Hochschild cohomology, the characteristic morphism and derived deformations

Ho c hsc hild cohomolog y , the c haracterist ic morphism and de riv ed deformations W endy Lo wen Abstract A notio n of Ho c hschild cohomol ogy H H ∗ ( A ) of an abelian categ ory A w as defined b y Lo w en and V an den Bergh (2005) and they show ed the existence of a c haracteristic mor- phism χ from the Ho c hsc hild cohomology of A into the graded centre Z ∗ ( D b ( A )) of the b ound ed deriv ed catego ry of A . An elemen t c ∈ H H 2 ( A ) corresp ond s to a first order deformation A c of A (Low en and V an d en Bergh, 2006). Th e problem of deforming an ob ject M ∈ D b ( A ) to D b ( A c ) wa s treated b y Lo wen (2005). In this pap er we sho w that the elemen t χ ( c ) M ∈ Ext 2 A ( M , M ) is p recisely the obstr u ction to deforming M to D b ( A c ). Hence this pap er provides a missing link b et we en the ab o ve w orks. Finally we d iscu ss some implications of these facts in the d ir ection of a “deriv ed deformation theory”. 1. Introduction Let k b e a comm utative ring. It is well kno wn that for a k -algebra A , ther e is a c h aracteristic morphism χ A of graded comm utativ e algebras fr om the Ho c hsc hild cohomology of A to the grad ed cen tre of the d eriv ed cate gory D ( A ). If k is a fi eld, this morphism is determined by the maps, for M ∈ D ( A ), M ⊗ L A − : H H ∗ k ( A ) ∼ = Ext ∗ A op ⊗ k A ( A, A ) − → Ext ∗ A ( M , M ) The charact eristic morph ism pla ys an imp ortan t rˆ ole for example in the theory of su pp ort v ari- eties ([1], [6], [25]). Characteristic morphisms were generalized to v arious situations where a go o d notion of Ho c hschild cohomology is at hand . Recen tly , Buc hw eitz and Flenn er d efined and studied Ho c hschild co homology for morphism s of sc hemes or analytic s paces, and pro v ed the existence of a c haracteristic morphism in this cont ext ([4]). I n [14], Keller defi ned the Ho c hschild cohomology of an exact catego ry as the Ho c hschild cohomology of a certain dg qu otien t. F or an ab elian categ ory A , this is pr ecisely the Ho chsc hild cohomology o f a “dg e nh ancemen t” o f the boun ded deriv ed category D b ( A ). Con s equen tly , the pro jectio n on the zero part of the Ho c hschild complex (see § 2.5 ) is itself a natural dg enhancement of a c haracteristic morphism χ A : H H ∗ ex ( A ) − → Z ∗ ( D b ( A )) where the right hand side d enotes the graded cen tre of D b ( A ) (see § 4.2). Explicitely , χ A maps a Ho c hschild n -co cycle c to a collectio n of elemen ts χ A ( c ) M ∈ Ext ∗ A ( M , M ) for M ∈ D b ( A ). The main pu rp ose of th is pap er is to giv e an interpretation of χ A ( c ) M in terms of d eform ation theory . In [22], a d eform ation theory of ab elian catego ries wa s dev elop ed. Its r elation w ith Ho chsc hild cohomology go es through an alternativ e d efinition of the latter give n by the authors in [21], and 2000 Math ematics Subje ct Classific ation 18G60 (primary), 13D10 (seco nd ary) Keywor ds: Hochschil d cohomology , B ∞ -algebra, chara cteristic morph ism, deformation, ab elian category , derived cat- egory , A [0 , ∞ [ -category The author is a p ostdo ctoral fello w with FWO/CNRS. S he ackno wledges the hospitality of the Institut de Math´ ematiques de Jussieu (IMJ) and of the Institu t des Hautes Etudes S cientifiques (IHES) d uring her p ostdo c- toral fello wship with CNRS. Wendy Lowen sho wn in the same p ap er to b e equiv alent to Keller’s defi n ition. Let us consider, from n o w on, an ab elian category A with enough in jectiv es, and let us assum e that k is a fi eld. T hen Inj ( A ) is a k -linear category and we put H H ∗ ab ( A ) = H H ∗ ( Inj ( A )) (1) The main adv an tage of Inj ( A ) is that, considering it as a ring with sev eral ob jects, its deform ation theory is entirely understo o d in the sense of Gerstenhab er ’s deformation theory of algebras ([9]). It is sho wn in [22] that the ab elian deformation theory of A is equiv alen t to the line ar deformation theory of Inj ( A ), ju stifying (1 ). An ab elian deformation B of A giv es rise to a morph ism D b ( B ) − → D b ( A ) (2) and an obstruction th eory for deforming ob jects M ∈ D b ( A ) to D b ( B ), whic h is the sub ject of [20]. The main theorem of the cur ren t pap er (see also Th eorem 4.8) states Theorem 1.1 . Consider c ∈ H H 2 ex ( A ) and le t A c b e the corresp ondin g (first o rd er) deformation of A . F or M ∈ D b ( A ) , the elemen t χ A ( c ) M ∈ Ext 2 A ( M , M ) is the obstruction against deforming M to an ob j ect of D b ( A c ) . Hence, the characte ristic morph ism χ A is a natur al ingredien t in a theory describing the simul- taneous d eformations of an ab elian categ ory toget h er with (families or diagrams of ) ob j ects in the ab elian (or deriv ed) category . The details of this theory remain to b e wo rked out. In [20], (2) is expr essed in terms of complexes of injectiv es in A and B , and the obstructions are exp ressed in terms of the elemen t c ∈ H H 2 ab ( A ) = H H 2 ( Inj ( A )) corresp onding to the ab elian deformation. Essen tially , our approac h for pro ving the ab o v e theorem is tighte nin g the relation b et ween H H ∗ ab ( A ) and H H ∗ ex ( A ). F or a differentia l graded catego ry a , let C ( a ) denote its Ho chsc hild complex ([14]). Let D b dg ( A ) b e a dg mo del of D b ( A ) constructed using complexes of in jectiv es. T he natural inclusion Inj ( A ) ⊂ D dg ( A ) in duces a pro jection morphism C ( D dg ( A )) − → C ( Inj ( A )) (3) whic h is pro ven in [21] to b e a quasi-isomorphism of B ∞ -algebras. T he B ∞ -structure of the Ho c hschild complexes captures all the op erations relev ant to deformation th eory , lik e the cup pro d uct and the Gerstenhab er brac ket , but also the more primitiv e brace op erations (see § 2.3). In § 3, w e explicitely construct a B ∞ -section em br δ : C ( Inj ( A )) − → C ( D dg ( A )) of (3) (Theorem 3.22). In the n otation, δ is the elemen t in C 1 ( D dg ( A )) determined b y the differen- tials of the complexes of injectiv es, and embr, short for “embrace ”, refers to the brace op erations. More concretely , f or c ∈ C n ( Inj ( A )), we ha ve em br δ ( c ) = n X m =0 c { δ ⊗ m } After in tro ducing the c h aracteristic morphism in § 4.3, w e use the morphism em br δ to pro ve Th eorem 1.1 in § 4.4. The morph ism em br δ also thro ws some ligh t on the follo wing question, wh ic h is p art of a researc h pro ject in progress: Question. Give n a n ab elian deformation B of an ab elian category A , in which sense can we int erp ret D b ( B ) as a “d er ived” deformation of D b ( A )? More precisely , the morphism embr δ giv es us a r ecip e to turn a linea r deformation of Inj ( A ) (and hence an abelian deformation of A ) in to a deformation of D b dg ( A )... as a c dg category! Here cdg, as 2 Hochschild cohomology, the characteristic morphism and derived deforma tions opp osed to dg, means that apart from comp ositions m and differen tials d , the category has “cur- v ature elemen ts” correcting the fact that d 2 6 = 0. T h is “small” alteration has ser ious consequences, ruining for example the classical cohomology theory . In Theorem 4.18 we sho w th at the cdg deformation of D b dg ( A ) con tains a maximal p artial dg deformation w hic h, at le ast morally , is precisely D b dg ( B ) (see also Remarks 4 .19 and 4.20 ). The part of D b dg ( A ) that gets dg deform ed in this w ay is spanned by th e “zero lo cus” of the c haracteristic elemen t ( M 7→ χ A ( c ) M ) ∈ Y Ob( D b ( A )) Ext 2 A ( M , M ) Hence, a n ob ject M ∈ D b ( A ) con tributes to the dg deformation of D b dg ( A ) if and only if it deforms, in the sense of [20], to an ob ject of D b ( B ). 2. A [0 , ∞ [ -categories A ∞ -algebras and categories are b y now widely used as algebraic m o dels for triangulated categories (see [2], [13], [16], [17] and the references therein). Although the ge neralization to the A [0 , ∞ [ -setting causes s erious new issues, a large part of the theory can still b e dev elop ed “in the A ∞ -spirit”. In this section w e try to giv e a brief, reasonably self con tained account of the facts w e need. F or more detailed accoun ts we refer the reader to [12], [18] for A ∞ -algebras, to [19], [23] for A ∞ -categ ories and to [24 ] for A [0 , ∞ [ -algebras. 2.1 A w ord on signs and shifts Let k b e a commuta tive r ing. All the alge br aic constructions in this paper ta ke plac e in and around the category G ( k ) of Z -graded k -mo dules. F or M , N ∈ G ( k ), we ha ve the familiar tensor pr o duct ( M ⊗ N ) n = ⊕ i ∈ Z M i ⊗ N n − i and int ernal hom [ M , N ] p = Y i ∈ Z Hom( M i , M i + p ) o v er k . F or m ∈ M i , the de gr e e of m is | m | = i . W e adopt the Koszul sign c onvention , i.e. G ( k ) is endo wed with the w ell kno wn closed tensor structure with “sup er” comm utativit y isomorph isms M ⊗ N − → N ⊗ M : m ⊗ n 7− → ( − 1) | m || n | n ⊗ m (4) and the standard asso ciativit y and identi ty isomorph isms. Th e closed structure is d etermined by the ev aluatio n morphism [ M , N ] ⊗ M − → N : ( f , m ) 7− → f ( m ) F u rthermore, w e make a c hoice of sh ift fun ctors on G ( k ). F or i ∈ Z , let Σ i k ∈ G ( k ) b e the ob ject whose only nonzero comp onen t is (Σ i k ) − i = k . The shift functors are the fun ctors Σ i = Σ i k ⊗ − : G ( k ) − → G ( k ) : M 7− → Σ i M = Σ i k ⊗ M F or m ∈ M , w e p ut σ i m = 1 ⊗ m ∈ Σ i M . All the canonical isomorphisms (and in p articular th e signs) in this pap er are deriv ed from the ab o v e conv en tions. The most general canonical isomorphisms w e will use are of the form, for M 1 , . . . , M n , M ∈ G ( k ): ϕ : Σ i − i 1 −···− i n [ M 1 ⊗ · · · ⊗ M n , M ] − → [Σ i 1 M 1 ⊗ · · · ⊗ Σ i n M n , Σ i M ] (5) defined by ϕ ( σ i − i 1 −···− i n φ )( σ i 1 m 1 , . . . , σ i n m n ) = ( − 1) α σ i φ ( m 1 , . . . , m n ) wh ere α = ( i 1 + · · · + i n ) | φ | + i 2 | m 1 | + · · · + i n ( | m 1 | + · · · + | m n − 1 | ) 3 Wendy Lowen 2.2 The Ho c hschild ob ject of a (graded) quiv er In th is section and th e n ext one we will introd uce the Ho c hschild complex of an A [0 , ∞ [ -categ ory (see a lso [12], [18]) in t wo steps. O ur purp ose is to distiguish b etw een th e part of the structure that comes from the A [0 , ∞ [ -structure (next section) and th e p art that do es not (this section). This will b e usefu l later on when we will transp ort A [0 , ∞ [ -structures. Let k b e a comm utativ e ring. A gr ade d k - quiver is a quiv er enric hed in the category G ( k ). More precisely , a graded k -quiver a consists of a set of ob jects Ob( a ) and f or A, A ′ ∈ Ob( a ), a graded ob ject a ( A, A ′ ) ∈ G ( k ). Since w e will only use gr ade d k -quiv ers in this paper, w e will s ystematicall y call them simply quivers . The category of quivers with a fixed set of ob j ects admits a tensor pr o duct a ⊗ b ( A, A ′ ) = ⊕ A ′′ a ( A ′′ , A ′ ) ⊗ b ( A, A ′ ) and an in ternal hom [ a , b ]( A, A ′ ) = [ a ( A, A ′ ) , b ( A, A ′ )] W e put [ a , b ] = Q A,A ′ [ a , b ]( A, A ′ ). A morphism of de gr e e p fr om a to b is by defi n ition an elemen t of [ a , b ] p . The tensor c o c ate gory T ( a ) of a qu iver a is the qu iv er T ( a ) = ⊕ n > 0 a ⊗ n equip ed with the com ultiplication ∆ : T ( a ) − → T ( a ) ⊗ T ( a ) wh ich separates tensors. There are natu- ral notions of morp hisms and of co d eriv atio ns b et w een co categories and there is a G ( k )-isomorphism [ T ( a ) , a ] ∼ = Co der( T ( a ) , T ( a )) The ob ject [ T ( a ) , a ] is natur ally a b race algebra. W e recall the defin ition. Definition 2.1 . (see also [10 ]) F or V ∈ G ( k ), the structure of br ac e algebr a on V consists in the datum of (degree zero) op erations V ⊗ n +1 − → V : ( x, x 1 , . . . x n ) 7− → x { x 1 , . . . x n } satisfying the relation x { x 1 , . . . x m }{ y 1 , . . . y n } = X ( − 1) α x { y 1 , . . . , x 1 { y i 1 , . . . } , y j 1 . . . , x m { y i m , . . . } , y j m , . . . y n } where α = P m k =1 | x k | P i k − 1 l =1 | y l | . Th e asso ciate d Lie br acket of a brace algebra is h x, y i = x { y } − ( − 1) | x || y | y { x } A br ac e algebr a morphism (b et w een tw o b race algebras) is a graded m orphism p reserving all th e individual brace op erations. Pr op osition 2.2 . Let V b e a br ace algebra. The tensor coalgebra T ( V ) naturally b ecomes a (graded) bialgebra with the asso ciativ e multiplica tion M : T ( V ) ⊗ T ( V ) − → T ( V ) defined b y th e comp ositions M k ,l : V ⊗ k ⊗ V ⊗ l − → T ( V ) ⊗ T ( V ) − → T ( V ) − → V with M 1 ,l ( x ; x 1 , . . . x l ) = x { x 1 , . . . , x l } and all other comp onents equal to zero. T he un it for th e multiplica tion is 1 ∈ k = V ⊗ 0 . Pr o of. T his is sta n d ard (see [11]). A coalgebra morp h ism M is u n iquely determined b y the c omp o- nen ts M k ,l and the brace algebra axioms translate int o the associativit y of M . Put [ T ( a ) , a ] n = [ a ⊗ n , a ] = Y A 0 ,...,A n ∈ a [ a ( A n − 1 , A n ) ⊗ · · · ⊗ a ( A 0 , A 1 ) , a ( A 0 , A n )] 4 Hochschild cohomology, the characteristic morphism and derived deforma tions The brace algebra stru cture on [ T ( a ) , a ] = Q n > 0 [ T ( a ) , a ] n is giv en by the op erations [ T ( a ) , a )] n ⊗ [ T ( a ) , a )] n 1 ⊗ · · · ⊗ [ T ( a ) , a )] n k − → [ T ( a ) , a )] n − k + n 1 + ··· + n k with φ { φ 1 , . . . φ n } = X φ (1 ⊗ · · · ⊗ φ 1 ⊗ 1 ⊗ · · · ⊗ φ n ⊗ 1 ⊗ · · · ⊗ 1) The asso ciated Lie b rac k et corresp onds to th e comm utator of co deriv ations. W e put B a = T (Σ a ) and C br ( a ) = [ T (Σ a ) , Σ a ] = [ B a , Σ a ]. Summarizing, the quiver a yields to wering la y ers of (graded) algebraic stru cture: (0) the quive r a , i.e. the graded ob jects a ( A, A ′ ); (1) the c o c ate gory B a = T (Σ a ); (2) the br ac e algebr a C br ( a ) = [ B a , Σ a ] ∼ = Co der( B a , B a ) whic h is in particular a Lie algebr a and (0’) the asso ciated Ho chschild obje ct C ( a ) = Σ − 1 C br ( a ); (1’) the bialgebr a T ( C br ( a )) = B C ( a ); There is a natur al inclusion C br ( a ) − → T ( C br ( a ) = B C ( a ) (6) of (2) in to (1’). 2.3 The Ho c hschild complex of an A [0 , ∞ [ -category Definition 2.3 . ([12]) Let a b e a quiver. An A [0 , ∞ [ -structur e on a is an element b ∈ C 1 br ( a ) satisfying b { b } = 0 (7) The morphism s b n : Σ a ⊗ n − → Σ a defining b are sometimes called (T aylor) c o e fficients of b . Th e couple ( a , b ) is called an A [0 , ∞ ] - c ate gory . I f b 0 = 0, it is called an A ∞ -c ate gory . If b n = 0 for n > 3, it is called a c dg c ate gory . If b 0 = 0 and b n = 0 f or n > 3, it is called a dg c ate gory . R emark 2.4 . Consider b ∈ C 1 br ( a ). i) The equation (7) can b e written out completely in terms of th e co efficient s b n of b ([19], [24]). ii) If w e consider b as a cod eriv ati on inside [ B a , B a ] 1 , then (7) is equiv alen t to b 2 = b ◦ b = 0 iii) If we consider b as an elemen t of the bialgebra B C ( a ) through (6 ), then (7) is equiv alen t to b 2 = M ( b, b ) = 0 (8) where M is the multiplicatio n of B C ( a ). The easiest morph isms to consider b et we en A [0 , ∞ [ -categ ories a re those with a fixed set of ob j ects. T o capture m ore general morphisms, one could follo w th e approac h of [19 ] for A ∞ -categ ories. Definition 2.5 . ([12]) C on s ider A [0 , ∞ [ -categ ories ( a , b ) and ( a ′ , b ′ ) w ith Ob( a ) = Ob( a ′ ). A (fixe d ob- je ct) morp hisms of A [0 , ∞ [ -c ate gories is a (fixed ob j ect) m orphism of different ial graded co categories f : B a − → B a ′ (i.e. a morphism of quiv ers p reserving the com ultiplication and the differential ). It is determined b y morphisms f n : (Σ a ) ⊗ n − → Σ a ′ for n > 0. 5 Wendy Lowen An A [0 , ∞ [ -structure on a in tro d uces a load of additional algebraic structure on the to w er of § 2.2. The Ho chschild differ ential on C br ( a ) asso ciated to b is give n by d = h b, −i ∈ [ C br ( a ) , C br ( a )] 1 : φ − → h b, φ i and m akes C br ( a ) into a dg L ie algebra. Th e complex Σ − 1 C br ( a ) is (isomorphic to) the classical Ho c hschild complex of a . Similarly , considering b ∈ B C ( a ) 1 , we define a different ial D = [ b, − ] M ∈ [ B C ( a ) , B C ( a )] 1 : φ 7− → [ b, φ ] M where [ − , − ] M denotes the commutat or of the multiplicatio n M determined by the brace op erations. As D is a co d eriv atio n, it defines an A [0 , ∞ [ -structure on C ( a ). Let us examine the coefficients D n : Σ C ( a ) ⊗ n − → Σ C ( a ) of th is A [0 , ∞ [ -structure. By definition, D n ( φ 1 , . . . , φ n ) = M 1 ,n ( b ; φ 1 , . . . , φ n ) − M n, 1 ( φ 1 , . . . , φ n ; b ), so for n = 1 w e ha v e D 1 ( φ ) = h b, φ i whereas for n > 1 we ha ve D n ( φ 1 , . . . , φ n ) = b { φ 1 , . . . , φ n } The differenti al D make s B C ( a ) into a d g bialgebra. By definition, this mak es C ( a ) into a B ∞ - algebr a [11]. Su mmarizing, we obtain the follo wing to w er: (0) the A [0 , ∞ [ -c ate gory a ; (1) the dg c o c ate gory B a = T (Σ a ); (2) the B ∞ -algebr a C br ( a ) = [ B a , Σ a ] ∼ = Co der( B a , B a ) whic h is in p articular a dg Lie algebr a and (0’) the asso ciated Ho chschild c omplex C ( a ) = Σ − 1 C br ( a ); (1’) the dg b ialgebr a T ( C br ( a )) = B C ( a ); By a B ∞ -morphism (b et we en B ∞ -algebras B 1 and B 2 ) we will alw a ys mean a g raded morphism (sup er)commuting w ith all the individu al op erations on B 1 and B 2 . A B ∞ -morphism is a brace algebra morph ism, and a v ery particular case of a morph ism of A ∞ -algebras. 2.4 Limited functoriality The follo wing tautologica l prop osition will b e us ed later on to trans fer Ho chsc hild co chains: Pr op osition 2.6 . Consider quiv ers a and b and a brace algebra morphism Ψ : C br ( a ) − → C br ( b ) . If b is an A [0 , ∞ [ -structure on a , then Ψ( b ) is an A [0 , ∞ [ -structure on b and Ψ : C br ( a , b ) − → C br ( b , Ψ( b )) is a B ∞ -morphism. Pr o of. F or φ ∈ T ( C br ( a , b )), w e are to sho w that T (Ψ)([ b, φ ]) = [Ψ( b ) , T (Ψ)( φ )], w hic h immediate ly follo ws from the fact that Ψ pr eserv es the brace multiplicati on. Let b ⊂ a b e the in clus ion of a full sub quiv er, i.e. Ob( b ) ⊂ O b( a ) an d b ( B , B ′ ) = a ( B , B ′ ). Usin g the indu ced B b − → B a , there is a canonical restriction b race algebra morphism π b : C br ( a ) − → C br ( b ) If ( a , b ) is an A [0 , ∞ [ -categ ory , b can b e endo wed with the induced A [0 , ∞ [ -structure π b ( b ) and π b b ecomes a B ∞ -morphism (see Pr op osition 2.6). The A [0 , ∞ [ -categ ory ( b , π b ( b )) is called a ful l A [0 , ∞ [ - sub c ate gory of a . In particular, for ev ery ob ject A ∈ a , ( a ( A, A ) , π A ( b )) is an A [0 , ∞ [ -algebra. 6 Hochschild cohomology, the characteristic morphism and derived deforma tions 2.5 Pro je ction on the zero part Let a b e an A [0 , ∞ [ -categ ory . By the zer o p art of C br ( a ) we mean C br ( a ) 0 = [ T (Σ a ) , Σ a ] 0 = Y A ∈ a Σ a ( A, A ) Consider the graded morp hisms π 0 : C br ( a ) − → C br ( a ) 0 and σ 0 : C br ( a ) 0 − → C br ( a ). T h e morph ism b 1 ∈ [Σ a , Σ a ] 1 = Y A,A ′ ∈ a [Σ a ( A, A ′ ) , Σ a ( A, A ′ )] 1 determines degree 1 morp hisms ( b 1 ) A : Σ a ( A, A ) − → Σ a ( A, A ) and a pro d uct morphism b ∆ 1 : C br ( a ) 0 − → C br ( a ) 0 Let d : C br ( a ) − → C br ( a ) b e the Ho c hsc h ild differential. Pr op osition 2.7 . W e h a v e b ∆ 1 = π 0 dσ 0 In particular, if a is an A ∞ -categ ory , π 0 : ( C br ( a ) , d ) − → ( C br ( a ) 0 , b ∆ 1 ) is a morphism of differen tial graded ob j ects. Pr o of. F or an elemen t x ∈ Σ a ( A, A ), we ha ve π A ( d ( x )) = h b, x i = b 1 { x } = b 1 ( x ). 2.6 F rom Σ a to a The Ho c hschild complex of a and the B ∞ -structure on Σ C ( a ) are often expressed in terms of a rather than Σ a . Th is can b e d one using the canonical isomorphisms Σ 1 − n [ a ( A n − 1 , A n ) ⊗ · · · ⊗ a ( A 0 , A 1 ) , a ( A 0 , A n )]   [Σ a ( A n − 1 , A n ) ⊗ · · · ⊗ Σ a ( A 0 , A 1 ) , Σ a ( A 0 , A n )] (9) determined b y the conv en tions of § 2.1, thus introd u cing a lot of signs. W e define the bigraded ob ject C ( a ) by C i,n ( a ) = Y A 0 ,...A n [ a ( A n − 1 , A n ) ⊗ · · · ⊗ a ( A 0 , A 1 ) , a ( A 0 , A n )] i An element φ ∈ C i,n has de gr e e | φ | = i , arity ar( φ ) = n and Ho chschild de gr e e deg( φ ) = i + n . W e put C p ( a ) = Q i + n = p C i,n ( a ). T he B ∞ -stucture of C br ( a ) is translated in terms of op erations on C ( a ) th r ough (9). The complex C ( a ) will also b e called the Ho c hsc hild complex of a and its elemen ts are calle d Ho c hschild co c hains. F or a Ho c hsc hild co c hain φ ∈ C i,n ( a ), the corresp onding elemen t of C br ( a ) has σ 1 − n ( φ )( σ f n , . . . , σ f 1 ) = ( − 1) ni +( n − 1) | f n | + ··· + | f 2 | σ φ ( f n , . . . , f 1 ) This iden tification is d ifferen t from other ones, u sed for example in [11], [19], [24]. Neve rth eless, it allo ws us to reco v er many stand ard constructions (u p to minor mo difications). F or example, the op eration dot : C br ( a ) n ⊗ C br ( a ) m − → C br ( a ) n + m − 1 : ( x, y ) 7− → x { y } giv es r ise to the classical “dot p ro du ct” • : C i,n ( a ) ⊗ C j,m ( a ) − → C i + j,n + m − 1 7 Wendy Lowen on C ( a ) giv en b y φ • ψ = n − 1 X k =0 ( − 1) ǫ φ (1 ⊗ n − k − 1 ⊗ ψ ⊗ 1 ⊗ k ) (10) where ǫ = (deg ( φ ) + k + 1)(ar( ψ ) + 1) In the sequel, when no confusion arises, w e will not distinguish in notation b et wee n the op erations on C br ( a ) and the induced op erations on C ( a ). I n p articular, the brace op erations will alw a ys b e denoted using th e symbols { and } . An A [0 , ∞ [ -stucture b ∈ C 1 br ( a ) on a will often be translated into an elemen t µ ∈ C 2 ( a ), which will also b e called an A [0 , ∞ [ -stucture on a . Similarly , w e will sp eak ab out b race algebra and B ∞ -morphisms b et we en Ho c h s c hild complexes C ( a ), C ( a ′ ). One pr o v es: Lemma 2.8 . Cons id er φ ∈ C i,n ( a ) and δ ∈ C j, 0 ( a ) . W e ha v e φ { δ ⊗ n } = ( − 1) n ( i +(( n − 1) / 2) j ) φ ( δ , . . . , δ ) 2.7 The Ho c hschild complex of a cdg category By the previous section a cdg category is a graded quiv er a together with i) c omp ositions µ 2 = m ∈ Q A 0 ,A 1 ,A 2 [ a ( A 1 , A 2 ) ⊗ a ( A 0 , A 1 ) , a ( A 0 , A 2 )] 0 ii) differ entials µ 1 = d ∈ Q A 0 ,A 1 [ a ( A 0 , A 1 ) , a ( A 0 , A 1 )] 1 iii) curvatur e elements µ 0 = c ∈ Q A a ( A, A ) 2 satisfying the iden tities: i) d ( c ) = 0 ii) d 2 = − m ( c ⊗ 1 − 1 ⊗ c ) iii) dm = m ( d ⊗ 1 + 1 ⊗ d ) iv) m ( m ⊗ 1) = m (1 ⊗ m ) R emark 2.9 . Note that ii) differs from the conv en tional d 2 = m ( c ⊗ 1 − 1 ⊗ c ) ([12], [24]). Ho w ev er, it suffices to change c into − c to reco v er the other d efinition. Example 2.10 . L et a b e a linear catego ry . An example of a cdg categ ory is the cate gory PCom ( a ) of precomplexes of a -ob jects. A pr e c omplex o f a -obje cts is a Z -graded a -ob ject C (with C i ∈ a ) together with a Z -graded a -morphism δ C : C − → C of degree 1. W e ha v e PCom ( a )( C, D ) n = Q i a ( C i , D i + n ) and, for f : D − → E and g : C − → D : i) m ( f , g ) i = ( f g ) i = f i + | g | g i ii) d ( f ) = δ E f − ( − 1) | f | f δ D iii) c C = − δ 2 C Inside PCom ( a ), we ha ve the us ual dg category Com ( a ) of complexes C of a -ob jects, for whic h c C = δ 2 C = 0. W e w ill u se the notations Com + ( a ) and Com − ( a ) for the r esp ectiv e categories of b ound ed b elo w and b ounded ab o ve complexes. As an example of the passage from Σ a to a , let us u se (10) to compute the Ho chsc hild differen tial on C ( a ) for a cdg category a . Th e differen tial on C br ( a ) is giv en by h σ c + d + σ − 1 m, −i 8 Hochschild cohomology, the characteristic morphism and derived deforma tions Consider φ ∈ C i,n ( a ). By definition h σ c + d + σ − 1 m, σ 1 − n φ i = dot ( σ c + d + σ − 1 m, σ 1 − n φ ) − ( − 1) 1 − n + i dot ( σ 1 − n φ, σ c + d + σ − 1 m ) The corresp ond ing three term s in terms of C ( a ) are: i) [ c, φ ] = c • φ − ( − 1) deg ( φ )+1 φ • c whic h equals n − 1 X k =0 ( − 1) k +1 φ (1 ⊗ n − k − 1 ⊗ c ⊗ 1 ⊗ k ) ii) [ d, φ ] = d • φ − ( − 1) deg ( φ )+1 φ • d whic h equ als ( − 1) ar φ +1 ( dφ − ( − 1) | φ | n − 1 X k =0 φ (1 ⊗ n − k − 1 ⊗ d ⊗ 1 ⊗ k )) iii) [ m, φ ] = m • φ − ( − 1) deg ( φ )+1 φ • m whic h equals m ( φ ⊗ 1) + n − 1 X k =0 ( − 1) k +1 φ (1 ⊗ n − k − 1 ⊗ m ⊗ 1 ⊗ k ) + ( − 1) n +1 m (1 ⊗ φ ) If we lo ok at the bigraded ob j ect C i,n with i b eing the “ve rtical” grading and n b eing the “hor- izon tal” grading, then d h = [ m, − ] d efines a horizont al c ontribution whereas d v = [ d, − ] defin es a vertic al c ontribution to the H o chsc hild differen tial d . Clearly , up to a factor ( − 1) n +1 , th e horizon tal con tribution generaliz es the classical Ho c hschild differen tial for an associativ e algebra. If we lo ok at the “ n -th column” graded ob ject C ∗ ,n = Y A 0 ,...A n [ a ( A n − 1 , A n ) ⊗ · · · ⊗ a ( A 0 , A 1 ) , a ( A 0 , A n )] then the vertica l contribution on C ∗ ,n is ( − 1) n +1 times the canonical map induced from d . Compared to the dg case, we h a v e a new c urve d c ontribution d c = [ c, − ] w h ic h go es “t w o steps up and one step bac k”. The curved con tribution is zero on the zero part C ∗ , 0 . In the Ho c hschild complex of an arbitrary A [0 , ∞ [ -categ ory , there are additional cont rib utions going “ n steps do wn and n + 1 steps ahead” for n > 1. 3. A B ∞ -section to twisted ob jects Let a b e a qu iv er. As explained in § 2.4, an inclusion a ⊂ a ′ of a as a su b qu iv er of some a ′ induces a morp hism of br ace algebras π : C ( a ′ ) − → C ( a ). T his s ection is dev oted to the construction of certain quive rs a ′ = Tw ( a ) of “t wisted ob jects o v er a ” for which π has a certain br ace algebra section em br δ . The morp hism em br δ will b e used in § 3.3 to tr an s p ort A [0 , ∞ [ -structures from a to Tw ( a ). Quiv ers of twisted complexes encompass the classical t wisted complexes o ver a dg ca tegory ([3], [5], [15]), b u t also the “infinite” quiv ers of semifree dg mo d ules ([5]) a s well as qu iv ers of (pre)complexes o v er a linear category . T h e morphism embr δ is suc h that in those examples, it indu ces the correct A [0 , ∞ [ -structures on these quive rs, th us d efining a B ∞ -section of π . It will b e used in § 4.3 to defin e the char acteristic dg morphism of a linear category a , whic h allo ws us to pro ve T heorem 4.8 and hence Theorem 1.1. Th is c hapter is r elated to ideas in [7], [8], [19]. 3.1 Some quiv ers ov er a Let a b e a quiv er. In this section we define the quiv er Tw free ( a ) of formal copro du cts of sh ifts of a - ob jects twiste d by a morphism of degree 1. First w e d efine the qu iv er Free ( a ). An ob ject of F ree ( a ) is a formal expression M = ⊕ i ∈ I Σ m i A i with I an arb itrary ind ex set, A i ∈ a and m i ∈ Z . F or 9 Wendy Lowen another N = ⊕ j ∈ J Σ n i B i ∈ F ree ( a ), the graded ob ject F ree ( a )( M , N ) is b y definition F ree ( a )( M , N ) = Y i ⊕ j Σ n j − m i a ( A i , B j ) An elemen t f ∈ Fre e ( a )( M , N ) can b e repr esen ted b y a matrix f = ( f j i ), where f j i represent s th e elemen t σ n j − m i f j i . Definition 3.1 . F or M , N as ab o ve , consider a morp hism f ∈ F ree ( a )( M , N ). F or a sub set S ⊂ I , let Φ f ( S ) ⊂ J b e defin ed b y Φ f ( S ) = { j ∈ J | ∃ i ∈ S f j i 6 = 0 } W e sa y that f ∈ Free ( M , M ) is intrinsic al ly lo c al ly nilp otent (iln) if for ev ery i ∈ I th ere exists n ∈ N with Φ n f ( { i } ) = ∅ . Pr op osition 3.2 . The canonical isomorph isms [ a ( A n − 1 , A n ) ⊗ · · · ⊗ a ( A 0 , A 1 ) , a ( A 0 , A n )]   [Σ i n − i n − 1 a ( A n − 1 , A n ) ⊗ · · · ⊗ Σ i 1 − i 0 a ( A 0 , A 1 ) , Σ i n − i 0 a ( A 0 , A n )] (11) for A k ∈ a , n k ∈ Z d efine a morphism of brace algebras C ( a ) − → C ( Free ( a )) : φ 7− → φ (12) with φ ( f n , . . . , f 1 ) j i = X k n − 1 ,...,k 1 ( − 1) ǫ φ (( f n ) j k n − 1 , ( f n − 1 ) k n − 1 k n − 2 , . . . , ( f 2 ) k 2 k 1 , ( f 1 ) k 1 i ) (13) Lemma 3.3 . C onsider φ ∈ C ( a ) and ( f n , . . . , f 1 ) ∈ Free ( a )( M n − 1 , M n ) ⊗ · · · ⊗ Free ( a )( M 0 , M 1 ) . W r ite M 0 = ⊕ i ∈ I Σ α i A i and consider S ⊂ I . There is an inclusion Φ φ ( f n ,...,f 1 ) ( S ) ⊂ Φ f n (Φ f n − 1 ( . . . Φ f 1 ( S ))) Pr o of. S upp ose j is not con tained in the righ t hand side. Then for ev ery sequence j = k n , . . . , k 1 , k 0 = i with i ∈ S one of the en tries ( f p ) k p k p − 1 is zero. But then , looking at the expression (13), clearly φ ( f n , . . . , f 1 ) j i = 0 whence j is not cont ained in the left hand side. Next we d efine th e quiv er Tw free ( a ). An ob ject of Tw free ( a ) is a coup le ( M , δ M ) with M ∈ Fre e ( a ) and δ M ∈ F ree ( a )( M , M ) 1 F or ( M , δ M ) , ( N , δ N ) ∈ Tw free ( a ), Tw free ( a )(( M , δ M ) , ( N , δ N )) = F ree ( a )( M , N ). Consequently , the δ M determine an elemen t δ ∈ C 1 ( Tw free ( a )) The isomorphisms (11) also define a morphism of brace algebras C ( a ) − → C ( Tw free ( a )) : φ 7− → φ (14) whic h is a section of the canonical pro jection morp hism C ( Tw free ( a )) − → C ( a ). In the next section w e sh o w that for certain a ⊂ Tw ⊂ Tw free , π : C ( Tw ( a )) − → C ( a ) (15) has another section dep ending on δ , whic h can b e used to transp ort A [0 , ∞ [ -structures. 10 Hochschild cohomology, the characteristic morphism and derived deforma tions Definition 3.4 . A quiv er of lo c al ly nilp otent twiste d obje cts over a is b y definition a quiv er Tw ( a ) with a ⊂ Tw ( a ) ⊂ T w free ( a ) such that for ev ery φ ∈ C ( a ), for ev ery ( f n , . . . f 1 ) ∈ Tw ( a )( M n − 1 , M n ) ⊗ · · · ⊗ Tw ( a )( M 0 , M 1 ) with M 0 = ⊕ i ∈ I Σ α i A i , and for ev ery i ∈ I there exists m 0 ∈ N suc h th at for all m > m 0 , Φ g ( { i } ) = ∅ for g = φ m + n { δ ⊗ m } ( f n , . . . f 1 ) Example 3.5 . If a is concen trated in degree zero, then T w free ( a ) is a quive r of lo cally n ilp oten t t wisted ob j ects ov er a . In deed, for φ ∈ C ( a ), there is only a single m for whic h the comp onent φ m is different from zero. Pr op osition 3.6 . Let Tw ilnil ( a ) ⊂ Tw free ( a ) b e the quiv er with as ob jects the ( M , δ M ) for whic h δ M ∈ F ree ( a )( M , M ) is in trinsically lo cally nilp oten t. Then Tw ilnil ( a ) is a quiv er of lo cally nilp otent t wisted ob jects o v er a . Pr o of. C onsider φ , ( f n , . . . f 1 ) and i as in Definition 3.4 and pu t δ i = δ M i . F or m ∈ N , consider g m = φ m + n { δ ⊗ m } ( f n , . . . , f 1 ). Th is g m is a sum of expressions g m n ,...,m 0 = φ m + n ( δ ⊗ m n n , f n , δ ⊗ m n − 1 n − 1 , . . . , δ ⊗ m 1 1 , f 1 , δ ⊗ m 0 0 ) with m n + · · · + m 0 = m . F or Φ g m n ,...,m 0 ( { i } ) to b e emp ty , it su ffices by Lemma 3.3 that Φ m n δ n (Φ f n ( . . . (Φ f 1 (Φ m 0 δ 0 ( { i } ))) )) = ∅ (16) W e r ecur siv ely define num b ers p l and finite sets S l for l = 0 , . . . , n in the f ollo wing manner. Put S 0 = { i } . Once S l is defi ned, p l is su c h that Φ p l δ l ( S l ) = ∅ (such a p l exists since δ l is iln) and S l +1 = ∪ p ∈ N Φ f l +1 Φ p δ l ( S l ). By the p igeonhole prin ciple, if m > p n + · · · + p 0 , ev ery g m n ,...,m 0 with m n + · · · + m 0 = m has at least one m l > p l , and consequen tly (16) holds true. Hence in Definition 3.4, it suffices to take m 0 = p n + · · · + p 0 . 3.2 A w ord on top ology Although not strictly n ecessary , it will b e con v enient to use a b it of top ology to u nderstand and reform ulate d efinition 3.4. The language of this section will b e used in the pro of of Prop osition 3.11. All the top ologies we consid er will turn the und erlying k -mo dules in to top ologica l k -mo du les, so in particular w e can sp eak ab out completions. Put C = C br ( Tw ( a )) f or some arbitrary full sub category Tw ( a ) ⊂ Tw free ( a ). T o manipu late certain elemen ts of B Q C = Q n > 0 (Σ C ) ⊗ n that are not in B C , it will b e con v enient to consider a certain completion ˆ B C of B C . As a first step w e endow B C with a complete Hausdorff “p oint wise” top ology T 0 . T o d o so w e supp ose that a is naturally a complete Hausdorff top ologica l k -quiv er, i.e. th e a ( A, A ′ ) are complete Hausdorff top ological k -mo d ules (if there is no natural top ology , the a ( A, A ′ ) are endo wed with the discrete top ology). No w consider the algebra multiplica tion M : B C ⊗ B C − → B C defined b y the brace op erations. W e supp ose that M preserves Cauc h y nets with resp ect to T 0 . F or ev ery φ ∈ C br ( a ) ⊂ B C we consider th e map M φ = M ( φ, − ) : B C − → ( B C , T 0 ) Next w e end ow B C with the “wea k top ology” T ⊂ T 0 whic h is by definition the initial top ology for the collection ( M φ ) φ , and we let ˆ B C d enote the completion of B C with resp ect to T . The M φ ha v e natural con tin uous extensions ˆ M φ : ˆ B C − → B C (17) 11 Wendy Lowen Lemma 3.7 . F or ψ ∈ B C , the map M ψ = M ( − , ψ ) : B C − → B C preserves Cauc hy nets with resp ect to T . C on s equen tly , there is a natural con tin uou s extension ˆ M ψ : ˆ B C − → ˆ B C (18) Pr o of. S upp ose w e hav e a T -Cauch y net ( x α ) α in B C . W e ha ve to sh o w that M ( φ, M ( x α , ψ )) is T 0 -Cauc hy for ev ery φ ∈ C br ( a ). This follo w s since M is asso ciativ e and preserves C auc hy nets. Definition 3.8 . Let a b e a top ological k -quiver. A quiver of twiste d obje cts over a is b y defini- tion a qu iv er a ⊂ Tw ( a ) ⊂ Tw free ( a ) suc h that for the canonical δ ∈ C 1 ( Tw ( a )) th e sequence ( P m k =0 δ ⊗ k ) m > 0 con v erges in ˆ B C to a un ique elemen t e δ = ∞ X k =0 δ ⊗ k R emark 3.9 . W e noticed that the same su ggestiv e exp onent ial notation is used in [8]. Pr op osition 3.10 . Let a b e a k -quiver and consider a ⊂ T w ( a ) ⊂ Tw free ( a ) . T he follo wing are equiv ale nt: i) Tw ( a ) is a quiver of twiste d ob jects o v er a where a is en d o we d with the discrete top ology . ii) Tw ( a ) is a quiver of lo cally nilp oten t twisted ob jects o v er a . Pr o of. By definition of th e completion, the sequence con verge s in ˆ B C if and only if for every φ ∈ C br ( a ), the sequence ( P m k =0 φ { δ ⊗ k } ) m > 0 ) conv erges for the “point wise discrete” top ology T 0 on B C . By definition of this top ology , this means that for ev ery ( f n , . . . , f 1 ) and i ∈ I as in Defin ition 3.4, there exists an m 0 suc h that the general term (( P m k =0 φ { δ ⊗ k } ( f n , . . . , f 1 )( i )) b ecomes constan t for m > m 0 . This is clearly equiv alen t to the fact that the expressions φ { δ ⊗ k } ( f n , . . . , f 1 )( i ) b ecome zero for k > m 0 . 3.3 T ransp ort of A [0 , ∞ [ -structures to Tw ( a ) Let a b e a top ological quiver and consider the inclusion a ⊂ Tw ( a ) of a into a quiver of twisted ob jects o v er a (in p articular, T w ( a ) can b e a quiv er of lo cally nilp oten t t wisted ob jects ov er an arbitrary qu iv er a ). Let δ ∈ C 1 ( Tw ( a )) b e the canonical Ho chsc hild cochain of Tw ( a ). Pr op osition 3.11 . The canonical pro jectio n π : C ( Tw ( a )) − → C ( a ) has a br ace algebra s ection em br δ : C ( a ) − → C ( Tw ( a )) : φ 7− → ∞ X m =0 φ { δ ⊗ m } (19) F or M = ( M , δ M ) ∈ Tw ( a ) and φ ∈ C p ( a ) , the comp onent φ M = (e mbr δ ( φ )) M ∈ Tw ( a )( M , M ) 1 is giv en b y φ M = ∞ X m =0 ( − 1) α φ m ( δ ⊗ m M ) (20) with α = m (( p − m ) + ( m − 1) / 2) . R emark 3.12 . If char( k ) = 0, the map embr δ is th e Lie morphism e [ − ,δ ] . Pr o of. According to Definition 3.8, we disp ose of an element e δ = P ∞ k =0 δ ⊗ k ∈ ˆ B C . W e defin e em br δ to b e the r estriction of the morph ism ˆ M ( − , e δ ) : B C ( a ) − → B C ( Tw ( a )) (21) 12 Hochschild cohomology, the characteristic morphism and derived deforma tions whic h exists by § 3.2. I n particular, the right hand side of (19) should b e read as a p oint wise series, i.e. f or ( f n , . . . , f 1 ) ∈ Tw ( a )( M n − 1 , M n ) ⊗ · · · ⊗ Tw ( a )( M 0 , M 1 ) where M 0 = ⊕ i ∈ I Σ α i A i and M n = ⊕ j ∈ J Σ β j B j , we ha ve (( ∞ X m =0 φ { δ ⊗ m } )( f n , . . . , f 1 )) j i = ∞ X m =0 (( φ { δ ⊗ m } ( f n , . . . , f 1 )) j i ) and the right hand side conv erges for the top ology of a . Next we verify that (21) is a morph ism of algebras, i.e. preserves the multiplica tion M . Consider φ, ψ ∈ B C br ( a ). W e ha ve ˆ M ( ˆ M ( φ, ψ ) , e δ ) = ˆ M ( φ, ˆ M ( ψ , e δ )) = ˆ M ( ˆ M ( φ, e δ ) , ˆ M ( ψ , e δ )) where w e us ed asso ciativit y of M , con tinuit y of (17) and (18) and the fact that ˆ M ( e δ , ψ ) = ψ . Finally , the statemen t (20) follo ws f rom Lemma 2.8. Com binin g Prop osition 3.11 with Prop osition 2.6, we get: Pr op osition 3.13 . i) If µ is an A [0 , ∞ [ -structure on a , then em br δ ( µ ) is an A [0 , ∞ [ -structure on Tw ( a ) and em br δ : ( a , µ ) − → ( Tw ( a ) , embr δ ( µ )) is a B ∞ -morphism. ii) If µ = c + d + m is a cdg stru cture on a , then em br δ ( µ ) = ( c + d { δ } + m { δ , δ } ) + ( d + m { δ } ) + m is a cdg stru cture on Tw ( a ) . iii) If µ = d + m is a dg structure on a and δ ∈ C 1 ( Tw ( a )) satisfies d { δ } + m { δ , δ } = 0 then em br δ ( µ ) = ( d + m { δ } ) + m is a dg structure on Tw ( a ) . F r om no w on, quiv ers of twisted ob jects o v er an A [0 , ∞ [ -categ ory ( a , µ ) w ill alwa ys b e endow ed with the A [0 , ∞ [ -structure embr δ ( µ ). R emark 3.14 . A similar kin d of “transp ort” is us ed in [19, § 6] in order to constru ct A ∞ -functor catego ries. 3.4 Classical twisted complexes W e will no w discuss how some classical categories of twisted complexes fit in the fr amew ork of the previous sections. Definition 3.15 . Let a = ( a , µ ) b e an A [0 , ∞ [ -categ ory . The ∞ -p art of a is the f ull sub category a ∞ ⊂ a w ith as ob jects th e A ∈ a f or whic h µ A ∈ a ( A, A ) 2 is zero. Example 3.16 . Let a b e an A ∞ -categ ory and let tw ilnil ( a ) ⊂ Tw ilnil ( a ) b e the quiver with as ob jects the ( M , δ M ) wh ere M = ⊕ k i =0 Σ m i A i is “fin ite”. i) If a is a dg categ ory , then th e dg categ ory t w ilnil ( a ) ∞ is equiv alen t to the classica l d g cat egory of t wisted complexes o v er a ([3], [5], [15]). In d eed, the ∞ -part of t w ilnil ( a ) is its restriction to the ob jects ( M , δ M ) with d { δ M } + m { δ M , δ M } = 0 13 Wendy Lowen More generally , t w ilnil ( a ) ∞ is equiv alent to th e A ∞ -categ ory tw ( a ) of t wisted ob j ects o v er a ([19], and [8] for the algebra case). ii) The dg catego ry Tw ilnil ( a ) ∞ is equiv alen t to the classical dg category of semifree complexes o v er a ([5]) whic h is a dg-mod el f or D ( a ), i.e. there is an equiv alence of triangulated categories H 0 ( Tw ilnil ( a ) ∞ ) ∼ = D ( a ). R emark 3.17 . W e conjecture that for an A ∞ -categ ory a , the A ∞ -categ ory T w ilnil ( a ) ∞ is an A ∞ -mo del for the d eriv ed catego r y o f a , i.e. there is an equiv alence of triangulated catego ries H 0 ( Tw ilnil ( a ) ∞ ) ∼ = D ∞ ( a ), where for definitions of the righ t hand side, we refer the reader to [19]. The finite v ersion of this r esult has b een obtained in [19, § 7.4]. R emark 3.18 . F or an A [0 , ∞ [ -categ ory a , w e hav e t w ilnil ( a ) ∞ = t w ilnil ( a ∞ ) ∞ and similarly for Tw ilnil ( a ). In particular, an A [0 , ∞ [ -algebra A with µ 0 6 = 0 has Tw ilnil ( a ) ∞ = 0. This illustrates the p o or “deriv abilit y”, in general, of A [0 , ∞ [ -algebras. The follo wing theorem, whic h immediately follo ws f r om Prop osition 3.1 1 , is a refinemen t of [21, Theorem 4.4.1]. Theorem 3.19 . Let a b e a dg category . Then Tw ( a ) = T w ilnil ( a ) ∞ is a dg category whic h is dg equiv ale nt to th e category of semifree dg m o dules ov er a . T he canonical p ro jection π : C ( Tw ( a )) − → C ( a ) has a B ∞ -section em br δ : C ( a ) − → C ( Tw ( a )) : φ 7− → ∞ X m =0 φ { δ ⊗ m } whic h is an in verse in the h omotop y category of B ∞ -algebras. In particular, b oth π and em br δ are quasi-isomorphisms. 3.5 (Pre)complexes o ver linear categories Next w e apply Prop osition 3.11 to categories of (pre)complexes. Let ( a , m ) b e a linear category . Consider the quiv er Tw p re ( a ) with as ob jects ( M = ⊕ i ∈ Z Σ i A i , δ M ) with δ M ∈ F ree ( a )( M , M ) 1 . F or another ( N = ⊕ i ∈ Z Σ i B i , δ N ), since a is c oncentrat ed in degree zero, w e h a v e Tw p re ( a )( M , N ) n = Q i ∈ Z a ( A i , B i − n ). If we c hange to cohomologic al notation A i = A − i , w e h a v e Tw p re ( a )( M , N ) n = Y i ∈ Z a ( A i , B i + n ) By E xample 3.5, T w p re ( a ) is a qu iv er of lo cally n ilp oten t twiste d ob jects o ver a . According to Prop osition 3.13, the corresp ond in g A [0 , ∞ [ -structure on Tw p re ( a ) is em br δ ( m ) = m { δ, δ } + m { δ } + m with m { δ , δ } = − δ 2 and m { δ } = m ( δ ⊗ 1 − 1 ⊗ δ ) Hence Tw p re ( a ) is precisely th e cdg categ ory PCom ( a ) of precomplexes of a -ob jects of example 2.10. The category Tw com ( a ) = Tw p re ( a ) ∞ is the dg category Com ( a ) of complexes of a -ob jects. Consider the inclusions a ⊂ Co m + ( a ) ⊂ Com ( a ) The follo wing is implicit in [21]: 14 Hochschild cohomology, the characteristic morphism and derived deforma tions Pr op osition 3.20 . The canonical pro jection π : C ( Com + ( a )) − → C ( a ) is a B ∞ -quasi-isomorphism. Pr o of. C onsider the canonical morp hisms a op − → Com − ( a op ) − → Com − ( Mo d ( a op )) − → Com ( Mo d ( a op )) = Mod dg ( a op ) A complex in Com − ( a op ) gets mapp ed to a cofibrant ob j ect in M o d dg ( a op ). Consequently , by [21, Theorem 4.4.1], th e fir s t map in duces a B ∞ -quasi-isomorphism. T he r esult follo ws since π is indu ced b y the opp osite of th is map. Theorem 3.21 . Th e canonical pro jection π : C ( PCom ( a )) − → C ( a ) has a B ∞ -section em br δ : C ( a ) − → C ( PCom ( a )) : φ n 7− → n X m =0 φ n { δ ⊗ m } The restrictions of b oth maps to C ( Com + ( a )) are inv erse isomorphisms in the h omotop y category of B ∞ -algebras. In particular, they are b oth quasi-isomorphisms . 3.6 Ab elia n categories The results of the previous section hav e an immediate application to ab elian categories. L et A b e an ab elian categ ory . In [21], the Ho c hs c hild complex of A is defined to b e C ab ( A ) = C ( Inj ( Ind ( A )) Let A b e an ab elian category with enough injectiv es and pu t i = Inj ( A ). By [21, Theorem 6.6], w e h a v e C ab ( A ) ∼ = C ( i ) and it w ill b e conv enien t to actually tak e this as definition of C ab ( A ). The dg category Com + ( i ) of b ou n ded below complexes of injectiv es is a dg mo del for the b ound ed b elo w derived category D + ( A ) of A , whence the n otation D + dg ( A ) = Com + ( i ). In th e spirit of [14], put C ex ( A ) = C ( D + dg ( A )). With a = i , T heorem 3.21 no w yields: Theorem 3.22 . Th e canonical pro jection π : C ( Com + ( i )) − → C ( i ) h as a B ∞ -section em br δ : C ( i ) − → C ( Com + ( i )) : φ n 7− → n X m =0 φ n { δ ⊗ m } whic h is an in verse in the h omotop y category of B ∞ -algebras. In particular, b oth π and em br δ are quasi-isomorphisms establishing C ab ( A ) ∼ = C ex ( A ) . 4. Deformations This c hapter consists largely of applications of Theorem 3.21. W e first recall some facts on defor- mations and th e graded cen tre enablin g us to defin e, in § 4.3 , th e c haracteristic d g m orphism of a linear category , an d to sho w its relation to deformation theory in T heorem 4.8. The remainder of the chapter is dev oted to some app lications to deformations of (enhanced) derive d cate gories of ab elian categories. Throughout w e fo cus on first or der d eformations, i.e. d eformations along k [ ǫ ] − → k , since they are in the most direct corresp ondence with Ho c hsc hild cohomology . All d efinitions can b e giv en for arbitrary deformations, and in the classical setting of an Ar tin lo cal algebra R o ver a field k of c haracteristic zero (with maximal id eal m ), the deformation theory is go ve rn ed b y the Maurer- Cartan equation in the Ho c hschild complex (tensored by m ). F r om n o w on , k will b e a field. 15 Wendy Lowen 4.1 Deformations of linear and ab elian categories The d eformation th eory of linear and ab elian categories was devel op ed in [22] as a natur al extension of Gerstenhab er’s deformation th eory of algebras [9]. In this section w e recall the main defin itions. F or a commuta tive ring R , let cat ( R ) denote the (large) category of R -linear categories. The forgetful functor cat ( k ) − → cat ( k [ ǫ ]) has th e left adjoin t k ⊗ k [ ǫ ] − : cat ( k [ ǫ ]) − → cat ( k ) and the righ t adjoin t Hom k [ ǫ ] ( k , − ) : c at ( k [ ǫ ]) − → cat ( k ) where Hom k [ ǫ ] denotes the category of k [ ǫ ]-linear fu nctors. Clearly , for B ∈ cat ( k [ ǫ ]), there is a canonical inclus ion fun ctor Hom k [ ǫ ] ( k , B ) − → B iden tifying Hom k [ ǫ ] ( k , B ) with the fu ll s ub category of ob jects B ∈ B for wh ic h ǫ : B − → B is equal to zero. In [22], a notion of flatness for ab elian R -linear categories is defined which is su c h that an R - linear category a is flat (in the sense th at it has R -flat hom-mo dules) if and only if the mo dule catego ry Mo d ( a ) is fl at as an ab elian category . Definition 4.1 . i) Let a b e a k -linear category . A fi rst ord er line ar deformation of a is a flat k [ ǫ ]-linear category b together with an isomorph ism k ⊗ k [ ǫ ] b ∼ = a in cat ( k ). ii) Let A b e an ab elian k -linear category . A fi rst ord er ab elian deformation of A is a flat ab elian k [ ǫ ]-linear category B together with an equiv alence of catego ries A ∼ = Hom k [ ǫ ] ( k , B ). W e will denote the natural group oids of linear deformations of a and of abelian deformations of A b y Def a ( k [ ǫ ]) and a b − De f A ( k [ ǫ ]) resp ectiv ely . In [22], the n otations def s a and De f A are used and the terminology strict d eformation is used in the lin ear case. The follo wing p rop osition extends the w ell kn o wn result for algebras: Pr op osition 4.2 . Let a b e a k -linear category . There is a map Z 2 C ( a ) − → Ob(Def a ( k [ ǫ ])) whic h induces a bijection H H 2 ( a ) − → Sk(Def a ( k [ ǫ ])) Pr o of. C onsider φ ∈ Z 2 C ( a ). The co cycle φ describ es the corresp ondin g linear deformation of ( a , m ) in the follo wing w a y . Cons id er the quiv er a [ ǫ ] = k [ ǫ ] ⊗ k a ov er k [ ǫ ]. T he linear deformation of a is a φ [ ǫ ] = ( a [ ǫ ] , m + φǫ ). Finally we mention the follo wing fundamenta l resu lt of [21 ], where the Ho chsc hild cohomology of the ab elian category A is as d efined in § 3.6. Pr op osition 4.3 . Let A b e a k -linear ab elian category . There is a bijection H H 2 ab ( A ) − → ab − Def A ( k [ ǫ ]) 4.2 The cen tre of a graded category W e recall the defin ition of the cen tre of a graded category (see also [4, § 3]). Definition 4.4 . Let a b e a graded category . The c entr e of a is the centre of a as a catego ry enric hed in G ( k ), i.e. Z ( a ) = Hom(1 a , 1 a ) where 1 a : a − → a is the identit y functor and Hom denotes the graded mo dule of graded natural transformations. 16 Hochschild cohomology, the characteristic morphism and derived deforma tions R emark 4.5 . Ex p licitely , an elemen t in Z ( a ) is giv en by an element ( ζ A ) A ∈ Q A ∈ a a ( A, A ) with the naturalit y prop erty that for all A, A ′ ∈ a , th e follo wing diagram comm utes: a ( A, A ′ ) ζ ⊗ 1 / / 1 ⊗ ζ   a ( A ′ , A ′ ) ⊗ a ( A, A ′ ) m   a ( A, A ′ ) ⊗ a ( A, A ) m / / a ( A, A ′ ) In other w ords, for f ∈ a ( A, A ′ ), ζ A ′ f = ( − 1) | f || ζ | f ζ A R emark 4.6 . Let T b e a susp ende d linear categ ory with susp ension Σ T : T − → T . There is an asso ciated graded category T g r with T g r ( T , T ′ ) n = T ( T , Σ n T T ′ ) and the gr ade d c entr e of T is the cen tre of the graded c ategory T g r . If t is an exact d g ca tegory with associated triangulated ca tegory T = H 0 t , we ha v e T g r = H ∗ t . 4.3 The c haracteristic dg morphism It is w ell known that f or a k -algebra A , there is a charact eristic morph ism of graded commutativ e algebras from the Ho c hschild cohomology of A to the graded cent re of the deriv ed category D ( A ). This morph ism is determined by the maps, for M ∈ D ( A ), M ⊗ L A − : H H ∗ k ( A ) ∼ = Ext ∗ A op ⊗ A ( A, A ) − → Ext ∗ A ( M , M ) The charact eristic morphism o ccurs for example in the theory of supp ort v arieties ([1], [6], [25]). Recen tly , Buch w eitz and Flenner pro ve d the existence of a c haracteristic morph ism in the cont ext of morphisms of sc hemes or analytic spaces ([4]). In [21], it is observe d th at a characte ristic morphism also exists for ab elian cat egories. Let A be an ab elian categ ory with en ou gh injective s, i = Inj ( A ) and Com ( i ) the dg category of complexes of injectiv es. As asserted in Pr op osition 2.7, there is a morp hism of d ifferen tial graded ob jects π 0 : C ( Com ( i )) − → Y E ∈ Com ( i ) Com ( i )( E , E ) T aking cohomology of π 0 (where w e restrict to Com + ( i )) and comp osing with the isomorp hisms H H ∗ ab ( A ) ∼ = H H ∗ ex ( A ) of Theorem 3.22, w e obtain the c haracteristic morphism χ A : H H ∗ ab ( A ) − → Z ∗ D + ( A ) Using the B ∞ -section of Prop osition 3.21 , w e can actuall y lift the c haracteristic m orphism to the lev el of dg ob jects. In fact we can construct this lifted c haracteristic morphism for an arbitrary k -linear category a in stead of i . Definition 4.7 . Let a b e a k -linear category . The char acteristic dg morphism C ( a ) − → Y C ∈ Com ( a ) Com ( a )( C, C ) is the comp osition of the B ∞ -morphism C ( a ) − → C ( Com ( a )) of Theorem 3.21 and the pro jection on the zero part of Prop osition 2.7. T aking cohomology , w e obtain the c har acteristic morphism H H ∗ ( a ) − → Z ∗ K ( a ) where K ( a ) is the h omotop y categ ory of complexes of a -ob jects. In the n ext section we will interpret the charact eristic morphism in terms o f deformatio n theory . 17 Wendy Lowen 4.4 The c haracteristic morphism and obstructions Let a b e a k -linear category . In [20], an obstru ction theory is established for deforming ob jects of the h omotop y category K ( a ). L et c ∈ Z 2 C ( a ) b e a Ho chsc hild co cycle and a c [ ǫ ] the corresp ondin g linear deformation. Consider the fun ctor k ⊗ k [ ǫ ] − : K ( a c [ ǫ ]) − → K ( a ) and consider C ∈ K ( a ). W e w ill say that a (homoto py) c -deform ation of C is a lift of C al ong k ⊗ k [ ǫ ] − . According to [20, T heorem 5.2], first ord er c -deformations of C are go v erned by an obstru ction theory in vo lving K ( a )( C, C [2]) and K ( a )( C, C [1]). In particular, the obstruction against c -deforming C is an elemen t o c ∈ K ( a )( C, C [2]) dep end ing on c , whereas K ( a )( C, C [2]) itself is indep en den t of c . In the remainder of th is section we sho w th at the wa y in w hic h the obstruction o c dep end s on c is enco ded in the charac teristic morp hism. Theorem 4.8 . Let a b e a linear category and consider the characte ristic morphism χ a : H H 2 ( a ) − → Z 2 K ( a ) W e ha ve χ a ( c ) = ( o C ) C ∈ K ( a ) where o C ∈ K ( a )( C, C [2]) is the ob s truction to c -deforming C into an ob ject of K ( a c [ ǫ ]) . Pr o of. Let ¯ χ a b e the c haracteristic dg morph ism C 2 ( a ) − → Q C ∈ Com ( a ) Com ( a ) 2 ( C, C ) enhancing χ a . C on s ider C = ( C, δ C ) ∈ Co m ( a ) and φ ∈ C 2 ( a ). According to T h eorem 3.21, w e ha v e ( ¯ χ a ( φ )) C = − φ ( δ C , δ C ) According to [20, Theorems 3.8, 4.1], [ φ ( δ C , δ C )] is the obstru ction to c -deforming C . Corollar y 4.9 . Let A b e an ab elian category with enough injectiv es. Th e charac teristic morphism χ A : H H 2 ab ( A ) − → Z 2 D + ( A ) satisfies χ A ( c ) = ( o C ) C ∈ D + ( A ) where o C ∈ Ext 2 A ( C, C ) is the obstruction to d eforming C into an ob j ect of D + ( A c ) . F or C ∈ D b ( A ) , this is equally the obstruction to deforming C into an ob ject of D b ( A c ) . Pr o of. T his easily follo ws from Theorem 4.8 since D + ( A ) ∼ = K + ( Inj ( A )). The last s tatemen t follo ws from [20, § 6.3] . 4.5 A [0 , ∞ [ -deformations In this section, we discuss the s en se in wh ic h the Ho chsc hild cohomology of an A [0 , ∞ [ -categ ory a describ es its first order A [0 , ∞ [ -deformations. The easiest (though not n ecessarily th e b est, see Remark 4.17 and § 4.7) deformations to hand le are those with fi x ed set of ob jects. Note that the “flatness” automatica lly imp osed in this definition is graded freeness, w hic h do es not imp ly cofibrancy in the dg-case! Definition 4.10 . Let a b e a k -linear A [0 , ∞ [ -categ ory where, in the en tire defin ition, A [0 , ∞ [ - can b e replaced by A ∞ -, cdg, d g or blanc. i) A first or der A [0 , ∞ [ -deformation of a is a structure of k [ ǫ ]-linear A [0 , ∞ [ -categ ory on a k [ ǫ ]-quiver b ∼ = k [ ǫ ] ⊗ k a , suc h that its r eduction to a coincides with the A [0 , ∞ [ -structure of a (in other w ords, the canonical k ⊗ k [ ǫ ] b ∼ = a is an A [0 , ∞ [ -isomorphism). 18 Hochschild cohomology, the characteristic morphism and derived deforma tions ii) A p artial first or der A [0 , ∞ [ -deformation of a is an A [0 , ∞ [ -deformation of a ′ for some full A [0 , ∞ [ - sub catego ry a ′ ⊂ a . iii) Let b and b ′ b e (partial) deformations of a . An isomorphism of (p artial) deformations is an isomorphism g : b − → b ′ of A [0 , ∞ [ -categ ories, of which the redu ction to a (resp. a ′ in case of partial deformations) is the iden tit y morphism. A morphism of p artial deformations is an isomorphism of deformations b et wee n b and a f u ll A [0 , ∞ [ -sub category of b ′ . iv) A p artial deformation b of a is called maximal if ev ery morp hism b − → b ′ of p artial deforma- tions is an isomorphism . v) The group oid A [0 , ∞ [ − Def a ( k [ ǫ ]) has as ob jects firs t ord er A [0 , ∞ [ -deformations of a . Its m or- phisms are isomorphisms of deformations. vi) The category A [0 , ∞ [ − pDef a ( k [ ǫ ]) h as as ob jects first ord er partial A [0 , ∞ [ -deformations of a . Its morph isms are morphisms of p artial deformations. vii) The group oid A [0 , ∞ [ − mpDef a ( k [ ǫ ]) has as ob jects maximal p artial A [0 , ∞ [ -deformations of a . Its morph isms are isomorphisms of partial deformations. viii) The group oid M C a ( k [ ǫ ]) has as ob jects Z C 2 ( a ). F or c, c ′ ∈ Z C 2 ( a ), a morphism c − → c ′ is an elemen t h ∈ C 1 ( a ) w ith d ( h ) = c ′ − c . Pr op osition 4.11 . Le t a b e a k -linear A [0 , ∞ [ -categ ory . i) There is an equiv alence of categories M C a ( k [ ǫ ]) − → A [0 , ∞ [ − Def a ( k [ ǫ ]) ii) Consequently , there is a bij ection H H 2 ( a ) − → S k( A [0 , ∞ [ − Def a ( k [ ǫ ])) Pr o of. Let µ b e the A [0 , ∞ [ -structure on a an d consider φ ∈ Z 2 C ( a ). The image of φ is the A [0 , ∞ [ - catego ry A φ [ ǫ ] = ( a [ ǫ ] , µ + φǫ ). T o s ee that µ + φǫ is an A [0 , ∞ [ -structure, it suffices to compute µ + φǫ { µ + φǫ } = µ { µ } + [ µ { φ } + φ { µ } ] ǫ whic h is zero since µ is an A [0 , ∞ [ -structure and φ is a Ho chsc hild co cycle. Next consider a morph ism of co cycles h : φ − → φ ′ . The image of h is the morphism 1 + hǫ : B a φ [ ǫ ] − → B a φ ′ [ ǫ ]. The i d entit y d ( h ) = φ ′ − φ easily implies t h e compatibilit y of 1 + hǫ with the resp ectiv e A [0 , ∞ [ -structures. Definition 4.12 . Consider a k -linear A ∞ -categ ory a and φ ∈ H H 2 ( a ). The φ − ∞ -part of a is the full su b category a φ −∞ ⊂ a w ith Ob( a φ −∞ ) = { A ∈ a | 0 = H 2 ( π 0 )( φ ) ∈ H 2 ( a ( A, A )) } where π 0 is as in § 2.5. Example 4.13 . Consid er a linear category a and φ ∈ H H 2 ( a ). Put φ ′ = [em br δ ]( φ ) ∈ H H 2 ( Com ( a )). W e ha ve Ob( Com ( a ) φ ′ −∞ ) = { C ∈ Com ( a ) | 0 = χ a ( φ ) C ∈ K ( a )( C, C [2]) } Pr op osition 4.14 . Le t a b e a k -linear A ∞ -categ ory . i) There is a morphism A [0 , ∞ [ − Def a ( k [ ǫ ]) − → A ∞ − pDef a ( k [ ǫ ]) : b 7− → b ∞ where b ∞ is as in Definition 3.15. ii) There is a m orphism H H 2 ( a ) − → Sk( A ∞ − pDef a ( k [ ǫ ])) mapping φ ∈ H H 2 ( a ) to an A ∞ -deformation of a φ −∞ ⊂ a . 19 Wendy Lowen Pr o of. F or (ii), let ¯ φ ∈ Z 2 C ( a ) b e a Ho c hschild co cycle with [ ¯ φ ] = φ and let ¯ φ ′ b e its restric- tion to Z 2 C ( a φ −∞ ). Th en ( ¯ φ ′ ) 0 ∈ Q A ∈ a φ −∞ a 2 ( A, A ) is a cob oun dary , h ence there exists h ∈ Q A ∈ a φ −∞ a 1 ( A, A ) with d a ( h ) = ( ¯ φ ′ ) 0 . If we consider h as an element of C 1 ( a ), then ¯ φ ′′ = ¯ φ ′ − d ( h ) is a represen tativ e of φ with ( ¯ φ ′ + d ( h )) 0 = 0. Consequen tly , ( A φ −∞ ) ¯ φ ′′ [ ǫ ] is a partial A ∞ -deformation of a corresp onding to φ . 4.6 Deformations of categories of ( pre)complexes Let a b e a k -linear category . In this section we use Theorem 3.21 to asso ciate to a linear d eformation of a , a cdg deformatio n of the cdg catego ry PCom ( a ) of pr ecomplexes of a -ob j ects, and a partial dg deformation of the dg catego ry Com ( a ) of complexes of a -ob jects. Com binin g Theorem 3.21 and Prop osition 4.11 w e obtain a functor M C a ( k [ ǫ ]) − → M C PCom ( a ) ( k [ ǫ ]) − → A [0 , ∞ [ − Def PCom ( a ) ( k [ ǫ ]) factoring through a “realization” functor R : M C a ( k [ ǫ ]) − → cdg − Def PCom ( a ) ( k [ ǫ ]) whose restriction to cdg − Def Com + ( a ) ( k [ ǫ ]) is an equiv alence. Similarly , using Pr op osition 4.14(2), there is a map ρ ′ : H H 2 ( a ) − → H H 2 ( Com + ( a )) − → Sk(dg − pDef Com + ( a ) ( k [ ǫ ])) Theorem 4.15 . Consid er φ ∈ Z 2 C ( a ) and the corresp onding linear deformation a φ [ ǫ ] . i) The cdg deformation R ( φ ) of PCom ( a ) is (isomorph ic to) the su b category PCom triv ( a φ [ ǫ ]) ⊂ PCom ( a φ [ ǫ ]) consisting of the “trivial” pr ecomplexes ¯ C = k [ ǫ ] ⊗ k C for C ∈ PCo m ( a ) . ii) F or ev ery collecti on of precomplexes Γ = { ¯ C } C ∈ PCom ( a ) where k ⊗ k [ ǫ ] ¯ C = C , the sub cate- gory PCom Γ ( a φ [ ǫ ]) ⊂ PCom ( a φ [ ǫ ]) spanned b y Γ is a cd g d eformation of PCom ( a ) w hic h is isomorphic to R ( φ ) . iii) F or ev ery collection of complexes Γ = { ¯ C } C ∈ Com + ( a ) φ −∞ where k ⊗ k [ ǫ ] ¯ C = C , the su b category Com + Γ ( a φ [ ǫ ]) ⊂ Com + ( a φ [ ǫ ]) sp anned by Γ is a maximal partial dg deformation of Com + ( a ) represent ing ρ ′ ([ φ ]) . Consequent ly , ρ ′ factors o v er an injection ρ : H H 2 ( a ) − → Sk(dg − mpDef Com + ( a ) ( k [ ǫ ])) The image consists of th ose maximal partial dg deformations that are dg deformations of s ome a ′ with a ⊂ a ′ ⊂ Com + ( a ) . R emark 4.16 . According to Theorem 4.15 iii), the part o f Com + ( a ) that “dg- deform s” with resp ect to φ ∈ H H 2 ( a ) is spanned by the ob jects { C ∈ Com + ( a ) | 0 = χ a ( φ ) C ∈ K ( a )( C, C [2]) } R emark 4.17 . Morally , Th eorem 4.15 ii) suggests that we ma y consider PCom ( a φ [ ǫ ]) as a represen- tativ e of the class of cdg deformations of PCom ( a ) corresp ond ing to the element [ φ ] ∈ H H 2 ( a ). T o mak e this s tatemen t m athematical ly tru e, one needs a somewh at more relaxed notion of d eforma- tion (and isomorp hism) in w hic h the ob ject set is not n ecessarily pr eserv ed. C learly , the statemen t is true for any such notion of wh ic h Defin ition 4.10 with the isomorp h isms r elaxed to fully faithful morphisms that are surj ectiv e on ob jects is a sp ecial case. In the same spirit, iii) s uggests that w e ma y consider Com + ( a φ [ ǫ ]) as a represen tativ e of the class of maximal p artial dg deform ations of Com + ( a ) corresp ondin g to [ φ ]. 20 Hochschild cohomology, the characteristic morphism and derived deforma tions Pr o of. T here is a canonical morphism of k [ ǫ ]-quiv ers F : PCom ( a φ [ ǫ ]) − → k [ ǫ ] ⊗ k PCom ( a ) defined in the follo wing manner. A precomplex ¯ C of a φ [ ǫ ]-ob jects gets m app ed to C = k ⊗ k [ ǫ ] ¯ C ∈ PCom ( a ). F or t wo precomplexes ¯ C an d ¯ D , PCom ( a φ [ ǫ ]) n ( ¯ C , ¯ D ) = Q i ∈ Z a φ [ ǫ ]( C i , D i + n ) ∼ = k [ ǫ ] ⊗ PCom ( a ) n ( C, D ). T his d efines F . F rom now on we will tacitly us e F to identify the left and the righ t hand side. Let us den ote the comp osition of a b y m . By definition, the co mp osition of a φ [ ǫ ] is m + φǫ . W rite δ for the predifferentia ls in PCom ( a ) and ¯ δ = δ + δ ′ ǫ for the pr ed ifferen tials in (a fu ll sub category of ) PCom ( a φ [ ǫ ]). By Examples 2.10, 3.5, the cdg structure on PCom ( a ) is giv en by µ = m { δ , δ } + m { δ } + m and the cdg structure on PCom ( a φ [ ǫ ]) is giv en by ˜ µ = ( m + φǫ ) { δ + δ ′ ǫ, δ + δ ′ ǫ } + ( m + φǫ ) { δ + δ ′ ǫ } + m . This expression can b e rewritten as ˜ µ 0 = m { δ , δ } + [ m { δ , δ ′ } + m { δ ′ , δ } + φ { δ , δ } ] ǫ ˜ µ 1 = m { δ } + [ m { δ ′ } + φ { δ } ] ǫ ˜ µ 2 = m + φǫ On the other hand, we ha v e em br δ ( φ ) = φ { δ, δ } + φ { δ } + φ so the cdg structure on PCom ( a ) em br δ ( φ ) [ ǫ ] is ¯ µ with ¯ µ 0 = m { δ , δ } + φ { δ , δ } ǫ ¯ µ 1 = m { δ } + φ { δ } ǫ ¯ µ 2 = m + φǫ Comparing ˜ µ and ¯ µ , it b ecomes clear that on trivial precomplexes (where δ ′ = 0), th ey coincide. This already p ro ve s i). T o pro d uce a deformati on isomorph ic to ¯ µ , it is, b y Pr op osition 4.11, allo w ed to c hange em br δ ( φ ) up to a Ho chsc hild cob oun dary . F or a coll ection Γ as in (2), consider the cor- resp ond ing δ ′ ∈ C 1 ( PCom ( a )). Using Definition 2.1 and the defi nition of the Hoc hschild differen tial d ( § 2.3), it b ecomes clear that ˜ µ = ¯ µ + d ( δ ′ ) ǫ th us pro ving ii). No w consider a collection Γ of complexes as in iii). Obvio usly Com + Γ ( a φ [ ǫ ]) defin es a dg de- formation of Com + ( a ) φ −∞ hence a p artial deform ation of Com + ( a ). By the r easoning ab o ve, d g deformations of a ′ ⊂ Com + ( a ) isomorphic to ¯ µ | a ′ are precisely giv en by ¯ µ η = ¯ µ | a ′ + d ( η ) ǫ for some η ∈ C 1 ( a ′ ). The existence of an η for whic h ( ¯ µ η ) 0 = 0 (and hence f or wh ic h the d eformation is dg) is equiv alen t to th e existence of δ ′ ∈ Q C ∈ a ′ a ′ ( C, C ) 1 with ( d ( δ ′ )) 0 = φ ( δ , δ ), in other words to the fact that 0 = χ a ( φ ) C ∈ H 2 Com + ( a )( C, C ) for eve ry C ∈ a ′ . C learly , a ′ = Com + ( a ) φ −∞ is m aximal with this prop ert y . Finally , the statement concerning ρ easily follo ws from the observ ation that for every [ φ ] ∈ H H 2 ( a ), a ⊂ Com + ( a ) φ −∞ . 4.7 Deformations of derived categories Let A b e a n ab elian c ategory with e n ou gh injectiv es. Putting a = Inj ( A ) in t he previous section, w e obtain a bijection Sk( R ) : H H 2 ab ( A ) − → Sk(cdg − Def D + dg ( A ) ( k [ ǫ ])) 21 Wendy Lowen and the morphism ρ ′ translates in to ρ ′ : H H 2 ab ( A ) − → Sk(dg − pDef D + dg ( A ) ( k [ ǫ ])) The follo wing no w immediately follo ws from Th eorem 4.15: Theorem 4.18 . Consider φ ∈ Z 2 C ab ( A ) and the corresp onding ab elian d eformation A φ of A . Consider the sub catego ry D + dg ( A ) φ −∞ ⊂ D + dg ( A ) sp anned by the complexes C with 0 = χ A ( φ ) C ∈ D + ( A )( C, C [2]) F or every col lection Γ = { ¯ C } C ∈ D + dg ( A ) φ −∞ of b ounded b elo w complexes of A φ -injectiv es with k ⊗ k [ ǫ ] ¯ C = C , the sub catego ry D + dg , Γ ( A φ ) ⊂ D + dg ( A φ ) spanned by Γ is a dg deformation of D + dg ( A ) φ −∞ and a maximal partial dg d eformation of D + dg ( A ) represent ing ρ ′ ([ φ ]) . Consequen tly , ρ ′ factors o v er an injection ρ : H H 2 ab ( A ) − → Sk(dg − mpDef D + dg ( A φ ) ( k [ ǫ ])) The image consists of those maximal partial dg deformations that are dg deformations of some a with Inj ( A ) ⊂ a ⊂ D + dg ( A ) . R emark 4.19 . Theorem 4.18 suggests that we ma y consider D + dg ( A φ ) as a representa tiv e of the cla ss of maximal p artial dg deformations of D + dg ( A ) corresp onding to [ φ ] (see also Remark 4.17). R emark 4.20 . A “b ound ed” v ersion of Theorem 4.1 8 also holds true: simp ly replace ev ery dg cat egory D + dg in the theorem by its b oun d ed v ersion D b dg ⊂ D + dg spanned by the complexes with b oun ded cohomology (see also [20, § 6.3] ). As the maps S k( R ) and ρ are not entirely satisfactory , we prop ose another sense in whic h to deform (exa ct) dg categ ories, that seems more adapted to (mo dels of ) deriv ed categ ories of ab elian catego ries. F or a commutati ve rin g R , let dgcat ( R ) d enote the (large) category of R -linear dg categories. In [28], T abuada defi n ed a mo del str u cture on dgcat ( R ) f or whic h the weak equiv alences are the quasi-equiv ale n ces of d g categories. Let ho dgcat ( R ) denote the h omotop y category for this mo del structure. In [2 9], T o ¨ en sho wed th at hodgcat ( R ) is a closed tensor catego ry , w ith the deriv ed tensor pro du ct ⊗ L R of dg categorie s, and with an inte rn al hom b et we en dg categories a and b , w h ic h we will denote RH om R ( a , b ), but wh ic h is not a derive d ve r s ion of the in ternal hom of dgcat ( R ) for the ab ov e m o del stru cture (i n fact it does ha v e a derive d in terpretation for another mo del structure defined in [27]). Th e forgetful fu nctor ho dgcat ( k ) − → ho dgcat ( k [ ǫ ]) h as the left adjoint k ⊗ L k [ ǫ ] − : ho dgcat ( k [ ǫ ]) − → ho dgcat ( k ) and the righ t adjoin t RH om k [ ǫ ] ( k , − ) : hodgcat ( k [ ǫ ]) − → ho dgcat ( k ) Definition 4.21 . Let a b e a k -linear dg category . i) A fir st order homot opy dg deformation of a is a k [ ǫ ]-linear d g category b together with an isomorphism k ⊗ L k [ ǫ ] b ∼ = a in ho dgcat ( k ). ii) If a is exac t, a first order exact homo topy dg deformat ion of a is a k [ ǫ ]-linear exact d g category b together with an isomorphism a ∼ = RH om k [ ǫ ] ( k , b ) in ho dgcat ( k ). Using the tec hniques of [29], it is not hard to s h o w th e follo win g Pr op osition 4.22 . Let A b e an abelian k -linear category and supp ose B is a fl at ab elian deforma- tion of A . Then D dg ( B ) is an exact homotop y d g deformation of D dg ( A ) . 22 Hochschild cohomology, the characteristic morphism and derived deforma tions The f u rther in vesti gation of Definition 4.21 (and its v ariati ons with resp ect to other mo del structures on dg categories ([26, 27, 28]) is part of a work in progress. A cknowledgement s The author is grateful to Bernh ard Keller for his contin uous inte rest in, and many pleasan t discus- sions on the topic of this pap er. Referen ces 1 Luchezar L. Avr amov and Ragnar -Olaf Buch weitz, Supp ort varieties and c ohomolo gy over c omplete in- terse ctions , In ven t. Math. 142 (2000), no. 2, 285– 318. MR MR179406 4 (2001j:13 017) 2 Y uri B espalov, V olo dymyr Ly ubashenko, and O leksandr Manzyuk, Close d multic ate gory of pr etriangulate d A ∞ -c ate gories , bo ok in progress, 200 6. 3 A. I. Bondal a nd M. M. Kapranov, F r ame d triangulate d c ate gories , Mat. Sb. 181 (1990), no. 5 , 66 9–68 3. MR MR1055981 ( 91 g:180 10) 4 Ragnar -Olaf Buc hw eitz and Hubert Flenner , Glob al Ho chschild (c o-)homolo gy of singular sp ac es , preprint math.A G/0 6065 93v1. 5 V. Drinfeld, DG quotients of DG c ate gories , J. Algebra 27 2 (2004), no. 2, 643–691. MR MR20280 75 6 Kar in Erdmann, Miles Hollow ay , Ra chel T aillefer, Nicole Snashall, and Øyvind So lbe r g, Supp ort varieties for selfinje ctive algebr as , K -Theory 33 (20 04), no. 1, 67–87. MR MR219 9789 (20 07f:160 14) 7 K. F uk aya, Y. G. O h, O hta H., and Ohno K., L agr angia n int erse ction Flo er the ory - anomaly and ob- struction , preprint http://www.math.ky oto -u.ac.jp/ fuk ay a/fuk ay a.html. 8 Kenji F uk ay a, Deformation t he ory, homolo gic al algebr a and mirr or symmetry , Geometry and physics of branes (Como, 20 0 1), Ser. High E nergy P hys. Cosmol. Gravit., IOP , Bristol, 2003 , pp. 12 1–20 9. MR MR19509 58 (200 4c:140 15) 9 M. Gerstenhab er, On the deforma tion of rings and algeb r as , Ann. o f Ma th. (2) 79 (1964 ), 59–10 3. MR MR01718 07 (30 #2034) 10 Murray Gerstenha b er and Alexander A. V oronov, Homotopy G -algebr as and mo duli sp ac e op er ad , In ter- nat. Math. Res. Notices (1995), no. 3, 141– 153 (electronic). MR MR1321701 (9 6 c:1800 4) 11 E. Getzler and J. D. S. Jones, Op er ads, homotopy algebr a and iter ate d inte gr als for double lo op sp ac es , preprint hep-th/94 0305 5. 12 Ezra Getzler and John D. S. Jones, A ∞ -algebr as and the cyclic b ar c omplex , Illinois J. Math. 34 (1990), no. 2, 2 56–2 83. MR MR1046565 ( 9 1 e:1900 1) 13 Alastair Hamilton and Andrey La zarev, Homotopy algebr as and nonc ommutative ge ometry , prepr int math.QA/041 0621 v1. 14 B. Keller, Derive d invarianc e of higher structu r es on the Ho chsch ild c omplex , pr eprint http:/ /www. math. jussieu.fr/~keller/publ/dih.dvi . 15 , On the cyclic homolo gy of exact c ate gories , J. Pur e Appl. A lgebr a 136 (19 99), no. 1, 1–5 6. MR MR16675 58 (99m:1 8012 ) 16 Bernhard Keller, A -infinity algebr as, mo dules and functor c ate gorie s , T rends in representation theory of algebras and related topics, Con temp. Ma th., vol. 406, Amer. Math. So c., Providence, RI, 2006, pp. 6 7–93 . MR MR2258042 ( 20 07g:1 8002) 17 Maxim Kontsevic h and Y an Soib elman, Notes on A ∞ -algebr as, A ∞ -c ate gories and n on- c ommutative ge ometry I , preprint math.RA/06062 41v2. 18 A. Lazarev, Ho chschild c ohomolo gy and mo duli sp ac es of str ongly homotop y asso ciative algebr as , Homol- ogy Homotop y Appl. 5 (2003 ), n o . 1, 73–100 (electro nic). MR MR198 9615 ( 20 04k:1 6 018) 19 Kenji Lef` evre-Hasegawa, Sur les A ∞ -c at ´ egories , Th` ese de do ctora t, Universit´ e Denis Diderot Paris 7, Nov ember 2003, av ailable at the homepage o f B . Keller. 23 Hochschild cohomology, the characteristic morphism and derived deforma tions 20 W. Low en, Obstruct ion the ory for obje cts in ab elian and derive d c ate gories , Co mm. Algebra 33 (2 005), no. 9, 3 195– 3223, preprint math.KT/0 4070 19. 21 W. Low en and M. V a n den Bergh, Ho chschild c ohomo lo gy of ab elian c at e gories and ringe d sp ac es , Ad- v ances in Math. 198 (2005), no. 1, 172–221 , pr e print math.KT/04 0522 7. 22 , Deformation the ory of ab elian c ate gorie s , T rans. Amer. Ma th. Soc. 35 8 (2006), no . 12, 54 4 1–54 83, preprint math.KT/0 4052 27. 23 V olo dymyr Lyubashenko, Cate gory of A ∞ -c ate gories , Homolo gy Homotopy Appl. 5 (2 0 03), no. 1, 1–48 (electronic). MR MR1989611 (200 4e:180 0 8) 24 Pedro Nicol´ as, The Bar derive d c ate gory of a cu rve d dg algebr a , preprint math.R T/070 2449 v1. 25 Nicole Snashall and Øy vind Solber g, S u pp ort varieties and Ho chschild c ohomolo gy rings , Pro c . London Math. Soc. (3) 88 ( 20 04), no. 3, 705–7 32. MR MR2044054 (20 05a:16 014) 26 Gon¸ c alo T abuada, Homotopy the ory of wel l-gener ate d algebr aic t riangulate d c ate gories , preprint math.KT/070 3172 v1. 27 , A new Quil len mo del for the Morita homotop y the ory of dg c ate gories , prepr int math.KT/070 1205 v2. 28 , Une structur e de c at´ egorie de mo d` eles de Quil len sur la c at ´ egorie des dg-c at´ egories , C. R. Math. Acad. Sci. P aris 340 (2005 ), no . 1, 15– 19. MR MR211 2034 (200 5h:180 33) 29 Bertrand T o¨ en, The homotopy the ory of dg -c ate gories and derive d Morita the ory , Inv ent. Math. 167 (2007), no. 3 , 615–667. MR MR2276 263 W en d y Low en wlo w en@vub.ac.b e Departemen t of Mathematics, F aculty of Sciences, V rije Unive r s iteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium 24