Agnieszka Bodzenta. Uniwersytet Warszawski. DG categories of exceptional collections. Praca semestralna nr 2 (semestr letni 2010/11)

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1 Agnieszka Bodzenta Uniwersytet Warszawski DG categories of exceptional collections Praca semestralna nr 2 (semestr letni 2010/11) Opiekun pracy: Sławomir Cynk

2 DG categories of exceptional collections Agnieszka Bodzenta-Skibi«ska October 3, 2011 Abstract Given a full exceptional collection we recall an equivalence between a bounded derived category of coherent sheaves on a smooth projective variety X and a derived category of modules over a DG category C. We propose two explicit ways of calculating the category C. The rst one uses A categories and Massey products. The second is based on mutations of exceptional collections. Introduction The derived category of coherent sheaves D b (Coh(X)) on a smooth projective variety X has been for a long time an object of interest. Full and strong exceptional collections provide probably the easiest known description of it. The next step is understanding the structure of D b (Coh(X)) given a full exceptional collections with nontrivial higher Ext groups. A structure theorem was proved by Bondal and Kapranov in the beginning of 90's. It requires nding a DG category D with nitely many objects. However, we are not aware of any known algorithm allowing to actually calculate the category D. With this paper, we are trying to ll in the gap by proposing two ways of explicit calculations of the DG category D σ associated to a full exceptional collection σ = E 1,..., E n on X. One method is based on the existence of the minimal model for a DG category regarded as an A category. If due to degree reasons there are not many possibly nontrivial higher multiplications on End( E i ) then the A structure on H(D σ ) can be calculated by means of the Massey products. It is also possible to determine the category D σ via mutations from the collection σ to a collection τ for which the DG category D τ is known. In this approach the collection τ can either be a strong one or a collection in which the Massey products fully determine the A structure. The paper is organised as follows. In the rst section we recall the denitions of DG categories and the theory of enhanced triangulated categories. We state after [BK] the equivalence of a bounded derived category of coherent sheaves on a smooth projective variety and the category of twisted complexes over a DG category C. Following [BLL] we also present the category of twisted complexes as a subcategory of the derived category of C. In the second section we recall the denition of an A category after [K1] and following [M] we give an algorithm for nding the A structure on a cohomology category of a DG category. After [L-H] we describe the bar and cobar construction to obtain for any A category a quasi-isomorphic DG category. We end this section with a description of modules over A categories. 1

3 We state after [L-H] the equivalence of derived categories of modules over A and DG categories. This allows to substitute the category C described by Bondal and Kapranov with a category D in which the morphisms spaces between objects are nite-dimensional. In the third section we notice the similarity between the A operations on H(D) and the Massey products in End( E i ). We use it to determine the higher multiplications m n 's on H(D) in some cases. Later, we describe an alternative approach. We recall after [B] mutations of exceptional collections and generalise this notion to mutations of associated DG categories. Finally, we use both methods to calculate the DG category associated to a full exceptional collection on P 2 blown-up in three points. Acknowledgements. I am grateful to professor Alexey Bondal for many fruitful discussions and helpful comments. 1 Equivalence of D b (Coh(X)) and D b (C σ ) Let X be a smooth projective variety over C. A coherent sheaf E on X is exceptional if Hom(E, E) = C and Ext i (E, E) = 0 for i > 0. An ordered collection σ = E 1,..., E n of exceptional objects is an exceptional collection if Ext k (E i, E j ) = 0 for i > j and k 0. Moreover the collection σ is strong if also Ext k (E i, E j ) = 0 for k > 0 and all i, j. An exceptional collection σ is full if the smallest strictly full triangulated subcategory of D b (Coh(X)) containing σ is equal to D b (Coh(X)). A strong full exceptional collection provides a description of the category D b (Coh(X)). In [B] Bondal has shown the following theorem Theorem 1.1. Let E 1,..., E n be a full strong exceptional collection on a smooth projective variety X. Then D b (Coh(X)), the bounded derived category of coherent sheaves on X is equivalent to D b (mod A) a bounded derived category of nite dimensional right modules over an algebra A = End( n i=1 E i). Moreover the algebra A can be presented as a path algebra of a quiver Q with n vertices and no cycles. In [BK] Bondal and Kapranov proved an analogous theorem for a collection σ which is full but not strong. Before formulating the result we will recall some denitions after [BK]. Denition 1.2. A DG category is a preadditive category C in which the abelian groups Hom C (A, B) are endowed with a Z-grading and a dierential of degree one. The composition of morphisms Hom C (A, B) Hom C (B, C) Hom C (A, C) is a morphism of complexes and for any object C C the identity morphism id C is a closed morphism of degree zero. For an element x in a graded vector space we will denote by x the grading of x. For simplicity we shall write C(A, B) instead of Hom C (A, B). A DG category C is ordered if there exists a partial order on the set of objects such that C(A, B) = 0 for B A. 2

4 To a DG category C we associate two graded categories. The category C gr obtained from C by forgetting the dierentials on morphisms and the homotopy category H(C) with the same objects as C and morphisms H(C)(A, B) given by cohomology of C(A, B). A further restriction to the zeroth cohomology gives a preadditive category H 0 (C). Two object C, C of a DG category C are homotopy equivalent if they become isomorphic in H(C). A DG functor between two DG categories C and C is an additive functor F : C C that preserves the grading and dierential on morphisms. DG functors between DG categories form a DG category. Let F, G : C C be DG functors. To construct a DG category DG-Fun(C, C ) we put Hom k (F, G) to be the set of natural transformations t : F gr G gr [k]. The dierential is dened pointwise; for t C : F (C) G[k](C) we have ( (t)) C = (t C ) : F (C) G[k 1](C). The category of contravariant DG functors is denoted by DG-Fun o (C, C ). A functor F : C C is a quasi-isomorphism if H(F ) : H(C) H(C ) is an isomorphism. The category DGVect C of complexes of vector spaces with homogenous morphisms f : V W, f(v i ) W i+k and a dierential (f) i = f i+1 V ( 1) f W f i+1 is a DG category. Its homotopy category H(DGVect C ) is the homotopy category of complexes. A right DG module over a DG category C is a DG functor M DG-Fun o (C, DGVect C ). After [K2] we dene a derived category D(C) as a localization of H 0 (DG-Fun o (C, DGVect C )) with respect to the class of quasiisomorphisms. By D b (C) we will denote the subcategory of D(C) formed by compact objects. The Yoneda embedding gives a functor h : C D b (C) which assigns to every C C a free module h C = Hom C (, C). For a DG category C we dene the category Ĉ of formal shifts. The objects of Ĉ are of the form C[n] where C C and n N. For elements B[k] and C[n] of Ĉ we put Homḽ (B[k], C[n]) = Homl+n k C C (B, C). For appropriate sign convention see [BLL]. Let f Hom C (B, C) be a closed morphism of degree zero in a DG category C such that B[1] is an object of C. Then an object D C is the cone of f i there exist morphisms B[1] i D p B[1], C j D s C, such that pi = 1, sj = 1, si = 0, pj = 0, ip + js = 1. It is proved in [BLL] that the cone of closed degree zero morphism is uniquely dened up to a DG isomorphism. One can formally add cones of closed morphisms to a DG category C by considering the category C pre-tr of one-sided twisted complexes over C. Denition 1.3. A one-sided twisted complex over a DG category C is an expression ( n i=1 C i[r i ], q i,j ) where C i 's are objects of C, r i Z, n 0 and q i,j Hom 1 (C i [r i ], C j [r j ]) such that q i,j = 0 for i j and q + q 2 = 0. 3

5 One-sided twisted complexes over C form a DG category C pre-tr with the morphism space between C = ( C i [r i ], q) and C = ( C j [r j ], q ) given by the set of matrices f = (f i,j ), f i,j : C i [r i ] C j [r j ]. With a dierential dened as (f) = ( f i,j ) + q f ( 1) degf fq the category C pre-tr is a DG category. We will denote its zeroth homotopy category H 0 (C pre-tr ) by C tr. Let C = ( C i [r i ], q i,j ) be an object of C pre-tr. The shift of C is dened as C[1] = ( C i [r i + 1], q i,j ). The category C pre-tr is closed under formal shifts. Let f : C D be a closed morphism of degree 0 in C pre-tr, where C = ( n i=1 C i[r i ], q i,j ), D = ( m i=1 D i[s i ], p i,j ). The cone of f is a twisted complex C(f) = ( n+m i=1 E i[t i ], u i,j ), where { Ci for i n E i = D i n for i > n { ri + 1 for i n t i = s i for i > n q i,j for i, j n u i,j = f i,j n for i n < j for i, j > n p i n,j n The functor Tot : (C pre-tr ) pre-tr C pre-tr establishes a quasi-isomorphism between (C pre-tr ) pre-tr and C pre-tr. Let C i = ( n i j=1 Di j [ri j ], qi jk ) be objects of C pre-tr and let C = ( n i=1 C i[r i ], q ij ) be a twisted complex in (C pre-tr ) pre-tr. Then the convolution Tot(C) is equal to ( n n i j i=1 j=1 Di j [ri j + r i], qjk i + q ij). The DG category C is pretriangulated if the embedding H 0 (C) C tr is an equivalence. The category H 0 (C) for a pretriangulated category is triangulated. If for a triangulated category A there exists a pretriangulated category C such that A is equivalent to H 0 (C) the category A is called enhanced with the enhancement C. Recall, that an additive category A is Karoubian if every projector splits. The category C tr needs not to be Karoubian. As showed in [BLL] the category D b (C) is the Karoubization of C tr. A standard example of an enhanced triangulated category is the bounded derived category D b (A) of an abelian category A with enough injectives I. Its enhancement is Kom b (I), a DG subcategory of Kom(I) consisting of complexes bounded from below and with almost all cohomology groups equal to zero. With these denitions we are ready to state the following theorem after [BK]. Theorem 1.4. Let D be a pretriangulated category, E 1,..., E n objects in D and C D the full DG subcategory on the objects E i. Then the smallest triangulated subcategory of H 0 (D) containing E 1,..., E n is equivalent to C tr as a triangulated category. It follows that a full exceptional collection σ on a smooth projective variety X leads to an equivalence of D b (Coh(X)) and Cσ tr for some DG category C σ. The category C σ is a subcategory of the enhancement of D b (Coh(X)) and thus H i Hom Cσ (E, F ) = Ext i D b (Coh(X))(E, F ) for any elements E, F of σ. 4

6 Remark 1.5. The category D b (Coh(X)) for a smooth projective variety X is Karoubian. Hence, the category D b (C σ ) is equivalent to Cσ tr. We thus obtain an equivalence of D b (Coh(X)) and D b (C σ ). Remark 1.6. As shown in [K2] for two quasi-isomorphic DG categories C, C the derived categories D(C) and D(C ) are equivalent. Hence for a full exceptional collection σ on X the category C σ is determined up to a quasi-isomorphism. Remark 1.7. The group Z n acts on the set of exceptional collections in D b (A) by translations. For (a 1,..., a n ) Z n and an exceptional collection σ = E 1,..., E n the collection σ[a 1,..., a n ] = E 1 [a 1 ],..., E n [a n ] is also exceptional. The composition in the category C σ[a1,...,a n] is the same as in the category C σ and for f Hom(E i [a i ], E j [a j ]) we put σ[a1,...,a n](f) = ( 1) aj ai σ (f). Remark 1.8. Let C σ (E 1, E n ) = H(C σ (E 1, E n )) P be any decomposition of the vector space C σ (E 1, E n ). Morphisms in P form an ideal and C σ is quasi-isomorphic to a DG category C with C (E 1, E n ) = H(C σ (E i, E i+1 )). Moreover, let C σ (E i, E i+1 ) = H(C σ (E i, E i+1 )) P i be such a decomposition for morphisms between adjacent objects. Then the category C with C (E i, E i+1 ) = H(C σ (E i, E i+1 )) is a subcategory of C σ quasi-isomorphic to it. It proves that the category C σ is always quasi-isomorphic to a DG category C σ with C σ (E 1, E n ) = H( C σ (E 1, E n )), C σ (E i, E i+1 ) = H( C σ (E i, E i+1 )). Remark 1.9. If a DG category C has only zeroth cohomology then it is quasi-isomorphic to H(C) = H 0 (C). Indeed, for C 1, C 2 C let C(C 1, C 2 ) = n Z Cn (C 1, C 2 ) with dierential n C 1,C 2 : C n (C 1, C 2 ) C n+1 (C 1, C 2 ) and let C I be a DG category such that ob C I = ob C and C I (C 1, C 2 ) = C n (C 1, C 2 ) ker C 0 1,C 2. n 1 Then the natural inclusion functor C I C is a quasi-isomorphism. Let also J C1,C 2 = Im( 1 C 1,C 2 ) ( n<0 C n (C 1, C 2 )) for any C 1 and C 2 in C and consider a category C I/J with ob C I/J = ob C and C I/J (C 1, C 2 ) = C I (C 1, C 2 )/J C1,C 2. Then C I/J is isomorphic to H(C) and the natural functor C I C I/J is a quasiisomorphism. Hence, Theorem 1.4 is a generalisation of the Theorem 1.1. The denition of C σ from the theorem 1.4 requires taking injective resoultions and thus is not convenient for calculations. In the remaining part of the paper we will present two ways of nding C σ. We will always assume that X is a smooth projective variety and σ = E 1,..., E n is a full exceptional collection in D b (Coh(X)). 5

7 2 A categories We will use A categories. We start by recalling denitions and facts we shall use after [K1] and [L-H]. Remark 2.1. The graded Leibniz rule usually has the form (ab) = (a)b + ( 1) a a (b). However, in the case of morphisms in the category DGVect C we have (fg) = f (g) + ( 1) degg (f)g. Thus, all the denitions below dier from the ones given in [K1] by an appropriate sign change. Denition 2.2. An A category A over C consists of ˆ objects ob(a), ˆ for any two A, B ob(a) a Z - graded C-vector space A(A, B), ˆ for any n 1 and a sequence A 0, A 1,..., A n ob(a) a graded map: m n : A(A n 1, A n )... A(A 0, A 1 ) A(A 0, A n ) of degree 2 n such that for any n ( 1) rs+t m r+1+t (id r m s id t ) = 0. r+s+t=n Note that when these formulae are applied to elements additional signs appear because of the Koszul sign rule: (f g)(x y) = ( 1) y f f(x) g(y). An A algebra is an A category with one object. An A category A is ordered if there exists a partial order on the set ob(a) such that A(A, A ) = 0 for A A. The operation m 1 gives for any A, B ob(a) a structure of a complex on A(A, B). The homotopy category with respect to m 1 is denoted by H(A). The A category is minimal if the operation m 1 is trivial. Any DG category can be regarded as an A category with m 1 given by the dierential, m 2 given by composition and trivial m i 's for i > 2. For any set S there exists an A category C{S}. The objects of C{S} are elements of S and { C if s1 = s C{S}(s 1, s 2 ) = 2, 0 otherwise All operations m n in C{S} are trivial. Denition 2.3. A functor of A categories F : A B is a map F 0 : ob(a) ob(b) and a family of graded maps F n : A(A n 1, A n )... A(A 0, A 1 ) B(F 0 (A 0 ), F 0 (A n )) of degree 1 n such that ( 1) rs+t F r+1+t (id r m s id t ) = r+s+t=n i i r=n ( 1) p m r (F i1... F ir ), 6

8 where p = (r 1)(i r 1) + (r 2)(i r 1 1) (i 3 1) + (i 2 1). Composition of functors is given by (F G) n = F s (G i1... G is ). i i s=n A functor F : A B is called a quasi-isomorphism if F 1 is a quasiisomorphism. An A category A is strictly unital if for any object A ob(a) there exists 1 A A(A, A) of degree 0 such that for any objects B, C in A and any morphisms φ A(A, B), ψ A(C, A) we have m 2 (φ, 1 A ) = φ and m 2 (1 A, ψ) = ψ. Moreover for n 2 the operation m n equals 0 if any of its argument is equal to 1 A. In particular, for any set S the category C{S} is strictly unital. The category A is homologically unital if there exist units for the homotopy category H(A). Here, we cite the theorem proved by Lefevre-Hasegawa in [L-H] about a connection between strictly and homologicaly unital categories. Theorem 2.4. Minimal homolgically unital A category is quasi-isomorphic to a minimal strictly unital A category. Remark 2.5. A minimal A category is equal to its homotopy category. Hence, a quasi-isomorphism F given by the above theorem satises F 1 = id. An A category A is augmented if there exists a strict unit preserving functor ɛ : C{ob(A)} A. Then A decomposes as A = C{ob(A)} Ā. 2.1 Minimal model Any A category A is quasi-isomorphic to its homotopy category H(A) as stated in [K1]. Theorem 2.6. If A is an A category, then H(A) admits an A category structure such that 1. m 1 = 0 and m 2 is induced from m A 2 and 2. there is an A -quasi-isomorphism A H(A) inducing the identity on cohomology. Moreover, this structure is unique up to a non unique A -isomorphism. The A category H(A) is called the minimal model of A. If C is a DG category then its minimal model H(C) is quasi-isomorphic to a strictly unital A category. Indeed, the category C is strictly unital and hence homologically unital. It follows that its homotopy category is also homologically unital and the Theorem 2.4 guarantees that there exists a stricly unital minimal category quasi-isomorphic to H(C). Following [M] we describe an explicit construction of a minimal model for a DG category. Let C be a DG category with dierential and composition µ. In order to calculate its minimal model, the A structure on H(C) one has to dene 7

9 maps p : C H(C) and i : H(C) C such that both p and i are identities on objects of C and for any C, C C maps p : C(C, C ) H(C)(C, C ) and i : H(C)(C, C ) C(C, C ) are morphisms of complexes of degree 0. Also a homogeneous map h : C(C, C ) C(C, C ) of degree -1 is needed. These have to satisfy the following conditions pi = id, ip = id (h), h 2 = 0. The choice of i, p and h satisfying the above conditions is equivalent to the choice of splittings of the following exact sequences: 0 Ker n C n (C i, C j ) C n (C i, C j ) 0 ijs n 1 0 C n 1 (C i, C j ) n 1 Ker n H n C(C i, C j ) 0 for all C i, C j ob(c) and all n. Denition 2.7. The triple i : H(C) C, p : C H(C), h : C C is compatible with a nite family { ij s n k 1, ij s m l 2 } of splittings if i H m l (C i,c j) = ij s m l 2 for all m l, h Ims2 = 0 = h Ims1 and Im(h) = Im(s 1 ) for all s 1 in the family. We dene the operations λ n of degree 2 n on the category C as follows. We put λ 1 = h 1, λ 2 = µ and ijs n 2 n 1 λ n = ( 1) k+1 µ(h(λ n k ), h(λ k )). k=1 For n > 1 the higher multiplications m n on H(C) are dened as m n = pλ n i n. 2.2 The universal DG category For any A category A there exists a DG category U(A) and an A quasiisomorphism A U(A). To dene the category U(A) we need the following denitions (see further [L-H]). Denition 2.8. A DG cocategory B consists of ˆ the set of objects B i ob(b), ˆ for any pair of objects B i, B j ob(b) a complex of C-vector spaces B(B i, B j ) with a dierential d ij of degree 1 and ˆ a coassociative cocomposition a family of linear maps : B(B i, B j ) B(B k, B j ) B(B i, B j ). B k ob(b) These data have to satisfy the condition d = (d id + id d). 8

10 For any set S the A category C{S} is also a DG cocategory. A functor Φ between DG cocategories B and B preserves the grading and dierentials on morphisms and satises the condition Φ = (Φ Φ). A DG cocategory B is counital if it admits a counit, a functor η : B C{ob(B)}. The category B is coaugmented if it is counital and admits a coaugmentation functor ε : C{ob(B)} B such that the composition ηε is the identity on C{ob(B)}. Let B be a coaugmented DG cocategory. The category B has the same set of objects as B and B(B i, B j ) = ker ε. For an augemnted A category A one can dene its bar DG cocategory B (A). Recall that A = Ā C{ob(A)}. Then B c (A) = T (SĀ) is a tensor cocategory of the suspension of Ā. Here SĀ denotes the category Ā with a shift in a morphisms spaces (SĀ)n (A, A ) = Ān+1 (A, A ). SĀ is not an A category, however the operations m n is Ā dene graded maps of degree 1 in SĀ. b n : SĀ(A n 1, A n )... SĀ(A 0, A 1 ) SĀ(A 0, A n ), b n = s m n ω n, where s : V SV is a suspension of a graded vector space V and ω = s 1. The cocategory B (A) has the same objects as A and B (A)(A, A ) = SĀ(A, A ) SĀ(A 1, A ) SĀ(A, A 1) A 1,A 2 ob(a) A 1 ob(a) for A A. In the case A = A we have B (A)(A, A) = 1 A SĀ(A, A) A 1,A 2 ob(a) SĀ(A 2, A ) SĀ(A 1, A 2 ) SĀ(A, A 1)... A 1 ob(a) SĀ(A 1, A) SĀ(A, A 1) SĀ(A 2, A) SĀ(A 1, A 2 ) SĀ(A, A 1)... where the degree of 1 A is zero. To simplify the notation we shall write (α n,..., α 1 ) for α n... α 1. The dierential in B (A) is given by d(α n,..., α 1 ) = n n k+1 ( 1) α l α 1 (α n,... α l+k, b k (α l+k 1,..., α l ), α l 1,... α 1 ), k=1 l=1 for (α n,..., α 1 ) B (A)(A, A ). The cocomposition is given by (α n,..., α 1 ) = n 1 1 A (α n,..., α 1 ) + (α n,..., α 1 ) 1 A + (α n,..., α l+1 ) (α l,..., α 1 ). With these denitions B (A) is an augmented DG cocategory. l=1 9

11 Remark 2.9. Let A be an ordered A category with nitely many objects, nitely dimensional morphisms spaces and such that Ā(A, A) = 0 for any object A oba. Then the DG cocategory B (A) also satises these conditions; i.e. is ordered, has nitely many objects, nitely dimensional morphisms spaces and B (A)(A, A) = 0 for any object A. Analogously, to an augmented DG cocategory B via a cobar construction one can assign a DG category Ω(B). Let B be a DG cocategory with a dierential d and cocomposition. It's cobar DG category is equal to T (S 1 B). Here S 1 B denotes the shift of the cocategory B and T (S 1 B) is the tensor DG category of it. As before the morphisms spaces in T (S 1 B) are given by Ω(B)(B, B ) = S 1 B(B, B ) S 1 B(B1, B ) S 1 B(B, B1 ) for B B and by B 1,B 2 ob(b) Ω(B)(B, B) = 1 B S 1 B(B, B) B 1,B 2 ob(b) B 1 ob(b) S 1 B(B2, B ) S 1 B(B1, B 2 ) S 1 B(B, B1 )... B 1 ob(b) S 1 B(B1, B) S 1 B(B, B1 ) S 1 B(B2, B) S 1 B(B1, B 2 ) S 1 B(B, B1 )... for 1 B a morphism in degree zero. The composition in Ω(B) is dened by concatenation and the dierential on the morphisms spaces is = (d + ) Remark If an ordered DG cocategory B has nitely many objects, nitely dimensional morphisms spaces between the objects and satises the condition B(B, B) = 0 then the same is true about Ω(B). For an augmented A category A its universal DG category U(A) is dened as Ω(B (A)). There is a natural map A U(A). It is proved in [L-H] that this map extends to a functor and is an A quasi-isomorphism (Lemme of [L-H]). Moreover, for an A quasi-isomorphisms φ the functor U(φ) is a quasi-isomorphism of DG categories. 2.3 A modules Denition An A module over an A category A is an A functor M : A DGVect C. A morphism of modules G : M N is given by a family {G A : M 0 (A) N 0 (A)} A oba of morphisms in DGVect C. Moreover, the diagram M 0 (A 0 ) Mn(αn 1... α0) M 0 (A n ) G A0 G An N 0 (A 0 ) Nn(αn 1... α0) N 0 (A n ) has to commute for any n N, A i oba and α i A(A i, A i+1 ). 10

12 The category of A modules over an A category A will be denoted as Mod strict A. This notation agrees with the notation in [L-H]. The morphism G of modules is a quasi-isomorphism of A modules if G A is a quasi-isomorphism of complexes for any A oba. The derived category D (A) of an A category A is dened as a localization of the category Mod strict A with respect to the class of quasi-isomorphisms. Remark 2.12 (Lemme of [L-H]). For a DG category C the derived categories D(C) and D (C) are equivalent. Remark In the case of A algebra there can be more morphism between modules, but there is no dierence on the level of derived categories; see [L-H]. For an A category A the derived category D (A) is equivalent to the derived category of the universal algebra D(U(A)) (see[l-h]). Let A and A be A categories and F : A A an A quasi-isomorphism. Then U(F ) : A A is a quasi-isomorphism of DG categories. It follows that D(U(A)) is equivalent to D(U(A )) (see [K2]) and hence D (A) is equivalent to D (A ). Thus, we have proved, after [L-H] the Proposition Let A, A be quasi-isomorphic A categories. Then categories D (A) and D (A ) are equivalent. Let us come back to the case of exceptional collections. Recall, that X is a smooth projective variety and σ a full exceptional collection on X. We know already that there exist a DG category C σ such that the derived category of coherent sheaves on X, D b (Coh(X)) is equivalent to D b (C σ ), where D b (C σ ) D(C σ ) is a subcategory spanned by compact objects. Moreover, the morphisms in the homotopy category H(C σ ) can be calculated via Ext groups in the category D b (Coh(X)). In particular, the minimal model of C σ, H(C σ ) is an ordered A category with nitely many objects and nitely dimensional morphisms space. According to the theorem 2.4 and remark 2.5 there exists a quasi-isomorphic strictly unital A category C σ with the same objects and morphism spaces. By remarks 2.9 and 2.10 the DG category U( C σ ) is an ordered DG category with nitely many objects and nitely dimensional morphisms spaces. It is A quasi-isomorphic to the original category C σ. From the proposition 2.14 and the remark 2.12 we deduce the following Theorem Let X be a smooth projective variety and let σ = E 1,..., E n be a full exceptional collection on X. There exists a DG category D σ with objects E 1,..., E n and nitely dimensional morphisms spaces such that (i) the bounded derived category of coherent sheaves on X, D b (Coh(X)) is equivalent to the bounded derived category of D σ and (ii) for any i, j we have H (D σ (E i, E j )) = Ext X(E i, E j ). Remark The category D pre-tr σ is pretriangulated and gives an enhancement of D b (Coh(X)). In the next sections we will propose two methods of calculating the category D σ. 11

13 3 Calculating the category D σ by Massey products The triangulated category D b (Coh(X)) might be enough to determine the A category H(C σ ) and hence the DG category D σ. To see it we will use Massey products. We start by recalling some denitions after [GM]. Denition 3.1. Let X 0 d 0 X 1 d 1 d... n 2 dn 1 n 1 X X n be a complex in a triangulated category D. A convolution of this complex is any object T such that there exists the following diagram X 0 d 0 X 1 Xn 1 d n 1 id d 0 Y 0 = X 0 +1 Y 1 Y n 1 X n Y n = T where all triangles Y i X i+1 Y i+1 Y i [1] are distinguished and the composition of X i Y i X i+1 is equal to d i : X i X i+1. Note, that for a convolution T we have a map of degree 0 from X n to T and a map of degree n from T to X 0. Knowing convolutions one can dene Massey products µ n. Denition 3.2. For a pair of morphisms +1 X 0 f X 1 g X 2 the double Massey product µ 2 is dened as µ 2 (g, f) = gf. Let X 0 d 0 X 1 d 1 d... n 2 dn 1 n 1 X X n be a complex in a triangulated category D and let T be a convolution of X 1... X n 1. The Massey product µ n (d n 1,..., d 0 ) is dened as a set {qr} where q and r are morphisms X 0 d 0 X r [2 n] 1 d 1... X n 1 X n [n 2] q T such that the composition X 0 T X 1 is d 0 and X n 1 T X n is d n 1. The morphisms X n 1 T and T X 1 [n 2] arise from the denition of the convolution. Proposition 3.3 (see [GM]). If the set µ n (d n 1,..., d 0 ) is not empty then 0 µ n 1 (d n 2,..., d 0 ) and 0 µ n 1 (d n 1,..., d 1 ). Moreover, the complex X 0... X n has at least one convolution if and only if 0 µ n (d n 1,..., d 0 ). d n 1 12

14 3.1 Massey products and minimal model Let A be a C-linear abelian category with enough injectives I. The bounded derived category of A, D = D b (A) has an enhancement given by the DG category C = Kom b (I). Lemma 3.4. Let f 1,... f n, for f i : X i 1 X i, be a sequence of morphisms in D and let v µ n (f n,..., f 1 ). Then there exists a choice of i 1,i+k 1s λ k(f j,...,f j k+1 ) 2 and i 1,i+k 1 s λ k(f j,...,f j k+1 ) 1 1 for 1 < k < n and k j n such that for any triple i, p, h compatible with these splittings we have v = pλ n (i(f n ),..., i(f 1 )) and 0 = pλ k (i(f j ),..., i(f j k+1 )). Proof. We will show that for a complex X 0 f0 X 1 [n 1 ] f 1... X k [n k ] f k X k+1 [n k+1 ] of closed morphisms of degree 0 in C the convolution T of it is given by a complex T = X 1 [n 1 + k 1] X 2 [n 2 + k 2]... X k [n k ] with a dierential = ( 1,..., k ), l : T X l [n l + k l] equal to l 1 l (x 1,..., x l ) = Xl [n l +k l](x l )+( 1) k l ( 1) ( λi 1)n l i hλ i ( f l 1,..., f l i )(x l i ). A closed morphism of degree 0 r : X 0 T [1 k], r = ( r 1,..., r k ) is given by i=1 r l (x 0 ) = ( 1) l+1 hλ l ( f l 1,..., f 0 )(x 0 ) and a morphism q : T X k+1 [n k+1 ], q = k 1 l=0 q l, q k l : X l [n l + k l] X k+1 [n k+1 ] is equal to q k l (x l ) = ( 1) ( λ k l+1 1)n l hλ k l+1 ( f k,..., f l ). Hence, q r = ( 1) k+1 fk+1 hλ k ( f k,..., f 1 ) ( 1) ( λ k 1) f 0 hλ k ( f k+1,..., f 2 ) f 1 ( 1) i+1+( λ k i+1 1)( f f i ) hλ k i+1 ( f k+1,..., f i+1 )hλ i ( f i,..., f 1 ) k 1 + i=2 = λ k+1 ( f k+1,..., f 1 ). r and q are closed morphisms of degree 0 in C and can be treated as morphisms in D. Together with T they dene the Massey product in the category D. Maps r and q determine also the value of λ l ( f i+l 1,..., f i ) for any l k. Moreover, pλ l ( f i+l 1,..., f i ) = 0 µ l (f l+i 1,..., f i ) as only with such a choice the Massey product µ k+1 is dened. We will proceed by induction. Assume that v µ n (f 2, f 1, f 0 ). Then, according to the denition v = q r, for a pair of morphisms in D. Let us write f j = i(f j ) for any inclusion i : D C. In the category C we have X 0 f0 X 1 [n 1 ] f 1 X 2 [n 2 ] f 2 X 3 [n 3 ]. 13

15 The convolution T of X 1 [n 1 ] X 2 [n 2 ] is the cone of f 1, T = X 1 [n 1 + 1] X 2 [n 2 ] with a dierential (x 1, x 2 ) = ( X1[n 1+1](x 1 ), f 1 (x 1 ) + X2[n 2](x 2 )). Let r : X 0 T [ 1] be a morphism in D lifting r. It is a closed morphism of degree 0 and commutes with f 0 : X 0 X 1 [n 1 ] hence is given by with r(x 0 ) = ( f 0 (x 0 ), r 2 (x 0 )), r 2 X0 = f 1 f0 + X2[n 2 1], ( r 2 ) = λ 2 ( f 1, f 0 ). Thus, one can choose 0,2 s n2 2 and 0,2 s n2 1 1 in such a way that for any compatible h the equality hλ 2 ( f 1, f 0 ) = r 2 will be satised. Analogously, a morphism of degree 0, q : T X 3 [n 3 ] is given by q(x 1, x 2 ) = q 1 (x 1 ) + f 2 (x 2 ). It is closed if q T = X2[n 2] q which leads to q 1 X1[n 1+1](x 1 ) + f 2 f1 (x 1 ) + f 2 X 2 [n 2 ](x 2 ) = X3[n 3]q 1 (x 1 ) + X3[n 3] f 2 (x 2 ), q 1 X1[n 1+1] + f 2 f1 = X3[n 3] q 1, ( q 1 ) = ( 1) n1 λ 2 ( f 2, f 1 ). Again, there exists such a choice of 1,3 s n3 n1 2 and 1,3 s n3 n1 1 1 that for any compatible h we will have h(λ 2 ( f 2, f 1 )) = q 1. Finally µ 3 (f 2, f 1, f 0 ) =p( q r) = p( q 1 r 1 + q 2 r 2 ) Now, let us consider = p(( 1) n1 hλ 2 ( f 2, f 1 ) f 0 f 2 hλ 2 ( f 1, f 0 )) = p(λ 3 ( f 2, f 1, f 0 )). X 0 f0 X 1 [n 1 ]... X k [n k ] in C lifting a complex in the category D. We have f k X k+1 [n k+1 ] f k+1 X k+2 [n k+2 ] T (X 1 [n 1 ]... X k+1 [n k+1 ]) = Cone(T (X 1 [n 1 ]... X k [n k ]) X k+1 [n k+1 ]) and For l = 1,..., k the dierential is = X 1 [n 1 + k] X 2 [n 2 + k 1]... X k+1 [n k+1 ]. l 1 l = Xl [n+l+k+1 l] + ( 1) k+1+l ( ( 1) ( λ i 1)n l i hλ i ( f l 1,..., f l i ). k+1 = Xk+1 [n k+1 ] + = Xk+1 [n k+1 ] + i=1 k ( 1) ( λ k l+1 1)n l hλ k l+1 ( f k,..., f l ) l=1 k ( 1) ( λi 1)n k+1 i hλ i ( f k,..., f k+1 i ). i=1 14

16 A morphism r : X 0 X 1 [n 1 ]...X k+1 [n k+1 k] is given by ( r 1,..., r k+1 ), r l : X 0 X l [n l l + 1]. For l = 1,..., k r k = ( 1) l+1 hλ l ( f l 1,..., f 0 ). r is closed so r X0 = T [ k] r, in particular r k+1 X0 = k+1 r. This gives k r k+1 X0 = Xk+1 [n k+1 k] r k+1 + ( 1) k ( 1) ( λ k l+1 1)n l hλ k l+1 ( f k,..., f l ) r l = Xk+1 [n k+1 k] r k+1 k + ( 1) k ( 1) ( λ k l+1 1)n l +l 1 hλ k l+1 ( f k,..., f l )λ l ( f l 1,..., f 0 ), l=1 l=1 r k+1 X0 Xk+1 [n k+1 k] r k+1 = ( 1) k λ k+1 ( f k,..., f 0 ), r k+1 = ( 1) k λ k+1 ( f k,..., f 0 ). Hence, as before one can choose such splittings s 1 and s 2 that r k+1 = ( 1) k hλ k+1 ( f k,..., f 0 ) for any compatible h. A morphism q : X 1 [n 1 + k]... X k+1 [n k+1 ] X k+2 [n k+2 ] is a sum of morphisms q k l : X l+1 [n l+1 + k l] X k+2 [n k+2 ]. For l = 1,..., k q k l = ( 1) ( λ k l+1 1)n l+1 hλ k l+1 ( f k+1,..., f l+1 ). To nd q k we again use the fact that q is a morphisms of complexes, that is q T = Xk+2 [n k+2 ] q. Writing down this equality for elements of the form (x 1, 0,..., 0) T we obtain Xk+2 [n k+2 ] q k = q k X1[n 1+k] k + ( 1) ( λ k l+1 1)n l+1 +k l+( λ l 1)n 1 hλ k l+1 ( f k+1,..., f l+1 )hλ l ( f l,..., f 1 ), l=1 q k X1 Xk+2 [n k+2 k] q k = ( 1) k+n1 k l=1 ( 1) ( λ k l+1 1)n l+1 +k l+( λ l 1)n 1 hλ k l+1 ( f k+1,..., f l+1 )hλ l ( f l,..., f 1 ) = ( 1) k+ λ k l+1 n 1+ λ l n 1+n 1 k ( 1) ( λ k l+1 1)( f f l )+k l hλ k l+1 ( f k+1,..., f l+1 )hλ l ( f l,..., f 1 ) l=1 = ( 1) kn1 µ k+1 ( f k+1,..., f 1 ), q k = ( 1) ( λ k+1 1)n 1 λ k+1 ( f k+1,..., f 1 ) 15

17 and there exists splittings s 1 and s 2 such that for any compatible h. Finally µ k+2 (f k+1,..., f 0 ) = p( q r) = p( = p( = p( q k = ( 1) ( λ k+1 1)n 1 hλ k+1 ( f k+1,..., f 1 ) k q k l r l+1 ) l=0 k ( 1) l+( λ k l+1 1)n l+1 hλ k l+1 ( f k+1,..., f l+1 )hλ l+1 ( f l,..., f 0 )) l=0 k ( 1) l+( λ k l+1 1)( f f l ) hλ k l+1 ( f k+1,..., f l+1 )hλ l+1 ( f l,..., f 0 )) l=0 = p(λ k+2 ( f k+1,..., f 0 )). With this lemma we are ready to formulate the Theorem 3.5. Let D be an enhanced triangulated category, E 1,..., E n objects in D and C the full DG subcategory of the enhancement of D with objects E i. Let also (f1 1,..., fn 1 1 ),... (f j 1,..., f n j j ) be a family of morphisms in H( C) such that for all i and 1 < l < n i any two of λ l (fk+l 1 i,..., f k i ) do not lie in the same space C n (E i1, E i2 ) and nally let v i µ ni (fn i i,..., f1). i Then, up to an A isomorphism, in the minimal model for C one has v i = m ni (fn i i,..., f1) i and 0 = m l (fk+l 1 i,..., f k i) for l < n i. Proof. The lemma 3.4 tells us that for morphisms f k 1,..., f k n k and v k there exists a choice of splittings s 1 and s 2 such that v k = pλ nk (i(f k n k ),..., i(f k 1 )) for any compatible triple i, p and h. The conditions of the theorem imply that for all k = 1,..., j the choice of these splittings is independent. Combining it with the result of Merkulov recalled in 2.1 we obtain the proof of the theorem. Remark 3.6. In the case when a triple Massey product is not dened, one can still deduce something about the operation m 3. Let A α B β C γ D be morphisms in H( (C) such that βα 0 and γβ = 0. Then m 3 (γ, β, α) = pλ 3 (i(γ), i(β), i(α)) = p(( 1) α h(i(γ)i(β))i(α) i(γ)h(i(β)i(α))). Fixing h(i(β)i(α)) = 0. Then where λ 3 (i(γ), i(β), i(α)) = ( 1) α i(γ)i(βα). If the category C has innite dimensional morphisms spaces and nite A,C s β + α 2 one can assume that i(β)i(α) = i(βα) and hence dimensional cohomology groups then there exists a choice of A,Cs α + β 2 and C,D s γ 2 such that λ 3 (i(γ), i(β), i(α)) 0. Finally, there exists a choice of A,D s α + β + γ 1 1 such that λ 3 (i(γ), i(β), i(α)) Kerp which gives m 3 (γ, β, α) = 0. 16

18 4 Calculating the category D σ via mutations In the previous section we have proved that in some cases the DG category D σ can be determined by means of Massey products. However, the theorem 3.5 does not determine the A category H(C σ ) for any collection σ and moreover calculating Massey products of arbitrary morphisms in a triangulated category is not always an easy task. Therefore, in this section we will present another algorithm allowing to determine the DG category D σ. It uses a mutation of the collection σ to a collection ρ for which the DG category D ρ is known. In particular, it determines the DG categories of all exceptional collections obtained via mutations from a strong one. We begin with recalling the denitions from [B]. Let E, F be an exceptional pair in D b (Coh(X)) for a smooth projective variety X. Then L E F, E and F, R F E are also exceptional pairs for L E F and R F E dened by means of distinguished triangles in D b (Coh(X)). L E F Hom(E, F) E F L E F[1], E Hom(E, F) F R F E E[1]. Here Hom(E, F) is treated as a complex of C-vector spaces with trivial dierential, Hom(E, F) = Hom k (E, F) for Hom k (E, F) = Hom Db (Coh(X))(E, F[k]). For an element E of a D b (Coh(X)) and a complex V the tensor product is dimv k i=0 E[ k]. dened by E V = k Z If ρ = E 1,..., E n is an exceptional collection in D b (Coh(X)) then the left i-th mutation L i ρ and the right i-th mutation R i ρ are dened as follows L i ρ = E 1,..., E i 1, L Ei E i+1, E i, E i+2,..., E n, R i ρ = E 1,..., E i 1, E i+1, R Ei+1 E i, E i+2,..., E n. Twisted complexes provide a description of the categories D Liρ and D Riρ by means of D ρ. To see it we need to dene a tensor product of a twisted complex with a complex of vector spaces. Let C be a DG category with nitely dimensional vector spaces, C C pre-tr be a twisted complex and let V be a nite dimensional complex of vector spaces with the dierential i : V i V i+1. C V is dened as ( {i V i 0} C[ i]dimv i, q i,j ) (C pre-tr ) pre-tr. The morphisms q i,i+1 are induced by the dierential i tensored with the identity on C and q i,j = 0 for j i + 1. Now, let C, D C pre-tr be twisted complexes. There exist closed morphisms of degree 0 φ :C C pre-tr (C, D) D, ψ :C C pre-tr (C, D) D. The morphism φ i,0 : C[ i] dim Homi (C,D) D is given by morphisms of degree i between C and D and ψ is dened analogously. Now we dene new twisted complexes over C L C D = Tot(C(φ)[ 1]), R D C = Tot(C(ψ)). 17

19 Let ρ = E 1,..., E n be a full exceptional collection on a smooth projective variety X and let D ρ with objects E 1,..., E n be the category described in the theorem We dene two full subcategories of Dρ pre-tr ; Dρ Li with objects E 1,..., E i, L Ei E i+1, E i+2,... E n and Dρ Ri with E 1,..., E i 1, R Ei+1 E i, E i+1,..., E n. Proposition 4.1. The category Dρ Li satises the condition of the theorem 2.15 for the exceptional collection L i ρ. Analogous statement is true for D Ri Proof. The theorem 2.15 guarantees that the category D ρ has nite dimensional vector spaces. Hence, mutations of twisted complexes over D ρ are well dened. Furthermore, according to the remark 2.16 the category D b (Coh(X)) is equivalent to (D ρ ) tr. Under this equivalence the object L Ei E i+1 corresponds to L Ei E i+1. This proves that condition (ii) of the theorem 2.15 is satised. Finally, the condition (ii), the theorem 1.4 and the fact that the category D b (Coh(X)) is Karoubian show that the condition (i) of the theorem 2.15 is also satised. 5 Example Let X be a toric surface with a fan presented in the Figure 1. ρ. Figure 1: The fan of X. X is obtained from P 2 by blowing-up two points and then blowing up one point on an exceptional divisor. The Picard group of X has basis H, E 1, E 2 and E 3 with H 2 = 1, E 2 1 = 1, E 2 2 = 2, E 2 3 = 1, HE i = 0, E 1 E i = 0, E 2 E 3 = 1. In this basis the toric divisors are: D 1 = E 1, D 2 = H E 1 E 2 E 3, D 3 = E 2, D 4 = E 3, D 5 = H E 2 2E 3, D 6 = H E 1. In [HP] Hille and Perling prove that the collection O, O(E 1 ), O(H E 2 E 3 ), O(H E 3 ), O(H), O(2H E 2 2E 3 ) is full. It is not strong, there is one nontrivial Ext 1 group between O(H E 2 E 3 ) and O(H E 3 ). The quiver of this collection has the form presented in the Figure 2. There is one relation 18

20 Figure 2: The quiver of the collection O, O(E 1 ), O(H E 2 E 3 ), O(H E 3 ), O(H), O(2H E 2 2E 3 ) on X. edcba = icf. The dotted arrow represents the nontrivial Ext group. Calculating Massey product for morphisms in this collection requires nding a nontrivial extension 0 O(H E 3 ) E O(H E 2 E 3 ) 0. After one mutation we obtain a collection in which calculating Massey products is signicantly easier. We have a short exact sequence 0 O(E 1 ) O(H E 2 E 3 ) O H E1 E 2 E 3 0 so up to a translation, the 2-nd left mutation of the considered collection is O, O H E1 E 2 E 3, O(E 1 ), O(H E 3 ), O(H), O(2H E 2 2E 3 ). This collection has nontrivial rst Ext groups from O H E1 E 2 E 3 to O(E 1 ), O(H E 3 ), O(H) and O(2H E 2 2E 3 ). The quiver of the this collection is presented in the Figure 3. Figure 3: The quiver of the collection O, O H E1 E 2 E 3, O(E 1 ), O(H E 3 ), O(H), O(2H E 2 2E 3 ) on X. The arrows α 1 and α 2 are of degree 1. The relations in the quiver are d c b e = g h, α 1 a = 0, α 2 a = 0, b α 1 = 0, g α 2 = 0, f α 1 = c α 2. The values of m 3 (b, α 1, a ), m 3 (d, f α 1, a ), m 3 (f, α 1, a ), m 3 (g, α 2, a ) and m 3 (c, α 2, a ) determine the A structure of the quiver presented in the Figure 19

21 3. Using the theorem 3.5 and the remark 3.6 we get that m 3 (b, α 1, a ) µ 3 (b, α 1, a ), m 3 (d, f α 1, a ) = 0, m 3 (g, α 2, a ) µ 3 (g, α 2, a ), m 3 (f, α 1, a ) = 0, m 3 (c, α 2, a ) = 0. Calculating µ 3 for morphisms in this collection is not dicult. For example h µ 3 (b, α 1, a ). a O O α 1 H E1 E 2 E O(E 3 1 )[1] O(H E 3 ) p[ 1] q O(H E 2 E 3 )[1] In the language of toric divisor the zeros of b are Z(b ) = D 2 + D 3. For the map p we have Z(p) = D 4 + D 5. Finally, for the zero locus of q is Z(q) = D 3. It follows that Z(qp) = D 3 + D 4 + D 5 and hence qp = h. Similarly one can show that µ 3 (g, α 2, a ) = d f e. Finally we obtain the following A structure m 3 (b, α 1, a ) = h, m 3 (c b, α 1, a ) = c h, m 3 (d c b, α 1, a ) = d c h, m 3 (d f, α 1, a ) = 0, m 3 (d, f α 1, a ) = 0, m 3 (f, α 1, a ) = 0, m 3 (d c, α 2, a ) = 0, m 3 (g, α 2, a ) = d f e, m 3 (c, α 2, a ) = 0, m 3 (d, f, α 1 ) = 0, m 3 (c, b, α 1 ) = 0, m 3 (d c, b, α 1 ) = 0, m 3 (d, c b, α 1 ) = 0, m 3 (d, c, α 2 ) = 0. The operation having as the last argument α 1 or α 2 are trivial because there are no morphism of degree 0 from B to neither E nor F. With these information we can calculate the DG category of the collection O, O H E1 E 2 E 3, O(E 1 ), O(H E 3 ), O(H), O(2H E 2 2E 3 ). It is presented in the Figure 4. Again, arrows α 1 and α 2 are in degree 1. The relations and dierentials in the quiver are d c b e = g b u 1 g u 3 a (= g h ), f α 1 = c α 2, d f e = g u 2 u 4 a, f u 1 = c u 2, (u 1 ) = α 1 a, (u 2 ) = α 2 a, (u 3 ) = b α 1, (u 4 ) = g α 2. Now we can mutate the above DG category to obtain the DG category of the collection O, O(E 1 ), O(H E 2 E 3 ), O(H E 3 ), O(H), O(2H E 2 2E 3 ). Here, O(H E 2 E 3 ) is a right mutation of O H E1 E 2 E 3 over O(E 1 ) and can be written as a twisted complex, where the morphism is α 1. O(H E 2 E 3 ) O H E1 E 2 E 3 O(E 1 ), b 20

22 Figure 4: The DG category of the collection O, O H E1 E 2 E 3, O(E 1 ), O(H E 3 ), O(H), O(2H E 2 2E 3 ) on X. Up to a quasi-isomorphism we obtain the DG category in the Figure 5. Figure 5: The DG category of the collection O, O(E 1 ), O(H E 2 E 3 ), O(H E 3 ), O(H), O(2H E 2 2E 3 ) on X. The arrow c 1 is in degree 1. We have following relations and dierentials References ehba = ij, edcba = icf, edga = lf, ig = lb, dj = hf, edc 1 ba = ic 1 f, (h) = dc 1, (j) = c 1 f, (l) = ic 1, (g) = c 1 b. [B] A. Bondal, Representations of associative algebras and coherent sheaves, Math. USSR Izviestiya, 34(1), 23-42, [BK] A. Bondal, M. Kapranov, Enhanced triangulated categories, Mat. Sb. 181 (1990), no. 5, ; translation in Math. USSR-Sb. 70, no. 1, 93107, [BLL] A. Bondal, M. Larsen, V. Lunts, Grothendieck ring of pretriangulated categories [GM] S. Gelfand, Y. Manin, Methods of Homological Algebra, Springer, Berlin,

23 [HP] L. Hille, M. Perling, Exceptional sequences of invertible sheaves on rational surfaces, arxiv: , [K1] B. Keller, A-innity algebras, modules and functor categories, Trends in representation theory of algebras and related topics, 6793, Contemp. Math., 406, Amer. Math. Soc., Providence, RI, [K2] B. Keller, Deriving DG categories, Ann. Scient. Ec. Norm. Sup. 27, , [M] S. Merkulov, Strong homotopy algebras of a Kähler manifold, Intern. Math. Research Notices, 3, (math/ ). [L-H] Kenji Lefevre-Hasegawa, Sur les A -categories, These de doctorat, Universite Denis Diderot 22