( A X 1 × X a 3 ) . Pr o of. Since the question is lo cal in X 1 and X 3 , w e may assume fro m the b eginning that A X 1 and A X 3 are DQ-algebras. W e shall ﬁrst sho w that K 1 L ⊗ A 2 K 2 is cohomologically complete. (3.2.1) Since this statemen t is a lo cal statemen t on X 1 × X 2 × X 3 , we ma y assume that A X 2 is a DQ-algebra. Since K 1 and K 2 ma y b e lo cally r epresen ted b y ﬁnite complexes of free modules o f ﬁnite rank, in order to see (3.2.1), w e ma y assume K i ≃ A X i × X a i +1 ( i = 1 , 2) . Then K 1 L ⊗ A 2 K 2 ≃ A X 1 × X 2 × X a 3 is cohomo- logically complete b y Theorem 1.6.1. Hence K 1 ◦ K 2 = R p 13 ∗ ( K 1 L ⊗ A 2 K 2 ) is also cohomolo gically complete b y Proposition 1.5.12. On the other hand, gr ~ ( K 1 ◦ K 2 ) ≃ R p 13 ∗  p ∗ 12 gr ~ K 1 L ⊗ O X 1 × X 2 × X 3 p ∗ 23 gr ~ K 2  b elongs to D b coh ( O X 1 × X 3 ) by Grauert’s direct image theorem ([29]). Hence Theorem 1.6.4 implies that K 1 ◦ K 2 b elongs to D b coh ( A X 1 × X a 3 ). Q.E.D. Remark 3.2.2. In [4], its authors use a v arian t of Theorem 3 .2.1 in the symplectic case . The y assert that the proo f follo ws from Houzel’s ﬁnitenes s theorem on mo dules o v er shea v es of m ultiplicativ ely conv ex nucle ar F r´ ec het algebras (see [32]). How ev er, they do no t give a ny pro o f , details being qual- iﬁed of “routine”. Corollary 3.2.3. L et M and N b e two obje cts o f D b coh ( A X ) and assume that Supp( M ) ∩ Supp( N ) is c omp act. Then the obje ct RHom A X ( M , N ) b elongs to D b f ( C ~ ) . Prop osition 3.2.4. L et K i ∈ D b coh ( A X i × X a i +1 ) ( i = 1 , 2 , 3) and let L ∈ D b coh ( A X 4 ) . Set Λ i = supp( K i ) and ass ume that Λ i × X i +1 Λ i +1 is pr op er ove r X i × X i +2 ( i = 1 , 2 ) . (i) Ther e is a c anonic al isomorphism ( K 1 ◦ X 2 K 2 ) L ⊠ L ∼ − → K 1 ◦ X 2 ( K 2 L ⊠ L ) . (ii) Ther e is a c anonic al isomo rphism ( K 1 ◦ X 2 K 2 ) ◦ X 3 K 3 ≃ K 1 ◦ X 2 ( K 2 ◦ X 3 K 3 ) . 3.3. CONV OLUTION OF KERNELS: DUALITY 95 Pr o of. The morphism ( K 1 ◦ X 2 K 2 ) L ⊠ L − → K 1 ◦ X 2 ( K 2 L ⊠ L ) is deduced f rom the morphism (w e do not write the functors p − 1 i , p − 1 ij for short): A 13 a 4 ⊗ A 13 a ⊠ A 4   A 12 a 23 a ⊗ A 12 a ⊠ A 23 a ( K 1 L ⊠ K 2 )  L ⊗ A 22 a C 2  L ⊠ L  ≃  ( A 13 a 4 ⊗ A 13 a ⊠ A 4 A 12 a 23 ) ⊗ A 12 a ⊠ A 23 a ⊠ A 4 K 1 L ⊠ K 2 L ⊠ L  L ⊗ A 22 a C 2 − →  A 12 a 23 a 4 ⊗ A 12 a ⊠ A 23 a ⊠ A 4  K 1 L ⊠ K 2 L ⊠ L  L ⊗ A 22 a C 2 . Applying the functor gr ~ to this morphism in D b coh ( A X 1 × X a 3 × X 4 ), we get an isomorphism. This prov es the result in view of Corollary 1 .4.6. (ii) By (i), w e ha v e ( K 1 ◦ X 2 K 2 ) ◦ X 3 K 3 ≃  ( K 1 ◦ X 2 K 2 ) L ⊠ K 3  ◦ X 3 × X a 3 C X 3 ≃  K 1 ◦ X 2 ( K 2 L ⊠ K 3 )  ◦ X 3 × X a 3 C X 3 ≃  C X 2 ◦ X 2 × X a 2 ( K 1 L ⊠ K 2 L ⊠ K 3 )  ◦ X 3 × X a 3 C X 3 . Then this o b j ect is isomorphic to ( K 1 L ⊠ K 2 L ⊠ K 3 ) ◦ X 2 × X a 2 × X 3 × X a 3 ( C X 2 L ⊠ C X 3 ). Similarly , K 1 ◦ X 2 ( K 2 ◦ X 3 K 3 ) is isomorphic to ( K 1 L ⊠ K 2 L ⊠ K 3 ) ◦ X 2 × X a 2 × X 3 × X a 3 ( C X 2 L ⊠ C X 3 ). Q.E.D. 3.3 Con v oluti on of k ernels : dualit y The dualit y morphism for kerne ls Denote as usual by p 13 : X 1 × X 2 × X a 3 − → X 1 × X a 3 the pro jection. Lemma 3.3.1. F or K i ∈ D b ( A X i × X a i +1 ) ( i = 1 , 2) , we have a natur al mor- phism in D b ( A X a 1 × X 3 ) : (D ′ A X 1 × X a 2 K 1 ) ◦ X a 2 ω A X a 2 ◦ X a 2 (D ′ A X 2 × X a 3 K 2 ) − → D ′ A X 1 × X a 3 ( K 1 ◦ X 2 K 2 ) . (3.3.1) 96 CHAPTER 3. KERNELS Pr o of. W e hav e D ′ A K 1 L ⊗ A 2 a ω A 2 a L ⊗ A 2 a D ′ A K 2 ≃ (D ′ A K 1 L ⊠ D ′ A K 2 ) L ⊗ A 2 a 2 ω A 2 a ≃ (D ′ A K 1 L ⊠ D ′ A K 2 ) L ⊗ A 12 a 23 a ω A 12 a 3 a / 13 a ≃ D ′ A ( K 1 L ⊠ K 2 ) L ⊗ A 12 a 23 a ω A 12 a 3 a / 13 a ≃ R H o m A 12 a 23 a ( K 1 L ⊠ K 2 , ω A 12 a 3 a / 13 a ) . Hence w e hav e morphisms D ′ A K 1 L ⊗ A 2 a ω A 2 a L ⊗ A 2 a D ′ A K 2 ≃ R H om A 12 a 23 a ( K 1 L ⊠ K 2 , ω A 12 a 3 a / 13 a ) − → R H om p − 1 13 A 13 a  ( K 1 L ⊠ K 2 ) L ⊗ A 22 a C 2 , ω A 12 a 3 a / 13 a L ⊗ A 22 a C 2  − → R H om p − 1 13 A 13 a ( K 1 L ⊗ A 2 K 2 , p − 1 13 A 13 a [2 d 2 ]) . The la st arro w is induced b y (2.5.7). T aking R p 13 ! , w e obtain (D ′ A K 1 ) ◦ X a 2 ω A X a 2 ◦ X a 2 (D ′ A K 2 ) ≃ R p 13 !  (D ′ A K 1 ) L ⊗ A 2 a ω A 2 a L ⊗ A 2 a (D ′ A K 2 )  − → R p 13 ∗ R H om p − 1 13 A 13 a ( K 1 L ⊗ A 2 K 2 , p − 1 13 A 13 a [2 d 2 ]) ∼ − → R H om A 13 a ( K 1 ◦ X 2 K 2 , A 13 a ) . Here t he last isomorphism is given by the P oincar´ e dualit y . Q.E.D. Serre dualit y Let us recall the Serre dualit y for O - mo dules. Let X and Y b e complex manifolds. Denote b y f : X × Y − → X the pro jection, b y ω Y = Ω d Y Y [ d Y ] the dualizing complex on Y and by ω X × Y /X := O X L ⊠ ω Y the relative dua lizing complex. F or G ∈ D b coh ( O X ), we set f ! G = f − 1 G L ⊗ f − 1 O X ω X × Y /X . Theorem 3.3.2. F or F ∈ D b coh ( O X × Y ) and G ∈ D b coh ( O X ) , we have a m o r- phism R f ∗ R H om O X × Y ( F , f ! G ) − → R H om O X (R f ! F , G ) . (3.3.2) If the supp ort of F is pr op er over X , then this morphism is an isomorphism. 3.3. CONV OLUTION OF KERNELS: DUALITY 97 This result is classical and w e shall only recall a construction of the mor- phism (3.3.2) adapted to o ur study . Since Ω Y has a D op Y -mo dule structure, w e ma y regard ω X × Y /X as a n ob ject of D b ( O X ⊠ D op Y ). By the de Rham theorem, w e hav e an isomorphism: ω X × Y /X L ⊗ D Y O Y ≃ f − 1 O X [2 d Y ] . By comp osing with the morphism ω X × Y /X − → ω X × Y /X L ⊗ D Y O Y , w e get a mor- phism in D b ( f − 1 O X ): ω X × Y /X − → f − 1 O X [2 d Y ] . No w we hav e a chain o f morphisms in D b ( f − 1 O X ) R H om O X × Y ( F , f ! G ) = R H om O X × Y ( F , f − 1 G L ⊗ f − 1 O X ω X × Y /X ) − → R H om f − 1 O X ( F , f − 1 G L ⊗ f − 1 O X f − 1 O X [2 d Y ]) ≃ R H om f − 1 O X ( F , f − 1 G [2 d Y ]) . On the other hand, the P oincar´ e duality giv es an isomorphism R f ∗ R H om f − 1 O X ( F , f − 1 G [2 d Y ]) ≃ R H om O X (R f ! F , G ) . Dualit y for k ernels Let X i b e complex manifo lds of dimens ion d i and let A X i b e DQ-alg ebroids on X i ( i = 1 , 2 , 3). As in Notat io n 3.1.1, w e often write for short X ij instead of X i × X j , X ij a instead o f X i × X a j , etc. W e a lso write A ij instead of A X ij , etc. and ij /i instead of X ij /X i etc. Theorem 3.3.3. L et K i ∈ D b coh ( A X i × X a i +1 ) ( i = 1 , 2) . We assume that Supp( K 1 ) × X 2 Supp( K 2 ) is pr op er over X 1 × X a 3 . Then the natur al morphism ( se e (3.3.1)) (D ′ A K 1 ) ◦ X a 2 ω A X a 2 ◦ X a 2 (D ′ A K 2 ) − → D ′ A ( K 1 ◦ X 2 K 2 ) (3.3.3) is an isomorph i sm in D b coh ( A X a 1 × X 3 ) . 98 CHAPTER 3. KERNELS Pr o of. Since the question is lo cal on X 1 × X a 3 , we may assume that gr ~ ( A X 1 ) and gr ~ ( A X 3 ) are isomorphic to O X 1 and O X 3 , resp ectiv ely . Applying the functor g r ~ , w e get gr ~ (D ′ A ( K 2 ) ◦ ω A X 2 ◦ D ′ A ( K 1 )) ≃ R p 13 ! (R H om O 123 ( p ∗ 12 gr ~ ( K 1 ) L ⊗ O 123 p ∗ 23 gr ~ ( K 2 ) , ω X 123 /X 13 )) ≃ R H om O X 13 (R p 13 !  p ∗ 12 gr ~ ( K 1 ) L ⊗ O 123 p ∗ 23 gr ~ ( K 2 )  , O 13 ) ≃ gr ~ (D ′ A ( K 1 ◦ K 2 )) . Here the second isomorphism follow s from Theorem 3.3.2. Hence (3.3.3) is an isomorphism b y Corollary 1.4.6. Q.E.D. Recall that D ′ X denotes the dualit y functor for C ~ X -mo dules: (see (1 .1.1)) and ( • ) ⋆ the dualit y functor o n D b f ( C ~ ) (see (1.1.2)). Corollary 3.3.4. L et M and N b e two obje cts of D b coh ( A X ) . (i) Ther e is a natur al mo rp hism in D b ( C ~ ) RHom A X ( N , ω A X L ⊗ A X M ) − → (R Hom A X ( M , N )) ⋆ . (3.3.4) (ii) If Supp( M ) ∩ Supp ( N ) is c omp act, then (3.3 .4) is an isomorphism in D b f ( C ~ ) . Pr o of. ( i) In Lemma 3.3.1, tak e X 1 = X 3 = pt, X 2 = X , K 1 = N and K 2 = D ′ A M . (ii) f o llo ws from Theorem 3 .3 .3. Q.E.D. In particular, if X is compact, then M 7→ ω A X ⊗ A X M is a Serre functor on the t riangulated category D b coh ( A X ). Remark 3.3.5. F or K i ∈ D b ( A lo c X i × X a i +1 ) ( i = 1 , 2), one can deﬁne their pro d- uct K 1 L ⊗ A loc 2 K 2 similarly as in Deﬁnition 3.1.2 and their con v olution similarly as in D eﬁnition 3.1.3. (Details are left to the reader.) One in tro duces ω A loc X := C ~ , lo c ⊗ C ~ ω A X (3.3.5) and fo r M ∈ D b ( A lo c X ), o ne deﬁnes its dual by setting D ′ A M := R H om A loc X ( M , A lo c X ) ∈ D b ( A lo c X a ) . (3.3.6) Then Theorems 3.2.1 and 3.3.3 exte nd to go o d A lo c -mo dules. 3.4. AC TION OF KERNELS ON GROTHE ND IECK GROUP S 99 Theorem 3.3.6. L et Λ i b e a clo s e d subset of X i × X i +1 ( i = 1 , 2 ) and assume that Λ 1 × X 2 Λ 2 is pr op er o v er X 1 × X 3 . Set Λ = Λ 1 ◦ Λ 2 . L et K i ∈ D b gd , Λ i ( A lo c X i × X a i +1 ) ( i = 1 , 2) . Then the obje ct K 1 ◦ K 2 b elongs to D b gd , Λ ( A lo c X 1 × X a 3 ) and we have a natur al isomorphism D ′ A ( K 1 ) ◦ X a 2 ω A loc X a 2 ◦ X a 2 D ′ A ( K 2 ) ∼ − → D ′ A ( K 1 ◦ X 2 K 2 ) . Pro of of Theorem 2.5.7 W e are no w ready t o giv e a pro of of Theorem 2.5.7. In Theorem 3.3.3, set X 1 = X 2 = X 3 = X a and K 1 = K 2 = C X a . Then w e obt a in D ′ A C X a ◦ X ω A X ◦ X D ′ A C X a ≃ D ′ A ( C X a ◦ X a C X a ) ≃ D ′ A ( C X a ) . By applying ◦ (D ′ A C X a ) ⊗− 1 , w e obtain D ′ A C X a ◦ X ω A X ≃ C X . 3.4 Action of k ernels on G rothendi e c k gro u ps Grothendiec k group F or an ab elian or a triangulated category C , w e denote as usual b y K( C ) its Grothendiec k group. F or an ob ject M of C , w e denote b y [ M ] its image in K( C ). Recall that if C is ab elian, then K( C ) ≃ K(D b ( C )). If A is a ring, w e write K( A ) instead of K(Mo d( A )) and write K coh ( A ) instead of K(Mo d coh ( A )). In this subsection, w e will adapt to DQ-mo dules well-kno wn argumen ts concerning the Grothendiec k group of ﬁltered ob jects. Referenc es are ma de to [37, Ch. 2.2]. F or a closed subs et Λ o f X , w e shall write for short: K coh , Λ ( A X ) := K(D b coh , Λ ( A X )) , K coh , Λ (gr ~ A X ) := K ( D b coh , Λ (gr ~ A X )) , K gd , Λ ( A lo c X ) := K(D b gd , Λ ( A lo c X )) . Recall that for an o p en subse t U of X and M ∈ Mo d coh ( A lo c X ), an A U - submo dule M 0 of M | U is called a lattice of M on U if M 0 is coheren t o v er A U and generates M | U . Lemma 3.4.1. L et 0 − → L − → M − → N − → 0 b e an exact se quenc e in Mo d coh ( A lo c X ) . Then ther e lo c al ly exist lattic es L 0 , M 0 and N 0 of L , M and N r esp e ctive l y, such that this se quenc e induc es an exact se quenc e of A X -mo dules: 0 − → L 0 − → M 0 − → N 0 − → 0 . 100 CHAPTER 3. KERNELS Pr o of. ( i) Let M 0 b e a lattice o f M and let N 0 b e its image in N . W e set L 0 := M 0 ∩ L . These A X -mo dules give rise t o the exact sequence of the statemen t and it remains to c hec k that L 0 and N 0 are lattices of L and N , resp ectiv ely . (ii) Clearly , N 0 generates N , and b eing ﬁnitely generated, it is coheren t ov er A X . (iii) Let us show that L 0 is a lattice of L . Being the kernel of the morphism M 0 − → N 0 , L 0 is coheren t. Since the functor ( • ) lo c is exact, the sequence 0 − → L lo c 0 − → M lo c 0 − → N lo c 0 − → 0 is exact. Therefore, L lo c 0 ≃ L . Q.E.D. Lemma 3.4.2. L et M ∈ Mo d coh ( A lo c X ) , let U b e a r elatively c omp a ct op en subset of X and as sume that ther e exists a lattic e M 0 of M in a neighb orho o d of the cl o s ur e U of U . Then the image of M 0 in K coh (gr ~ A U ) dep ends only on M . Pr o of. ( i) Recall that [gr ~ M 0 ] denotes the image of gr ~ M 0 in K coh (gr ~ A U ). First, remark tha t fo r N ∈ N , the t w o gr ~ A X -mo dules gr ~ M 0 and gr ~ ~ N M 0 are isomorphic, whic h implies [gr ~ M 0 ] = [gr ~ ~ N M 0 ] . (ii) No w consider another lattice M ′ 0 of M on U . Since M is an A lo c X - mo dule of ﬁnite ty p e and M ′ 0 generates M , there exists n > 1 suc h that M 0 ⊂ ~ − n M ′ 0 . Similarly , there exis ts m > 1 with M ′ 0 ⊂ ~ − m M 0 , so tha t we ha v e the inclusions ~ m + n M 0 ⊂ ~ m M ′ 0 ⊂ M 0 . Using (i) we may replace M ′ 0 with ~ m M ′ 0 . Hence, changing our notatio ns, w e ma y assum e ~ m M 0 ⊂ M ′ 0 ⊂ M 0 . (3.4.1) (iii) Assume m = 1 in (3.4.1). Using ~ m M ′ 0 ⊂ ~ m M 0 , w e g et the exact sequence s 0 − → M ′ 0 / ~ M 0 − → M 0 / ~ M 0 − → M 0 / M ′ 0 − → 0 , 0 − → ~ M 0 / ~ M ′ 0 − → M ′ 0 / ~ M ′ 0 − → M ′ 0 / ~ M 0 − → 0 , and the r esult follo ws in this case. (iv) No w we argue by induction on m in (3 .4 .1) and w e assume the result is true for m − 1 with m ≥ 2. Set M ′′ 0 := ~ m − 1 M 0 + M ′ 0 . Then ~ M ′′ 0 ⊂ M ′ 0 ⊂ M ′′ 0 and ~ m − 1 M 0 ⊂ M ′′ 0 ⊂ M 0 . Then the result follo ws from (iii) and the induction hypothesis. Q.E.D. 3.4. AC TION OF KERNELS ON GROTHE ND IECK GROUP S 101 W e set b K coh , Λ (gr ~ A X ) := lim ← − U K coh , Λ (gr ~ A U ) . (3.4.2) where U ranges o ve r the family of relativ ely compact o p en subsets of X . Using Lemma 3.4.2, we get: Prop osition 3.4.3. Ther e is a natur al morphism of gr oups gr ~ : K gd , Λ ( A lo c X ) − → b K coh , Λ (gr ~ A X ) . Remark that when X = pt, the mor phism in Prop osition 3.4.3 reduces to the isomorphism K f ( C ~ , lo c ) ∼ − → K f ( C ) , (3.4.3) and b oth are isomorphic to Z b y [ M ] 7→ dim M . Kernels Consider the situatio n o f The orem 3.2.1 . Let Λ i b e a closed subset of X i × X i +1 ( i = 1 , 2) and assume that Λ 1 × X 2 Λ 2 is prop er ov er X 1 × X 3 . Set Λ = Λ 1 ◦ Λ 2 . Since the con volution of k ernels comm utes with distinguishe d triangles, it factors through the Gro thendiec k groups. Moreov er, one can deﬁne the con v olution of gr ~ A X -k ernels and a v arian t of Theorem 3.2.1 with A X replaced with gr ~ A X is w ell-kno wn. Since the functor gr ~ comm utes with the con v olution of kerne ls, the diagram b elo w comm utes: Ob  D b coh , Λ 1 ( A 12 a )  × Ob  D b coh , Λ 2 ( A 23 a )  ◦ / /   Ob  D b coh , Λ ( A 13 a )    K coh , Λ 1 ( A 12 a ) × K coh , Λ 2 ( A 23 a ) ◦ / / gr ~ × gr ~   K coh , Λ ( A 13 a ) gr ~   K coh , Λ 1 (gr ~ A 12 a ) × K coh , Λ 2 (gr ~ A 23 a ) ◦ / / K coh , Λ (gr ~ A 13 a ) . (3.4.4) Similarly to (3.4.4), the dia g ram b elow commutes : Ob  D b gd , Λ 1 ( A lo c 12 a )  × Ob  D b gd , Λ 2 ( A lo c 23 a )  ◦ / /   Ob  D b gd , Λ ( A lo c 13 a )    K gd , Λ 1 ( A lo c 12 a ) × K gd , Λ 2 ( A lo c 23 a ) ◦ / / gr ~ × gr ~   K gd , Λ ( A lo c 13 a ) gr ~   b K coh , Λ 1 (gr ~ A 12 a ) × b K coh , Λ 2 (gr ~ A 23 a ) ◦ / / b K coh , Λ (gr ~ A 13 a ) . (3.4.5) 102 CHAPTER 3. KERNELS Chapter 4 Ho c hsc hild clas s es 4.1 Ho c hs c hild ho mology and Ho c hsc hild classes Let X b e a complex manifold and let A X b e a D Q-algebroid. Recall that δ X : X − → X × X a is the diag o nal embedding. W e deﬁne the Ho chs child homology HH ( A X ) of A X b y: HH ( A X ) := δ − 1 X ( C X a L ⊗ A X × X a C X ) , an ob ject of D b ( C ~ X ) . (4.1.1 ) Note that b y The orem 2 .5 .7, w e get the isomorphisms: HH ( A X ) ≃ δ − 1 X R H om A X × X a (D ′ A X a × X C X a , C X ) ≃ δ − 1 X R H om A X × X a ( ω A ⊗− 1 X , C X ) . W e hav e also the isomorphisms R H om A X × X a ( ω A ⊗− 1 X , C X ) ≃ R H om A X × X a ( ω A X ◦ X ω A ⊗− 1 X , ω A X ◦ X C X ) ≃ R H om A X × X a ( C X , ω A X ) . One shall b e aw are that t he comp o sition of these isomorphisms do es not coincide in general with the comp osition of R H om A X × X a ( ω A ⊗− 1 X , C X ) ≃ R H om A X × X a ( ω A ⊗− 1 X ◦ X ω A X , C X ◦ X ω A X ) ≃ R H om A X × X a ( C X , ω A X ) . W e shall see that they diﬀer up to hh X ( ω X ) ◦ (see Lemma 4.3.4 b elo w). F o r that reason, w e shall no t iden tify H H ( A X ) and R H om A X × X a ( C X , ω A X ). 103 104 CHAPTER 4. HOCHSCHILD CLASSES Lemma 4.1.1. L et M ∈ D b coh ( A X ) . Ther e ar e natur al morphisms in D b coh ( A X × X a ) : ω A ⊗− 1 X − → M L ⊠ D ′ A M , (4.1.2) M L ⊠ D ′ A M − → C X . (4.1.3) Pr o of. ( i) W e ha v e R H om A X ( M , M ) ≃ (D ′ A M ) L ⊗ A X M ≃ C X a L ⊗ A X × X a ( M L ⊠ D ′ A M ) ≃ R H om A X × X a ( ω A ⊗− 1 X , M L ⊠ D ′ A M ) . The identit y of Hom A X ( M , M ) deﬁnes the desired morphism. (ii) Applying the duality f unctor D ′ A X × X a to (4.1.2), w e g et (4.1.3). Q.E.D. Let M ∈ D b coh ( A X ). W e hav e the c hain of morphisms R H om A X ( M , M ) ∼ ← − D ′ A M L ⊗ A X M ≃ C X a L ⊗ A X × X a ( M L ⊠ D ′ A M ) − → C X a L ⊗ A X × X a C X = H H ( A X ) . (4.1.4) W e g et a map Hom A X ( M , M ) − → H 0 Supp( M ) ( X ; H H ( A X )) . F or u ∈ End( M ), the image of u giv es an elemen t hh X (( M , u )) ∈ H 0 Supp( M ) ( X ; H H ( A X )) . (4.1.5) Notation 4.1.2. F or a closed subs et Λ of X , we set HH Λ ( A X ) := H 0 RΓ Λ ( X ; H H ( A X )) . (4.1.6) Deﬁnition 4.1.3. Let M ∈ D b coh , Λ ( A X ). W e set hh X ( M ) = hh X (( M , id M )) ∈ HH Λ ( A X ) and call it the Ho chsc hild class of M . Lemma 4.1.4. L et M ∈ D b coh ( A X ) . The c omp osition of the two morphism s (4.1.2) and (4.1.3) : ω A ⊗− 1 X − → M L ⊠ D ′ A M − → C X c oincides with the Ho chschild cla s s hh X ( M ) when identifying H H ( A X ) with R H om A X × X a ( ω A ⊗− 1 X , C X ) . 4.1. HOCHSCHILD HOMOLOGY AND HOCH SCHILD CLAS SES 105 Pr o of. The Ho chs child class hh X ( M ) is the imag e of id M b y the comp osition R H om A X ( M , M ) ≃ R H om A X × X a ( ω A ⊗− 1 X , M L ⊠ D ′ A M ) − → R H om A X × X a ( ω A ⊗− 1 X , C X ) ≃ H H ( A X ) . Q.E.D. Theorem 4.1.5. The Ho chschild class is additive w i th r esp e ct to distin- guishe d triangles. In other w or ds, for a distinguishe d triangle M ′ − → M − → M ′′ +1 − → in D b coh ( A X ) , we have hh X ( M ) = hh X ( M ′ ) + hh X ( M ′′ ) . (4.1.7) Pr o of. Altho ugh the bifunctor • L ⊗ A X • is not inte rnal to our category , the theorem of Ma y [4 9] is easily adapted to this situation. Q.E.D. By this re sult, the Ho c hsc hild class factorizes through t he Grothendiec k group. Therefore, if Λ is a closed subset o f X , w e ha ve the morphisms D b coh , Λ ( A X ) − → K coh , Λ ( A X ) − → HH Λ ( A X ) . (4.1.8) Dualit y Denote by s : X × X a − → X a × X the map ( x, y ) 7→ ( y , x ) and recall that δ X is the diagonal em bedding. Then s ◦ δ X = δ X , s − 1 C X ≃ C X a , s − 1 A X × X a ≃ A X a × X and we obta in the isomorphisms HH ( A X ) = δ − 1 X ( C X a L ⊗ A X × X a C X ) ≃ δ − 1 X s − 1 ( C X a L ⊗ A X × X a C X ) ≃ δ − 1 X ( s − 1 C X a L ⊗ s − 1 A X × X a s − 1 C X ) ≃ δ − 1 X ( C X L ⊗ A X a × X C X a ) = H H ( A X a ) . After iden tifying H H ( A X ) and HH ( A X a ) by the isomorphism ab ov e, we ha v e: hh X a (D ′ A M ) = hh X ( M ) . (4.1.9) Remark 4.1.6. Let A b e a DQ-algebroid and let P b e a n inv erstible C ~ - algebroid on X . Then A P := A ⊗ C ~ X P (4.1.10) 106 CHAPTER 4. HOCHSCHILD CLASSES is a D Q-algebroid on X . W e ha v e the natural equiv alences ( A op ) P op ≃ ( A P ) op , δ − 1 X ( A P ⊠ ( A P ) op ) ≃ δ − 1 X ( A ⊠ ( A P )) . W e deduce the isomorphism HH ( A X ) ≃ HH ( A P X ) . (4.1.11) 4.2 Comp osi t ion of Ho c hsc hild classe s Let X i b e complex manifolds endo w ed with DQ-algebroids A X i ( i = 1 , 2 , 3 ) and denote as usual b y p ij the pro jection from X 1 × X 2 × X 3 to X i × X j (1 ≤ i < j ≤ 3). Prop osition 4.2.1. Ther e is a natur al morphism ◦ : R p 13 ! ( p − 1 12 HH ( A X 1 × X a 2 ) L ⊗ p − 1 23 HH ( A X 2 × X a 3 )) − → HH ( A X 1 × X a 3 ) . Pr o of. ( i) Set Z i = X i × X a i . W e shall denote b y the same letter p ij the pro jection from Z 1 × Z 2 × Z 3 to Z i × Z j . W e hav e HH ( A X i × X a j ) ≃ ( C X a i L ⊠ C X j ) L ⊗ A Z i × Z a j ( C X i L ⊠ C X a j ) ≃ R H om A Z i × Z a j ( ω A ⊗− 1 X i L ⊠ ω A ⊗− 1 X a j , C X i L ⊠ C X j a ) ≃ R H om A Z i × Z a j  ( ω A ⊗− 1 X i L ⊠ ω A ⊗− 1 X a j ) L ⊗ A X a j ω A X a j , ( C X i L ⊠ C X j a ) L ⊗ A X a j ω A X a j  ≃ R H om A Z i × Z a j ( ω A ⊗− 1 X i L ⊠ C X a j , C X i L ⊠ ω A X j a ) . Set S ij := ω A ⊗− 1 X i L ⊠ C X a j ∈ D b coh ( A Z i × Z a j ) and K ij := C X i L ⊠ ω A X a j ∈ D b coh ( A Z i × Z a j ). Then we get HH ( A X i × X a j ) ≃ R H om A Z i × Z a j ( S ij , K ij ) . Th us w e obt a in a mor phism in D b ( C ~ Z 1 × Z 2 × Z 3 ) p − 1 12 HH ( A X 1 × X a 2 ) L ⊗ p − 1 23 HH ( A X 2 × X a 3 ) ≃ p − 1 12 R H om A Z 1 × Z a 2 ( S 12 , K 12 ) L ⊗ p − 1 23 R H om A Z 2 × Z a 3 ( S 23 , K 23 ) − → p − 1 13 R H om A Z 1 × Z a 3  S 12 L ⊗ A Z 2 S 23 , K 12 L ⊗ A Z 2 K 23  . 4.2. COMPOSITION OF HOCHSCHILD CLASS ES 107 W e g et a morphism R p 13 ! ( p − 1 12 HH ( A X 1 × X a 2 ) L ⊗ p − 1 23 HH ( A X 2 × X a 3 )) − → R p 13 ! R H om A Z 1 × Z a 3  S 12 L ⊗ A Z 2 S 23 , K 12 L ⊗ A Z 2 K 23  . (4.2.1) (ii) W e ha ve a morphism C ~ X 2 − → R H om A Z a 2 ( C X a 2 , C X a 2 ) ≃ C X a 2 L ⊗ A Z 2 ω A ⊗− 1 X 2 , whic h induces the morphism: p − 1 13 ( ω A ⊗− 1 X 1 L ⊠ C X a 3 ) − → ( ω A ⊗− 1 X 1 L ⊠ C X a 2 ) L ⊗ A Z 2 ( ω A ⊗− 1 X 2 L ⊠ C X a 3 ) , that is, the morphism in D b ( A Z 1 × Z a 3 ): S 13 − → R p 13 ∗ ( S 12 L ⊗ A Z 2 S 23 ) . (4.2.2) (iii) W e ha ve a morphism ( see (2.5.7 )): ( C X 1 L ⊠ ω A X a 2 ) L ⊗ A Z 2 ( C X 2 L ⊠ ω A X a 3 ) − → p − 1 13 ( C X 1 L ⊠ ω A X a 3 )[2 d 2 ] , whic h induces the morphism in D b ( A Z 1 × Z a 3 ): R p 13 ! ( K 12 L ⊗ A Z 2 K 23 ) − → K 13 . (4.2.3) (iv) Using (4.2.2) and (4.2.3 ) we obta in R p 13 ! R H om A Z 1 × Z a 3  S 12 L ⊗ A Z 2 S 23 , K 12 L ⊗ A Z 2 K 23  − → R H om A Z 1 × Z a 3  R p 13 ∗ ( S 12 L ⊗ A Z 2 S 23 ) , R p 13 ! ( K 12 L ⊗ A Z 2 K 23 )  − → R H om A Z 1 × Z a 3 ( S 13 , K 13 ) ≃ H H ( A X 1 × X a 3 ) . (4.2.4) Com bining (4.2.1) a nd (4.2.4), w e get the result. Q.E.D. Let us denote b y X R the real underlying manifold to X and by ω top X R the top ological dualizing complex of the space X R with coeﬃcien ts in C ~ . Note that X b eing smo oth and oriented , ω top X R is isomorphic to C ~ X [2 d X ]. Corollary 4.2.2. T h e r e is a c anonic al m orphism HH ( A X ) ⊗ HH ( A X ) − → ω top X R . Pr o of. L et us apply Prop osition 4.2.1 with X 2 = X , X 1 = X 3 = pt. Denoting b y a X the map X − → pt, we get the morphism R a X ! ( HH ( A X ) ⊗ HH ( A X )) − → C ~ pt . By a djunction w e get the desired morphism. Q.E.D. 108 CHAPTER 4. HOCHSCHILD CLASSES 4.3 Main the o rem Consider ﬁv e manif o lds X i endo w ed with D Q -algebroids A X i ( i = 1 , . . . , 5) . Notation 4.3.1. In the sequel and un til the end of this section, when there is no risk of confusion, w e use the follo wing conv en tions. (i) F or i, j ∈ { 1 , 2 , 3 , 4 , 5 } , we set X ij := X i × X j , X ij a := X i × X a j and similarly with X ij k , etc. (ii) W e sometimes o mit the sym b ols p ij , p ij ∗ , p − 1 ij , etc. (iii) W e write A i instead of A X i , A ij a instead of A X ij a and similarly with C i , ω A i , etc., and w e write ◦ i instead of ◦ X i , ∗ i instead of ∗ X i , H om i instead of H om A i and ⊗ i instead of ⊗ A i and similarly with ij a , ij k , etc. (iv) W e write D ′ instead of D ′ A and ω X instead of ω A X . (v) W e often iden tify an inv ertible ob ject of D b ( A X ⊗ A X a ) with an ob ject of D b ( A X × X a ) supp orted b y the diago na l. (vi) W e iden tify ( X i × X a j ) a with X a i × X j . Let Λ ij ⊂ X ij ( i = 1 , 2, j = i + 1) b e a closed subset and assume that Λ 12 × X 2 Λ 23 is prop er o v er X 1 × X 3 . Using Prop osition 4.2.1, w e get a map ◦ 2 : HH Λ 12 ( A X 12 a ) × HH Λ 23 ( A X 23 a ) − → HH Λ 12 ◦ Λ 23 ( A X 13 a ) . (4.3.1) F or C ij ∈ HH Λ ij ( A X ij a ) ( i = 1 , 2, j = i + 1), w e obtain a class C 12 ◦ 2 C 23 ∈ HH Λ 12 ◦ Λ 23 ( A X 13 a ) . (4.3.2) The morphism ( C 1 a L ⊗ 11 a C 1 ) L ⊠ ( C 2 a L ⊗ 22 a C 2 ) − → ( C 1 a 2 a L ⊗ 121 a 2 a C 12 ) induces the exterior pro duct ⊠ : HH Λ 1 ( A X 1 ) × HH Λ 2 ( A X 2 ) − → HH Λ 1 × Λ 2 ( A X 1 × X 2 ) (4.3.3) for Λ i ⊂ X i ( i = 1 , 2) . Lemma 4.3.2. L et Λ ij ⊂ X ij ( i = 1 , 2 , 3 , j = i + 1 ) and assume that Λ ij × X j Λ j k is pr op er over X ik ( i = 1 , 2 , j = i + 1 , k = j + 1 ) . L et C ij ∈ HH Λ ij ( A X ij a ) ( i = 1 , 2 , 3 , j = i + 1) . (a) One has ( C 12 ◦ 2 C 23 ) ◦ 3 C 34 = C 12 ◦ 2 ( C 23 ◦ 3 C 34 ) . 4.3. MAIN THEOREM 109 (b) F or C 245 ∈ HH ( A X 245 a ) we have ( C 12 ⊠ C 34 ) ◦ 24 C 245 = C 12 ◦ 2 ( C 34 ◦ 4 C 245 ) . (c) Set C ∆ i = hh X ii a ( C X i ) . T hen C 12 ◦ 2 C ∆ 2 = C ∆ 1 ◦ 1 C 12 = C 12 . (d) ( C 12 ⊠ C ∆ 3 ) ◦ 23 a C 23 = C 12 ◦ 2 C 23 . Her e C 12 ⊠ C ∆ 3 ∈ HH Λ 12 × ∆ 3 ( A X 12 a 33 a ) is r e gar de d as an element of HH Λ 12 × ∆ 3 ( A X (13 a )(23 a ) a ) . Pr o of. The pro of of (a) and (b) is left to the reader and (c) follows from Lemma 4.3.4 b elo w. Indeed, Φ K in (4.3 .7) is equal to the iden tity when K = C X since the functor L 7→ K ∗ 2 L ◦ 2 ω 2 ∗ 2 D ′ K is isomorphic to the iden tit y functor. (d) follows from (b) and (c). Q.E.D. In order to prov e Theorem 4.3.5 b elow, w e nee d some lemmas. Lemma 4.3.3. L et K ∈ D b coh ( A X 12 a ) . The n , ther e ar e natur al morphisms in D b ( A X 11 a ) : ω ⊗− 1 1 − → K ∗ 2 D ′ A K , (4.3.4) K ◦ 2 ω 2 ◦ 2 D ′ A K − → C 1 . (4.3.5) Pr o of. ( i) By (4.1 .2), we hav e a morphism in D b ( A X 12 a 21 a ) ω ⊗− 1 12 a − → K ⊠ D ′ A K . Applying the functor • L ⊗ 22 a C 2 , we obta in p − 1 11 a ω ⊗− 1 1 − → ω ⊗− 1 1 ⊠ (D ′ A C 2 L ⊗ 22 a C 2 ) − → ω ⊗− 1 12 a L ⊗ 22 a C 2 − → ( K L ⊠ D ′ A K ) ⊗ 22 a C 2 . By adjunction, w e get (4.3.4). (ii) By (4.1.3), we ha v e a morphism in D b ( A X 12 a 21 a ) K ⊠ D ′ K − → C 12 a . Applying the functor • L ⊗ 22 a ω 2 , w e obtain ( K L ⊠ D ′ A K ) ⊗ 22 a ω 2 − → C 12 a L ⊗ 22 a ω 2 − → C 1 L ⊗ C ~ X 2 [2 d 2 ] . Here t he last arrow is giv en b y (2.5.7). By adjunction, w e g et (4.3.5). Q.E.D. 110 CHAPTER 4. HOCHSCHILD CLASSES F or the sak e of brevity , w e shall write Γ Λ Hom instead of H 0 (RΓ Λ R H om ). Let Λ 12 b e a closed subset of X 1 × X a 2 and Λ 2 a closed subset of X 2 . Let K ∈ D b coh ( A X 12 a ) with support Λ 12 . W e assume Λ 12 × X 2 Λ 2 is prop er ov er X 1 . (4.3.6) W e deﬁne the map Φ K : HH Λ 2 ( A X 2 ) − → HH Λ 12 ◦ Λ 2 ( A X 1 ) (4.3.7) as the comp osition of the sequence of maps HH Λ 2 ( A 2 ) ≃ Γ Λ 2 Hom 22 a ( ω ⊗− 1 2 , C 2 ) − → Γ Λ 12 × X 2 Λ 2 Hom 11 a ( K L ⊗ 2 ( ω ⊗− 1 2 ◦ 2 ω 2 ◦ 2 D ′ A K ) , K L ⊗ 2 ( C 2 ◦ 2 ω 2 ◦ 2 D ′ A K )) − → Γ Λ 12 ◦ Λ 2 Hom 11 a  R p 1 ∗ ( K L ⊗ 2 ( ω ⊗− 1 2 ◦ 2 ω 2 ◦ 2 D ′ A K )) , R p 1 ! ( K L ⊗ 2 ( C 2 ◦ 2 ω 2 ◦ 2 D ′ A K ))  ≃ Γ Λ 12 ◦ Λ 2 Hom 11 a ( K ∗ 2 D ′ A K , K ◦ 2 ω 2 ◦ 2 D ′ A K ) − → Γ Λ 12 ◦ Λ 2 Hom 11 a ( ω ⊗− 1 1 , C 1 ) ≃ HH Λ 12 ◦ Λ 2 ( A 1 ) . The ﬁrst arrow is obtained by applying the f unctor L 7→ K L ⊗ 2 ( L ◦ 2 ω 2 ◦ 2 D ′ A K ), The la st arro w is asso ciated with the morphisms in Lemma 4.3.3. Lemma 4.3.4. The map Φ K : HH Λ 2 ( A X 2 ) − → HH Λ 12 ◦ Λ 2 ( A X 1 ) in (4.3.7) is the map hh X 12 a ( K ) ◦ given in (4.3.2) . Pr o of. In the pro of, w e do not write Λ 12 and Λ 2 . Let λ = hh 12 a ( K ) ∈ HH ( A 12 a ) and let λ 2 ∈ HH ( A 2 ). W e regard λ as a morphism on X 12 a 21 a : λ : ω ⊗− 1 12 a − → K ⊠ D ′ A K − → C 12 a . W e regard λ 2 as a morphism ω ⊗− 1 2 − → C 2 . Then Φ K ( λ 2 ) is obtained as the comp osition ω ⊗− 1 1 − → K ∗ 2 D ′ A K − → R p 1 ∗  K L ⊗ 2 ( ω ⊗− 1 2 ◦ 2 ω 2 ◦ 2 D ′ A K )  λ 2 − − → R p 1 !  K L ⊗ 2 ( C 2 ◦ 2 ω 2 ◦ 2 D ′ A K )  − → K ◦ 2 ω 2 ◦ 2 D ′ A K − → C 1 . In the followin g diagram in the category D b ( A 11 a ⊠ C ~ X 2 × X a 2 ), w e write D ′ 4.3. MAIN THEOREM 111 instead of D ′ A for sak e of brevity : p − 1 11 a ω ⊗− 1 1   ( ω ⊗− 1 1 ⊠ C 2 a ) L ⊗ 22 a ω ⊗− 1 2 λ 2 / / ≀   ( ω ⊗− 1 1 ⊠ C 2 a ) L ⊗ 22 a C 2 ≀     (( ω ⊗− 1 1 ⊠ ω ⊗− 1 2 a ) ◦ 2 a ω 2 a ) L ⊗ 22 a ω ⊗− 1 2 λ 2 / /   (( ω ⊗− 1 1 ⊠ ω ⊗− 1 2 a ) ◦ 2 ω 2 a ) L ⊗ 22 a C 2   ) ) T T T T T T T T T T T T T T T ( ω 2 ◦ 2 ω ⊗− 1 12 a ) L ⊗ 22 a ω ⊗− 1 2 λ 2 / /   ( ω 2 ◦ 2 ω ⊗− 1 12 a ) L ⊗ 22 a C 2    ( K ⊠ D ′ K ) ◦ 2 a ω 2 a  L ⊗ 22 a C 2   ∼ u u j j j j j j j j j j j j j j j ( K ⊠ ω 2 ◦ 2 D ′ K ) L ⊗ 22 a ω ⊗− 1 2 λ 2 / / ≀   ( K ⊠ ω 2 ◦ 2 D ′ K ) L ⊗ 22 a C 2 ≀    C 12 a ◦ 2 a ω 2 a  L ⊗ 22 a C 2 ≀   K L ⊗ 2 ( ω ⊗− 1 2 ◦ 2 ω 2 ◦ 2 D ′ K ) λ 2 / / K L ⊗ 2 ( C 2 ◦ 2 ω 2 ◦ 2 D ′ K )   ( C 1 ⊠ ω 2 a ) L ⊗ 22 a C 2   K L ⊗ 2 D ′ K / / C 1 ⊠ C ~ X 2 [2 d 2 ] . Here we used ω 2 ◦ 2 L ≃ L ◦ 2 a ω 2 a . This diagram comm utes, and the ro ws on the top and the righ t columns p − 1 11 a ω ⊗− 1 1 − →  ω 2 ◦ 2 ( K ⊠ D ′ A K )  L ⊗ 22 a C 2 − → p ! 11 a C 1 induce λ ◦ λ 2 : ω ⊗− 1 1 − → C 1 b y a dj unction. Therefore, the diagram ω ⊗− 1 1 / / λ ◦ λ 2 P P P P P P P P P P P P P P P P P ' ' P P P P P P P P P P P P P P P P P K ∗ 2 D ′ A K ∼ / / K ∗ 2 ( ω ⊗− 1 2 ◦ 2 ω 2 ◦ 2 D ′ A K ) λ 2   K ◦ 2 C 2 ◦ 2 ω 2 ◦ 2 D ′ A K   C 1 comm utes, which g ives the result since t he comp o sition of the rows on the top a nd the v ertical a rro ws is Φ K ( λ 2 ). Q.E.D. Theorem 4.3.5. L et Λ i b e a clo s e d subset of X i × X i +1 ( i = 1 , 2 ) and assume that Λ 1 × X 2 Λ 2 is pr op er o v er X 1 × X 3 . Set Λ = Λ 1 ◦ Λ 2 . L et 112 CHAPTER 4. HOCHSCHILD CLASSES K i ∈ D b coh , Λ i ( A X i × X a i +1 ) ( i = 1 , 2) . Then hh X 13 a ( K 1 ◦ K 2 ) = hh X 12 a ( K 1 ) ◦ hh X 23 a ( K 2 ) ( 4 .3.8) as el e m ents of HH Λ ( A X 1 × X a 3 ) . Pr o of. F o r the sak e of simplicit y , we assume that X 3 = pt. Consider the diagram in whic h we set λ 2 = hh 2 ( K 2 ) ∈ HH ( A X 2 ) ≃ Hom ( ω ⊗− 1 2 , C 2 ) and w e write D ′ instead of D ′ A : ω ⊗− 1 1 / / @A / / K 1 ◦ 2 ω ⊗− 1 2 ◦ 2 ω 2 ◦ 2 D ′ K 1 λ 2 / /   K 1 ◦ 2 C 2 ◦ 2 ω 2 ◦ 2 D ′ K 1 / / C 1 K 1 ◦ 2 ( K 2 ⊠ D ′ K 2 ) ◦ 2 ω 2 ◦ 2 D ′ K 1 4 4 i i i i i i i i i i i i i i i i ≀   ( K 1 ◦ 2 K 2 ) ⊠ D ′ K 2 ◦ 2 ω 2 ◦ 2 D ′ K 1 ≀   ( K 1 ◦ 2 K 2 ) ⊠ D ′ ( K 1 ◦ 2 K 2 ) BC O O Here, the left horizontal arrow on the top is the comp osition of the morphisms ω ⊗− 1 1 − → K 1 ◦ 2 D ′ A K 1 − → K 1 ◦ 2 ω ⊗− 1 2 ◦ 2 ω 2 ◦ 2 D ′ A K 1 . The comp osition of the arro ws on the b ottom is hh 1 ( K 1 ◦ K 2 ) by Lemma 4.1.4 and the comp osition of the arrow s on the to p is Φ K 1 (hh 2 ( K 2 )). Hence, the assertion follows from the comm utativit y of the diagram b y Lemma 4.3.4. Q.E.D. Recall Diagram 3.4.4. Using (4.1.8), we g et the commutativ e diagram K coh , Λ 1 ( A 12 a ) × K coh , Λ 2 ( A 23 a ) ◦ / / hh 12 a × hh 23 a   K coh , Λ ( A 13 a ) hh 13 a   HH Λ 1 ( A 12 a ) × HH Λ 2 ( A 23 a ) ◦ / / HH Λ ( A 13 a ) . (4.3.9) Remark 4.3.6. (i) The fact that Ho c hsc hild homo lo gy of O -mo dules is func- torial seems to b e well-kno wn, although we do not know an y pap er in whic h it is explicitly stated (fo r closely relat ed results, see e.g., [58 , 14, 33]). (ii) In [15], its author s interpret Ho c hsc hild homology as a morphism o f func- tors and the a ction o f k ernels as a 2-mo r phism in a suitable 2-category . Its authors claim that the the relation Φ K 1 ◦ Φ K 1 = Φ K 1 ◦ K 2 follo ws by gen- eral argumen ts on 2-categories. Their result applies in a general framew ork 4.3. MAIN THEOREM 113 including in particular O -mo dules in the algebraic case and presumably DQ- mo dules but the precise a xioms are not sp eciﬁed in lo c. cit. See also [58] for related results. Note that, as far a s we understand, these authors do not in- tro duce the con volution of Ho c hsc hild homologies and they did not consider Lemma 4.3.4 nor Theorem 4 .3 .5. Index Let K b e a ﬁeld, let M ∈ D b f ( K ) and let u ∈ End( M ). One sets tr( u, M ) = X i ∈ Z ( − 1) i tr( H i ( u ) : H i ( M ) − → H i ( M )) , χ ( M ) = X i ∈ Z ( − 1) i dim K ( H i ( M )) . If X = pt, then HH ( A X ) is isomorphic to C ~ , and D b coh ( A X ) = D b f ( C ~ ). Recall that we ha v e set M lo c = C ~ , lo c ⊗ C ~ M . F o r M ∈ D b f ( C ~ ) and u ∈ End( M ), w e ha v e hh pt (( M , u )) = tr( u lo c , M lo c ) . (4.3.10) In particular, hh pt ( M ) = χ ( M lo c ) . Moreo v er, w e ha ve χ ( M lo c ) = χ (gr ~ ( M )) = X i ∈ Z ( − 1) i  dim C ( C ⊗ C ~ H i ( M )) − dim C T or C ~ 1 ( C , H i ( M ))  . In the sequel, w e set χ ( M ) := χ ( M lo c ) . As a particular case of Theorem 4.3.5, consider t w o ob jects M and N in D b coh ( A X ) a nd assume that Supp ( M ) ∩ Supp ( N ) is compact. Then RHom A X ( M , N ) belongs to D b f ( C ~ ) a nd χ (RHom A X ( M , N )) = hh pt (D ′ A M ◦ X N ) = hh X a (D ′ A M ) ◦ X hh X ( N ) = hh X ( M ) ◦ X hh X ( N ) . Note that w e hav e χ  RHom A X ( M , N )  = χ  RHom A loc X ( M lo c , N lo c )  = χ  RHom gr ~ ( A X ) (gr ~ ( M ) , gr ~ ( N ))  . 114 CHAPTER 4. HOCHSCHILD CLASSES 4.4 Graded and lo calized Ho c hsc hild class e s Graded Ho chsc hild classes Similarly to the case of A X , one deﬁnes HH (g r ~ ( A X )) := gr ~ ( C X a ) L ⊗ gr ~ ( A X × X a ) gr ~ ( C X ) . Note that H H (g r ~ ( A X )) ≃ C L ⊗ C ~ HH ( A X ) a nd there is a natural morphism gr ~ : H H ( A X ) − → HH (gr ~ ( A X )) . Notation 4.4.1. F or a closed subs et Λ of X , we set HH Λ (gr ~ A X ) := H 0 RΓ Λ ( X ; H H (gr ~ ( A X ))) . (4.4.1) W e a lso need to introduce d HH Λ (gr ~ A X ) := lim ← − U HH Λ (gr ~ A U ) , (4.4.2) where U ranges o v er t he family of r elat ively compact op en subsets of X . F or F ∈ D b coh (gr ~ ( A X )), one deﬁ nes its Ho c hsc hild class hh X ( F ) b y the same construction as fo r A X -mo dules. F or M ∈ D b coh ( A X ), we hav e: gr ~ (hh X ( M )) = hh X (gr ~ ( M )) . Theorem 4.3.5 obv iously also holds when r eplacing A X with gr ~ ( A X ). Corollary 4.4.2. L et Λ i b e a close d subset of X i × X i +1 ( i = 1 , 2 ) and assume that Λ 1 × X 2 Λ 2 is pr op er over X 1 × X 3 . Set Λ = Λ 1 ◦ Λ 2 . L et K i ∈ D b coh , Λ i (gr ~ ( A X i × X a i +1 )) ( i = 1 , 2) . Then hh X 13 a ( K 1 ◦ K 2 ) = hh X 12 a ( K 1 ) ◦ hh X 23 a ( K 2 ) ( 4 .4.3) as el e m ents of HH Λ (gr ~ A X 1 × X a 3 ) . It follow s that the diag ram b elo w commu tes K coh , Λ 1 (gr ~ A 12 a ) × K coh , Λ 2 (gr ~ A 23 a ) ◦ / / hh   K coh , Λ (gr ~ A 13 a ) hh   HH Λ 1 (gr ~ A 12 a ) × HH Λ 2 (gr ~ A 23 a ) ◦ / / HH Λ (gr ~ A 23 a ) . (4.4.4) W e shall study the Ho c hsc hild class of O -mo dules with some details in Chapter 5. 4.4. GR ADED AND LOCALIZED HOCHSCHILD CLASSES 115 Ho ch schild classes for A lo c X One deﬁnes HH ( A lo c X ) := C lo c X a L ⊗ A loc X × X a C lo c X . W e hav e HH ( A lo c X ) ≃ C ~ , lo c ⊗ C ~ HH ( A X ). and there is a natural morphism ( • ) lo c : HH ( A X ) − → HH ( A lo c X ) . F or F ∈ D b coh ( A lo c X ), one deﬁnes its Ho c hsc hild class hh X ( F ) b y the same construction as for A X -mo dules. F o r M ∈ D b coh ( A X ), setting M lo c = C ~ , lo c ⊗ C ~ M , we ha ve (hh X ( M )) lo c = hh X ( M lo c ) . Recall that the notion o f go o d mo dules a nd the categor y D b gd ( A lo c X ) ha v e b een giv en in D eﬁnition 2.3.16. O ne immediately deduces from Theorem 4.3.5 the follow ing: Corollary 4.4.3. L et Λ i b e a close d subset of X i × X i +1 ( i = 1 , 2) and assume that Λ 1 × X 2 Λ 2 is pr op er o v er X 1 × X 3 . Set Λ = Λ 1 ◦ Λ 2 . L et K i ∈ D b gd , Λ i ( A lo c X i × X a i +1 ) ( i = 1 , 2) . Then hh X 13 a ( K 1 ◦ K 2 ) = hh X 12 a ( K 1 ) ◦ hh X 23 a ( K 2 ) (4.4.5) as elements of HH Λ ( A lo c X 1 × X a 3 ) . Using Prop osition 3.4.3 and the additivity of t he Ho c hsc hild class in The- orem 4.1.5, w e ﬁnd that there is a natura l map b K coh , Λ (gr ~ A X ) − → d HH Λ (gr ~ A X ) . (4.4.6) F or M ∈ D b gd , Λ ( A lo c X ), w e denote by c hh gr X ( M ) the image of M by the se- quence o f maps D b gd , Λ ( A lo c X ) − → b K coh , Λ (gr ~ A X ) − → d HH Λ (gr ~ A X ) . Let Λ i b e a closed subset o f X i × X i +1 ( i = 1 , 2 ) and assume that Λ 1 × X 2 Λ 2 is prop er ov er X 1 × X 3 . Set Λ = Λ 1 ◦ Λ 2 . 116 CHAPTER 4. HOCHSCHILD CLASSES Using the commutativit y of D iagrams 3.4.5 , w e get that the diagram b elo w comm utes Ob  D b gd , Λ 1 ( A lo c 12 a )  × Ob  D b gd , Λ 2 ( A lo c 23 a )  ◦ / / gr ~   Ob  D b gd , Λ ( A lo c 13 a )  gr ~   b K coh , Λ 1 (gr ~ A 12 a ) × b K coh , Λ 2 (gr ~ A 23 a ) ◦ / /   b K coh , Λ (gr ~ A 13 a )   d HH Λ 1 (gr ~ A 12 a ) × d HH Λ 2 (gr ~ A 23 a ) ◦ / / d HH Λ (gr ~ A 23 a ) . (4.4.7) In other words, c hh gr 13 a ( K 1 ◦ K 2 ) = c hh gr 12 a ( K 1 ) ◦ c hh gr 23 a ( K 2 ) . (4.4.8) Corollary 4.4.4. L et M , N ∈ D b gd ( A lo c X ) and as s ume that Supp ( M ) ∩ Supp( N ) is c omp a ct. Then RHom A X ( M , N ) b elongs to D b f ( C ~ ) and χ (RHom A loc X ( M , N )) = c hh gr X a (D ′ A M ) ◦ c hh gr X ( N ) = c hh gr X ( M ) ◦ c hh gr X ( N ) . Pr o of. O ne has by (3.4.3) χ (RHom A loc X ( M , N )) = hh pt (D ′ A M ◦ N ) = c hh gr pt (D ′ A M ◦ N ) = c hh gr X a (D ′ A M ) ◦ c hh gr X ( N ) and the la st equality fo llo ws from (4.1.9 ). Q.E.D. Remark 4.4.5. In the algebraic case, that is, in the situation of § 2 .7, one should replace b K coh , Λ with K coh , Λ and d HH Λ (gr ~ A X ) with HH Λ (gr ~ A X ). W e shall ex plain ho w to calculate c hh gr X in Chapter 5. Chapter 5 The comm utativ e case W e shall mak e the link b et w een the Ho c hsc hild class and the Chern and Euler classes of coheren t O X -mo dules, follow ing [35], an unpublished letter from the ﬁrst named author (M.K) to the second (P .S), dated 18 / 11/1991. 5.1 Ho c hs c hild class o f O -mo dules In this section, w e shall study the Ho c hsc hild class in the particular case of a trivial deformation. In this case, the formal parameter ~ doesn’t pla y any role, and w e ma y w ork with O -mo dules. W e shall use the same notations for O X -mo dules as for ( O X [[ ~ ] ] , ⋆ )-mo dules whe re ⋆ is the usual commutativ e pro duct. Note that the results of this se ction are well kno wn from the sp ecialists. Let us quote in particular [14 , 15, 33, 48, 53, 58, 63]. Let ( X , O X ) b e a complex manifold of complex dimension d X . As usual, w e denote by δ X : X ֒ → X × X the diagonal embedding. W e denote by Ω i X the sheaf of holomorphic i -forms and one sets Ω X := Ω d X X . W e set ω X := Ω X [ d X ] . W e denote by D ′ O and D O the dualit y functors D ′ O ( F ) = R H om O X ( F , O X ) , D O ( F ) = R H om O X ( F , ω X ) . When there is no risk of confusion, w e write D ′ and D instead of D ′ O and D O , resp ectiv ely . Let f : X − → Y b e a morphism of complex manifolds. F or G ∈ D b ( O Y ), w e set f ∗ G := O X L ⊗ f − 1 O Y f − 1 G . 117 118 CHAPTER 5. THE COMMUT A TIVE CASE W e use the notation H 0 ( f ∗ ) : Mo d( O Y ) − → Mo d( O X ) for the (non deriv ed) in v erse image functor. The Ho c hsc hild homology of O X is giv en by: HH ( O X ) := δ ∗ X δ X ∗ O X , an ob ject o f D b ( O X ). (5.1.1) Note that δ X ! ≃ δ X ∗ ≃ R δ X ∗ , and moreo v er δ X ∗ HH ( O X ) ≃ δ X ∗ ( O X L ⊗ δ ∗ X ( δ X ∗ O X )) ≃ δ X ∗ O X L ⊗ O X × X δ X ∗ O X . (5.1.2) By reform ulating the construction of the Ho chsc hild class for mo dules o ve r DQ-algebroids, w e get Deﬁnition 5.1.1. F or F ∈ D b coh ( O X ), w e deﬁne its Ho chsc hild class hh X ( F ) ∈ H 0 Supp F ( X ; δ ∗ X δ X ∗ O X ) a s t he composition O X − → R H om O X ( F , F ) ∼ − → δ ∗ X ( F ⊠ D ′ F ) − → δ ∗ X δ X ∗ O X . (5.1.3) Here the morphism F ⊠ D ′ F − → δ X ∗ O X is deduced from the morphism δ ∗ X ( F ⊠ D ′ F ) ∼ − → F L ⊗ O X D ′ F − → O X b y a dj unction. Applying Theorem 4.3.5, w e get that for t w o complex manifolds X and Y and f o r F ∈ D b coh ( O X ) and G ∈ D b coh ( O Y ), we hav e hh X × Y ( F ⊠ G ) = hh X ( F ) ⊠ hh Y ( G ) . Let f : X − → Y be a morphism of complex m anif o lds and denote b y Γ f ⊂ X × Y its graph. W e denote b y hh X × Y ( O Γ f ) the Ho chs child class of the coheren t O X × Y -mo dule O Γ f . Hence hh X × Y ( O Γ f ) ∈ H 0 ( X × Y ; HH ( O X × Y )) . Applying Theorem 4.3.5, w e get Corollary 5.1.2. ( i) L et G ∈ D b coh ( O Y ) . T hen hh X ( f ∗ G ) = hh X × Y ( O Γ f ) ◦ hh Y ( G ) . (ii) L et F ∈ D b coh ( O X ) and assume that f is pr op er on Supp( F ) . T hen hh Y (R f ! F ) = hh X ( F ) ◦ hh X × Y ( O Γ f ) . In Prop osition 5.1.3 and 5.2 .3 below, w e giv e a more direct description of the maps hh X × Y ( O Γ f ) ◦ and ◦ hh X × Y ( O Γ f ). 5.1. HOCHSCHILD CLASS OF O -MOD ULES 119 Prop osition 5.1.3. L et f : X − → Y b e a morphis m of c omplex manifo l d s. (i) Ther e is a c anonic al morphi sm f ∗ δ ∗ Y δ Y ∗ O Y − → δ ∗ X δ X ∗ O X . (5.1.4 ) (ii) This morphism to gether w i th the isomorphis m O X ∼ ← − f ∗ O Y induc es a morphism f ∗ : H 0 (RΓ( Y ; δ ∗ Y δ Y ∗ O Y )) − → H 0 (RΓ( X ; δ ∗ X δ X ∗ O X )) (5.1.5) and for G ∈ D b coh ( O Y ) , we have hh X ( f ∗ G ) = f ∗ hh Y ( G ) . (5.1.6) Pr o of. ( i) Consider the diagram X δ X / / f   X × X e f   Y δ Y / / Y × Y . (5.1.7) Then w e hav e mo r phisms f ∗ δ ∗ Y δ Y ∗ O Y ≃ δ ∗ X e f ∗ δ Y ∗ O Y − → δ ∗ X δ X ∗ f ∗ O Y ≃ δ ∗ X δ X ∗ O X . Here t he arro w e f ∗ δ Y ∗ − → δ X ∗ f ∗ is deduced by adjunction from δ Y ∗ − → δ Y ∗ R f ∗ f ∗ ≃ R e f ∗ δ X ∗ f ∗ . (ii) The diagram e f ∗ ( G ⊠ D ′ G ) ∼   / / e f ∗ δ Y ∗ O Y   f ∗ G ⊠ f ∗ D ′ G ∼   δ X ∗ f ∗ O Y ∼   f ∗ G ⊠ D ′ f ∗ G / / δ X ∗ O X 120 CHAPTER 5. THE COMMUT A TIVE CASE comm utes. It fo llo ws that the diagram b elow comm utes. f ∗ O Y / / ∼   f ∗ δ ∗ Y ( G ⊠ D ′ G ) ∼   / / f ∗ δ ∗ Y δ Y ∗ O Y ∼   δ ∗ X e f ∗ ( G ⊠ D ′ G ) ∼   / / δ ∗ X e f ∗ δ Y ∗ O Y   δ ∗ X ( f ∗ G ⊠ f ∗ D ′ G ) ∼   δ ∗ X δ X ∗ f ∗ O Y ∼   O X / / δ ∗ X ( f ∗ G ⊠ D ′ f ∗ G ) / / δ ∗ X δ X ∗ O X . Therefore, the image of hh Y ( G ) ∈ Hom O Y ( O Y , δ ∗ Y δ Y ∗ O Y ) by t he ma ps Hom O Y ( O Y , δ ∗ Y δ Y ∗ O Y ) − → Hom O X ( f ∗ O Y , f ∗ δ ∗ Y δ Y ∗ O Y ) − → Hom O X ( O X , δ ∗ X δ X ∗ O X ) is hh X ( f ∗ G ). Q.E.D. Remark 5.1.4. Although w e omit t he pro of, the map in (5 .1 .5) coincides with hh X × Y ( O Γ f ) ◦ . Ring structure F or an exp osition on tensor categories, w e refer to [41 ]. Prop osition 5.1.5. (i) The obje ct δ ∗ X δ X ∗ O X is a ring in the tensor c ate- gory (D b coh ( O X ) , L ⊗ O X ) . Mor e pr e cisely, (a) the map µ obtaine d as the c omp o sition δ ∗ X δ X ∗ O X L ⊗ O X δ ∗ X δ X ∗ O X ∼ − → δ ∗ X ( δ X ∗ O X L ⊗ O X × X δ X ∗ O X ) − → δ ∗ X δ X ∗ O X . is asso ciative. Her e the last arr ow is induc e d by δ X ∗ O X ⊗ δ X ∗ O X − → δ X ∗ O X . (b) hh X ( O X ) is a unit of this ring . Mor e pr e cisely, the natur al mor- phism ε deﬁ n e d as the c omp osition ε : O X ∼ − → δ ∗ X O X × X − → δ ∗ X δ X ∗ O X 5.1. HOCHSCHILD CLASS OF O -MOD ULES 121 has the pr op erty that the c omp osition δ ∗ X δ X ∗ O X ≃ δ ∗ X δ X ∗ O X L ⊗ O X O X ε − − → δ ∗ X δ X ∗ O X L ⊗ O X δ ∗ X δ X ∗ O X µ − − → δ ∗ X δ X ∗ O X is the identity. (ii) The ring ( δ ∗ X δ X ∗ O X , µ ) is c ommutative. Mor e pr e cisely, we have µ ◦ σ = µ , wher e σ ∈ Aut D b ( O X ) ( δ ∗ X δ X ∗ O X L ⊗ O X δ ∗ X δ X ∗ O X ) is the morphism asso ciate d with x ⊗ x ′ 7→ x ′ ⊗ x . (iii) The obje ct δ ! X δ X ! ω X has a structur e of a δ ∗ X δ X ∗ O X -mo dule. Mor e pr e- cisely, the c omp osition δ ∗ X δ X ∗ O X L ⊗ O X δ ! X δ X ! ω X − → δ ! X ( δ X ∗ O X L ⊗ O X × X δ X ! ω X ) − → δ ! X δ X ! ω X . is a s s o ciative and p r eserves the unit. Her e, the last arr ow is induc e d by δ X ∗ O X L ⊗ O X × X δ X ! ω X ≃ δ X ∗ ( δ ∗ X δ X ∗ O X ⊗ O X ω X ) − → δ X ∗ ( O X ⊗ O X ω X ) ≃ δ X ∗ ω X by adjunction. Pr o of. The v eriﬁcation of these assertions is left to the reader. W e only remark that the commutativit y and asso ciativit y are consequences of the corresp onding prop erties of δ X ∗ O X . F o r example, the comm utativity is the consequenc e of the commutativit y of δ X ∗ O X L ⊗ O X × X δ X ∗ O X − → δ X ∗ O X . Q.E.D. Notation 5.1.6. F or λ i ∈ H 0 Λ i ( X ; δ ∗ X δ X ∗ O X ) ( i = 1 , 2), w e deﬁne their pro duct λ 1 • λ 2 as the comp osition O X ∼ − → O X L ⊗ O X O X λ 1 ⊗ λ 2 − − − → δ ∗ X δ X ∗ O X L ⊗ O X δ ∗ X δ X ∗ O X µ − → δ ∗ X δ X ∗ O X . Prop osition 5.1.7. L et F i ∈ D b coh ( O X ) ( i = 1 , 2 ) . T hen hh X ( F 1 L ⊗ O X F 2 ) = hh X ( F 1 ) • hh X ( F 2 ) . (5 .1.8) 122 CHAPTER 5. THE COMMUT A TIVE CASE Pr o of. Consider the comm utative diag ram b elo w (in whic h ⊗ stands for ⊗ O ): O X / / @A / / O X ⊗ O X   δ ∗ X ( F 1 ⊠ D ′ F 1 ) ⊗ δ ∗ X ( F 2 ⊠ D ′ F 2 ) / / ∼   δ ∗ X δ X ∗ O X ⊗ δ ∗ X δ X ∗ O X ≀   δ ∗ X  ( F 1 ⊠ D ′ F 1 ) ⊗ ( F 2 ⊠ D ′ F 2 )    / / δ ∗ X ( δ X ∗ O X ⊗ δ X ∗ O X )   δ ∗ X  ( F 1 ⊗ F 2 ) ⊠ D ′ ( F 1 ⊗ F 2 )  / / δ ∗ X ( δ X ∗ O X ) . The comp o sition of the arrows on the top and the righ t give s hh X ( F 1 ) • hh X ( F 2 ) and the comp osition of the arrows on the left and t he b ottom giv es hh X ( F 1 L ⊗ O X F 2 ). Q.E.D. Note that hh X ( F 1 L ⊗ O X F 2 ) = δ ∗ X (hh X ( F 1 ) ⊠ hh X ( F 2 )) . 5.2 co-Ho c hs c hild class Deﬁnition 5.2.1. F or F ∈ D b coh ( O X ), w e deﬁne its co- Ho c hsc hild class thh X ( F ) ∈ H 0 Supp F ( X ; δ ! X δ X ! ω X ) as the comp osition O X − → R H om O X ( F , F ) ≃ δ ! X ( F ⊠ D O F ) − → δ ! X δ X ! ω X . (5.2.1) Here, the morphism ( F ⊠ D O F ) − → δ X ! ω X is induced from δ ∗ X ( F ⊠ D O F ) ≃ F L ⊗ O X D O F − → ω X b y adjunction. Consider the sequence of isomorphisms δ ∗ X δ X ∗ O X ∼ − → O X L ⊗ O X δ ∗ X δ X ∗ O X ∼ − → δ ! X ( O X ⊠ ω X ) L ⊗ O X δ ∗ X δ X ∗ O X ∼ − → δ ! X (( O X ⊠ ω X ) L ⊗ O X δ X ∗ O X ) ∼ − → δ ! X δ X ∗ ( δ ∗ X ( O X ⊠ ω X ) L ⊗ O X O X ) ∼ − → δ ! X δ X ! ω X . W e denote by td the isomorphism td : δ ∗ X δ X ∗ O X ∼ − → δ ! X δ X ! ω X (5.2.2) 5.2. CO-HOCHSCHILD CLASS 123 constructed ab o v e. F or a closed subset S ⊂ X , w e kee p the same notation td to denote the isomorphism td : H 0 S ( X ; δ ∗ X δ X ∗ O X ) ∼ − → H 0 S ( X ; δ ! X δ X ! ω X ) . (5.2.3) Prop osition 5.2.2. F or F ∈ D b coh ( O X ) , we h a ve thh X ( F ) = td ◦ hh X ( F ) . (5.2.4) Pr o of. The pro of follows fro m the comm utativit y of the diagra m b elo w in whic h we use the natural morphism O X − → δ ! X ( O X ⊠ ω X ) O X / / @A / / δ ∗ X ( F ⊠ D ′ F ) / /   δ ∗ X δ X ∗ O X ≀   δ ! X ( O X ⊠ ω X ) ⊗ δ ∗ X ( F ⊠ D ′ F ) / /   δ ! X ( O X ⊠ ω X ) ⊗ δ ∗ X δ X ∗ O X ≀   δ ! X (( O X ⊠ ω X ) ⊗ ( F ⊠ D ′ F )) / /   δ ! X (( O X ⊠ ω X ) ⊗ δ X ∗ O X ) ≀   δ ! X δ X ∗  δ ! X ( O X ⊠ ω X ) ⊗ O X  ≀   δ ! X ( F ⊠ D F ) / / δ ! X δ X ! ω X . Q.E.D. F or a morphism f : X − → Y of complex manifolds, w e denote by Γ f − pr ( X ; • ) the functor of global sections with f -prop er supp orts. Prop osition 5.2.3. L et f : X − → Y b e a morphis m of c omplex manifo l d s. (i) Ther e is a c anonic al morphi sm R f ! δ ! X δ X ! ω X − → δ ! Y δ Y ! ω Y . (5.2.5) (ii) This mo rphism to gether with the morphism O Y − → R f ∗ O X induc es a morphism f ! : H 0 (RΓ f − pr ( X ; δ ! X δ X ! ω X )) − → H 0 (RΓ( Y ; δ ! Y δ Y ! ω Y )) (5.2.6) and for F ∈ D b coh ( O X ) such that f is p r op er on Supp( F ) , we ha ve thh Y (R f ! F ) = f ! thh X ( F ) . (5.2 .7) 124 CHAPTER 5. THE COMMUT A TIVE CASE Pr o of. ( i) Consider the diagram (5.1.7). Then w e hav e morphisms R f ! δ ! X δ X ! ω X − → δ ! Y R e f ! δ X ! ω X ≃ δ ! Y δ Y ! R f ! ω X − → δ ! Y δ Y ! ω Y . Here, t he ﬁrst morphism is deduced by a djunction from δ ! X − → δ ! X e f ! R e f ! ≃ f ! δ ! Y R e f ! . (ii) The pro of is similar to tha t o f Prop osition 5.1.3 and follows from the comm utativit y of the diagram b elo w in whic h w e write for short f ! and f ∗ instead of R f ! and R f ∗ and similarly with e f . f ∗ O X / / f ∗ δ ! X ( F ⊠ D F ) ∼   / / f ! δ ! X δ X ! ω X   δ ! Y e f ! ( F ⊠ D ′ F ) ∼   / / δ ! Y e f ! δ X ! ω X ∼   δ ! Y ( f ! F ⊠ f ! D F ) ∼   δ ! Y δ Y ! f ! ω X   O Y / / O O δ ! Y ( f ! F ⊠ D f ! F ) / / δ ! Y δ Y ! ω Y . Therefore, the image of thh X ( F ) ∈ Hom O X ( O X , δ ! X δ X ! ω X ) b y the maps Γ f − pr ( X ; H om O X ( O X , δ ! X δ X ! ω X )) − → Hom O X (R f ∗ O X , R f ! δ ! X δ X ! ω X ) − → Hom O Y ( O Y , δ ! Y δ Y ! ω Y ) is thh Y ( f ! F ). Q.E.D. Remark 5.2.4. Although w e omit t he pro of, the map in (5 .2 .6) coincides with ◦ hh X × Y ( O Γ f ). 5.3 Chern and Euler classe s of O -mo du les The Ho dge cohomology of O X is giv en by: HD ( O X ) := d X M i =0 Ω i X [ i ] , an ob ject of D b ( O X ) . (5.3.1) 5.3. CHERN AND EULER CLASSES OF O -MODULES 125 Lemma 5.3.1. L et f : X − → Y b e a mo rphism o f c omplex manifolds. Th er e ar e c anonic al morph i s m s ⊠ : HD ( O X ) ⊠ H D ( O Y ) − → HD ( O X × Y ) , (5.3.2) f ∗ : f ∗ HD ( O Y ) − → HD ( O X ) , (5.3.3) f ! : R f ! HD ( O X ) − → HD ( O Y ) . (5.3.4) Pr o of. The morphisms (5.3.2), (5.3 .3) and (5.3.4) are resp ective ly asso ciated with the morphisms Ω i X [ i ] ⊠ Ω j Y [ j ] − → Ω i + j X × Y [ i + j ] , f ∗ Ω i Y [ i ] − → Ω i X [ i ] , R f ! Ω i + d X X [ i + d X ] − → Ω i + d Y Y [ i + d Y ] . Q.E.D. Theorem 5.3.2. (a) Ther e is an isomorphism α X : δ ∗ X δ X ∗ O X ∼ − → HD ( O X ) which c omm utes wi th the f unctors ⊠ and f ∗ . (b) Ther e is an isomorphism β X : HD ( O X ) ∼ − → δ ! X δ X ! ω X which c omm utes wi th the f unctors ⊠ and f ! . Setting τ := β − 1 X ◦ td ◦ α − 1 X , w e g et a comm utative diagram in D b ( O X ): δ ∗ X δ X ∗ O X ∼ α X   ∼ td / / δ ! X δ X ! ω X HD ( O X ) ∼ τ / / HD ( O X ) . ∼ β X O O (5.3.5) The construction of α X and β X and the pro of are giv en in the next section. Deﬁnition 5.3.3. F or F ∈ D b coh ( O X ), we set c h( F ) = α X ◦ hh X ( F ) ∈ d X M i =0 H i Supp( F ) ( X ; Ω i X ) , (5.3.6) eu( F ) = β − 1 X ◦ thh X ( F ) ∈ d X M i =0 H i Supp( F ) ( X ; Ω i X ) . (5.3.7) W e call c h( F ) the Chern class of F and eu( F ) the Euler class of F . 126 CHAPTER 5. THE COMMUT A TIVE CASE Of course, c h ( F ) coincides with the classical Chern character a nd the morphism α X is the so-called Ho c hsc hild-Kostan t- Rosen b erg map. The followin g conjecture w as stated in [35]. Conjecture 5.3.4. One has eu( O X ) = td X ( T X ), where td X ( T X ) is t he T o dd class o f the ta ngen t bundle T X . This conjecture implies that eu( F ) = c h ( F ) ∪ td X ( T X ). Indeed, for a, b ∈ H ∗ ( X ; δ ∗ X δ X ∗ O X ), we ha ve td( a ◦ b ) = a ◦ td ( b ) b y Prop osition 5.1 .5 (iii) and Lemma 5.4.7 b elow. This conjecture has recen tly b een pro ve d by A. Ramadoss [53] in the algebraic case and b y J. Griv aux [30] in t he analytic case. An index t heorem Consider the particular case of t w o coherent O X -mo dules L i ( i = 1 , 2) suc h that Supp( L 1 ) ∩ Supp( L 2 ) is compact. In this case w e ha v e (see [3 3, 53]): hh pt ( L 1 ◦ L 2 ) = χ (RΓ( X ; L 1 L ⊗ O X L 2 )) = Z X (c h( L 1 ) ∪ c h( L 2 ) ∪ td X ( T X )) . (5.3.8) W e consider the situation of Corollary 4.4.4. Hence, A X is a DQ-alg ebroid on X . Corollary 5.3.5. L et M , N ∈ D b g d ( A lo c X ) and assume that K := Supp( M ) ∩ Supp( N ) is c omp act. L et U b e a r elatively c omp act op en subset of X c ontain- ing K . Then RHo m A loc X ( M , N ) b elongs to D b f ( C ~ , lo c ) and its Euler-Poinc ar´ e index is given by the formula χ (RHom A loc X ( M , N )) = Z U c h U ((gr U ~ D ′ A M )) ∪ c h U (gr U ~ ( N )) ∪ td U ( T U ) . Pr o of. Applying Corollary 4.4.4, w e hav e χ (RHom A loc X ( M , N )) = hh pt (D ′ A M ◦ N ) = hh pt (gr ~ D ′ A M 0 ◦ g r ~ N 0 ) , where M 0 (resp. N 0 ) is an o b j ect of D b coh ( A U ) whic h generates M (resp. N ) on U . Then, the result follows from (5.3.8) . Q.E.D. 5.4. PROOF OF THEOREM ?? 127 5.4 Pro of of Theo rem 5.3.2 As usual, w e den ot e by p i : X × X − → X the i -th pro jection ( i = 1 , 2). T he follo wing lemma is well-kno wn. Lemma 5.4.1. L et F b e an ( O X ⊠ O X ) -mo dule supp orte d by the diag o nal. Then the fol lowin g c onditions ar e e quivalent: (i) p 1 ∗ F is a c oher ent O X -mo dule, (ii) p 2 ∗ F is a c oher ent O X -mo dule. If these c onditions a r e satisﬁe d, then the ma p F − → O X × X ⊗ O X ⊠ O X F is an isomorphism. In p articular, the ( O X ⊠ O X ) -mo dule struct ur e on F extends uniquely to an O X × X -mo dule structur e. W e deﬁne the p − 1 1 O X -mo dule P k := δ X ∗ Ω k X ⊕ δ X ∗ Ω k +1 X for k ≥ 0 , P k = 0 for k < 0 . W e endow the P k ’s with a structure of p − 1 2 O X -mo dule b y setting p ∗ 2 ( a )( ω k ⊕ θ k +1 ) = aω k ⊕ ( aθ k +1 − da ∧ ω k ) for a ∈ O X , ω k ∈ Ω k X , θ k +1 ∈ Ω k +1 X . This deﬁnes an action of p − 1 2 O X since p ∗ 2 ( a 1 ) p ∗ 2 ( a 2 )( ω k ⊕ θ k +1 ) = p ∗ 2 ( a 1 )( a 2 ω k ⊕ ( a 2 θ k +1 − da 2 ∧ ω k )) = a 1 a 2 ω k ⊕ ( a 1 a 2 θ k +1 − a 1 da 2 ∧ ω k − da 1 ∧ a 2 ω k ) = a 1 a 2 ω k ⊕ ( a 1 a 2 θ k +1 − d ( a 1 a 2 ) ∧ ω k ) = p ∗ 2 ( a 1 a 2 )( ω k ⊕ θ k +1 ) . By Lemma 5.4.1, w e get tha t P k has a structure of O X × X -mo dule and we ha v e an exact sequence: 0 − → δ X ∗ Ω k +1 X α k − − − → P k β k − − − → δ X ∗ Ω k X − → 0 . (5.4.1) Hence δ X ∗ Ω k [ k ] ∼ ← − ( δ X ∗ Ω k +1 X − → P k ) − → δ X ∗ Ω k +1 X [ k + 1] deﬁnes the morphism ξ k : δ X ∗ Ω k [ k ] − → δ X ∗ Ω k +1 X [ k + 1] . It induces a morphism ξ : M k δ X ∗ Ω k X [ k ] − → M k δ X ∗ Ω k X [ k ] . (5.4.2) 128 CHAPTER 5. THE COMMUT A TIVE CASE Let d stan k : P k − → P k − 1 b e the comp osition d stan k : P k β k − − − → δ X ∗ Ω k X α k − 1 − − − − → P k − 1 . (5.4.3) W e deﬁne the complex P • whose diﬀerential d − k P : P k − → P k − 1 is given b y k d stan k . Then Im d stan k ≃ Im β k ≃ δ X ∗ Ω k X and Ker d stan k ≃ K er β k ≃ δ X ∗ Ω k +1 X . Therefore we hav e a quasi-isomorphism P • − → δ X ∗ O X . Lemma 5.4.2. The m orphism α X : δ ∗ X δ X ∗ O X − → H 0 ( δ ∗ X )( P • ) ≃ M k Ω k X [ k ] (5.4.4) is an isomorph i sm in D b ( O X ) . Pr o of of L emm a 5.4.2. Since the question is lo cal, w e ma y assume that X is a vec to r space V . Then w e ha v e a Koszul complex O X × X ⊗ • ^ V ∗ ≃  · · · − → O X × X ⊗ 2 ^ V ∗ − → O X × X ⊗ V ∗ − → O X × X  and an isomorphism O X × X ⊗ V • V ∗ − → δ X ∗ O X in D b ( O X × X ). Then applying H 0 ( δ ∗ X ), w e obtain an isomorphism in D b ( O X ): δ ∗ X δ X ∗ O X ∼ − → H 0 ( δ ∗ X )( O X × X ⊗ • ^ V ∗ ) . The C -linear maps V k V ∗ − → Ω k X ( V ) − → P k ( X × X ) induce a morphism o f complexes O X × X ⊗ V • V ∗ − → P • suc h that the diagram b elow commute s: O X × X ⊗ V • V ∗ + + X X X X X X   δ X ∗ O X . P • 2 2 f f f f f f f f f f f f Since H 0 ( δ ∗ X )( O X × X ⊗ V • V ∗ )[ d X ] − → H 0 ( δ ∗ X )( P • ) is an isomorphism, we obtain the desired result. Q.E.D. Remark 5.4.3. (i) Let I ⊂ O X × X b e the deﬁning ideal of the diagonal set δ X ( X ). Then the morphism ξ 0 : δ X ∗ O X − → δ X ∗ Ω 1 X [1] is give n b y the exact sequence 0 − → δ X ∗ Ω 1 X − → O X × X /I 2 − → δ X ∗ O X − → 0. Indeed, w e ha v e a comm utative diagram 0 / / I /I 2 / / ≀   O X × X /I 2 / / ≀   δ X ∗ O X / / id   0 0 / / δ X ∗ Ω 1 X β 0 / / P 0 α 0 / / δ X ∗ O X / / 0 . 5.4. PROOF OF THEOREM ?? 129 Here, the left v ertical isomorphism is giv en by I /I 2 ∋ p ∗ 1 ( a ) − p ∗ 2 ( a ) ← → da ∈ δ X ∗ Ω 1 X ( a ∈ O X ). (ii) Moreo v er the morphism ξ k : δ X ∗ Ω k X [ k ] − → δ X ∗ Ω k +1 X [ k + 1] coincides with the comp osition δ X ∗ Ω k X [ k ] ≃ δ X ∗ Ω k X [ k ] L ⊗ O X × X O X × X − → δ X ∗ Ω k X [ k ] L ⊗ O X × X δ X ∗ O X ξ 0 − − → δ X ∗ Ω k X [ k ] L ⊗ O X × X δ X ∗ Ω 1 X [1] − → δ X ∗ (Ω k X [ k ] L ⊗ O X Ω 1 X [1]) − → δ X ∗ Ω k +1 X [ k + 1] . (iii) Note that the morphism α X : δ ∗ X δ X ∗ O X ∼ − → M k Ω k X [ k ] coincides with the morphism obtained from δ X ∗ O X − → M k δ X ∗ Ω k X [ k ] exp( ξ ) − − − − − → M k δ X ∗ Ω k X [ k ] b y a dj unction. Lemma 5.4.4. The morphism α X in (5.4 .4) inter changes the c omp osition of the ring δ ∗ X δ X ∗ O X given in Prop osition 5.1.5 (a) with the c omp osition Ω i X [ i ] L ⊗ O X Ω j X [ j ] ≃ (Ω i X L ⊗ O X Ω j X )[ i + j ] ∧ − − → Ω i + j X [ i + j ] . Note t ha t the unit O X − → δ ∗ X δ X ∗ O X is given by O X ≃ δ ∗ X O X × X − → δ ∗ X δ X ∗ O X , where the last arrow is induced by O X × X − → δ X ∗ O X . Pr o of. W e deﬁne µ ij : P i ⊗ O X × X P j − → P i + j b y µ ij  (( ω i ⊕ θ i +1 ) ⊗ ( ω j ⊕ θ j +1 ))  = ( ω i ∧ ω j ) ⊕ ( θ i +1 ∧ ω j + ( − 1) i ω i ∧ θ j +1 ) . (5.4.5) This map is p − 1 2 ( O X )-bilinear since: µ ij   p ∗ 2 ( a )( ω i ⊕ θ i +1 )  ⊗ ( ω j ⊕ θ j +1 )  = µ ij   aω i ⊕ ( aθ i +1 − da ∧ ω i )  ⊗ ( ω j ⊕ θ j +1 )  = ( aω i ∧ ω j ) ⊕  ( aθ i +1 − da ∧ ω i ) ∧ ω j + ( − 1) i aω i ∧ θ j +1  = p ∗ 2 ( a )  ( ω i ∧ ω j ) ⊕ ( θ i +1 ∧ ω j + ( − 1) i ω i ∧ θ j +1 )  = p ∗ 2 ( a ) µ ij  ( ω i ⊕ θ i +1 ) ⊗ ( ω j ⊕ θ j +1 )  , 130 CHAPTER 5. THE COMMUT A TIVE CASE and µ ij  ( ω i ⊕ θ i +1 ) ⊗ p ∗ 2 ( a )( ω j ⊕ θ j +1 )  = µ ij   ω i ⊕ θ i +1  ⊗  aω j ⊕ ( aθ j +1 − da ∧ ω j   =  aω i ∧ ω j  ⊕  θ i +1 ∧ aω j + ( − 1) i ω i ∧ ( aθ j +1 − da ∧ ω j )  = ( aω i ∧ ω j ) ⊕ ( aθ i +1 ∧ ω j + ( − 1) i aω i ∧ θ j +1 − da ∧ ω i ∧ ω j ) = p ∗ 2 ( a )  ω i ∧ ω j ⊕ ( θ i +1 ∧ ω j + ( − 1) i ω i ∧ θ j +1 )  = p ∗ 2 ( a ) µ ij  ( ω i ⊕ θ i +1 ) ⊗ ( ω j ⊕ θ j +1 )  . The morphism µ comm utes with the diﬀerentials since: µd  ( ω i ⊕ θ i +1 ) ⊗ ( ω j ⊕ θ j +1 )  = µ i − 1 ,j  (0 ⊕ iω i ) ⊗ ( ω j ⊕ θ j +1 )  + ( − 1) i µ i,j − 1  ( ω i ⊕ θ i +1 ) ⊗ (0 ⊕ j ω j )  = 0 ⊕  iω i ∧ ω j + ( − 1) i ( − 1) i j ω i ∧ ω j  = 0 ⊕ ( i + j ) ω i ∧ ω j = dµ  ( ω i ⊕ θ i +1 ) ⊗ ( ω j ⊕ θ j +1 )  . Hence w e hav e a commutativ e diagram in D b ( O X × X ) δ X ∗ O X L ⊗ O X × X δ X ∗ O X / /   δ X ∗ O X P • ⊗ O X × X P • µ / / P • . O O Therefore, a pplying δ ∗ X , the morphism δ ∗ X δ X ∗ O X L ⊗ δ ∗ X δ X ∗ O X − → δ ∗ X δ X ∗ O X is represen ted by H 0 ( δ ∗ X ) P • ⊗ O X H 0 ( δ ∗ X ) P • − → H 0 ( δ ∗ X ) P • . Th us w e obt a in the desired result. Q.E.D. Lemma 5.4.5. Consider a morphism f : X − → Y . T hen the diagr am b elow c ommutes: f ∗ δ ∗ Y δ Y ∗ O Y / / α Y   δ ∗ X δ X ∗ O X α X   f ∗ ( M k Ω k Y [ k ]) / / M k Ω k X [ k ] . 5.4. PROOF OF THEOREM ?? 131 Pr o of. L et ˜ f : X × X − → Y × Y b e t he morphism asso ciated with f . Let us denote b y P X • the complex on X constructed ab ov e. Then we easily construct a commutativ e diagr a m H 0 ( ˜ f ∗ ) P Y • / / ϕ   H 0 ( ˜ f ∗ ) δ Y ∗ O Y   P X • / / δ X ∗ O X suc h that H 0 ( δ ∗ X ˜ f ∗ ) P Y • / / δ ∗ X ϕ   H 0 ( f ∗ δ ∗ Y ) P Y • ∼ / / f ∗ ( M k Ω k Y [ k ]) ψ   H 0 ( δ ∗ X ) P X • / / M k Ω k X [ k ] comm utes wh ere ψ is g iv en in (5.3.3). Q.E.D. No w we set Q k =      P k − 1 for 1 ≤ k ≤ d X , δ X ∗ O X for k = 0, 0 otherwise. (5.4.6) and deﬁne the diﬀeren tia l d Q with d Q k = ( k − 1 − d X ) d stan k − 1 , where d stan k − 1 is giv en by (5.4.3) a nd d stan 0 : O X ⊕ Ω 1 X − → O X is the canonical morphism. Then Q • is a complex of O X × X -mo dules and the canonical homomorphism Ω d X X − → Ω d X − 1 X ⊕ Ω d X X induces a mo r phism of complexes δ X ∗ ω X − → Q • , whic h is an isomorphism in D b ( O X × X ). Let us denote by H 0 ( δ ! X ) the f unctor δ − 1 X H om O X × X ( δ ∗ O X , • ). Lemma 5.4.6. The m orphism β X : M k Ω k X ≃ H 0 ( δ ! X ) Q • − → δ ! X δ X ∗ ω X is an isomorph i sm in D b ( O X ) . Since the pro of is similar to that of Lemma 5.4.2 , we omit it. Note t hat t he morphism β X coincides with t he morphism obtained b y adjunction from M k δ X ! Ω k X exp( − ξ ) − − − − − − → M k δ X ! Ω k X − → δ X ! Ω n X [ n ] ≃ δ X ! ω X . 132 CHAPTER 5. THE COMMUT A TIVE CASE Lemma 5.4.7. The morphism δ ∗ X δ X ∗ O X L ⊗ O X δ ! X δ X ! ω X − → δ ! X δ X ! ω X in Prop o- sition 5.1.5 (d) c oincides with Ω i X [ i ] ⊗ O X Ω j X [ j ] V − − → Ω i + j X [ i + j ] . Pr o of. W e deﬁne the morphism µ ij : P i ⊗ O X × X Q j − → Q i + j b y the same form ula as in (5.4.5). Then it comm utes with the diﬀerential. Indeed the pro of is similar to that of Lemma 5.4.4 except when i + j = d X + 1. In this case, µd  ( ω i ⊕ θ i +1 ) ⊗ ( ω j − 1 ⊕ θ j )  = 0 ⊕ ( i + j − d X − 1) ω i ∧ ω j − 1 = 0 . With this morphism µ : P • ⊗ O X × X Q • − → Q • , the f o llo wing diagram in the category of complexes is comm utative: P • ⊗ O X × X Q • µ / /   Q •   P • ⊗ O X × X δ X ! ω X / / δ X ! ω X . Th us w e hav e a comm utat ive dia g ram in D b ( O X ): H 0 ( δ ∗ X ) P • ⊗ O X H 0 ( δ ! X ) Q • / / ≀   H 0 ( δ ! X )( P • ⊗ O X × X Q • ) / /   H 0 ( δ ! X )( Q • )   δ ∗ X δ X ∗ O X L ⊗ O X δ ! X δ X ! ω X / / δ ! X ( δ X ∗ O X L ⊗ O X × X δ X ! ω X ) / / δ ! X δ X ! ω X . Q.E.D. Recall that in Corollary 4.2 .2, w e ha v e constructed a morphism H H ( A X ) ⊗ HH ( A X ) − → ω top X R . Let us describ e its imag e via the isomorphisms α X and β X . Consider the diagra m HH ( O X ) ⊗ HH ( O X ) λ   u ( ( P P P P P P P P P P P P P P HD ( O X ) ⊗ HD ( O X ) v / / ω top X R . (5.4.7) Here, u is the map giv en b y Corollary 4 .2 .2, λ is the isomorphism α X ⊗ β − 1 X and v is the comp osition M k Ω k X [ k ] ⊗ M k Ω k Y [ k ] − → M k Ω k Y [ k ] − → ω top X R , where the ﬁrst morphism is g iv en b y the w edge pro duct a nd the last one by the map Ω d X X [ d X ] − → ω top X R . Then diagram (5.4.7) comm utes. Chapter 6 Symplectic case and D -mo dules 6.1 Deformation quan tizati o n o n c o tangen t bundles Consider the case where X is an op en subs et of the cotangent bundle T ∗ M of a complex manifold M . W e denote b y π : T ∗ M − → M the pro jection. As usual, w e denote by D M the C -algebra of diﬀeren tial op erators o n M . This is a r ig h t and left Noetherian sheaf o f rings. The space T ∗ M is endow ed with the ﬁltered sheaf o f C -algebras b E T ∗ M of formal micro diﬀerential op erators of [54], and its subsheaf b E T ∗ M (0) of op erators of order ≤ 0. On T ∗ M , there is also a D Q-algebra, denoted b y c W T ∗ M (0) and constructed in [51] a s follo ws. Consider the complex line C endow ed with the co o r dinate t a nd denote b y ( t ; τ ) the asso ciated symplectic co ordinates on T ∗ C . Let T ∗ τ 6 =0 ( M × C ) b e the op en subse t of T ∗ ( M × C ) deﬁned b y τ 6 = 0 and consider the map ρ : T ∗ τ 6 =0 ( M × C ) − → T ∗ M , ( x, t ; ξ , τ ) 7→ ( x ; τ − 1 ξ ) . Denote b y b E T ∗ ( M × C ) , b t (0) the subalgebra o f b E T ∗ ( M × C ) (0) consisting of o p erators not dep ending on t , tha t is, comm uting with ∂ t . Setting ~ = ∂ − 1 t , the DQ- algebra c W X (0) is deﬁned as c W X (0) = ρ ∗ b E T ∗ ( M × C ) , b t (0) . One denotes by c W T ∗ M the lo calizatio n of c W T ∗ M (0), that is, c W T ∗ M = C ~ , lo c ⊗ C ~ c W T ∗ M (0). 133 134 CHAPTER 6. SYMPLECTIC CASE AND D -MODULES Remark 6.1.1. One shall b e aw are that b E T ∗ M and b E T ∗ M (0) are denoted b y b E M and b E M (0), resp ectiv ely , in [54]. Similarly , c W T ∗ M and c W T ∗ M (0) a re denoted b y c W M and c W M (0), resp ectiv ely , in [51]. There are natural morphisms of alg ebras π − 1 M D M ֒ → b E T ∗ M ֒ → c W T ∗ M . (6.1.1) Lemma 6.1.2. (a) The algebr a c W T ∗ M (0) i s faithful ly ﬂat ove r b E T ∗ M (0) . (b) The algebr a c W T ∗ M is faithful ly ﬂat o ver b E T ∗ M . (c) b E T ∗ M is ﬂat over π − 1 M D M . Pr o of. In the sequel, we set X = T ∗ M . F or an b E X (0)-mo dule M , w e set M W := c W X (0) ⊗ b E X (0) M , gr E ( M ) =  b E X (0) / b E X ( − 1)  L ⊗ b E X (0) M . Note that the analogue of Corollary 1.4.6 holds for b E X (0)-mo dules, that is, the functor gr E ab o ve is conserv ativ e on D b coh ( b E X (0)). W e ha v e gr ~ ( M W ) ≃ O X ⊗ O X (0) gr E ( M ) , (6.1.2) where O X (0) denotes the subsheaf of O X of sections homogeneous of degree 0 in the ﬁb er v ariable of the v ector bundle T ∗ M , and O X is faithfully ﬂat o v er O X (0). (a) (i) Let us ﬁrst pro v e the result outside of t he zero-section, that is, on T ∗ M \ T ∗ M M . Let us sho w that H j ( c W X (0) L ⊗ b E X (0) M ) = 0 for an y j < 0 (6.1.3) holds for any coheren t b E X (0)-mo dule M . First assume that M is torsion- free, i.e., b E X ( − 1) ⊗ b E X (0) M − → M is a monomo r phism. Since O X is ﬂat ov er O X (0), gr ~ ( c W X (0) L ⊗ b E X (0) M ) ≃ O X L ⊗ O X (0) gr E ( M ) has zero cohomologies in degree < 0. Hence Prop osition 1.4.5 implies (6.1.3). 6.1. DEF ORMA TION QUANTIZA TION O N COT ANGENT BUNDLES 13 5 No w assume that b E X ( − 1) M = 0. Then we hav e c W X (0) L ⊗ b E X (0) M ≃ c W X (0) L ⊗ b E X (0) b E X (0) L ⊗ b E X (0) O X (0) L ⊗ O X (0) M ≃ c W X (0) L ⊗ b E X (0) O X (0) L ⊗ O X (0) M ≃ O X L ⊗ O X (0) M , whic h implies (6 .1 .3). Since an y coherent b E X (0)-mo dule is a success ive extension of torsion- free b E X (0)-mo dules and ( b E X (0) / b E X ( − 1))-mo dules, we obtain (6 .1 .3) fo r any coheren t b E X (0)-mo dule. Consider a coheren t b E X (0)-mo dule M and a ssume that M W ≃ 0. Then gr ~ ( M W ) ≃ 0 and this implies that gr E ( M ) ≃ 0 in view of (6.1.2 ) since O X is faithfully ﬂat ov er O X (0). Since gr E is conserv ative, the result follows . (a) (ii) T o prov e the result in a neigh b orho o d of the zero section, w e use the classical tric k o f the dumm y v ariable. Let ( t ; τ ) denote a homogeneous symplectic co ordinate system on T ∗ C . Consider the functors α : Mod coh ( O M ) − → Mo d coh ( b E X × T ∗ C (0) | τ 6 =0 ) , M 7→ M ⊠ ( b E C (0) / b E C (0) · t ) , β : Mo d coh ( c W X | M (0)) − → Mo d coh ( c W X × T ∗ C (0) | τ 6 =0 ) , M 7→ M ⊠ ( c W T ∗ C (0) / c W T ∗ C (0) · t ) . These t w o functors α and β are exact and faithful. Then the result follo ws from (a) (i). (b) (i) Here again, we prov e the r esult ﬁrst on T ∗ M \ T ∗ M M . In this case, it follo ws from the isomorphism c W X ≃ c W X (0) ⊗ b E T ∗ C (0) b E T ∗ C . (b) (ii) The case o f t he zero-section is deduced from (b) (i) similarly as for (a). (c) is pro ve d for example in [37, Th. 7.25]. Q.E.D. Recall that fo r a coheren t D M -mo dule M , the supp ort of b E T ∗ M ⊗ π − 1 M D M π − 1 M M is called its characteristic v ariet y and denoted b y char( M ). It is a closed C × - conic complex analytic inv olutiv e subset of T ∗ M . 136 CHAPTER 6. SYMPLECTIC CASE AND D -MODULES No w assume that M is op en in some ﬁnite-dimensional C -v ector space. Denote b y ( x ) a linear co o r dina t e system o n M and by ( x ; u ) the asso ciated symplectic coor dina t e system on T ∗ M . Let f , g ∈ O X [[ ~ ] ]. In this case, the DQ-algebra c W X (0) is isomorphic to the star algebra ( O X [[ ~ ] ] , ⋆ ) where: f ⋆ g = X α ∈ N n ~ | α | α ! ( ∂ α u f )( ∂ α x g ) . (6.1.4) This pro duct is similar to the pro duct of the total sym b ols of diﬀerential op erators on M and indeed, the morphism of C -alg ebras π − 1 M D M − → c W X is giv en b y f ( x ) 7→ f ( x ) , ∂ x i 7→ ~ − 1 u i . Note that there also exists an analytic v ersion of b E T ∗ M and c W T ∗ M , ob- tained by using the C - subalgebra of ( O X [[ ~ ] ] , ⋆ ) consisting of sections f = P j ≥ 0 f j ~ j of O X [[ ~ ] ]( U ) ( U op en in T ∗ M ) satisfying: ( for an y compact subset K of U there exists a p o sitiv e con- stan t C K suc h that sup K | f j | ≤ C j K j ! for all j > 0. (6.1.5) They are the total sym b ols of the analytic (no mor e f o rmal) micro diﬀeren tial op erators of [54]. Remark 6.1.3. (i) Let X be a complex symplectic manifold. Then X is lo cally isomorphic t o an op en subset of a cotang en t bundle T ∗ M , for a com- plex manifold M (Darb oux’s theorem), and it is a w ell-kno wn fact that if A X is a DQ - algebra and the asso ciated P oisson structure is the symplectic structure of X , then A X is lo cally isomorphic to c W T ∗ M (0). (ii) On X , there is a canonical DQ- algebroid, still denoted b y c W X (0). It has b een constructed in [51], after [36] had ﬁrst tr eat ed the contact case. Clearly , any DQ-algebroid A is equiv alen t to c W X (0) ⊗ C ~ X P , where P is a n in v ertible C ~ X -algebroid. It fo llo ws that the DQ-alg ebroids on X are classiﬁed b y H 2 ( X ; ( C ~ X ) × ). See [50] for a detailed study . (iii) Using (4.1.11), w e get the isomorphism HH ( A X ) ≃ H H ( c W X (0)) . (6 .1 .6) 6.2 Ho c hsc hi l d homology of A Throughout this section, X denotes a complex manifold endo w ed with a DQ-algebroid A X suc h that the asso ciated Poiss on structure is symplectic. Hence, X is symplectic and w e denote by α X the symplectic 2-for m on X . 6.2. HOCHSCHILD HOMOLOGY OF A 137 W e set 2 n = d X , Z = X × X a and we denote b y dv the volume form on X given by d v = α n X /n !. Lemma 6.2.1. L et Λ b e a smo oth L agr angian submanifold of X and let L i ( i = 0 , 1 ) b e simple A X -mo dules along Λ . Then: (i) L 0 and L 1 ar e lo c al ly isomo rp hic, (ii) the natur al morphism C ~ − → H om A X ( L 0 , L 0 ) is an isomorphism. Note that the lemma ab ov e do es not hold if one remo v es the hypothesis that X is symplectic (see Example 2.5 .9). Pr o of. ( i) W e ma y assume that X = T ∗ M f or a complex manifo ld M , A X = c W T ∗ M (0). Cho ose a lo cal co ordinate system ( x 1 , . . . , x n ) on M , and denote b y ( x ; u ) the asso ciated co ordinates on X . W e shall iden tify the section u i of A X with the diﬀeren tial operat o r ~ ∂ i . W e ma y assume that Λ is the zero-section T ∗ M M and L 0 = O M [[ ~ ] ] ≃ A X / I 0 , where I 0 is t he left ideal generated b y ( ~ ∂ 1 , . . . , ~ ∂ n ). Since L 1 is simple, it lo cally admits a generator, sa y u . Denote b y I 1 the annihilator ideal of u in A X . Since I 1 / ~ I 1 is reduced, there exist sections ( P 1 , · · · , P n ) of A X suc h that { ~ ∂ 1 + ~ P 1 , . . . , ~ ∂ n + ~ P n } ⊂ I 1 . By identifying c W T ∗ M (0) with the sheaf of micro diﬀeren tial op erators of order ≤ 0 in the v ariable ( x 1 , . . . , x n , t ) not dep ending on t a nd ~ with ∂ − 1 t , a classical result o f [54] (see also [56 , Th 6.2 .1] for an exp osition) sho ws that there exists an inv ertible section P ∈ A X suc h that I 0 = I 1 P . Hence, L 1 ≃ L 0 . (ii) W e ma y a ssume L 0 = O M [[ ~ ] ]. Then H om A X ( O M [[ ~ ] ] , O M [[ ~ ] ]) is iso- morphic to the k ernel of the map u : O M [[ ~ ] ] − → ( O M [[ ~ ] ]) n , u = ( ~ ∂ 1 , . . . , ~ ∂ n ) . Q.E.D. Recall that the ob jects Ω A X and ω A X are deﬁned in § 2.5. Lemma 6.2.2. Ther e exi sts a lo c al system L of r ank one over C ~ X such that Ω A X ≃ L ⊗ C ~ X C X in Mo d( A X × X a ) . Pr o of. Bo th Ω A X and C X are simple A X × X a -mo dules along the diagonal ∆. By Lemma 6.2.1, L := H om A Z ( C X , Ω A X ) is a lo cal system of rank one o ve r C ~ and we hav e Ω A X ≃ L ⊗ C ~ X C X . Q.E.D. 138 CHAPTER 6. SYMPLECTIC CASE AND D -MODULES Note that this implies the isomorphisms D ′ A X × X a C X ≃ L ⊗− 1 ⊗ C X [ − d X ] . (6 .2.1) Hence w e obtain the chain of morphisms L − → L ⊗ R H om A Z ( C X , C X ) ≃ L ⊗ D ′ A X × X a C X L ⊗ A Z C X ≃ C X L ⊗ A Z C X [ − d X ] = HH ( A X ) [ − d X ] ≃ L ⊗− 1 ⊗ Ω A X L ⊗ A Z C X [ − d X ] − → L ⊗− 1 ⊗ Ω A X L ⊗ D A X C X [ − d X ] ≃ L ⊗− 1 . Therefore, we g et the morphism: L ∼ − → H − d X ( HH ( A X )) − → L ⊗− 1 . (6.2.2) Lemma 6.2.3. ( i) gr ~ ( L ) − → H om gr ~ ( A Z ) (gr ~ ( C X ) , gr ~ (Ω A X )) ≃ Ω X gives an isomorphism gr ~ ( L ) ∼ − → C X · dv . (ii) The morphism L ⊗ 2 − → C ~ X induc e d by (6.2.2) de c omp oses as L ⊗ 2 ϕ − → ~ 2 n C ~ X ֒ → C ~ X and ϕ is an is o morphism. (iii) The diagr am b elow c omm utes: gr ~ ( L ⊗ 2 ) ∼ / / ≀   gr ~ ( ~ 2 n C ~ X ) gr ~ ( C ~ X ) ∼ ~ 2 n o o ≀   (gr ~ ( L )) ⊗ 2 ∼ / / C ⊗ 2 X ∼ C X Pr o of. The question b eing lo cal, we may assume to b e giv en a lo cal co or- dinate system x = ( x 1 , . . . , x 2 n ) on X and a scalar-v alued no n-degenerate sk ew-symmetric matr ix B = ( b ij ) 1 ≤ i,j ≤ 2 n suc h that the symplectic form α X is giv en by α X = X i,j b ij dx i ∧ dx j . W e set A = ( a ij ) 1 ≤ i,j ≤ 2 n = B − 1 . W e may assume that A X = ( O X [[ ~ ] ] , ⋆ ) is a star-alg ebra with a star pro duct f ⋆ g =  exp  P ij ~ a ij 2 ∂ 2 ∂ x i ∂ x ′ j  f ( x ) g ( x ′ )  x ′ = x . 6.2. HOCHSCHILD HOMOLOGY OF A 139 Set δ i = 2 n X j =1 a ij ∂ x j ( i = 1 , . . . , 2 n ) . Then, the C ~ -linear morphisms from O X [[ ~ ] ] to D X [[ ~ ] ] Φ l : f 7→ f ⋆, Φ r : f 7→ ⋆ f (6.2.3) are giv en by Φ l ( x i ) = x i + ~ 2 δ i , Φ r ( x i ) = x i − ~ 2 δ i . These mo r phisms deﬁne the morphism Φ : A X ⊗ A X a − → D X [[ ~ ] ] (6.2.4) x i 7→ x i + ~ 2 δ i , y i 7→ x i − ~ 2 δ i . where w e denote b y y = ( y 1 , . . . , y 2 n ) a cop y of the lo cal co ordinate system on X a . W e iden tify Ω A X with the ( D X [[ ~ ] ]) op -mo dule Ω X [[ ~ ] ]. Then, regarding Ω X [[ ~ ] ] as an A Z -mo dule through A Z | X − → A op Z | X − → ( D X [[ ~ ] ]) op , w e hav e x i ( a dv ) = ( a dv )Φ r ( x i ) = ( a d v )( x i − ~ 2 δ i ) = (( x i + ~ 2 δ i ) a ) dv and similarly y i ( adv ) = (( x i − ~ 2 δ i ) a ) dv . Hence, a 7→ a d v giv es an A Z -linear isomorphism C X ≃ O X [[ ~ ] ] ∼ − → Ω X [[ ~ ] ] ≃ Ω A X . Hence it giv es an isomorphism L := H om A Z ( C X , Ω A X ) ≃ H om A Z ( C X , C X ) ≃ C ~ X , and the induced morphism gr ~ ( L ) − → H om gr ~ ( A Z ) (gr ~ ( C X ) , gr ~ (Ω A X )) ≃ Ω X giv es an isomor phism gr ~ ( L ) ∼ − → C X dv . Hence we obta in (i). F or a sheaf of C ~ -mo dules F , we set F ( p ) =  p ^ ( C ~ X ) 2 n  ⊗ C ~ X F . 140 CHAPTER 6. SYMPLECTIC CASE AND D -MODULES Let ( e 1 , . . . , e 2 n ) b e the basis of ( C ~ ) 2 n . Consider the Ko szul complex K • ( A Z ; b ) where b = ( b 1 , . . . , b 2 n ), b i = ( x i − y i ) is the righ t m ultiplication b y ( x i − y i ) on A Z : K • ( A Z ; b ) := 0 − → A (0) Z b − → · · · b − → A (2 n ) Z − → 0 , b = X i • ∧ b i e i : K p ( A Z ; b ) − → K p +1 ( A Z ; b ) . On the o ther hand, consider the Koszul complex K • ( D X [[ ~ ] ]; δ ) where δ = ( δ 1 , . . . , δ 2 n ): K • ( D X [[ ~ ] ]; δ ) := 0 − → ( D X [[ ~ ] ]) (0) δ − → · · · δ − → ( D X [[ ~ ] ]) (2 n ) − → 0 , δ = ( δ 1 , . . . , δ 2 n ) . There is a quasi-isomorphism K • ( A Z ; b ) q is − → C X [ − 2 n ] in t he category of complexes in Mo d( A Z ). Set δ = ( δ l 1 − δ r 1 , . . . , δ l 2 n − δ r 2 n ). Then the morphism Φ in (6.2.4) sends ( x i − y i ) to ~ δ i . Consider the Koszul complex K • ( D X [[ ~ ] ]; δ ): K • ( D X [[ ~ ] ]; δ ) := 0 − → ( D X [[ ~ ] ]) (0) δ − → · · · δ − → ( D X [[ ~ ] ]) (2 n ) − → 0 , δ = ( δ 1 , . . . , δ 2 n ) . There is a quasi-isomorphism K • ( D X [[ ~ ] ]; δ ) q is − → O X [[ ~ ] ] [ − 2 n ]. Therefore w e get a comm utative diagram in Mo d( A Z ): 0 / / A (0) Z b / / ~ 2 n Φ   · · · / / A (2 n − 1) Z b / / ~ Φ   A (2 n ) Z ~ 0 Φ   / / 0 0 / / ( D X [[ ~ ] ]) (0) δ / / · · · / / ( D X [[ ~ ] ]) (2 n − 1) δ / / ( D X [[ ~ ] ]) (2 n ) / / 0 . The o b j ect Ω A X L ⊗ A Z C X is obtained by applying the functor Ω A X ⊗ A Z • to the ro w on the top a nd the ob j ect Ω A X L ⊗ D A X C X is obtained b y applying the functor Ω A X ⊗ D A X • to the row on the b o ttom. By identifyin g Ω A X with Ω X [[ ~ ] ], the morphism Ω A X L ⊗ A Z C X [ − d X ] − → Ω A X L ⊗ D A X C X [ − d X ] is describ ed b y the morphism o f complexes: 0 / / Ω 0 X [[ ~ ] ] ~ d / / ~ 2 n   · · · / / Ω 2 n − 1 X [[ ~ ] ] ~ d / / ~   Ω 2 n X [[ ~ ] ] ~ 0   / / 0 0 / / Ω 0 X [[ ~ ] ] d / / · · · / / Ω 2 n − 1 X [[ ~ ] ] d / / Ω 2 n X [[ ~ ] ] / / 0 . (6.2.5) 6.2. HOCHSCHILD HOMOLOGY OF A 141 Here d denotes the usual ex terior deriv ativ e. Therefore, w e ﬁnd the comm utative dia g ram with exact ro ws: L ⊗ 2 ≀   0 / / C ~ X / / ~ 2 n   Ω 0 X [[ ~ ] ] ~ d / / ~ 2 n   Ω 1 X [[ ~ ] ] ~ 2 n − 1   0 / / C ~ X / / Ω 0 X [[ ~ ] ] d / / Ω 1 X [[ ~ ] ] in whic h the morphism L ⊗ 2 − → C ~ X corresp onds to the morphism L [ d X ] − → L ⊗− 1 ⊗ Ω A X ⊗ A Z C X . This completes the pro of. Q.E.D. Theorem 6.2.4. Assume that X is symple c tic. (i) L et L b e the lo c al system given by L emma 6.2.2. T hen ther e is a c anoni- c al C ~ -line ar isomorphism L ∼ − → ~ d X / 2 C ~ X , henc e, a c anonic al A Z -line ar isomorphism Ω A X ∼ − → ~ d X / 2 C ~ X ⊗ C ~ X C X . (6.2.6) (ii) The iso m orphism (6.2.6) to gether with (6.2.2) induc e c anonic al mor- phisms ~ d X / 2 C ~ X [ d X ] ι X − → HH ( A X ) τ X − → ~ − d X / 2 C ~ X [ d X ] (6.2.7) and the c omp osition τ X ◦ ι X is the c anonic al morphism ~ d X / 2 C ~ X [ d X ] − → ~ − d X / 2 C ~ X [ d X ] . (iii) H j ( HH ( A X )) ≃ 0 unless − d X ≤ j ≤ 0 and the morphism ι X induc es an isomorphism ι X : ~ d X / 2 C ~ X ∼ − → H − d X ( HH ( A X )) . (6.2.8) In p articular, ther e is a c anonic al non - z e r o se ction in H − d X ( X ; H H ( A X )) . Pr o of. ( i) By Lemm a 6.2.3, w e ha v e an isomorphism ( ~ − d X / 2 L ) ⊗ 2 ≃ C ~ X together with a compatible isomorphism gr ~ ( ~ − d X / 2 L ) ≃ C X . This implies ~ − d X / 2 L ≃ C ~ X since the only inv ertible elemen t a ∈ C ~ satisfying a 2 = 1, σ 0 ( a ) = 1 is a = 1. 142 CHAPTER 6. SYMPLECTIC CASE AND D -MODULES (ii)–(iii) D enote by (Ω • X [[ ~ ] ] , ~ d ) and (Ω • X [[ ~ ] ] , d ) the complexes giv en b y the top ro w and the b ottom ro w of (6.2.5), resp ectiv ely . The mor phism ι X is represen ted by L [ d X ] − → L ⊗− 1 ⊗ (Ω • X [[ ~ ] ] , ~ d )[ d X ] and the mor phism τ X is the comp osition L ⊗− 1 ⊗ (Ω • X [[ ~ ] ] , ~ d )[ d X ] − → L ⊗− 1 ⊗ (Ω • X [[ ~ ] ] , d )[ d X ] ∼ − → L ⊗− 1 [ d X ] . Q.E.D. Applying Theorem 6.2.4 together with Corolla r y 3 .3 .4, we obta in: Corollary 6.2.5. L et X b e a c omp act c omplex symple ctic manifold . T hen D b gd ( A lo c X ) is a Cala b i-Y au triangulate d c ate g ory of dimens i o n d X over C ~ , lo c . Remark 6.2.6. The statemen t in Theorem 9.2 (ii) of [42] is not correct. If Y is a compact complex contact manifold of dimension d Y , then the dimension of the Calabi-Y au category asso ciated to it in lo c. cit. is d Y , not d Y − 1. 6.3 Euler class es of A lo c -mo du l es Theorem 6.3.1. The c omplex HH ( A lo c X ) is c onc entr ate d in de gr e e − d X and the morphisms ι X and τ X in Th e or em 6.2.4 in duc e isomorphisms C ~ , lo c X [ d X ] ∼ − → ι X HH ( A lo c X ) ∼ − → τ X C ~ , lo c X [ d X ] . (6 .3 .1) Pr o of. This fo llo ws from t he fact that (Ω • X [[ ~ ] ] , ~ d ) − → (Ω • X [[ ~ ] ] , d ) b ecomes a quasi-isomorphism af ter applying the functor ( • ) lo c = C ~ , lo c ⊗ C ~ ( • ). Q.E.D. Deﬁnition 6.3.2. Let M ∈ D b coh ( A lo c X ). W e set eu X ( M ) = τ X (hh X ( M )) ∈ H d X Supp( M ) ( X ; C ~ , lo c X ) (6.3.2) and call eu X ( M ) t he Euler class of M . Remark 6.3.3. (i) The existence of a canonical section in H − d X ( X ; H H ( A lo c X )) is we ll kno wn whe n X = T ∗ M is a cotangen t bundle, see in particular [12, 25, 62]. It is in tensiv ely used in [1 1 ] where these authors call it the “trace density map”. (ii) The Ho ch sc hild a nd cyclic homology of an algebroid stac k ha v e b een de- ﬁned in [9] where the Chern c haracter of a p erfect complex is constructed in the negative cyclic homology . It give s in particular a n alternativ e con- struction of the Hochsc hild class of a coheren t DQ-mo dule, but it is not clear whether the t wo constructions give t he same class. 6.3. EULER CLASSES OF A lo c -MODULES 143 Consider the diagram p 13 ! ( p − 1 12 HH ( A lo c X 1 × X a 2 ) ⊗ p − 1 23 HH ( A lo c X 2 × X a 3 )) ⋆ / / τ 12 a ⊗ τ 23 a   HH ( A lo c X 1 × X a 3 ) τ 13 a   p 13 ! ( p − 1 12 C ~ , lo c X 12 [ d 12 ] ⊗ p − 1 23 C ~ , lo c X 23 [ d 23 ]) R 2 ( ·∪· ) / / C ~ , lo c X 13 [ d 13 ] . (6.3.3) Here, the horizontal arro w in the b ottom denoted b y R 2 ( • ∪ • ) is obtained b y taking the cup pr o duct and in tegrating on X 2 (P oincar ´ e dualit y), using the fact that the manifold X 2 has real dimension 2 d 2 and is oriented . The arro w in the top denoted b y ⋆ is obtained by Prop osition 4.2.1. Prop osition 6.3.4. Diag ram 6.3.3 c ommutes. Pr o of. Since X 1 and X 3 pla y the role of para meter spaces, w e ma y a ssume that X 1 = X 3 = { pt } . W e set X 2 = X and denote b y a X the pro jection X − → { pt } . W e are reduce to pro v e the commu ta t ivity of the diagram b elow : a X ! ( HH ( A lo c X ) ⊗ HH ( A lo c X )) τ ⊗ τ   ⋆ * * U U U U U U U U U U U U U U U U U U U U U a X ! ( C ~ , lo c X [ d X ] ⊗ C ~ , lo c X [ d X ]) R X ( ·∪· ) / / C ~ , lo c . (6.3.4) This will follo w b y applying the functor a X ! to Diagram 6.3.5 b elo w. Q.E.D. Lemma 6.3.5. The d i a gr am b elow c ommutes. HH ( A lo c X ) ⊗ HH ( A lo c X ) ⋆ ) ) S S S S S S S S S S S S S S τ ⊗ τ   C ~ , lo c X [ d X ] ⊗ C ~ , lo c X [ d X ] / / C ~ , lo c X [2 d X ] . (6.3.5) Pr o of. The morphism L ⊗ L [2 d X ] ≃ C ~ , lo c X [ d X ] ⊗ C ~ , lo c X [ d X ] − → C ~ , lo c X [2 d X ] is giv en by L ⊗ L [2 d X ] − → L [ d X ] ⊗ R H om A Z ( C X , Ω A X )[ d X ] ≃ L ⊗ D ′ A C X [ d X ] ⊗ A Z ω A X ≃ C X a ⊗ A Z ω A X − → C ~ X [2 d X ] . On the other hand, L ⊗ L [2 d X ] − → HH ( A X ) ⊗ HH ( A X ) − → C ~ X [2 d X ] is giv en b y L ⊗ L [2 d X ] − → R H om A Z a (D ′ A C X , C X a ) ⊗ R H om A Z ( C X , ω A X ) ≃ R H om A Z a (D ′ A C X , C X a ) ⊗  D ′ A ( C X ) ⊗ A Z ω A X  − → C X a ⊗ A Z ω A X − → C ~ X [2 d X ] . 144 CHAPTER 6. SYMPLECTIC CASE AND D -MODULES These t w o morphisms giv e the same morphism from L ⊗ L [2 d X ] to C ~ X [2 d X ]. Q.E.D. Corollary 6.3.6. L et K i ∈ D b coh ( A lo c X i × X a i +1 ) ( i = 1 , 2) . Assume that the pr o- je ction p 13 deﬁne d on X 1 × X 2 × X 3 is p r op er on p − 1 12 Supp( K 1 ) ∩ p − 1 23 Supp( K 2 ) . Then eu X 13 a ( K 1 ◦ 2 K 2 ) = Z X 2 eu X 12 a ( K 1 ) ∪ eu X 23 a ( K 2 ) . (6.3.6) Remark 6.3.7. Consider an ob ject M ∈ D b coh ( A lo c X ). Then, according to Deﬁnition 6.3.2, it s Euler class is well-deﬁne d in the de R ha m cohomology of X with v alues in C ~ , lo c . No w assume that M is generated b y M 0 ∈ D b coh ( A X ) and consider g r ~ ( M 0 ). Assume for simplicit y that gr ~ ( A X ) = O X (the general case can b e treated with suitable mo diﬁcations). Then gr ~ ( M 0 ) ∈ D b coh ( O X ) and w e ma y consider its Chern class in de Rham cohomology . A natural question is to compare these tw o classes. A precise conjecture had b een made in the case of D -mo dules b y one of the authors (PS) and J-P . Sc hneiders in [57] and prov ed b y P . Bressler, R. Nest and B. Tsygan in [11]. These authors, together with A. Gorokhovs ky , recen tly treated the general case of DQ-algebroids in the sym plectic setting in [10]. The form ula they obtain mak es use of a cohomolog y class naturally asso ciated to t he deforma t io n A X . 6.4 Ho c hsc hi l d classes o f D -mo du les W e shall apply the preceding result to the study of the Euler class of D - mo dules. Recall after [37] that a cohere nt D M -mo dule M is go o d if, for an y op en relativ ely compact set U ⊂ M , t here exists a coheren t sub- O U -mo dule F of M | U whic h g enerates it on U as a D M -mo dule. One denotes b y D b gd ( D M ) the full sub-triangulated category of D b coh ( D M ) consisting of o b j ects with go o d cohomology . F rom now on, we set X = T ∗ M . W e intro duce the functor ( • ) W : Mo d( D M ) − → Mo d( c W X ) (6.4.1 ) M 7→ c W X ⊗ π − 1 M D M π − 1 M M . The next result sho ws that one can, in some sense, reduce the study of D -mo dules to that of c W X -mo dules. 6.4. HOCHSCHILD CLASSES OF D -MODULES 145 Prop osition 6.4.1. The functor M 7→ M W | T ∗ M M is exact an d faithful. Pr o of. The morphism M − → ( b E T ∗ M ⊗ π − 1 M D M π − 1 M M ) | T ∗ M M . is an isomor phism, and hence the result is a particular case of Lemma 6.1.2. Q.E.D. It follows that ( • ) W sends D b coh ( D M ) to D b coh ( c W X ) and D b gd ( D M ) to D b gd ( c W X ). Deﬁnition 6.4.2. Let M ∈ D b gd ( D M ). W e set hh gr X ( M ) = hh gr X ( M W ) ∈ HH c har( M ) ( O X ) . (6.4.2) F or Λ a closed subset of T ∗ M , w e denote by K gd , Λ ( D M ) the G rothendiec k group of the full ab elian sub category o f Mo d gd ( D M ) consisting of D -mo dules whose c haracteristic is contained in Λ. Let V b e an op en relativ ely compact subset of M . By slightly mo difying the pro of of Prop osition 3 .4 .3, we get morphisms of gro ups K gd , Λ ( D M ) − → K coh , Λ ( O π − 1 V ) . (6.4.3) Let M i ( i = 1 , 2 , 3 ) b e three complex manifo lds and set X i = T ∗ M i . Denote b y q ij the ij -t h pro jection deﬁned on M 1 × M 2 × M 3 and b y p ij the ij -th pro jection deﬁned on X 1 × X 2 × X 3 (1 ≤ i < j ≤ 3). W e set, as for DQ-algebras, D M a := ( D M ) op and w e write for sh or t M ij or M ij a instead of M i × M j or M i × M a j and similarly with X ij . W e also write D ij instead of D M ij and similarly with ij a , etc. F or example, D 12 a = O M 12 ⊗ ( O M 1 ⊠ O M 2 ) ( D M 1 ⊠ ( D M 2 ) op ) . Then D 1 ma y b e rega rded as a D 11 a -mo dule supp orted on the diagonal of X 1 × X 1 a . Let K i ∈ D b ( D ij a ) ( i = 1 , 2, j = i + 1). Set K 1 ◦ M 2 K 2 := R q 13 a !  D 2 ⊗ D 2 a 2 D 12 a 23 a L ⊗ D 12 a ⊠ D 23 a ( K 1 ⊠ K 2 )  . Theorem 6.4.3. L et Λ i b e a c lose d subse t of X i × X i +1 ( i = 1 , 2) a n d assume that the pr oje ction p 13 deﬁne d on X 1 × X 2 × X 3 is pr op er on p − 1 12 Λ 1 ∩ p − 1 23 Λ 2 . Set Λ = Λ 1 ◦ Λ 2 . L et K i ∈ D b gd ( D ij a ) ( i = 1 , 2 , j = i + 1) with ch ar ( K i ) ⊂ Λ i ( i = 1 , 2 ) . Then K 1 ◦ M 2 K 2 ∈ D b gd ( D 13 a ) , c har( K 1 ◦ M 2 K 2 ) ⊂ Λ and ( K 1 ◦ M 2 K 2 ) W ∼ − → K W 1 ◦ X 2 K W 2 . (6.4.4) 146 CHAPTER 6. SYMPLECTIC CASE AND D -MODULES The pro of is straigh tforward and is left to the r eader. By using Dia- gram 4.4.7, w e g et: Theorem 6.4.4. In the situation of T h e or em 6.4.3, let V ij b e a r elatively c omp act op en subset of M i × M j ( i = 1 , 2 , j = i + 1) a n d assume that π − 1 V 12 a × M 2 π − 1 V 23 a c ontains (Λ 1 × X 2 Λ 2 ) ∩ q − 1 13 a π − 1 V 13 a . Then the diag r am b elow c ommutes D b gd , Λ 1 ( D 12 a ) × D b gd , Λ 2 ( D 23 a ) ◦ / / gr ~   D b gd , Λ ( D 13 a ) gr ~   K coh , Λ 1 ( O π − 1 V 12 a ) × K coh , Λ 2 ( O π − 1 V 23 a ) ◦ / / hh × hh   K coh , Λ ( O π − 1 V 13 a ) hh   HH Λ 1 ( O π − 1 V 12 a ) × HH Λ 2 ( O π − 1 V 23 a ) ◦ / / HH Λ ( O π − 1 V 13 a ) . In p articular hh gr π − 1 V 13 a ( K 1 ◦ 2 K 2 ) = hh gr π − 1 V 12 a ( K 1 ) ◦ hh gr π − 1 V 23 a ( K 2 ) (6.4.5) in HH Λ ( O π − 1 V 13 a ) . As a particular case, and using Corollary 5.3.5, we recov er a theorem of Laumon [4 7] in t he a nalytic framew ork. 6.5 Euler class es of D -mo dules W e kee p the notations o f § 6.4 and we set X = T ∗ M . One deﬁnes t he Ho c hsc hild homology HH ( b E X ) of b E X and the Ho c hsc hild class hh X ( M ) of a coheren t b E X -mo dule M similarly as for HH ( A X ). In the sequel, w e identify a coheren t D M -mo dule M with b E X ⊗ π − 1 D M π − 1 M . In particular, w e deﬁne by this w a y the Ho chs child class hh X ( M ) of a coherent D -mo dule M . Hence hh X ( M ) ∈ H d X c har( M ) ( X ; H H ( b E X )) . (6.5.1) Lemma 6.5.1. Ther e is a natur al isomorphism HH ( b E X ) ∼ − → C X [ d X ] (6.5.2) which makes the diagr am b elow c ommutative: HH ( b E X )   ∼ / / C X [ d X ]   HH ( c W X ) ∼ τ / / C ~ , lo c X [ d X ] . 6.5. EULER CLASSES OF D -MODULES 147 Sketch o f pr o of. W e tak e co ordinates ( x 1 , . . . , x n , u 1 , . . . u n ), and set f O X := Q k ≤ 0 ~ − k O X ( k ), where O X ( k ) is the sheaf of holo mor phic functions on X homogeneous of degree k with resp ect to the v ariables ( u 1 , . . . , u n ). Then f O X is isomorphic to b E X (0) as a sheaf. Moreo v er, HH ( b E X ) is represen ted b y t he Koszul complex of ∂ /∂ x i , ~ ∂ /∂ u i ∈ E nd ( f O X ) ( i = 1 , . . . , n ). On the o ther hand, as we ha ve seen, HH ( c W X ) is represen ted by the Koszul complex of ~ ∂ /∂ x i , ~ ∂ /∂ u i ∈ E nd ( O X (( ~ ))) ( i = 1 , . . . , n ). Hence w e ha v e a comm utativ e diagram 0 / / f O X / / ~ − n   · · · / / f O X 2 n / /   f O X ~ 0   / / 0 0 / / O X (( ~ )) / / · · · / / O X (( ~ )) 2 n / / O X (( ~ )) / / 0 , in whic h the top ro w represen ts HH ( b E X ) and the b ottom ro w represen ts HH ( c W X ). Q.E.D. Deﬁnition 6.5.2. Let M ∈ D b coh ( b E X ). W e denote by eu X ( M ) t he image of hh X ( M ) in H d X c har( M ) ( X ; C X ) by the morphism in (6.5.2) and call it the Euler class of M . The next result immediately fo llows from L emma 6.5.1. Prop osition 6.5.3. F or M ∈ D b coh ( D M ) , eu X ( M W ) is the image of eu X ( M ) by the natur al map H d X c har( M ) ( X ; C X ) − → H d X c har ( M ) ( X ; C ~ , lo c X ) . Applying Theorem 4.3.5, we get: Theorem 6.5.4. In the situation of The or em 6.4.3, one has: eu 13 a ( K 1 ◦ 2 K 2 ) = eu 12 a ( K 1 ) ◦ eu 23 a ( K 2 ) (6.5.3) in H d 1 + d 3 Λ 1 ◦ Λ 2 ( X 13 ; C X 13 ) . This f orm ula is equiv alen t to the results of [57] on the functoria lity of the Euler class o f D -mo dules. No t e tha t the results of lo c. cit. also deal with constructible shea v es. 148 CHAPTER 6. SYMPLECTIC CASE AND D -MODULES Chapter 7 Holonomic DQ -mo dules The aim of this chapter is to study holonomic DQ - mo dules on symplec- tic manifolds. More precisely , we will prov e that, if L and M a re tw o holonomic A lo c X -mo dules on a s ymplectic ma nif o ld X , then the complex R H om A loc X ( M , L ) is p erv erse (hence, in part icular, C -constructible) ov er the ﬁeld C ~ , lo c . It follow s from the preceding results in Chapter 6 that if the in tersection of the supp orts of M and L is compact, t hen the Euler-P oincar ´ e index of this complex is giv en b y the in tegral R X eu X ( M ) · eu X ( L ). W e sho w here that the Euler class of a holonomic mo dule is a Lagrang ia n cycle, whic h mak es its calculation easy . If moreo ve r L and M a r e simple holonomic mo dules supp orted o n smo oth Lagrangian submanifolds Λ 0 and Λ 1 , then the microsupp ort of the complex R H om A loc X ( M , L ) is contained in the normal cone C(Λ 0 , Λ 1 ). This last result w as ﬁrst obtained in [42] in the analytic framew ork, that is, using W X -mo dules, not c W X -mo dules, whic h made the pro o f s m uch more intricate. Finally we prov e that, in some sens e, the complex R H om A loc X ( M , L ) is in v arian t by Hamiltonian sy mplectomorphism. 7.1 A -mo dules along a Lagrangian submani- fold Let X b e a complex symplectic manifold endow ed with a DQ-algebroid A X . The algebra A Λ /X Let Λ b e a smo oth La grangian submanifold o f X and let L b e a coheren t A X -mo dule simple along Λ. 149 150 CHAPTER 7. HOLONOMIC DQ - MODULES Lo cally , X is isomor phic a s a symplectic manifold to T ∗ Λ, t he cotangent bundle t o Λ. W e set for short O ~ Λ := O Λ [[ ~ ] ] , O ~ , lo c Λ := O Λ (( ~ ) ) . There are lo cal isomorphisms A X ≃ c W X (0) , L ≃ O ~ Λ . Then E nd C ~ ( L ) ≃ E nd C ~ ( O ~ Λ ) (see Lemma 2 .1 .12) a nd the subalgebroid of E n d C ~ ( L ) corr esp onding to t he subring D Λ [[ ~ ]] of E nd C ~ ( O ~ Λ ) is w ell-deﬁned. W e denote it b y D L . Lemma 7.1.1. ( i) D L is e quiva lent to D Λ [[ ~ ]] as a C ~ -algebr oid. (ii) The C ~ -algebr a D L satisﬁes (1.2.2) and (1.3.1) . In p articular, it is right and left No etherian. Pr o of. ( i) follow s by similar a r g umen ts as in Prop osition 2.5 .2 (ii). (ii) f o llo ws from Example 1.3 .1. Q.E.D. The functor A X | Λ − → E nd C ~ ( L ) fa ctorizes as A X | Λ − → D L , (7.1.1) and setting D lo c L := ( D L ) lo c , this functor induces a functor A lo c X | Λ − → D lo c L . (7.1.2) W e denote b y I Λ ⊂ O X the deﬁning ideal of Λ. Let I b e the k ernel of the comp osition ~ − 1 A X ~ − − → A X σ − − → O X − → O Λ . Then we hav e I / A X ≃ I Λ . Deﬁnition 7.1.2. W e denote b y A Λ /X the C ~ -subalgebroid of A lo c X generated b y I . Note that the a lg ebra A Λ /X is the analogue in the framew ork of D Q- algebras of the algebra E Λ constructed in [38]. The ideal ~ I is contained in A X , hence acts on L and one sees easily that ~ I sends L to ~ L . Hence, I acts on L and deﬁnes a functor A Λ /X − → D L . W e thus hav e t he functors o f alg ebroids A X | Λ / / $ $ I I I I I I I I I I A Λ /X | Λ   / / A lo c X | Λ   D L / / D lo c L . In particular, L is naturally an A Λ /X -mo dule. 7.1. A -MODULES ALONG A LA GR ANGIAN SUBMANIF OLD 151 Lemma 7.1.3. (i) I k = A Λ /X ∩ ~ − k A X for any k ≥ 0 , (ii) I k / I k − 1 ≃ I k Λ for k > 0 , (iii) A Λ /X is a right and left No etherian algebr oi d , (iv) gr ~ ( A Λ /X ) | Λ ∼ − → gr ~ D L ≃ D Λ , (v) ( A Λ /X ) lo c ≃ A lo c X and A lo c X is ﬂat over A Λ /X . Pr o of. Since the question is lo cal, w e may assume that X = T ∗ C n with co ordinates ( x, u ), Λ = { u = 0 } and A X is the star-algebra as in (6.1.4). Set A ′ := { X k f k ( x, u ) ~ k ∈ A lo c X ; f k ( x, u ) ∈ I − k Λ for k < 0 } . Then w e can ch eck that A ′ is a subalgebra o f A lo c X and it contains I . Hence it con tains A Λ /X . It is easy to see that the image of I k − → ~ − k A X / ~ − k +1 A X con tains ~ − k I k Λ . On the o t her ha nd, the imag e o f A ′ ∩ ~ − k A X − → ~ − k A X / ~ − k +1 A X coincides with ~ − k I k Λ . Hence, A Λ /X ∩ ~ − k A X and A ′ ∩ ~ − k A X ha v e the same image ~ − k I k Λ in ~ − k A X / ~ − k +1 A X . W e conclude that A Λ /X = A ′ and A Λ /X ∩ ~ − k A X ⊂ I k + ~ − k +1 A X . Hence, an induction o n k show s (i). (ii) is now ob vious. (iii) Considering the ﬁltration { A Λ /X ∩ ~ − k A X } k ≥ 0 of A Λ /X , the result follows b y [3 7, Theorem A.32]. (iv) is ob vious. (v) follow s fro m A X ⊂ A Λ /X ⊂ A lo c X . Q.E.D. By this lemma, for a coheren t A Λ /X -mo dule N , w e ma y regard gr ~ ( N ) as an ob ject of D b coh ( D Λ ). Recall that D b hol ( D Λ ) denotes the full triangulated category of D b coh ( D Λ ) consisting of ob jects with ho lo nomic coho mo lo gy . Lemma 7.1.4. The algebr oid D L is ﬂat over A Λ /X and D lo c L is ﬂat over A lo c X . Pr o of. It is enough to prov e the ﬁrst statemen t. Let us sho w that H j ( D L L ⊗ A Λ /X M ) ≃ 0 f o r any coheren t A Λ /X -mo dule M and any j < 0 . (i) Assume that M has no ~ -torsion. Using L emma 7.1.3 (iv), w e hav e for j < 0, H j gr ~ ( D L L ⊗ A Λ /X M ) ≃ H j gr ~ M ≃ 0, and henc e H j ( D L L ⊗ A Λ /X M ) ≃ 0 by Prop osition 1.4.5. 152 CHAPTER 7. HOLONOMIC DQ - MODULES (ii) Assume that ~ M = 0. Then D L L ⊗ A Λ /X M ≃ D L L ⊗ A Λ /X gr ~ A Λ /X L ⊗ gr ~ A Λ /X M ≃ gr ~ D L L ⊗ gr ~ A Λ /X M ≃ M . (iii) In the general case, set n N := Ker( ~ n : M − → M ) and M tor := S n n N . Note t hat this union is lo cally stationary . Deﬁning M tf b y t he exact se- quence, 0 − → M tor − → M − → M tf − → 0 , this mo dule has no ~ - torsion. It is th us enough to pro ve the r esult fo r the n N ’s a nd this follo ws fr om (ii) b y induction on n , using the exact se quence 0 − → n N − → n +1 N − → n +1 N / n N − → 0 . Q.E.D. Deﬁnition 7.1.5. An ob ject N of D b coh ( A Λ /X ) is holonomic if gr ~ ( N ) is Lagrangian in T ∗ Λ, that is, if gr ~ ( N ) b elongs t o D b hol ( D Λ ). Note that this condition is equiv alent to sa ying that H i ( N ) / ~ H i ( N ) and Ker( ~ : H i ( N ) − → H i ( N )) are holonomic D Λ -mo dules for any i (see Lemma 1.4.2). Microsupp or t and constructible shea ves Let us recall some notions and res ults of [39]. Let M b e a r e al analytic ma nif o ld and K a No etherian comm utative ring of ﬁnite global dimension. F or F ∈ D b ( K M ), w e denote by SS( F ) its microsupp o rt, a closed R + -conic ( i.e., inv ariant by the R + -action on T ∗ M ) subset of T ∗ M . Recall that this set is in v olutive (one also sa ys c o-isotr opic ), see [39 , Def. 6.5.1]. An ob ject F of D b ( K M ) is we akly R -c on structible if there exists a subana- lytic stratiﬁcation M = F α ∈ A M α suc h that H j ( F ) | M α is lo cally constan t for all j ∈ Z and all α ∈ A . The o b j ect F is R -c onstructible if moreo ver H j ( F ) x is ﬁnitely g enerated for a ll x ∈ M and all j ∈ Z . One denotes by D b R c ( K M ) the full sub category of D b ( K M ) consisting of R -constructible ob jects. Recall that the dualit y functor D ′ X ( • ) (see (1.1.1)) is an anti-auto-equiv a lence o f the category D b R c ( K M ). If M is complex ana lytic, one deﬁnes similarly the notions of (we akly) C -constructible sheaf, replacing “subanalytic” with “ complex analytic”. W e denote by D b w C c ( K M ) the full sub category of D b ( K M ) consisting of w eakly- C -constructible ob jects and b y D b C c ( K M ) the full sub category consisting of 7.1. A -MODULES ALONG A LA GR ANGIAN SUBMANIF OLD 153 C -constructible ob jects. Also recall ([39]) that F ∈ D b ( K M ) is weakly - C - constructible if and only if its microsupp ort is a closed C × -conic ( i.e., in- v arian t b y the C × -action on T ∗ M )) complex analytic Lagrangian subset of T ∗ M or, equiv alen tly , if it is con tained in a closed C × -conic complex analytic isotropic subset of T ∗ M . Prop osition 7.1.6. L et F ∈ D b ( Z M [ ~ ]) and assume that F is c ohomolo gi- c al ly c omplete. Then SS( F ) = SS(gr ~ ( F )) . (7.1.3) Pr o of. The inclusion SS(gr ~ ( F )) ⊂ SS( F ) follo ws from the distinguished triangle F ~ − → F − → gr ~ ( F ) +1 − → . Let us pro v e the con v erse inclusion. Using the deﬁnition of the microsupp ort, it is enough to pro v e that giv en t w o open subsets U ⊂ V of M , RΓ( V ; F ) − → RΓ( U ; F ) is an isomorphism as so on as RΓ( V ; g r ~ ( F )) − → RΓ( U ; gr ~ ( F )) is an isomorphism. Consider a dis- tinguished triangle RΓ( V ; F ) − → RΓ( U ; F ) − → G +1 − → . Then w e get a distin- guished triangle RΓ( V ; gr ~ ( F )) − → RΓ( U ; gr ~ ( F )) − → gr ~ ( G ) +1 − → . Therefore, gr ~ ( G ) ≃ 0. O n the other hand, G is cohomologically complete, thanks to Prop osition 1.5.12 and G ≃ 0 by Corolla ry 1 .5.9. Q.E.D. Prop osition 7.1.7. L et F ∈ D b R c ( C ~ X ) . Then F is c ohomolo gic al ly c omplete. Pr o of. O ne has “lim − → ” U ∋ x Ext j Z [ ~ ]  Z [ ~ , ~ − 1 ] , H i ( U ; F )  ≃ Ext j Z [ ~ ]  Z [ ~ , ~ − 1 ] , “lim − → ” U ∋ x H i ( U ; F )  ≃ Ext j Z [ ~ ]  Z [ ~ , ~ − 1 ] , F x  ≃ 0 where the last isomorphism follows from the fact that F x is cohomologically complete when taking X = pt. Hence, the h yp othesis (i) (c) of Prop osition 1.5.6 is satisﬁed. Q.E.D. Propagation for solutions of A Λ /X -mo dules Prop osition 7.1.8. L et N b e a c oher ent A Λ /X -mo dule. Then SS(R H om A Λ /X ( N , L )) ⊂ c har(gr ~ N ) . (7.1.4) 154 CHAPTER 7. HOLONOMIC DQ - MODULES Pr o of. By Lemm a 7.1.4, w e ha ve R H om A Λ /X ( N , L ) ≃ R H om D L ( D L ⊗ A Λ /X N , L ) . Since gr ~ ( D L ⊗ A Λ /X N ) = gr ~ ( N ), Prop osition 7.1.8 will follow from Prop o- sition 7.1.9 b elo w, alr eady obta ined in [17]. Q.E.D. Prop osition 7.1.9. L et N b e a c oher ent D L -mo dule. Then SS(R H om D L ( N , L )) = c har(gr ~ N ) . (7.1.5) Pr o of. Set F = R H om D L ( N , L ). Then F is cohomologically complete b y Corollar y 1.6.2 and SS( F ) = SS (gr ~ ( F )) b y Prop osition 7.1.6 . On the other hand, gr ~ ( F ) ≃ R H om D Λ (gr ~ N , O Λ ) b y Prop o sition 1.4.3 and the microsupp o rt of t his complex is equal to c har(gr ~ N ) by [39, Th 11.3.3]. Q.E.D. Constructibilit y of solutions Theorem 7.1.10 b elow has already b een obtained in [1 7] in the fra mework of D M [[ ~ ]]-mo dules. Recall that L is a coheren t A X -mo dule, simple along Λ. Theorem 7.1.10. L et N b e a holonomic A Λ /X -mo dule. (a) The obje cts R H om A Λ /X ( N , L ) and R H om A Λ /X ( L , N ) b elong to D b C c ( C ~ Λ ) and thei r micr osupp orts ar e c ontaine d in char(gr ~ N ) . (b) Ther e is a natur al isom orphism in D b C c ( C ~ Λ ) R H om A Λ /X ( N , L ) ∼ − → D ′ X  R H om A Λ /X ( L , N )  [ d X ] . (7 .1.6) The morphism in (b) is similar to the morphism in Lemma 3.3.1 and is asso ciated with R H om A Λ /X ( N , L ) ⊗ R H o m A Λ /X ( L , N ) − → R H om A Λ /X ( L , L ) − → R H om D L ( L , L ) ≃ C ~ Λ − → C ~ X [ d X ] . Pr o of. ( a) It is enough to treat F := R H om A Λ /X ( N , L ). In view of Prop o - sition 7.1.8, F is w eakly C -constructible and it remains to show that for eac h x ∈ Λ, F x b elongs to D b f ( C ~ ). If U is a suﬃcien tly small op en ball cen tered at x , then RΓ( U ; F ) − → F x is an isomorphism ([39]). The ﬁniteness of the complex gr ~ ( F x ) follow s 7.1. A -MODULES ALONG A LA GR ANGIAN SUBMANIF OLD 155 from the classical ﬁniteness theorem for holonomic D -mo dules of [34]. Since F is cohomolog ically complete, Prop osition 1.5.12 implies that R Γ( U ; F ) is cohomologically complete. Hence the res ult follo ws from Theorem 1.6.4. (b) follows f r o m Corollary 1.4.6, since w e kno w b y [3 4] that (7 .1.6) is an isomorphism after applying t he functor gr ~ . Q.E.D. A Λ /X mo dules and A lo c X -mo dules Deﬁnition 7.1.11. A coheren t A Λ /X -submo dule N of a coheren t A lo c X - mo dule M is called an A Λ /X -lattice of M if N generates M as an A lo c X - mo dule. Lemma 7.1.12. L et M b e a c oher ent A lo c X -mo dule and let N ⊂ M b e an A Λ /X -lattic e of M . Then char  gr ~ ( N )  ⊂ T ∗ Λ do es not dep end on the choic e of N . The pro of is similar to the one of Lemma 3.4.2, and w e shall not repeat it. Deﬁnition 7.1.13. Let M b e a coherent A lo c X -mo dule and let N ⊂ M b e an A Λ /X -lattice of M . W e set c har Λ ( M ) := c har(g r ~ N ) . Example 7.1.14. Let X = C 2 endo w ed with the symplectic co ordinates ( x ; u ) and let Λ b e the Lagrangia n manifold giv en by the equation { u = 0 } . In this case, A Λ /X = A X [ u ~ − 1 ]. No w let α ∈ C and consider the mo dules M = A lo c X / A lo c X ( xu − α ~ ) and N = A Λ /X / A Λ /X ( xu ~ − 1 − α ). Then N is an A Λ /X -lattice of M and gr ~ N ≃ D Λ / D Λ ( x∂ x − α ). Lemma 7.1.15. L et M b e a c oher e n t A lo c X -mo dule. (i) c har Λ ( M ) is a close d c onic c omplex a n alytic subset of T ∗ Λ a nd this set is involutive. (ii) L et 0 − → M ′ − → M − → M ′′ − → 0 b e an exact se quenc e of A lo c X -mo dules. Then c har Λ ( M ) = char Λ ( M ′ ) ∪ char Λ ( M ′′ ) . Pr o of. ( i) is a w ell-kno wn result of D - mo dule theory , se e [37]. (ii) Let N b e an A Λ /X -lattice of M . Set N ′ = M ′ ∩ N and N ′′ ⊂ M ′′ b e the imag e of N . Then N ′ and N ′′ are A Λ /X -lattices of M ′ and M ′′ , resp ectiv ely . Since we hav e an exact sequence 0 − → N ′ / ~ N ′ − → N / ~ N − → N ′′ / ~ N ′′ − → 0 , 156 CHAPTER 7. HOLONOMIC DQ - MODULES w e ha ve c har Λ ( M ) = char( N / ~ N ) = char( N ′ / ~ N ′ ) ∪ c har( N ′′ / ~ N ′′ ) = c har Λ ( M ′ ) ∪ char Λ ( M ′′ ). Q.E.D. Prop osition 7.1.16. F or a c oher ent A lo c X -mo dule M , we ha ve co dim c har Λ ( M ) ≥ co dim Supp ( M ) . Pr o of. In the course of the pro of, we shall hav e to consider t he analogue of the algebra A Λ /X but with A X a instead of A X . W e shall denote by A Λ a this alge- bra. W e shall show that codim Supp( M ) ≥ r implies co dim c har Λ ( M ) ≥ r b y descending induction on r . Applying Prop osition 2.3.15 (a), w e hav e R H om A loc X ( M , A lo c X ) ≃ τ ≥ r R H om A loc X ( M , A lo c X ), where τ ≥ r is the trunca- tion f unctor. Hence w e hav e a distinguishe d triangle in D b coh ( A lo c X a ): E xt r A loc X ( M , A lo c X )[ − r ] − → R H o m A loc X ( M , A lo c X ) − → K +1 − − − → , (7.1.7) where K = τ >r R H om A loc X ( M , A lo c X ). Note that co dim(Supp( K )) > r b y Prop osition 2.3.15 (b). Setting M ′ = E xt r A loc X ( M , A lo c X ), the distinguished triangle (7.1.7) induces a distinguished triangle in D b coh ( A lo c X ): R H om A loc X a ( K , A lo c X a ) − → M − → R H om A loc X a ( M ′ , A lo c X a )[ r ] +1 − − − → . Setting M 1 = E xt r A loc X a ( M ′ , A lo c X a ), w e obta in a morphism ϕ : M − → M 1 and Ker( ϕ ) has co dimension greater than r . Hence, co dim char Λ (Ker( ϕ )) > r b y the induction hypothesis. Since c har Λ ( M ) ⊂ c har Λ ( M 1 ) ∪ c har Λ (Ker( ϕ )), it is enough to sho w t ha t co dim char Λ ( M 1 ) ≥ r . Hence w e may assume from the b eginning that M = E xt r A loc X a ( M ′ , A lo c X a ) for a coheren t A lo c X a -mo dule M ′ . L et us take an A Λ a -lattice N ′ of M ′ . Set N 0 = E xt r A Λ a ( N ′ , A Λ a ). Then we hav e N lo c 0 ≃ M , and it induces a morphism N 0 − → M . Let N b e the image of the morphism N 0 − → M . Then N is an A Λ /X -lattice of M . Hence w e hav e c har Λ ( M ) = char( N / ~ N ), whic h implies c har Λ ( M ) ⊂ c har ( N 0 / ~ N 0 ) . (7.1.8) On the o ther hand, w e hav e an exact sequence E xt r A Λ a ( N ′ , A Λ a ) ~ − − → E xt r A Λ a ( N ′ , A Λ a ) − → E xt r A Λ a ( N ′ , gr ~ ( A Λ a )) . Since w e ha v e E xt r A Λ a ( N ′ , gr ~ ( A Λ a )) ≃ E xt r gr ~ ( A Λ a ) (gr ~ N ′ , gr ~ ( A Λ a )), w e ha v e a monomorphism N 0 / ~ N 0 ֌ E xt r gr ~ ( A Λ a ) (gr ~ N ′ , gr ~ ( A Λ a )) . 7.2. HOLONOMIC DQ -MODULES 157 Hence w e obtain c har( N 0 / ~ N 0 ) ⊂ char  E xt r gr ~ ( A Λ a ) (gr ~ N ′ , gr ~ ( A Λ a ))  . Since c har  E xt r gr ~ ( A Λ a ) (gr ~ N ′ , gr ~ ( A Λ a ))  has co dimension ≥ r b y e.g., [37, Theo- rem 2.19], w e conclude that co dim char( N 0 / ~ N 0 ) ≥ r . By (7.1.8), w e obta in co dim c har Λ ( M ) ≥ r . Q.E.D. 7.2 Holonomic DQ -mo dule s In a complex symplectic manifold X , an isotropic sub v ariet y Λ is a lo cally closed complex analytic sub v ariet y suc h that Λ reg is isotropic, i.e., the 2- form deﬁning the symplectic structure v a nishes on Λ reg . Here, Λ reg denotes the smo oth part of Λ. A Lag rangian sub v ariet y Λ is a n isotropic subv ariet y of pure dimension d X / 2. Equiv alently , Λ is a subv ariet y of pure dimension d X / 2 suc h that Λ reg is in v olutive . Deﬁnition 7.2.1. (a) An A lo c X -mo dule M is holonomic if it is coheren t and its supp ort is a Lag rangian sub v ariet y of X . (b) An A X -mo dule N is holonomic if it is coheren t, without ~ -torsion and N lo c is a holonomic A lo c X -mo dule. (c) Let Λ b e a smo oth Lag rangian submanifold of X . W e say that an A lo c X - mo dule M is simple holonomic along Λ if there exists lo cally a n A X - mo dule M 0 simple along Λ suc h that M ≃ M lo c 0 . Lemma 7.2.2. L et M b e a holonomic A lo c X -mo dule. Then D ′ A loc M [ d X / 2] is c on c entr ate d in de gr e e 0 and is holonomic. Pr o of. This follo ws from P rop osition 2.3.15 and the inv olutivit y the or em (Prop osition 2.3.18). Q.E.D. Let X b e a complex symplectic manifold a nd let M a nd L b e tw o holonomic A lo c X -mo dules. Using Lemma 2.4.10 (more precisely , an A lo c X -v arian t o f this lemma) and Theorem 6.2.4, we hav e R H om A loc X ( M , L ) ≃ R H om A loc X × X a ( M L ⊠ D ′ A L , C lo c X ) , R H om A loc X ( L , M ) ≃ R H om A loc X × X a ( L L ⊠ D ′ A M , C lo c X ) ≃ R H om A loc X × X a (D ′ A ( C lo c X ) , M L ⊠ D ′ A L ) ≃ R H om A loc X × X a ( C lo c X , M L ⊠ D ′ A L )[ d X ] . (7.2.1) 158 CHAPTER 7. HOLONOMIC DQ - MODULES Theorem 7.2.3. L et X b e a c om p lex symple ctic manifold and let M and L b e two holonomic A lo c X -mo dules. Then (i) the obje ct R H om A loc X ( M , L ) b elon g s to D b C c ( C ~ , lo c X ) , (ii) ther e is a c anonic al isomorphism : R H om A loc X ( M , L ) ∼ − →  D ′ X R H om A loc X ( L , M )  [ d X ] . (7.2.2) (iii) the obje ct R H om A loc X ( M , L )[ d X / 2] is p erverse. Pr o of. Using (7.2.1), w e may assume fr o m the b eginning that L is a simple holonomic A lo c X -mo dule supp orted o n a smo oth Lagr a ngian submanifold Λ of X . Let L 0 b e an A X -mo dule simple along Λ suc h that L ≃ L lo c 0 . (i)-(ii) Let N b e an A Λ /X -lattice of M . By Lemma 7.1.3 (v), w e hav e R H om A loc X ( M , L ) ≃ R H om A Λ /X ( N , L 0 ) lo c . Then the results follo w from Prop osition 7.1.16 a nd The orem 7.1.10. (iii) Since the pr o blem is lo cal, w e may assume that X = T ∗ M , A lo c X = c W X and L 0 = O ~ M . By (ii), it is enough to chec k the statemen t: H j  RΓ N  R H om A Λ /X ( N , L 0 )   v anishe s for j < l and for an y closed smo oth subm anif o ld N o f M o f co dimension l . (7.2.3) Since F := RΓ N  R H om A Λ /X ( N , L 0 )  is C - constructible, it is enough to sho w that H j (gr ~ ( F )) = 0 for j < l . This follow s f r o m t he we ll-known fact that H j (RΓ N ( O M )) = 0 for j < l . Q.E.D. Assume for simplicit y that X is op en in some cotangent bundle T ∗ M . W e shall compare the sheaf of solutions of holonomic b E X -mo dules and c W X - mo dules. Recall that c W X is faithfully ﬂat o ver b E X b y Lemma 6.1.2. Corollary 7.2.4. L et M and L b e two holo n omic b E X -mo dules. Then the obje ct R H om b E X ( M , L ) b elon g s to D b C c ( C X ) . Pr o of. L et t denote the co ordinate o n the complex line C , let E denote the ring b E T ∗ C | t =0 ,τ =1 and let L b e the E -mo dule E /E · t . Then we ha ve the em b edding C ~ , lo c ֒ → E , ~ 7→ ∂ − 1 t . 7.3. LAGRANGIAN CYCLES 159 Set for short F := R H om b E X ( M , L ). Then F ≃ R H om E ( L, R H o m b E X ( M , ( b E X × T ∗ C / b E X × T ∗ C · t ) | t =0 ,τ =1 L ⊗ b E X L )) ≃ R H om E ( L, R H o m c W X ( c W X ⊗ b E X M , c W X ⊗ b E X L )) . Set G := R H om c W X ( c W X ⊗ b E X M , c W X ⊗ b E X L ). Applying Theorem 7.2.3, w e ﬁnd that G ∈ D b C c ( C ~ , lo c X ) a nd it follows that F ∈ D b w C c ( C X ). Moreo v er, for eac h x ∈ X , G x is of ﬁnite type o ve r C ~ , lo c and is an E - mo dule. O ne easily deduces that F x ≃ RHom E ( L, G x ) is a C -v ector space o f ﬁnite dimension. Q.E.D. 7.3 Lagrangian c yc les Giv en tw o holonomic A lo c X mo dules M and L suc h that Supp( M ) ∩ Supp( L ) is compact, the Euler-P oincar´ e index is giv en by χ ( X ; M , L ) = χ (RHom A loc X ( M , L )) = P i ( − ) i dim Ext i A loc X ( M , L ) . (7.3.1) Applying ( 6 .3.6), we g et χ ( X ; M , L ) = Z X (eu X ( M ) · eu X ( L )) . (7.3.2) Recall that eu X ( M ) = ( − 1) d X / 2 eu X (D ′ A loc M ), and a lso recall tha t d X b eing ev en, eu X ( M ) · eu X ( L ) = eu X ( L ) · eu X ( M ). W e shall explain how to calculate the Euler classes by using the theory of Lagrangian cycle s. W e refer to [39, Ch. 9 § 3] for a detailed study of these cycles. Recall that K denotes a commutativ e No etherian unital ring of ﬁnite global dimension. Consider a closed Lagrang ia n sub v ariet y Λ of X . W e deﬁne the sheaf: L K Λ := H d X Λ ( K X ) , (7.3.3) and w e simply write L Λ instead of L Z Λ . The next results are obv ious and w ell-kno wn (see lo c. cit.). Lemma 7.3.1. (i) U 7→ H d X Λ ∩ U ( U ; K X ) ( U op en in X ) is a she af and this she af c oincides with L K Λ , 160 CHAPTER 7. HOLONOMIC DQ - MODULES (ii) H i Λ \ Λ reg (L K Λ ) ≃ 0 for i = 0 , 1 , (iii) if s is a se ction of L K Λ , then its supp ort is op en and close d in Λ , (iv) ther e is a c anonic al se ction in Γ(Λ ; L Λ ) which gives an isom orphism L Λ | Λ reg ∼ − → Z Λ reg . W e denote by [Λ] t he se ction giv en in (iv) ab ov e. Deﬁnition 7.3.2. W e call a section of L K Λ on an op en set U of Λ a Lagrangian cycle on U . Recall that K coh , Λ ( O X ) denotes the G rothendiec k group of the category D b coh , Λ ( O X ). W e denote by K coh , Λ ( O X ) the sheaf asso ciated with the presheaf U 7→ K coh , Λ ∩ U ( O U ). Then, there is a w ell deﬁned Z -linear map κ : K coh , Λ ( O X ) − → L Λ . (7.3.4) This map is c haracterized b y the prop erty that κ ( O Λ ) = [Λ] ∈ Γ(Λ; L Λ ) . (7.3.5) Let M ∈ D b hol ( A lo c X ) and let Λ b e a closed La grangian subv ariet y of X whic h con tains Supp( M ). Let M 0 b e an A X -lattice of M on an op en set U of X . Then gr ~ ( M 0 ) de- ﬁnes an elemen t [gr ~ ( M 0 )] ∈ K coh , Λ ( O X | U ), hence an elemen t of Γ( U ; K coh , Λ ( O X )). This elemen t dep ends only on M , and w e t hus hav e a morphism K coh , Λ ( A lo c X ) − → Γ(Λ; K coh , Λ ( O X )) . Comp osing with the map κ , we obtain a map K coh , Λ ( A lo c X ) − → Γ(Λ; L Λ ) . (7.3.6) Deﬁnition 7.3.3. W e denote by lc X ( M ) the image of M ∈ D b coh , Λ ( A lo c X ) b y the morphism in (7.3.6) and call it the Lagrangian cycle of M . On the other-hand, recall (see Deﬁnition 6.3 .2) that the Euler class eu X ( M ) of M b elongs to H d X Λ ( X ; C ~ , lo c X ). Hence, the Euler class of M is a Lagra ngian cycle supp orted b y Λ: eu X ( M ) ∈ Γ(Λ; L C ~ , lo c Λ ) . ( 7 .3.7) The map Z − → C ~ , lo c induces the morphism ι X : L Λ − → L C ~ , lo c Λ . (7.3.8) The next lemma is easily c hec ke d. 7.4. SIMPLE HOLONO MIC MODULES 161 Lemma 7.3.4. L et Λ b e a smo oth L agr angian submanifold of X and let L b e a c oher en t A lo c X -mo dule, simple along Λ . Then eu X ( L ) = ι X ([Λ]) . Theorem 7.3.5. One h a s eu X ( M ) = ι X ◦ lc X ( M ) . Pr o of. By Lemma 7.3.1 , it is enough t o prov e the result at the generic p o int of Λ. Hence, w e ma y assum e that Λ is smo oth. Let x ∈ Λ and let us c ho o se a smo oth L a grangian submanifold S x of X whic h interse cts Λ transv ersally at the single p oin t x . Let us also choose a simple A lo c X -mo dule L simple along S x . Using (7 .3.2), we ﬁnd χ (R H om A loc X ( L , M ) x ) = Z X (eu X ( L ) · eu X ( M )) . Let L 0 and M 0 b e A X -lattices of L and M , respectiv ely . W e also ha v e χ (R H om A loc X ( L , M ) x ) = χ (R H om gr ~ ( A X ) (gr ~ ( L 0 ) , gr ~ ( M 0 )) x ) = Z X ( κ ([gr ~ ( L 0 )]) · κ ([gr ~ ( M 0 )])) . Clearly , w e hav e κ ([gr ~ ( L 0 )]) = [ S x ] . (7.3.9) By Lemma 7.3.4, eu( L 0 ) = [ S x ]. Therefore, Z X ([ S x ] · eu X ( M )) = Z X ([ S x ] · lc X ( M )) (7.3.10) for any smo oth Lagrangian submanifold S x whic h in tersects Λ transv ersally at x . This completes the pro of . Q.E.D. Remark 7.3.6. The Euler class of a holo nomic A lo c X -mo dule supp orted b y a Lagrangian v ariet y Λ is easy to calculate, since it is enough to calculate it at generic p oints of Λ. Moreov er, the in tegra l in (7.3.2) is in v arian t by smo oth (real) ho motop y of the Lagrangian cycles lc X ( M ) and lc X ( L ) a nd one may deform them in order that they inters ect transv ersally at the smo oth part of their supp ort. See [39, Ch. 9 , § 3 ] for a detailed study . 7.4 Simple h olonomic mo dules When L 0 and L 1 are simple along smo oth La g rangian manifolds, one can giv e an estimate on the microsupp ort of R H om A loc X ( L 1 , L 0 ). It follo ws from Lemma 6.2.1 that t wo simple holonomic mo dules along Λ are lo cally isomor- phic. 162 CHAPTER 7. HOLONOMIC DQ - MODULES Example 7.4.1. Assume X = T ∗ M for a complex manifold M and A X = c W X (0). Then O ~ , lo c M is a simple holonomic A lo c X -mo dule along M . Recall that on a complex symplectic manifold X , the symplectic form giv es the Hamiltonian isomorphism from the cotangen t bundle to the tangen t bundle: H : T ∗ X ∼ − → T X , h θ , v i = ω ( v , H ( θ )) , v ∈ T X , θ ∈ T ∗ X . (7.4.1) F or a smo o th Lagrangian submanifold Λ of X the isomorphism (7.4.1) induces an isomorphism b et w een the normal bundle to Λ in X and it s cotangent bundle T ∗ Λ. F or the notion of normal cone, see e.g., [3 9, Def. 4.1 .1 ]. The next result is pro v ed in [42, Prop. 7 .1]. Prop osition 7.4.2. L et X b e a c omplex symple ctic manifold and let Λ 0 and Λ 1 b e two close d c omplex analytic i s otr o pic subvarieties of X . Then, after identifying T X and T ∗ X by (7.4.1) , the normal c one C(Λ 0 , Λ 1 ) is a c omplex analytic C × -c onic isotr opic subvariety o f T ∗ X . Theorem 7.4.3. L et L i b e a simple holonomic A lo c X -mo dule along a smo oth L agr an g ian manifold Λ i ( i = 0 , 1 ) . Then SS  R H om A loc X ( L 1 , L 0 )  ⊂ C(Λ 0 , Λ 1 ) . (7 .4.2) Ide a of the p r o of of The or em 7.4.3. ( i) By iden tifying R H om A loc X ( L 1 , L 0 ) with a sheaf supp ort ed b y Λ 0 , the estimate (7.4 .2) is equiv alen t to the estimate SS(R H om A loc X ( L 1 , L 0 )) ⊂ C Λ 0 (Λ 1 ) . ( 7 .4.3) (ii) The problem b eing lo cal, w e ma y assume X = T ∗ M , A X = c W X (0), Λ 0 = M , L 0 = O ~ , lo c M . If Λ 1 = Λ 0 , Theorem 7.4 .3 is immediate. Hence, we assume Λ 0 6 = Λ 1 . Then there exists a non constan t holomorphic function ϕ : M − → C suc h that Λ 1 = { ( x ; u ) ∈ X ; u = grad ϕ ( x ) } Consider the ideal I W = n X i =1 c W X · ( ~ ∂ x i − ϕ ′ i ) . (7.4.4 ) 7.5. INV ARIANCE BY DEFORMA TION 163 W e ma y assume that L 1 = c W X / I W . Let u ∈ L 1 b e the image o f 1 ∈ c W X and denote b y N the A Λ 0 /X -submo dule of L 1 generated b y u . T o conclude , it r emains to pro v e the inclusion c har(gr ~ ( N )) ⊂ C(Λ 1 , T ∗ M M) . (7.4.5) W e shall not give the pro of of (7.4 .5 ) here and refer to [42]. Let us simply men tion t ha t the pro of uses [37, Th. 6.8]. Q.E.D. Remark 7.4.4. Consider a smo oth L agrangian submanifold Λ of X a nd denote b y c h (Ω Λ ) ∈ H 1 (Λ; O × Λ ) the class corresp onding to the line bundle Ω Λ . T o t he exact sequence 1 − → C × Λ − → O × Λ d l og − − → d O Λ − → 0 one asso ciates the maps β and γ : H 1 (Λ; O × Λ ) β − → H 1 (Λ; d O Λ ) γ − → H 2 (Λ; C × Λ ) . W e shall denote b y C 1 / 2 Λ the in v ertible C Λ -algebroid associated with the co- homology class γ ( 1 2 β (c h(Ω Λ )) ∈ H 2 (Λ; C × Λ ) (see (2.1.13)). Consider an in vertible C ~ Λ -algebroid A on Λ and denote by Inv( A ) the category of in v ertible A -mo dules (see Deﬁnition 2.1.4). On the other hand, denote by Simple(Λ) the catego r y of simple A X -mo dules along Λ. It can b e easily deduced fr o m Lemma 6 .2.1 that, given a DQ-algebroid A X , t here exist an in ve rtible C ~ Λ -algebroid A and an equiv alence of categories Simple(Λ) ≃ In v( A ) . (7.4.6) When A X is the canonical algebroid c W X (0) (see remark 6.1.3), it is pro v ed in [19] that one has an equiv alence A ≃ C ~ Λ ⊗ C Λ C 1 / 2 Λ . 7.5 In v ariance b y de formation W e shall sho w that in the situatio n of Theorem 7.2.3, R H om A loc X ( M , L ) is, in some sense, in v arian t b y Hamiltonian sy mplectomorphism. First, w e nee d a lemma. Lemma 7.5.1. L et M b e a c omplex manifol d , X = T ∗ M and let M b e a holonomic c W X -mo dule. Assume that the pr oje ction π M : X − → M is pr op er ( henc e, ﬁnite ) on Supp( M ) . Then π M ∗ M is a lo c a l l y fr e e O ~ , lo c M -mo dule of ﬁnite r an k . 164 CHAPTER 7. HOLONOMIC DQ - MODULES Pr o of. ( i) In the sequel, w e write A X and A lo c X instead of c W X (0) and c W X , resp ectiv ely . Since π M is ﬁnite on Supp( M ) , R π M ∗ M is concen trated in degree 0. Let us pr ov e that this sheaf is O ~ , lo c M -coheren t. Denote b y Γ π the graph of the pro jection π M and consider the diagra m M × X z z v v v v v v v v v $ $ I I I I I I I I I Γ π ? _ s o o p   M X . Using the morphism of C ~ -algebras π − 1 M O ~ M ֒ → A X , we may regard L := s ∗ p − 1 A X a as a coheren t A M × X a -mo dule simple along Γ π . Then R π M ∗ M ≃ L lo c ◦ X M . W e may apply Theorem 3 .3.6 a nd w e get tha t R π M ∗ M is O ~ , lo c M -coheren t. (ii) Let n = d M = 1 2 d X . By Lemma 7.2.2, D ′ A loc ( M ) [ n ] is concen trated in de- gree 0 and it fo llo ws from a similar argumen t as in (i) that D ′ A ( M ) ◦ L ′ [ n ] is O ~ , lo c M -coheren t and concen trated in degree 0 for an y coheren t A lo c X × M -mo dule L ′ simple along Γ π . Denote b y D ′ O ~ , lo c the dualit y functor o ver O ~ , lo c M . Ap- plying again Theorem 3.3.6, w e get D ′ O ~ , lo c ( M ◦ L lo c ) ≃ D ′ A loc ( L lo c ) ◦ ω A loc X ◦ D ′ A loc ( M ) ≃ R π M ∗  R p ∗ (D ′ A ( L ) ◦ ω A X ) L ⊗ A X D ′ A loc ( M )  . Since ω A X ◦ D ′ A ( L ) ≃ L ′ [ n ] for an A M × X -mo dule L ′ simple along Γ π and D ′ A loc ( M ) is concen trated in degree n , D ′ O ~ , lo c ( π M ∗ M ) is concentrated in degree zero. Therefore, π M ∗ M is a lo cally pro jectiv e O ~ , lo c M -mo dule o f ﬁnite rank. T o conclude, note t ha t, for x ∈ M , an y ﬁnitely g enerated pro j ectiv e O ~ , lo c M ,x -mo dule is free, by a result of [52] ( see [59]) . Q.E.D. Recall t he situation of (3 .1.9): w e ha ve three symplectic ma nif o lds X i ( i = 1 , 2 , 3) and closed subsets Λ i of X i × X i +1 ( i = 1 , 2). Assume that the Λ i ( i = 1 , 2) are closed sub v arieties a nd the pro jection p 13 is prop er on p − 1 12 Λ 1 ∩ p − 1 23 Λ 2 . Then Λ 1 ◦ Λ 2 is a closed sub v ariet y o f X 1 × X 3 . No w assume that Λ i ( i = 1 , 2) is isotropic in X i × X a i +1 . Then Λ 1 ◦ Λ 2 is isotropic in X 1 × X a 3 b y classical results (see e.g., [39, Prop. 8.3.1 1]). In the sequel, w e denote b y D the op en unit disc in the complex line C , endo w ed with the co ordinate t . W e set fo r short Y := T ∗ D , 7.5. INV ARIANCE BY DEFORMA TION 165 and w e consider the pro jections X × Y p 1 { { w w w w w w w w w p   p 2 / / q # # G G G G G G G G G Y π   X X × D s / / D . Assume t o be give n a Lagra ngian subv ariet y Λ ⊂ X × Y satisfying the restriction p | Λ : Λ − → X × D is ﬁnite. (7.5.1) F or a ∈ D , writing for short T ∗ a D instead of T ∗ { a } D , we set Λ a := Λ ◦ T ∗ a D = p 1 (Λ ∩ q − 1 ( a )) , and this set is a Lagr a ngian sub v ariet y of X . W e intro duce the “skyscrap er” A lo c Y -mo dule C a := A lo c Y / A lo c Y · ( t − a ) . (7.5.2) Theorem 7.5.2. L et X b e a c omplex symple ctic manifo l d , let Λ b e a L a- gr angian subvariety of X × Y sa tisfying (7.5.1) , and let V b e a L agr a ngian subvariety of X . L et L b e a holo nomic A lo c X × Y -mo dule such that Supp( L ) ⊂ Λ and let N b e a holonomic A lo c X -mo dule such that Supp( N ) ⊂ V . Assume that the m ap q : Λ ∩ ( p − 1 1 V ) − → D is pr op er. F or a ∈ D , we set L a := L ◦ Y C a and M := R p 2 ∗ R H om p − 1 1 A loc X ( p − 1 1 N , L ) . T h en (i) L a is c onc entr a te d in de gr e e 0 and is a holonomic A lo c X -mo dule sup- p orte d by Λ a , (ii) M is a c oher ent A lo c Y -mo dule supp orte d by V ◦ X Λ , (iii) F a := RHom A loc X ( N , L a ) ≃ RΓ( Y ; ( O D / O D ( t − a )) L ⊗ O D M ) is an obje ct of D b f ( C ~ , lo c ) , an d F a and F b ar e isomorphic for any a, b ∈ D . Pr o of. ( i) First note that t − a : L − → L is a monomorphism. Indeed for an y s ∈ Ker( t − a : L − → L ), A lo c X × Y s ⊂ L is a coheren t A lo c X × Y -mo dule whose supp ort is in v olutiv e and of co dimension > d X × Y / 2, hence empt y . Therefore L a = L ◦ Y ( A lo c Y / A lo c Y · ( t − a )) ≃ R p 1 ∗  ( O D / O D ( t − a )) ⊗ O D L  , and (i) fo llo ws immediately from the hypothesis (7.5.1). (ii) W e ha ve R p 2 ∗ R H om p − 1 1 A loc X ( p − 1 1 N , L ) ≃ D ′ A ( N ) ◦ L . 166 CHAPTER 7. HOLONOMIC DQ - MODULES By t he hypothesis, the pro jection Λ ∩ ( V × Y ) − → Y is pro p er. It fo llo ws from Theorem 3.2.1 t hat M b elongs to D b coh ( A lo c Y ) and is supported b y the isotropic v ariet y Λ Y := V ◦ X Λ. (iii) By the h yp othesis, the pro jection π : Λ Y − → D is prop er, hence ﬁnite. It follows easily that H i ( M ) is a holonomic A lo c Y -mo dule and H i (R π ∗ M ) ≃ π ∗ H i ( M ) is a lo cally free O ~ , lo c D -mo dule of ﬁnite rank by Lemma 7.5.1. Hence H i  RΓ( Y ; ( O D / O D ( t − a )) L ⊗ O D M )  ≃ Γ( Y ; H i ( M ) / ( t − a ) H i ( M )) is a ﬁnite-dimensional C ~ , lo c -v ector space whos e dimension do es no t dep end on a ∈ D . Q.E.D. W e shall make a link b etw een the h yp otheses in Theorem 7.5.2 a nd the Hamiltonian deformations of a Lagr angian v ariet y Λ 0 . Assume t o be give n a holomorphic map Φ( x, t ) : X × D − → X suc h that Φ( · , a ) : X − → X is a symplectomorphism for eac h a ∈ D and is the iden tit y for a = 0. Set Γ := { ( x, t, Φ( x, t )) } , the gr a ph of Φ in X × X a × D . Consider the diﬀeren tial ∂ Φ ∂ t : X × D − → T X ≃ T ∗ X . W e make the hy p ot hesis: there exists f : X × D − → C suc h that ∂ Φ ∂ t = H f , (7.5.3) where H f denotes as usual t he Hamiltonian ve ctor ﬁeld. In this case, w e can deﬁne (iden tifying T ∗ D with D × C ) e Γ := { (( x, Φ( x, t )) ; ( t, f ( x, t ))) } ⊂ X × X a × T ∗ D and e Γ is Lagrangia n. Let Λ 0 b e a Lagrangian sub v ariet y of X . W e set: Λ := Λ 0 ◦ e Γ . Then Λ will satisfy h yp otheses (7.5.1) and Λ a = Φ( x, a )(Λ 0 ). 7.5. INV ARIANCE BY DEFORMA TION 167 Example 7.5.3. Let X = T ∗ M , V = T ∗ M M a nd let ϕ : M × D − → C b e a holomorphic function. Set Y = T ∗ D and let Λ = { ( x, t ; u, τ ) ∈ X × Y ; ( u, τ ) = grad x,t ϕ ( x, t ) } , Λ a = { ( x ; u ) ∈ X ; u = gr a d x ϕ ( x, a ) } . Consider the family of symplectomorphisms Φ( x, u , t ) = ( x, u + ϕ ′ x ( x, t ) − ϕ ′ x ( x, 0)) . Then ∂ Φ ∂ t = − H ∂ t ϕ and Λ a = Φ( x, u, a )Λ 0 . Set Z = { ( x, t ) ∈ M × D ; grad x ϕ ( x, t ) = 0 } and assume that the pro jection Z − → D is prop er. Consider the ideals I = n X i =1 A lo c X × Y · ( ~ ∂ x i − ϕ ′ x i ) + A lo c X × Y · ( ~ ∂ t − ϕ ′ t ) , I a = n X i =1 A lo c X · ( ~ ∂ x i − ϕ ′ x i ( · , a )) . Set N = A lo c X ⊗ D M O M and L = A X × Y / I . Hence w e hav e L a = A lo c X / I a and H i  RHom A loc X ( L a , N )  do es not dep end on a ∈ D . 168 CHAPTER 7. 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Index of notations A , 16 A 0 , 16 A Λ /X , 15 0 A Λ a , 15 6 c har Λ ( M ), 1 55 C ~ = C [[ ~ ]], 60 c h( F ), 125 C ~ , lo c = C (( ~ )), 60 c har, 135 C − ( C ), 84 C( C ), 11 C + ( C ), 11 C − ( C ), 11 C b ( C ), 11 ◦ X , 92 C + ( C ), 84 C X , 76 D M , 133 d X (dimension), 60 D( A ), 11 D b ( A ), 1 1 D b coh ( A ), 11 D A X , 65 D ′ A M , 70 , 98 E n d C ~ ( A X ), 78 D ′ A loc M , 72 D b C c ( K M ), 152 D b R c ( K M ), 152 D b w C c ( K M ), 152 D( C ), 11 D + ( C ), 11 D − ( C ), 11 D b coh , Λ ( A X ), 9 3 D b ( C ), 11 D( R lo c ) ⊥ r , 38 D b gd , Λ( A loc X ) , 93 D X [[ ~ ] ], 65 D ′ O ~ , lo c M , 16 4 DQ-mo dule, 91 b E T ∗ M , 133 eu( F ), 125 eu X ( M ), 1 42 HH ( O X ), 118 hh X ( F ), 118 HH Λ (gr ~ A X ), 1 1 4 d HH Λ (gr ~ A X ), 1 1 4 hh X ( M ), 1 04 hh gr X ( M ), 1 45 HH (gr ~ ( A X )), 114 HH ( A X ), 1 0 3 HH ( A lo c X ), 1 1 5 c hh gr X ( M ), 1 15 HD ( O X ), 124 K , 11 K( C ), 99 [ M ], 99 [Λ], 160 K coh , Λ ( O X ), 1 6 0 b K coh , Λ (gr ~ A X ), 1 0 1 L Λ , 15 9 Λ 1 ◦ Λ 2 , 93 [Λ reg ], 15 7 175 176 INDEX OF NOT A TIONS L K Λ , 159 lc X ( M ), 1 60 N lo c , 72 Mo d( A ), 1 1 Mo d af ( A ), 84 Mo d af ( A ), 84 M W , 14 4 O ~ M , 150 O ~ , lo c M , 150 Ω A X , 81 ω A X , 81 ω A loc X , 98 ω A X × Y / Y , 82 Ω ~ M , 150 ω top X R , 107 { pt } , 12 SS, 152 τ ≥ n , 11 ⊠ , 68 L ⊠ , 69 L ⊗ A X , 92 P ⊗− 1 , 77 thh X ( F ), 122 td X , 126 c W T ∗ M , 13 3 c W T ∗ M (0), 133 T erminologies A X -lattice, 72 algebraically go o d, 86 algebroid, 49 DQ-, 67 O X -, 58 R -, 57 in v ertible O X -, 58 in v ertible R -, 57 almost free A -mo dule, 8 4 bi-diﬀeren tial opera t o r, 60 bi-in v ertible, 54, 55, 76 canonical mo dule asso ciated with the diagonal, 76 C × -conic, 153 C -constructible, 152 c haracteristic v ariet y , 13 5 Chern class, 125 co-Ho c hsc hild class, 12 2 coheren t, 13 cohomologically complete, 38 conic C × -, 153 R + -, 152 constructible C -, 152 R -, 152 w eakly R - , 15 2 con v olution, 92 DQ-algebra, 62 DQ-algebroid, 67 dual of A X -mo dule, 70 of A lo c X -mo dule, 72 dualizing sheaf, 81 Euler class of A lo c X -mo dules, 142 of D -mo dules, 14 7 of O -mo dules, 125 external pro duct of DQ-algebras, 6 4 of DQ-algebroids, 68 go o d A X -mo dule, 73 D -mo dules, 14 4 algebraically , 86 mo dule, 30 Grothendiec k group, 99 ~ -complete, 16 ~ -separated, 16 ~ -torsion, 16 ~ -completion, 16 Ho c hsc hild class of an A X -mo dule, 104 of an D -mo dule, 14 5 of an O -mo dule, 118 Ho c hsc hild homology of O , 118 Ho c hsc hild-Kostant-Rosen b erg map, 126 Ho dge cohomology , 124 holonomic, 15 7 A Λ /X -mo dule, 152 177 178 TERMINOLOGIES simple, 157 in v ertible, 51 O X -algebroid, 58 R -alg ebroid, 57 isomorphism standard, 63 isotropic sub v ariet y , 157 Lagrangian cycle, 160 Lagrangian sub v ariet y , 157 lattice, 72 A Λ /X -lattice, 155 lo cally ﬁnitely generated, 12 lo cally of ﬁnite presen tation, 12 lo cally pro jectiv e, 34 microsupp o rt, 152 Mittag-Leﬄer condition, 13 mo dule bi-in v ertible, 54, 55 coheren t, 13 in v ertible, 51 lo cally ﬁnitely generated, 12 lo cally of ﬁnite presen tation, 12 No etherian, 13 pseudo-coheren t, 12 simple, 71 mo dules o v er a n algebroid, 5 1 No etherian, 13 no ~ - torsion, 1 6 O X -algebroid, 58 in v ertible, 58 pseudo-coheren t, 12 R -alg ebroid, 57 in v ertible, 57 R + -conic, 152 R -constructible, 152 righ t or thogonal, 38 ring No etherian, 13 section standard, 63 simple holonomic, 157 mo dule, 71 standard isomorphism, 63 standard section, 63 star pro duct, 60 submo dule of A lo c X -mo dule, 72 thic k sub category , 73 T o dd class, 126 w eakly R - constructible, 152 179 Masaki Kashiwara Researc h Institute for Mathematical Sciences, Kyo to University , Kyo to 606-01 Japan email: masaki@kuri ms.kyo to-u.ac.jp Pierre Schapira Institut de Math´ ematiques Unive r sit ´ e Pierre et Marie Curie e-mail: schapira@math.jussieu.fr h ttp://www.math.jussieu.f r/˜ schapira/

Deformation quantization modules

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