Linearisations of triangulated categories with respect to finite group actions

LINEARISA TIONS OF TRIANGULA TED CA TEGORIES WITH RESPECT TO FINITE GR OUP ACTIONS P A WEL SOSNA Abstra ct. G iven an action of a finite g roup on a triangulated category , we in vestiga te u n der whic h conditions one can construct a linearised triangulated category using DG-enhan cements. In particular, if the group is a finite group of automorphisms of a smo oth pro jectiv e va riety and th e category is the b ounded derived category of coherent sheav es, t h en our construction prod uces the boun ded d eriv ed category of coheren t sheav es on the smooth quotient vari ety resp. stac k. W e also consider the action giv en by th e tensor pro duct with a t orsion canonical bundle and the action of a finite group on the category generated b y a sph erical ob ject. 1. Introduction T riangulated categories are u biquitous in sev eral areas of mathematics. Ho w ev er, it is also w ell-kno wn that th ese categorie s are not r igid enough to p erform certain op erations: F or ex- ample, the category of exact functors b et w een triangulated categories is n ot itself triangulated. This note is concerned with defining a linearised triangulated category T G when the action of a group on a triangulated category T is giv en. One p ossib le motiv ation for this comes f rom geometry . The b ounded derived category D b ( X ) of a sm o oth pro jectiv e v ariet y X has dra wn a lot of atten tion in recen t y ears since it enco des a lot of interesting geomet ric information whic h is not visible when w orking with the ab elian catego ry of shea v es. The group of auto equiv alence s Aut(D b ( X )) is also a very in teresting ob ject to study . It is completely u ndersto o d for v arieties with ample canonical (or antic anonical) bund le, b ut its stru cture is unknown in general. Of course, similar questions can b e asked ab out an y triangulated catego ry . If the triangulated category happ ens to b e D b ( X ) and the group G is con tained in Au t( X ) and is fin ite, then a r easonable construction should pro duce D b ([ X/G ]) , where [ X/G ] is the smo oth quotien t v ariet y r esp . stac k. On the other h and, if G is not some group of auto- morphisms one migh t hop e to extract some interesti ng geomet ry out of its action on D b ( X ). Unfortunately , w e do not h a v e man y examples of fi nite groups acting on D b ( X ) at th e momen t, but since th ere are v arious sources for triangulated categories it is r easonable to hop e that the tec hniques dev elop ed in this note will also b e applicable in other situations. The basic idea of the construction is to u se linearisations. T his is we ll-established for sh ea v es (or m o dules): I f G ⊂ Aut( X ) is fi nite, then the ab elian category Coh G ( X ) of linearised sh ea v es 2010 Mathematics Subje ct Classific ation. 18E30 , 14F05. Key wor ds and phr ases. triangulated categories, auto equiv alences, enh ancemen ts, linearisations. This work was sup p orted by the researc h grant SO 1095/1 -1 of the DFG ( German R esearc h F oun d ation). 1 2 P . SOSNA on X is equiv ale nt to Coh ([ X/G ]). W e w ould lik e to ha v e something s imilar for tr iangulated catego ries. Ho w ev er, th ere are tw o b asic problems. The fir st is that a reasonable n otion of a group acting on a category has to assign an autoequ iv a lence to an y group elemen t (and the assignmen t is sub ject to some conditions), but the group Au t( T ) is th e set of auto equiv alences mo dulo isomorphisms. F urther m ore, it is fairly easy to see th at the category of linearised ob jects (with resp ect to a reasonable group action) of T is not necessarily triangulated, b ecause cones are not f unctorial (a pr ob lem alluded to in the first p aragraph). The remedy is to consider triangulated categories which are homotop y categories of pretri- angulated DG-categories and auto equiv alences which come from equiv alences on the DG-lev el. Our main r esult can b e rough ly summarised as follo ws, see Prop osition 4.3 and Corollary 5.3. Theorem 1.1. L et G b e a finite gr oup acting on a triangulate d c ate gory T ≃ H 0 ( A ) which is the homoto py c ate gory of a pr etriangulate d DG-c ate gory A and such that the action of G c omes fr om an action on A . Then a line arise d triangulate d c ate gory T G A c an b e c onstructe d. If T ≃ D b ( X ) and G is a finite gr oup of automorphisms of X , then D b ( X ) G ≃ D b ([ X/G ]) . Given a v ariety S with a c anonic al bund le ω S which is torsion of or der n , the triangulate d c ate gory line arise d with r esp e ct to the action of Z /n Z , wher e one identifies 1 with the auto e quivalenc e given by the tensor pr o duct with ω S , is e quivalent to the b ounde d derive d c ate g ory of the c anonic al c over. It is in general not clea r w h ether the ab ov e result dep end s on the choice of the category A (so the ab o v e statemen ts ha ve to b e read as in v olving sp ecific c hoices of A ), but see Prop osition 3.12 for a p artial result. In an y case, choosing a fairly natural A we can p ro v e the follo wing. F or example, giv en a F ourier–Muk ai partner Y of X and a group G acting on Y , the lin earised catego ry D b ( X ) G with resp ect to the action of G induced b y the F ourier–Muk ai equiv alence is equiv alen t to D b ([ Y /G ]), see Corollary 4.4. Lastly , the category generated b y a spherical ob ject adm its actions of fi nite groups and w e p ro v e that the spherical ob ject b ecomes an exceptional one in the linearised category , s ee Prop osition 5.6. Remark 1.2. If the group w as not fin ite, one w ould ha ve to adjus t certain things: F or example, w e w ould ha ve to w ork with (the deriv ed catego ry of ) quasi-coheren t shea ves, but the approac h w ould still w ork. The n ote is organised as follo ws. In Section 2 we recall some basic facts ab out DG-categ ories, define the linearised triangulated catego ry in the follo wing section, consider the ab o v e men - tioned geometric situation in Section 4 and lo ok at new examples in the last section. Con v entions. F rom Section 3 on we work o v er the field of complex n umb ers (although most of the results hold o v er an arbitrary field p ro vided the order of the group is prime to the c haracteristic). All functors b et ween d eriv ed categories are assumed to b e exact. Unless stated otherwise all considered groups are fin ite. Ac kno wledgemen ts. I thank Davi d Plo og and Paol o Stellari for useful discussions and for commen ts on a preliminary v ersion of th is pap er and the dep artmen t of mathematics and the complex geometry group of the Unive rsit` a d egli Studi di Milano for their hospitalit y . LINEARISA TIONS OF TRIANGULA TED CA TEGORIES... 3 2. Differential grad ed ca t e gories In this section w e recall the necessary notions and facts f rom the theory of differential graded catego ries. F or details see e.g. [5], [8] or [11]. Definition 2.1. A differ ential gr ade d c ate gory or DG-c ate g ory o ver a field K is a K -linear additiv e category A suc h that for any t w o ob jects X , Y ∈ A the space of morphisms Hom( X , Y ) is a complex, the comp osition of morp hisms Hom( X, Y ) ⊗ Hom( Y , Z ) / / Hom( X , Z ) is a c hain map and the iden tit y w ith resp ect to the comp osition is closed of degree 0. Example 2.2. The most basic example of a K -linear DG-categ ory is the category of complexes of K -ve ctor s paces. F or tw o complexes X and Y we d efine Hom( X , Y ) n to b e the K -vec tor space formed b y families α = ( α p ) of m orp hisms α p : X p / / Y p + n , p ∈ Z . W e define Hom D G ( X, Y ) to b e the graded K -vect or space with comp onents Hom( X , Y ) n and whose differen tial is given b y d ( α ) = d Y ◦ α − ( − 1) n α ◦ d X . The DG-catego ry C D G ( K ) h as as ob jects complexes and the morph isms are defined by C D G ( K )( X , Y ) = Hom D G ( X, Y ) . Of cour se, sta rting w ith the category of complexes o v er an arb itrary K -linear ab elian (or additiv e) cate gory one can asso ciate a DG-cate gory to it in a similar m anner. Clearly , w e get bac k the usual category of complexes by taking as morphisms only th e closed morphisms of degree zero and we get the usual homotop y category if we r eplace Hom D G ( X, Y ) b y ker( d 0 ) / im( d − 1 ). A DG- functor Φ : A / / B b et w een DG-catego ries A and B is by definition required to b e compatible with the str u cture of complexes on the spaces of morp hisms. If Φ , Ψ : A / / B are t w o DG-functors, then we d efi ne the c omplex of gr ade d morp hisms Hom(Φ , Ψ) to b e the complex whose n th comp onen t is the space formed b y families of morph isms φ X ∈ Hom B (Φ( X ) , Ψ( X )) n suc h that (Ψ α )( φ X ) = ( φ Y )(Φ α ) for all α ∈ Hom A ( X, Y ), where X, Y ∈ A . The different ial is giv en b y that of Hom B (Φ( X ) , Ψ( X )). Using this w e define the DG-ca tegory of DG-functors from A to B , denoted by Hom( A , B ), to b e the category with DG-functors as ob jects and the ab o v e describ ed spaces as morphisms. Note that th e DG-functors b et we en A and B are precisely the close d morphisms of degree zero in Hom( A , B ). T o an y DG-category A on e can naturally asso ciate t wo other catego ries: Firstly , there is the gr ade d c ate gory H o • ( A ) = H • ( A ) ha ving the same ob j ects as A and where the s pace of morphisms b etw een t wo ob jects X , Y is by defi nition the direct sum of the cohomolog ies of the complex Hom A ( X, Y ). Secondly , restricting to the cohomology in degree zero we get the homoto py c ate gory H o ( A ) = H 0 ( A ). Definition 2.3. A DG-functor Φ : A / / B is quasi ful ly faithful if for an y t wo ob j ects X , Y in A the m ap Hom( X, Y ) / / Hom(Φ( X ) , Φ( Y )) 4 P . SOSNA is a quasi-isomorphism and Φ is a quasi-e quivalenc e if in addition the induced functor H 0 (Φ) is essen tially su rjectiv e. Two DG-categ ories A and B are called qu asi-e quivalent if there exist DG- catego ries C 1 , . . . , C n and a c hain of quasi-equiv alences A C 1 o o / / · · · C n o o / / B . A DG-functor Φ : A / / B is a DG-e quivalenc e if it is fully faithful and for every ob ject B ∈ B there is a closed isomorphism of degree 0 b et we en B and an ob ject of Φ( A ). W e also h a v e to recall the follo wing construction from [3]. Definition 2.4. Let A b e a DG-ca tegory . Define the pr etriangulate d hul l A pr etr of A to b e the follo wing catego ry . Its ob jects are formal expressions ( ⊕ n i =1 C i [ r i ] , q ), where C i ∈ A , r i ∈ Z , n ≥ 0, q = ( q ij ), q ij ∈ Hom( C j , C i )[ r i − r j ] is homogeneous of degree 1, q ij = 0 for i ≥ j , dq + q 2 = 0. If C = ( ⊕ n j = 1 C j [ r j ] , q ) and C ′ = ( ⊕ m i =1 C ′ i [ r ′ i ] , q ′ ) are ob jects in A pr etr , then the Z - graded K -mo dule Hom( C, C ′ ) is th e space of m atrices f = ( f ij ), f ij ∈ Hom( C j , C ′ i )[ r ′ i − r j ] and the comp osition map is matrix m ultiplication. Th e differen tial d : Hom( C , C ′ ) / / Hom( C, C ′ ) is defined by d ( f ) = ( d f ij ) + q ′ f − ( − 1) l f q if deg f ij = l . The category A is called pr etriangulate d if the natural fu lly faithful functor Ψ : A / / A pr etr is a quasi-equiv alence and A is str ongly pr etriangulate d if Ψ is a DG-equiv alence. The reason f or introdu cing th e pretriangulated hull is that its homotop y category is alw a ys triangulated. Thus, w e ha ve the follo wing Definition 2 .5. Let A b e a DG-categ ory . The asso ciated triangulated catego ry is A tr := H 0 ( A pr etr ). Finally we h a ve the follo wing notion. Definition 2.6. Let T b e a triangulated cate gory . An enhanc ement of T is a p air ( A , ǫ ), where A is a pretriangulated DG-category and ǫ : H 0 ( A ) ∼ / / T is an equiv alence of triangulated catego ries. The categ ory T is said to ha ve a u nique enh ancemen t if it has one and for t wo en hance- men ts ( A , ǫ ) and ( A ′ , ǫ ′ ) there exists a quasi-functor (see [11]) φ : A / / A ′ whic h induces an equiv alence H 0 ( φ ) : H 0 ( A ) / / H 0 ( A ′ ). One th en call s the t wo enhancemen ts e quivalent . Two enhancemen ts are called str ongly e quivalent if th ere exists a quasi-functor φ suc h that ǫ ′ ◦ H 0 ( φ ) and ǫ are isomorphic. If T ≃ D b ( X ) for X a smo oth pro jectiv e v ariet y (or a smo oth stac k), the en hancemen t one usually wo rks with is (2.1) A := D b D G ( X ) := C D G ( I ( X )) , where C D G ( I ( X )) is the DG-ca tegory of b ounded-b elo w complexes of injectiv e shea v es with b ound ed coherent cohomology . Denote the t wo pro jections fr om X × X to X by q and p . Let F : D b ( X ) / / D b ( X ) b e an equiv alence. By results of Orlo v ([12 ], [13]) w e kno w that F is of F ourier–Mukai typ e , that is, there exists a unique (up to isomorph ism) ob j ect P ∈ D b ( X × X ), called the kernel , such th at F ≃ Φ P , wh ere Φ P ( E ) = p ∗ ( q ∗ E ⊗ P ) for an y E ∈ D b ( X ). It is clear that an y equiv alence LINEARISA TIONS OF TRIANGULA TED CA TEGORIES... 5 of FM-t yp e lifts to a DG-e nd ofunctor of D b D G ( X ). In fact, the DG-lifts of the three standard auto equiv alences (shifts, automorphisms and line bu ndle twists) d o lift to DG-equiv alences of the ab o ve describ ed enhancement , but for a general auto equiv alence this is not clear. Giv en an arbitrary triangulated catego ry th e existence and /or uniqueness of an enhan cement is not kno wn (but see [11] for sev eral results) an d the question w hether an exact functor lifts to the DG-enhancemen t is also op en. 3. Linearisa tions Let G b e a group and C b e any category . The follo wing notions are b ased on Deligne’s article [4]. A we ak action of G on C is the assignmen t of an auto equiv alence g ∗ to an y element g ∈ G suc h that th ere exist isomorphisms of fun ctors c g ,h : ( g h ) ∗ ≃ h ∗ g ∗ for all g , h ∈ G . Note that this, in particular, implies that 1 ∗ ≃ id C . An action of G on C is a we ak action suc h that the isomorphisms c g ,h satisfy an asso ciativit y condition: ( g hi ) ∗   / / i ∗ ( g h ) ∗   ( hi ) ∗ g ∗ / / i ∗ h ∗ g ∗ . Of course, there is a “co v arian t” version of the ab ov e d efinition. Definition 3.1. A line arisation of an ob ject A of a catego ry A consists of a collection of morphisms λ g : A / / g ∗ ( A ) in A for eac h g ∈ G which satisfy the f ollo win g: λ 1 = id and λ g h = h ∗ ( λ g ) λ h , that is A λ gh 3 3 λ h / / h ∗ ( A ) h ∗ ( λ g ) / / h ∗ ( g ∗ ( A )) . A pair ( A, λ g ) w ill b e called a line arise d obje ct . W e define a morphism b et wee n t w o linearised ob jects ( A, λ g ) and ( A ′ , λ ′ g ) to b e a G -inv ariant morp hism, that is, a morphism ϕ : A / / A ′ in A such that the follo wing diagram comm u tes in A for all g ∈ G : A λ g   ϕ / / A ′ λ ′ g   g ∗ ( A ) g ∗ ( ϕ ) / / g ∗ ( A ′ ) . Th u s , w e hav e a category of linearised ob jects A G . The linearised catego ry inherits p rop erties of A if A is “rigid” enough, f or example w e h a v e the follo win g Prop osition 3.2. If G acts on an ab elian c ate gory A b y exact auto e quivalenc es, then A G is also an ab elian c ate g ory. Pr o of. Th e existence of direct sums an d the zero ob ject is ob vious. Give n a G -inv ariant mor- phism ϕ : ( A, λ g ) / / ( A ′ , λ ′ g ) the un iv ersal prop erties of the ke rn el and the cok ernel in A en - sure that ke r ( ϕ ) and coke r ( ϕ ) are canonically linearised and the r esp ectiv e morph isms are 6 P . SOSNA G -in v ariant . Hence, kernels and cok ernels exist. Th e most inte resting part is the equalit y of the image and the coimage: By th e ab ov e arguments these ob jects are linearised and it can b e c hec k ed, u s ing that the k ernel is a monomorp hism, that the isomorphism in A b et wee n im( ϕ ) and coim( ϕ ) is G -in v arian t.  If G acts on a triangulated category T b y exact auto equiv alences, then, giv en a G -in v ariant map ϕ , it is not clear ho w to linearise a cone of this map. T herefore we will tak e th e detour via DG-categ ories. There is the follo wing easy result. Prop osition 3.3. If A is a DG-c ate gory with an action by a gr oup G , then the c ate gory of line arise d obje cts as define d ab ove is a DG-c ate gory. Pr o of. W e only need to pro ve that the sp ace of morphisms has the stru cture of a complex. This b oils d o wn to pr o v in g that for any morp hism ϕ : ( A, λ g ) / / ( A ′ , λ ′ g ) the morphism d ( ϕ ) : A / / A ′ is also compatible with the linearisations. Since λ g is a DG-isomorphism for all g ∈ G we, in particular, ha ve that an y λ g is closed, that is, d ( λ g ) = 0 for all g ∈ G . No w one only has to use the Leibniz rule, the fact that λ ′ g has degree 0 and that any g ∈ G defin es a DG-functor and therefore is compatible with the differential s: λ ′ g ◦ ϕ = g ∗ ( ϕ ) ◦ λ g = ⇒ λ ′ g ◦ d ( ϕ ) = d ( λ ′ g ◦ ϕ ) = d ( g ∗ ( ϕ ) ◦ λ g ) = g ∗ ( d ( ϕ )) ◦ λ g .  Remark 3.4. Giv en an action of an algebraic group G on a v ariet y X den ote the action by σ : G × X / / X and the m ultiplication by µ : G × G / / G . On e defines a linearisation of a sheaf F in this case to b e an isomorphism λ : σ ∗ F / / p ∗ 2 F of O G × X -mo dules s ub ject to the co cycle condition ( µ × id X ) ∗ λ = p ∗ 23 λ ◦ ( σ × id G ) ∗ λ , where p 2 : G × X / / X and p 23 : G × G × X / / G × X are the pro jections. F or a fi nite (or discrete) group this r educes to isomorphisms F / / g ∗ F as ab o v e. Hence, we sh ould not exp ect the ab o ve construction to b e compatible with geometry in the case of an arb itrary group (there is a notion of an equiv ariant deriv ed category in th is case, see [1 ]). Definition 3.5. W e define the for getful functor F org as the functor A G / / A wh ic h forgets the linearisatio ns. T he inflation functor I nf fr om A to A G is the fun ctor w h ic h on ob jects is defined by A  / / ⊕ g g ∗ ( A ). Remark 3.6. Giv en a s u bgroup H ⊂ G , w e ha v e an ob vious DG-functor A G / / A H . This functor is clearly faithful, but the case H = { 1 } s h o ws that it is not essen tially sur jectiv e in general. Note th at, since th ere alwa ys exists an extension of a DG fu nctor to a DG functor on the pretriangulated h ulls, the action of G on A naturally extends to an actio n on A pr etr . Explicitly , the act ion of G on the formal shifts is clea r and for an ob ject ( ⊕ n i =1 C i [ r i ] , q ) in A pr etr w e simply apply an element g in G to the comp onents of q . One th en c h ec ks that the condition d ( q )+ q 2 = 0 induces th at d ( g ∗ ( q )) + g ∗ ( q ) 2 = 0 (to see this note that g ∗ ( αβ ) = g ∗ ( α ) g ∗ ( β ) and similarly for LINEARISA TIONS OF TRIANGULA TED CA TEGORIES... 7 sums; hence g ∗ is compatible with matrix multiplicat ion) and hence the action is well-defined. The definition of the action on morph isms is similarly straigh tforward. Prop osition 3.7. If G is a gr oup acting on a str ongly pr etriangulate d D G-c ate gory A , then A G is also str ongly pr etriangulate d. Pr o of. Cons id er the formal shif t of a linearised ob j ect ( A, λ g )[1]. W e kno w that A [1] is DG- isomorphic to an ob ject A ′ of A and λ g induces a linearisation λ ′ g of A ′ . It is then clear th at ( A, λ g )[1] is DG-isomorphic to ( A ′ , λ ′ g ). The reasoning for the cone of a closed degree zero morphism is similar.  Remark 3.8. W e cannot sho w that the s tatemen t of the lemma h olds tru e without the “strongly” assump tion. One would need to sho w that for ev ery k ∈ Z and an y ( A, λ g ) ∈ A G the ob ject ( A, λ g )[1] is homotop y equiv alen t to an ob ject in A G and similarly th at the cone of any closed degree zero morphism also has this prop ert y . Let ϕ : ( A, λ g ) / / ( A ′ , λ ′ g ) b e a closed degree zero morphism in A G . F or simp licit y write e A f or ( A, λ g ) and similarly f A ′ for ( A ′ , λ ′ g ). By definition the cone C ( ϕ ) is the ob ject ( f A ′ [1] ⊕ e A, q ), wh er e q is the (2 × 2)-matrix with q 12 = ϕ and 0 otherwise. On th e other hand the cone of ϕ considered in A pr etr , wh ic h w e will d enote by C ′ ( ϕ ), h as a linearisation γ g giv en by  λ ′ g 0 0 λ g  Using that A is p retriangulated we know that C ′ ( ϕ ) is homotop y equ iv alen t to an ob ject D of A , but the linearisation γ g do es not n ecessarily induce a linearisation δ g on D . W e can now give our defin ition of the linearised triangulated category . Definition 3.9. Let T b e the homotop y category of a pretriangulated DG-ca tegory A and let G b e a group acting on A and hence on T . The line arisation of T by G , denoted by T G A , is defined to b e H 0 (( A G ) pr etr ). Since the enhancement will usually b e clear from th e context, w e will simply write T G instead of T G A . Remark 3.10. Giv en an additiv e category T and an au tomorp h ism Φ there is an “orbit catego ry” T / Φ, w h ic h has the s ame ob jects as T and the morph isms b et we en tw o ob jects A and B are give n b y M n ∈ Z Hom T ( A, Φ n B ) . If T is a triangulated category , then T / Φ is not triangulated in general, but under some fairly strong assumptions it do es hav e a triangulated structure, see [9]. In particular, the mentioned assumptions are not satisfied b y D b ( X ) for X a smo oth pro jectiv e v ariety . In general, the orb it catego ry do es n ot seem to capture quotien ts: Consider a sm o oth pro jectiv e v ariet y X w ith an action by an automorphism such that X/G is smo oth and the category Coh ( X ), considered for simplicit y as an add itiv e category . Th en the orb it category is π ∗ Coh ( X ) and not Coh ( X/G ). 8 P . SOSNA Under certain assump tions the construction is functorial. Namely , assume that G acts on t w o pr etriangulated DG-categ ories A and A ′ and that w e are given an equ iv arian t DG-functor F : A / / A ′ meaning that th e follo wing d iagram comm utes for all g ∈ G : A g ∗   Φ / / A ′ g ∗   A Φ / / A ′ . Then Φ induces a fu nctor Φ G : A G / / ( A ′ ) G b y sending ( A, λ g ) to (Φ( A ) , Φ( λ g )) and similarly for maps. Using the induced functor on the pretriangulated hulls, we get an exact fun ctor H 0 ((Φ G ) pr etr ) : T G / / ( T ′ ) G . As a particular example of this consider the situation of a triangulated sub catego ry . So, supp ose we are giv en a triangulated category T = H 0 ( A ) with a group act ion and a triangulated sub category T ′ suc h that the action of G restricts to an action on the natural enh ancemen t of T ′ (i.e. th e ob j ects of th e enhancemen t are those ob jects A ′ in A s uc h that H 0 ( A ′ ) ∈ T ′ ). Then ( T ′ ) G is a triangulated su b category of T G . It is, of course, an inte resting question whether the constru ction dep ends on the choic e of the en hancemen t, that is, if T can b e written as th e homotop y category of tw o distinct DG- catego ries A and B , is then T G A equiv alen t to T G B . W e will no w see that this can b e indeed sho wn, alb eit under some assumptions. Lemma 3.11. L et A and B b e two pr etriangulate d DG-c ate gories and let Φ : A / / B b e an e quivariant quasi-e quiv alenc e. Assume that for al l ( A, λ g ) , ( A ′ , λ ′ g ) ∈ A G and for al l ϕ ∈ Hom A ( A, A ′ ) we have that g ∗ ( dϕ ) λ g = λ ′ g dϕ implies that g ∗ ( ϕ ) λ g = λ ′ g ϕ (we c al l this c ondition ( ∗ ) ) and similarly for B . Then Φ G is quasi ful ly faithful. Pr o of. Cond ition ( ∗ ) en s ures that H i A G (( A, λ g ) , ( A ′ , λ ′ g )) em b eds int o H i A ( A, A ′ ) for all i . T h e comm utativit y of the follo wing diagram (for all i ∈ Z ) H i A G (( A, λ g ) , ( A ′ , λ ′ g ))   H i (Φ G ) / / H i B G ((Φ( A ) , Φ( λ g )) , (Φ( A ′ ) , Φ ( λ ′ g )))   H i A ( A, A ′ ) H i (Φ) / / H i B (Φ( A ) , Φ( A ′ )) com bined with the injectivit y of the left ve rtical map and the fact that the low er map is an isomorphism implies the injectivit y of H i (Φ G ). Next, tak e an elemen t ψ in H i B G ((Φ( A ) , Φ( λ g )) , (Φ( A ′ ) , Φ ( λ ′ g ))). Since b y ( ∗ ) the right ve r- tical map is inj ective this giv es an element in H i B (Φ( A ) , Φ( A ′ )) and hence a unique element ϕ in H i A ( A, A ′ ). W e need to c hec k that in fact ϕ is in H i A G (( A, λ g ) , ( A ′ , λ ′ g )). By assu mption and quasi-faithfulness of Φ w e h a ve that g ∗ ( ϕ ) λ g − λ ′ g ϕ = d ( α g ) LINEARISA TIONS OF TRIANGULA TED CA TEGORIES... 9 for some α g (here ϕ is a r ep resen tativ e of ϕ ). Differentia ting the ab o ve equation gives th at d ( ϕ ) commutes with the linearisations and therefore, by ( ∗ ), ϕ do es. Hence, Φ G is quasi fully faithful.  Prop osition 3.12. In addition to the assumptions of the pr evious lemma let furthermor e Ψ : B / / A b e an adjoint e quivariant quasi-e quivalenc e. Then T G A and T G B ar e e quivalent. Pr o of. Sin ce Φ and Ψ are adj oint, so are Φ G and Ψ G . Both these fu nctors are quasi fully faithfu l and hence defin e equiv alences on th e homotop y categories. No w use that if a DG-functor F is a quasi-equiv alence, then so is F pr etr .  Remark 3.13. One probably s hould n ot exp ect that our construction is indep end en t of the c hoice of the enhancemen t in general, since linearisations do use the DG-structur e. Of course, it ma y then b e ask ed what the relation b et w een T G A and T G B is. It is difficult to pro duce new int eresting auto equiv alences of finite order in the geometric setting. The case of a finite group of au tomorp h isms will b e settled in the next section. T he next case is the action of the group generated b y a line bu ndle of fi nite ord er, which will b e partially dealt with in Section 5. Basical ly , these are the only examples w e ha v e at our disp osal. One might hop e to p ro duce new ones by conjugating the action of one of the ab o ve mentio ned groups bu t we w ill no w s ee that this d o es not giv e anything new. Let G act on a pretriangulated DG-categ ory A , let Φ b e a DG-equiv alence of A , consider the action of G give n by g  / / Φ − 1 ◦ g ∗ ◦ Φ and denote the DG-category linearised with r esp ect to this action by A e G . Then we h a ve the Prop osition 3.14. The c ate gories A G and A e G ar e DG-e q u ivalent. Pr o of. Th e functor F send ing an ob ject ( A, λ g ) in A G to (Φ − 1 ( A ) , Φ − 1 ( λ g )) is easily seen to b e a DG-equiv alence.  4. The case of automorphisms Of course, one has to c h ec k that the ab o ve pro cedur e pro d uces the deriv ed categ ory of [ X/G ] if G is a finite group of automorphisms of X . Denote the cardinalit y of G by n . There are canonical isomorphisms ( gh ) ∗ ≃ h ∗ g ∗ and these w ill b e used to d efine the action of G . W e fir st recall some useful facts. There is an equiv alence ( Q ) Coh ([ X/G ]) ≃ ( Q ) Coh G ( X ), where th e latter is the catego ry of linearised shea v es. The quotien t morp hism π : X / / [ X/G ] is flat, hence π ∗ is exact. Using the adjunction Hom( π ∗ ( − ) , − ) ≃ Hom( − , π ∗ ( − )) and the exactness of π ∗ w e see that the p ushforward of an injectiv e sheaf on X is an injectiv e sheaf on [ X/G ]. No w consider T = D b ( X ) as the homotop y categ ory of D b D G ( X ). Then an ob ject of A G is by definition a complex as ab ov e tog ether with c h ain isomorp h isms λ g satisfying the linearisation relation. Since A is strongly pretriangulated, so is A G (see Prop osition 3.7). W e need th e follo win g 10 P . SOSNA Lemma 4.1. L et F ′ = ( F , λ g ) b e an inje ctive she af on [ X/G ] . Then F = π ∗ ( F ′ ) is an inje ctive she af on X . Pr o of. Denote [ X/G ] by Y and the S erre functor by S X resp. S Y . The result follo ws from the follo w in g compu tation Hom X ( − , π ∗ ( F ′ )) ≃ Hom X ( π ∗ ( F ′ ) , S X ( − )) ∨ ≃ Hom Y ( F ′ , π ∗ S X ( − )) ∨ ≃ ≃ Hom Y ( π ∗ S X ( − ) , S Y ( F ′ )) ≃ Hom Y ( S − 1 Y π ∗ S X ( − ) , F ′ ) . Note th at the functor S − 1 Y π ∗ S X tak es s hea v es to shea ves, sin ce the shifts cancel out. F urther- more, F ′ is injectiv e b y assumption, π ∗ is exact and tensoring w ith line bu ndles is also exac t, hence we conclude that Hom ( − , π ∗ ( F ′ )) is an exact fun ctor and therefore F = π ∗ ( F , λ g ) is injectiv e.  Remark 4.2. In our situation one has that π ∗ ω Y ≃ ω X b y the r amification form ula ω X ≃ π ∗ ω Y ⊗ O ( R ). Hence the p ro jection formula giv es that S − 1 Y π ∗ S X ( F ) ≃ π ∗ ( F ⊗ ω X ) ⊗ ω − 1 Y ≃ π ∗ ( F ) . The forgetful fun ctor F org corr esp onds to π ∗ and the inflation fu n ctor Inf corresp onds to π ∗ . Hence, F org ◦ Inf ≃ ⊕ g ∈ G g ∗ ≃ π ∗ π ∗ . Using this, one can giv e another pro of of the ab o v e lemma: Con s ider the follo wing diagram in QCoh ( X ): π ∗ ( F ) 0 / / P ψ / / φ < < y y y y y y y y Q . W e need to constru ct a map θ : Q / / π ∗ ( F ) making the diagram comm utativ e. Ap ply π ∗ to the diagram. Sin ce π ∗ π ∗ ( F ) = F ⊗ π ∗ O X and π ∗ O X = L is lo cally free, π ∗ π ∗ ( F ) is an injectiv e sheaf again and therefore w e get a diagram F ⊗ L 0 / / π ∗ P π ∗ ( ψ ) / / π ∗ ( φ ) : : v v v v v v v v v π ∗ Q . α O O Applying π ∗ and using that π ∗ π ∗ = ⊕ g g ∗ w e get ( π ∗ ( F )) ⊕ n 0 / / ⊕ g g ∗ ( P ) ⊕ g g ∗ ( ψ ) / / ⊕ g g ∗ ( φ ) 8 8 q q q q q q q q q q ⊕ g g ∗ ( Q ) . π ∗ ( α ) O O Denoting th e inclusion of Q in ⊕ g g ∗ ( Q ) by ι and th e first p ro jection from ( π ∗ ( F )) ⊕ n to π ∗ ( F ) b y p 1 , the wan ted morphism θ is then p 1 ◦ π ∗ ( α ) ◦ ι . LINEARISA TIONS OF TRIANGULA TED CA TEGORIES... 11 Let D b D G ([ X/G ]) b e the enhancement of D b ([ X/G ]) by injectiv e shea ves. Using the ab ov e lemma w e can construct a f unctor from D b D G ([ X/G ]) to A G : If a complex of inj ectiv e lin earised shea v es ( F i , λ i g ) is giv en, then we can sen d it to the complex having the F i as terms and the linearisation of this complex is giv en term wise by the λ i g . A map in B is simply sent to itself. Clearly , this is fully faithful, so in particular w e get: Prop osition 4.3. Ther e exists a DG-e quivalenc e Φ : D b D G ([ X/G ]) / / A G . H enc e, D b ( X ) G ≃ D b [ X/G ] .  The previous d iscussion can b e us ed to deduce the follo wing Corollary 4.4. L et Y b e a F ourier–Mukai p artner of X , let G b e a finite gr oup of automor- phisms acting on Y and fix an e quivalenc e F : D b ( X ) ∼ / / D b ( Y ) . If we let the gr oup G acts on D b ( X ) by F − 1 ◦ g ∗ ◦ F for any g ∈ G , then D b ( X ) G is e quivalent to D b ([ Y /G ]) . Pr o of. If w e write D b ( X ) as the homotopy category of D b D G ( Y ), the result follo ws imm ediately .  Let us now consider the f ollo wing situation. Let G act on X without fixed p oints and consider the quotien t map π : X / / X/G =: Y . T ak e the stru cture sheaf O Z of a su b v ariety Z ⊂ Y and consider the full triangulated sub category T generated by O Z in D b ( Y ). Pulling O Z up to X giv es the d irect su m of O Z i , where ∪ Z i is the preimage of Z (for example, Z could b e a rational curv e on an Enriques surf ace and then we get t wo rational curves Z 1 and Z 2 on the co v ering K3 su rface). Consider the triangulated sub category of D b ( X ) generated by the shea v es O Z i and denote it b y T ′ . Then we h a ve Prop osition 4.5. Ther e exists an e quivalenc e ( T ′ ) G ≃ T . Pr o of. Clearly , ⊕ i O Z i can b e linearised, h en ce O Z ∈ ( T ′ ) G and therefore T ⊂ ( T ′ ) G . On the other h and, T ′ is bu ilt from the O Z i b y taking iterated extensions and shifts, whose sup p ort is alw a ys contai ned in the union of the O Z i . Th erefore the only ob jects whic h can b e linearised are generated by ⊕ i O Z i and therefore th e other inclusion holds as w ell.  5. Appl ica tions 5.1. Linearisations with resp e ct to a torsion canonical bundle. Ha ving chec ked the ab o v e we w ill lo ok at the case of a linearisation with resp ect to a line bu ndle t wist. Recall (Prop osition 3.2) th at giv en an action of a fi nite group G on an ab elian catego ry , the categ ory of linearised ob jects is also ab elian. W e will no w consider the follo wing situation. Let S b e a v ariety whose canonical bun dle is of finite order, fix an isomorp h ism f : ω n S ≃ O S and consider the global sp ectrum e S of the corresp ondin g sheaf of O S -algebras. Using f , the sheaf O S ⊕ ω S ⊕ · · · ω n − 1 S b ecomes an O S -algebra and e S has a fixed p oint -free automorphism τ of order n corresp ondin g to the action of ω S . Denote the quotien t morp hism e S / / S b y π . The isomorphism f indu ces an action of Z /n Z on th e cate gory of (quasi-)coheren t shea ve s on S b y sending 1 to the functor ( − ) ⊗ ω S . W e can consider s hea v es linearised w ith resp ect to th is 12 P . SOSNA action. Any suc h sheaf F has, in particular, the prop ert y that F ≃ F ⊗ ω S . This holds, for example, for F = ⊕ n − 1 k =0 ω k S . No w r ecall that b y [7, Ex. I I.5.17] there exists an equiv alence b et w een ( Q ) Coh ( e S ) and the category of (quasi-)coherent π ∗ O e S -mo dules. Note that the pullbac k of f is the identit y morphism of O e S and that th e canonical isomorphism π ∗ O e S ≃ ⊕ n − 1 k =0 ω k S is an isomorphism of O S -algebras. Lemma 5.1. L et F b e a (quasi-)c oher ent she af on e S . Then π ∗ ( F ) is line arise d with r esp e ct to the ab ove describ e d action of Z /n Z on the c ate gory of (quasi-)c oher ent she aves on S . Pr o of. W e use the pro jection formula to get an isomorphism α : π ∗ ( F ) ≃ π ∗ ( F ⊗ O e S ) ≃ π ∗ ( F ⊗ π ∗ ( ω S )) ≃ π ∗ ( F ) ⊗ ω S . Since the isomorphism in th is formula is canonical, the morphism ( α ⊗ id) ◦ α , resp . f u rther comp ositions are also canonical. On th e other hand , comp osing the n -fold comp osition with f giv es the ident ity b ecause f enters in the ve ry d efinition of e S and h ence of π . Alternativ ely , and b riefly assu m ing n = 2 to simp lify notation, one can use that π ∗ is a faithful functor and the fact that the pullbac k of f ◦ ( α ⊗ id) ◦ α is the iden tit y map of π ∗ π ∗ F ≃ F ⊕ τ ∗ ( F ).  No w, giv en a linearised sheaf ( F , α ) we can define a stru cture of π ∗ O e S -mo dule on it b y lo cally setting ( s, t ) · γ := s · γ + α ( t ⊗ γ ) , for sections of O S , ω S and F resp ectiv ely . Clearly , these t wo constructions are inv er s e to eac h other and π ∗ O e S -linearit y tr anslates to in v ariance with resp ect to the group action. Hence, we ha v e Prop osition 5.2. Ther e is an e quiv alenc e ( Q ) Coh ( e S ) and ( Q ) Coh Z /n Z ( S ) .  Corollary 5.3. Ther e exists an e qu ivalenc e D b ( S ) Z /n Z ≃ D b ( e S ) . Pr o of. Th is follo ws fr om the prop osition, the fact that π ∗ ( F ) of an injectiv e sh eaf F is an injectiv e π ∗ O e S -mo dule and with similar argumen ts as in Section 4.  5.2. The category generated by a spherical ob ject. Let X b e a smo oth p ro jectiv e v ariet y of dimension d . Recall that an ob ject E in D b ( X ) is d -sph erical if E ⊗ ω X ≃ E and if Hom( E , E ) = Hom( E , E [ d ]) = C and 0 otherwise. More generally , one can define a d -spherical ob ject in a triangulated category T to b e an ob ject E with the prop erty that th e graded endomorphism algebra B = M p ∈ Z Hom T ( E , E [ p ]) is isomorph ic to C [ s ] / ( s 2 ) and s is of degree d . F or example, an y line b undle on a Calabi–Y au v ariet y of dimension d (that is, ω X ≃ O X and H i ( X, O X ) = 0 f or i 6 = 1 , d ) is a d -spherical ob ject. The in terest in these ob jects stems, in particular, from the f act that one can associate an auto equiv alence of D b ( X ) to any sp h erical ob ject, the so-called spheric al twist (see [14]). LINEARISA TIONS OF TRIANGULA TED CA TEGORIES... 13 Recall that an ob j ect E in D b ( X ) is exc eptional if Hom( E , E ) = C and Hom ( E , E [ k ]) = 0 for all k 6 = 0. Note that spherical ob jects are usually studied on Calabi–Y au v arieties while the most n atur al environmen t for exceptional ob jects are probably F ano v arieties. Nev ertheless, there is a conn ection b etw een spherical ob jects and exceptional ob jects, see, f or example [14, Subsect. 3.3]. In [10, Thm . 2.1] the authors determined the structure of th e triangulated category generated b y a d -spherical ob ject ( d is arbitrary). Denoting this categ ory by T d , there exists an equiv alence b et wee n T d and the p erfect derive d cate gory P erf ( B ) of the algebra B int ro d uced ab o v e (with B corresp ond ing to the spherical ob ject), w h ic h is considered as a DG-algebra with trivial differen tial. Recall that P erf ( B ) is the smallest thic k triangulated su b category of the deriv ed catego ry of B wh ich con tains B . Since B is a lo cal ring, P erf ( B ) is ju st the homotop y category of b ounded complexes of finitely generated free B -mo d ules. T he enhancement one uses h ere is the DG-catego ry asso ciated to the additiv e category of finitely generated fr ee B -mo d ules. In [6, Lem. 2.3] the group of auto equiv alences of T d whic h admit DG-lifts was d etermined. Namely , it is isomorph ic to C ∗ × Z , where Z corresp onds to the action of p ow ers of the shift functor and an elemen t a in C ∗ acts by the functor induced by the ring isomorph ism ϕ a : B / / B , s  / / a · s . Note that for an y n ∈ Z we hav e th at G = Z /n Z acts on T d b y ident ifying G w ith the group of n -th ro ots of un it y (clearly , the group action is w ell-defined). Let us first sp ell out wh at a linearisation in a sp ecial case is, n amely for n = 2 and the B -mo dule B itself. Note that w e are therefore working in the category of mo d u les. Acting b y the non-trivial group element m eans c hanging the m o dule structure of B as a B -mo du le: The element s acts on an elemen t of B not by multiplic ation with s itself, but with − s . Let λ g : B / / ( − 1) ∗ B send 1 to α + β s . In order for λ g to b e B -linear, we then must h a ve λ g ( s ) = − αs . F urthermore, if w e wan t λ g to b e lin earisation, it has to b e order 2, so we conclude th at α 2 = 1 (and β is arb itrary). T h us, for an y β ∈ C we get linearisations λ 1 ,β g (1  / / 1 + β s ) and λ − 1 ,β g (1  / / − 1 + β s ) of B . How eve r, ther e is the follo w ing Lemma 5.4. The line arise d obje cts ( B , λ 1 ,β g ) and ( B , λ 1 ,β ′ g ) ar e isomorphic for β 6 = β ′ . A similar statement holds for λ − 1 , • g . Pr o of. A B -linear automorphism of B is a map sen d ing 1 to x + y s and s to xs for x 6 = 0. Denote β ′ − β by z . Th e m ap f will commute with the linearisations if w e c h o ose x, y such that z = 2 y x .  So, w e can w ork with β = 0. Denote λ ± 1 , 0 g b y λ ± 1 g . Lemma 5.5. The obje cts ( B , λ 1 g ) and ( B , λ − 1 g ) ar e not isomorph ic. The endomorphism ring of ( B , λ 1 g ) r esp. of ( B , λ − 1 g ) is isomorph ic to C . Pr o of. If f : B / / B , 1  / / x + y s is a map commuting with the linearisations, then w e h a v e x + y s = f λ 1 g (1) = λ − 1 g f (1) = λ − 1 g ( x + y s ) = − x + y s, hence x = 0 and f cannot b e an isomorphism. 14 P . SOSNA Concerning th e second state ment we only deal with the fi rst case. Give n an f as b efore we ha v e x + y s = f λ 1 g (1) = λ 1 g f (1) = x − y s, hence y = 0 and the endomorphisms of ( B , λ 1 g ) (resp. of ( B , λ − 1 g )) are therefore the morphism s f sending 1 to x ∈ C .  Com binin g everything w e h a v e Prop osition 5.6. L et T d b e the triangulate d c ate gory gener ate d by a d -spheric al obje ct E and c onsider the action of G = Z / 2 Z on it define d ab ove. Then E admits two distinct line arisations λ 1 g and λ − 1 g and the obje cts ( E , λ 1 g ) and ( E , λ 1 g ) ar e exc eptional i n the line arise d c ate gory. In p articular, the line arise d c ate gory T G d c ontains the derive d c ate gory of C -ve ctor sp ac e s as a f ul l admissible (the admissibility fol lows fr om [2, Th m. 3.2] ) triangulate d sub c ate gory.  Note th at the exceptio nal ob jects E 1 = ( E , λ 1 g ) and E 2 = ( E , λ − 1 g ) are n ot orthogonal. F or larger n the situation b ecomes more complicated. Referen ces [1] J. Bernstein and V. Lunts, Equivariant she aves and functors , S pringer, Berlin, 1994 [2] A. I. Bondal, R epr esentations of asso ciative algebr as and c oher ent she aves , Math. US SR Izv. 34 ( 1990), 23–42 [3] A. Bondal and M. K aprano v, Enhanc e d triangulate d c ate gories , Math. USSR Sb ornik 70 (1991), 93–10 7 [4] P . Deligne, A ction du gr oup e des tr esses sur une c at´ egorie , Inv ent. Math. 128 (1997), 159–175 [5] V. Drinfeld, DG quotients of DG c ate gories , J. Algebra 272 (2004), 643–691 [6] C. F u and D. Y ang, The Ringe l–Hal l algebr a of a spheric al obje ct , http://arxiv.org/abs/ 1103.1241 [7] R. Hartshorne, Algebr aic ge ometry , Springer, New Y ork-Heidelb erg, 1977 [8] B. Keller, O n differ ential gr ade d c ate gories , in: International Congress of Mathematicians, V ol. I I, 151–190 , Eur. Math. So c., Z¨ uric h, 2006 [9] B. Keller, On triangulate d orbit c ate gories , D oc. Math. 10 (2005), 551–581 [10] B. Keller, D. Y ang and G. Zhou, The Hal l algebr a of a spheric al obje ct , J. London Math. So c. 80 ( 2009), 771–784 [11] V. A. Lunts and D. O. Orlov, Uniqueness of enhanc ement for triangulate d c ate gories , J. AMS 23 (2010), 853–908 [12] D. Orlo v, Equivalenc es of derive d c ate gories and K3 surfac es , J. Math. Sci. 84 (1997), 1361–1381 [13] D. O rlo v, D erive d c ate gories of c oher ent she aves and e quivalenc es b etwe en them , Russian Math. Surveys 58 (2003), 511–591 [14] P . Seidel and R . Thomas, B r aid gr oup actions on derive d c ate gories of c oher ent she aves , Duke Math. J. 108 (2001), 37–109 Dip ar timento di Ma tema tica “F. Enriques”, Universit ` a degli Studi di Milano, V i a Cesare Saldini 50, 20133 Milano, It al y E-mail addr ess : pawel.sosna @guest.unimi.it