Hyperplane sections and derived categories

Izvestiya: Mathematics 70:3 447 547 Izvestiya RAN : Ser. Mat. 70:3 23 128 c 2006 RASDoM and LMS DOI 10.1070/IM2006v070n03ABEH002318 Hyperplane sections and derived categories A. G. Kuznetsov Abstract. We give a generalization of the theorem of Bondal and Orlov about the derived categories of coherent sheaves on intersections of quadrics, revealing the relation of this theorem to projective duality. As an application, we describe the derived categories of coherent sheaves on Fano 3-folds of index 1 and degrees 12, 16 and 18. Introduction One of the most promising directions in modern algebraic geometry is the systematic use of derived categories of coherent sheaves [5]. Derived categories were introduced by Verdier in 1967. Recently, they have attracted a lot of attention due to progress in the understanding of their role in modern geometry. It was realized that geometric similarity between different algebraic varieties is sometimes only the tip of the iceberg and is a consequence of a deeper relation which can be expressed as an equivalence of appropriate categories. Take for example the McKay correspondence. The story began in 1980 with McKay s discovery [19] of a correspondence between irreducible representations of a finite subgroup Γ SL2, C and vertices of an affine Coxeter Dynkin diagram of type A, D, E. A geometric interpretation of this correspondence incorporating a resolution of the simple surface singularity C 2 /Γ was given by Gonzalez-Sprinberg and Verdier [11] in 1983. Finally, Kapranov and Vasserot [15] showed that the derived category of coherent sheaves on the resolution of C 2 /Γ is equivalent to the derived category of coherent Γ-equivariant sheaves on C 2. See also [8] for a beautiful generalization. Another example of this sort is the theory of intersections of quadrics. Classically, it was formulated as a correspondence between intersections of quadrics and determinantal loci in the corresponding linear spaces of quadrics [26]. The geometric meaning of this correspondence was discovered in [10]. In the particular case of the intersection of a pencil of even-dimensional quadrics, one should consider the hyperelliptic curve obtained as a twofold covering of P 1 parametrizing quadrics in the pencil, with ramification at the points of P 1 corresponding to degenerate quadrics. As shown in [10], the moduli spaces of vector bundles of rank 2 on this hyperelliptic curve can be described in terms of linear subspaces in the intersection of quadrics. Bondal and Orlov [4] gave a categorical meaning to this by proving that the category of coherent sheaves on the hyperelliptic curve can be fully and This work was partially supported by the Russian Foundation for Basic Research grant nos. 05-01-01034 and 02-01-01041, a Russian Presidential grant for young scientists no. MK- 3926.2004.1, a CRDF Award no. RUM1-2661-MO-05 and the Russian Science Support Foundation. AMS 2000 Mathematics Subject Classification. 18E30, 14A22.

448 A. G. Kuznetsov faithfully embedded in the derived category of the intersection of quadrics. Finally, Bondal and Orlov [5] stated a theorem describing the structure of the derived category of a complete intersection of arbitrarily many quadrics. Theorem 0.1 [5]. For any family of quadrics L S 2 W in an even-dimensional vector space W there exists a sheaf of finite algebras A L on the twofold covering of PL ramified at the degeneration locus such that if 2 dim L < dim W, then there is a semiorthogonal decomposition D b X L = D b CohA L, O XL,..., O XL dim W 2 dim L 1, and if 2 dim L = dim W, then there is an equivalence of categories D b X L = D b CohA L. Here X L stands for the complete intersection of quadrics in the family L and D b CohA L stands for the derived category of sheaves of coherent A L -modules. Let us explain the relation of Theorem 0.1 to projective duality. Note that the intersection of quadrics X L can be regarded as a linear section of the Veronese subvariety X = PW PS 2 W by the linear space PL PS 2 W, where L S 2 W is the subspace orthogonal to L S 2 W. On the other hand, the sheaf of algebras A L on PL PS 2 W can be extended to a sheaf A PS2 W on the whole of PS 2 W. Actually, this sheaf is just the sheaf of even parts of the universal Clifford algebra, that is, A PS 2 W = O PS 2 W Λ 2 W O PS 2 W 1 Λ 2n W O PS 2 W n with the Clifford multiplication. We regard PS 2 W with the sheaf of algebras A PS2 W as a non-commutative algebraic variety. In fact, it is a non-commutative finite covering of PS 2 W. Note that the set of critical values of the projection PS 2 W, A PS 2 W PS 2 W coincides with the projectively dual variety X PS 2 W. Indeed, since the projection PS 2 W, A PS 2 W PS 2 W is flat, a point H PS 2 W is its critical value if and only if the fibre is singular. The singularity of an algebraic variety is equivalent to the Ext-boundedness of its derived category of coherent sheaves. Thus the set of critical values coincides with the set of those H PS 2 W such that the category D b CohA H is not Ext-bounded. It follows from Theorem 0.1 that this happens if and only if the category D b X H is not Ext-bounded, that is, X H is singular. But X H is the hyperplane section of X PS 2 W by the hyperplane H PS 2 W. Using the definition of projective duality, we see that the set of critical values coincides with the projectively dual variety. In fact, from a homological point of view, the non-commutative variety PS 2 W, A PS2 W is a better candidate for the projective dual of X than X. It not only describes the set of singular hyperplane sections but also remembers non-trivial parts of their derived categories. For example, the triangulated category of singularities [23] of X H is equivalent to the triangulated category of singularities

Hyperplane sections and derived categories 449 of the fibre of the projection PS 2 W, A PS 2 W PS 2 W at H. In particular, it contains a lot of information about their singularities. The non-commutative variety PS 2 W, A PS2 W can thus be regarded as a Homologically Projectively Dual variety of X. These considerations suggest the following definition. A Homologically Projectively Dual variety of a projective variety X PV is an algebraic variety Y with a morphism g : Y PV and a sheaf A Y of non-commutative algebras on Y such that, for any hyperplane H V, the non-trivial part of the derived category D b X H is equivalent to the derived category D b Y H, A Y. Roughly speaking, there exists a fully faithful embedding D b Y H, A Y D b X H, and the orthogonal to D b Y H, A Y in D b X H is Ext-bounded. I am convinced that Homological Projective Duality exists for a large class of algebraic varieties. In this paper we show how to construct a homological projective dual for a small class of varieties. Assume that X is a smooth projective variety admitting an exceptional pair E 1, E 2 of vector bundles such that E1 O PV X 1, E 2 O PV X 1,..., E 1 O PV X i, E 2 O PV X i 0.1 is an exceptional collection, where i is the index of X that is, ω X = OPV i X. In this case, the homologically projectively dual variety Y can be constructed as a moduli space of stable representations R 1, R 2 of the quiver Q = HomE1,E2 such that the cokernel of the corresponding map R 1 E 1 R 2 E 2 is supported on a hyperplane section of X. The moduli space Y carries a sheaf A Y of Azumaya algebras and a universal family of representations F 1, F 2 in the category of A Y -modules. The Azumaya algebra A Y corresponds to the element in the Brauer group which is the obstruction to the existence of a universal family in the category of O Y -modules, so the universal family exists in the category of A Y -modules. By sending every point of Y to the support hyperplane of the cokernel of the corresponding map R 1 E 1 R 2 E 2, we obtain a projection g : Y PV. Putting X L = X PL and Y L = Y PV PL = g 1 PL as above, we prove the following theorem. Theorem 0.2. Assume that X L and Y L are compact and dim X L =dim X dim L, dim Y L = dim Y dim L. If r = dim L i, then there is a semiorthogonal decomposition D b X L = D b Y L, CohA Y, E 1 1, E 2 1,..., E 1 i r, E 2 i r, and if r = dim L i, then there is a semiorthogonal decomposition D b Y L, A Y = D b X L, F 2 i r, F 1 i r,..., F 2 1, F 1 1. In particular, when dim L = i we have an equivalence of categories D b X L = D b Y L, CohA Y. Instead of giving a direct proof, we provide a list of conditions on X, E 1, E 2, Y, A Y and F 1, F 2 which imply the claim of the theorem. This enables us to skip

450 A. G. Kuznetsov the construction of Y in some cases when one can guess what Y is. The proof of the theorem splits into two principal parts. In the first part we consider the case dim L = 1, the case of hyperplane sections of X. In other words, we prove that Y is homologically projectively dual to X. It turns out that the second part the case dim L > 1 follows from the first part more or less formally. Thus Theorem 0.2 is in a sense a consequence of homological projective duality. Let us say a few words about the assumptions on L in Theorem 0.2. First of all, the compactness of X L is not a restriction since X is projective. On the other hand, the compactness of Y L is the most serious restriction. The problem is that Y may be non-compact. We construct Y as a moduli space of stable representations, and stability is an open condition. In fact, this problem arises because the construction of Y is imperfect see below. We actually expect that it is always possible to construct a compact variety Y such that the theorem holds. In this case the assumption of compactness of Y L will hold automatically. Finally, the assumptions on the dimension of X L and Y L can also be waived. However in this case we must be more accurate in the definition of X L and Y L. In fact, the correct definition of both X L and Y L is as a derived fibre product: X L = X L PV PL, Y L = Y L PV PL. As a topological space this coincides with the usual fibre product, but in contrast to the usual case it is ringed with a certain sheaf of DG-algebras instead of the usual sheaf of functions. Correspondingly, D b X L and D b Y L, A Y are appropriate categories of DG-modules. For instance, Theorem 0.2 enables us to describe a certain category of DG-modules on any incomplete intersection of quadrics. This point of view is of interest because every projective variety becomes an intersection of quadrics after a sufficiently ample embedding. Now let us discuss some formal consequences of Theorem 0.2. First of all, we recall that the exceptional collection 0.1 was not assumed to be full. However, taking L = 0, we get X L = X, Y L =, and Theorem 0.2 implies that 0.1 is full. Thus we obtain a rather unexpected corollary. Similarly, assume that Y is compact and take L = V. Then X L =, Y L = Y and it follows that F 2 i N, F 1 i N,..., F 2 1, F 1 1, where N = dim V, is a full exceptional collection in D b Y, A Y. It is also worth mentioning that Theorem 0.2 holds not only for individual linear sections of X and Y, but also for families. In fact, the proof is based on an investigation of families. For example, consider the universal family X 1 of hyperplane sections of X which is a divisor of bidegree 1, 1 in X PV. Then the family Y 1 Y PV of orthogonal linear sections of Y coincides with the graph of the projection g. The relative version of Theorem 0.2 yields a semiorthogonal decomposition D b X 1 = D b Y, A Y, D b PV E 1 1,..., D b PV E 2 i 1 if Y is compact. This decomposition may be regarded as a categorical definition of the compactification Y of Y : D b Y, A Y = D b PV E 1 1,..., D b PV E 2 i 1 D b X 1.

Hyperplane sections and derived categories 451 In fact, one could take this for the definition of the homologically projectively dual variety of X if only there were a possibility of proving that this category is geometric. Now let us describe what our approach yields in the case we started from, that is, the intersection of even-dimensional quadrics. In this case X = PW, where W is an even-dimensional vector space. We consider the Veronese embedding X PS 2 W = PV and the exceptional pair E 1, E 2 = O PW, O PW 1. Note that O PV X 1 = O PW 2, so that 0.1 is the standard exceptional collection on PW. The moduli space of representations of the quiver which is used in the construction of the homologically projectively dual variety gives an open subset Y = PS 2 W \Z, where Z is the locus of quadrics of corank 2. Thus, at the moment our method gives a rigorous proof of Theorem 0.1 only for generic subspaces L with dim L 3. One reason why the quiver approach gives only an open subset in PS 2 W is the fact that the sheaf of algebras A PS 2 W fails to be Azumaya over Z, while the quiver approach can give only Azumaya algebras. However, we expect that there is an algebraic stack Y with underlying algebraic variety PS 2 W such that the sheaf of algebras A PS 2 W comes from a sheaf of Azumaya algebras on Y. This suggests that it might generally be better to consider an appropriate moduli stack of quiver representations instead of the moduli space. In the general situation, one can also try to find an appropriate stack compactification Y of the moduli space Y such that the sheaf of algebras A Y and the universal family F 1, F 2 can be extended to a sheaf A Y of Azumaya algebras. Another possible approach to the construction of a Homologically Projectively Dual variety is to avoid the use of quivers altogether and construct Y as a moduli space of appropriate objects for example, spinor bundles in the derived category of coherent sheaves on X which are supported on hyperplane sections of X. In both cases, the main obstacle is the absence of a good categorical theory of moduli spaces. The output of a good theory of moduli spaces should in general be a non-commutative algebraic variety. We now briefly describe the contents of the paper. In 1 we introduce notation and state our main result, Theorem 1.2. In 2 we introduce the homological background. We define exact cartesian squares and faithful base changes in 2.4 and prove the main technical results: Theorem 2.46 on faithful base change and Proposition 2.51 a relative version of Bridgeland s trick. We take the first step in the proof of Theorem 1.2 in 3 and finish it in 4. In 5 we describe the general construction of Y from X via quiver moduli spaces. In 6 we give some applications of Theorem 1.2, in particular, a description of the derived categories of Fano threefolds of index 1 and degrees 12, 16 and 18. In 7 9 we work out some details related to these examples. Finally, in 10 we check the basic facts about derived categories of coherent modules over sheaves of Azumaya algebras. We pay special attention to the relation of complexes of finite Tor- and Ext-amplitude to perfect complexes. I am grateful to A. Bondal, D. Orlov, D. Kaledin and A. Samokhin for useful discussions. Some of the results of this paper were presented at the conference Noncommutative algebra and algebraic geometry at the University of Warwick. I would like to thank organizers for that opportunity.

452 A. G. Kuznetsov 1. Statement of results As indicated in the introduction, we begin by listing a set of conditions on a pair of algebraic varieties X and Y that imply their homological projective duality. Let k be an algebraically closed field of characteristic zero. Assume that we have the following data: D.1 a smooth connected projective variety X with an algebraic morphism f : X PV to a projective space; D.2 a pair E 1, E 2 of vector bundles on X; D.3 an algebraic variety Y with a sheaf A Y of Azumaya algebras and a projective morphism g : Y P = PV Z, where Z PV is a closed subset in the dual projective space; D.4 a pair F 1, F 2 of locally projective A Y -modules on Y ; D.5 a linear morphism φ: HomE 1, E 2 Hom AY F 1, F 2. In all examples considered in this paper, φ is an isomorphism. We use the following notation. Consider the vector space W = HomE 1, E 2 and the integer N = dim V. Let Q PV PV be the incidence quadric, that is, the divisor of bidegree 1, 1 whose fibre over any point H PV is the corresponding hyperplane in PV. We consider the subscheme QX, Y = X Y PV PV Q in X Y. Let i: QX, Y X Y be the corresponding embedding. The fibre of QX, Y over a point y Y is the corresponding hyperplane section X H = X H of the variety X, where H is the hyperplane in PV corresponding to the point gy PV. We also call QX, Y the incidence quadric. Condition C.1 below implies that the incidence quadric QX, Y is a divisor of bidegree 1, 1 on X Y. The map φ: W Hom AY F 1, F 2 = Hom AY F2, F1 induces evaluation homomorphisms W F 1 F 2 and W F2 F1. Tensoring the first of these by F2 and the second by F 1 and adding, we obtain a homomorphism adφ: W F 2 F 1 F 2 F 2 F 1 F 1 on Y Y. On the other hand, composing the evaluation homomorphism W F 1 F 2 tensored by E 2 with the coevaluation homomorphism E 1 W E 2 tensored by F 1, we obtain a homomorphism e: E 1 F 1 E 2 F 2 on X Y. Assume that the data D.1 D.5 satisfy the following conditions: C.1 the image fx is not contained in a hyperplane in PV ; C.2 the canonical class of X satisfies ω X = OX i := f O PV i, where i > 0; C.3 the collection of bundles E 1 1, E 2 1,..., E 1 i, E 2 i is exceptional that is, Ext E s, E s = k for s = 1, 2 and Ext E s k, E t l = 0 for i k > l 1 or k = l and s > t and also Ext >0 E 1, E 2 = 0; C.4 the map g is projective; C.5 the cokernel of the map adφ is isomorphic to Coker adφ = A Y, where : Y Y Y is the diagonal embedding;

Hyperplane sections and derived categories 453 C.6 Ker e = 0 and Coker e = i E, where E is a coherent A Y -module on QX, Y ; C.7 we have dim X+N codim Y Y Y PV Y =2i and Y is Cohen Macaulay; C.8 we have dimz H N i 2 for any hyperplane H PV. The integer i defined in condition C.2 is called the index of X. If the subset Z is irreducible and is not contained in a hyperplane in PV, then C.8 is equivalent to the condition dim Z N i 1. For every linear subspace L V we put X L = X PV PL and Y L = Y PV PL. Definition 1.1. A subspace L V is said to be admissible if a dim X L = dim X dim L; b dim Y L = dim Y + dim L N; c PL Z =. Note that condition c together with C.4 implies that Y L is compact. Theorem 1.2. If conditions C.1 C.8 hold, then 1 the variety Y is smooth and ω Y = OY i N; 2 for any admissible subspace L V there exist semiorthogonal decompositions D b X L = D b Y L, A Y, E 1 1, E 2 1,..., E 1 i r, E 2 i r if r = dim L < i, D b Y L, A Y = D b X L, F 2 i r, F 1 i r,..., F 2 1, F 1 1 if r = dim L > i, and an equivalence of categories if dim L = i. D b X L = D b Y L, A Y We now sketch a proof of Theorem 1.2, assuming for simplicity that Z =. First of all, to avoid the problems that arise when one works with derived categories of singular varieties, we replace the individual linear sections X L, Y L of X, Y by the universal families of linear sections X r X Grr, V, Y r Y Grr, V, which are smooth although their projections to Grr, V are not. We note that the natural projection X r Grr,V Y r X Y factors through QX, Y. Let E r be the pullback of the sheaf E defined by condition C.6. Then the objects E r on X r Grr,V Y r X r Y r, regarded as kernels, provide us with functors Φ r : D b Y r, A Y D b X r. We also consider their left and right adjoint functors Φ r, Φ! r : D b X r D b Y r, A Y. The proof of Theorem 1.2 consists of several steps. Step 1. We note that Y 1 = Y and prove that the functor Φ 1 : D b Y, A Y D b X 1 is fully faithful. This is done by an explicit computation of Φ 1 Φ 1. The convolution of the kernels of these functors is shown to be isomorphic to the sheaf Coker adφ on Y Y, which gives the identity functor on D b Y, A Y by condition C.4; see the details in 3.

454 A. G. Kuznetsov Step 2. We prove that the functor Φ r : D b Y r, A Y D b X r is fully faithful for all r i by induction on r and that the collection D b Y r, A Y, D b Grr, V E 1 1,..., D b Grr, V E 2 i r 1.1 is semiorthogonal in D b X r ; see the details in 4.3. Step 3. We use Bridgeland s trick in a relative situation. Note that the relative canonical bundle of X i over Gri, V is given by ω Xi / Gri,V = det L i, where L i is the tautological subbundle of rank i on Gri, V. Thus X i is a family of Calabi Yau varieties over Gri, V. On the other hand, the functor Φ i is Gri, V -linear that is, it commutes with tensoring by pullbacks of bundles on Gri, V. We show in 2.8 that a relative version of Bridgeland s trick Proposition 2.51 works in this case, so that Φ i is an equivalence and ω Yi / Gri,V = det L i ; see the details in 4.4. Step 4. We use descending induction on r to see that ω Yr/ Grr,V = det L ri r, where L r is the tautological subbundle of rank r on Grr, V ; see the details in 4.4. Step 5. We use descending induction on r to check that semiorthogonal collection 1.1 is full in D b X r. In other words, we prove that we have the semiorthogonal decompositions D b X r = D b Y r, A Y, D b Grr, V E 1 1,..., D b Grr, V E 2 i r for all r i; see the details in 4.4. Step 6. Using induction on r, we show that the functors Φ! r : D b X r D b Y r, for all r i, take the exceptional pair E 0 1, E 1 1 to the pair F2, F1 up to a twist and a shift, where E 0 is the mutation of E 2 through E 1 that is, E 0 is defined from the exact triangle E 0 W ev E 1 E 2. It follows that D b Grr, V F 2 i r,..., D b Grr, V F 1 1 is a semiorthogonal collection in D b Y r, A Y for all r i. In particular, it follows that F2 i N, F1 i N,..., F2 1, F1 1 is an exceptional collection in D b Y, A Y if Z = ; see the details in 4.5. Step 7. We use induction on r to check that D b Y r, A Y = D b X r, D b Grr, V F 2 i r,..., D b Grr, V F 1 1 for all r i; see the details in 4.6. The induction steps are based on the following construction. Consider the partial flag variety Flr 1, r; V and the tautological subbundles L r 1 L r V O Flr 1,r;V of ranks r 1 and r on it. Let X Lr 1 = X r 1 Grr 1,V Flr 1, r; V, X Lr = X r Grr,V Flr 1, r; V, Y Lr 1 = Y r 1 Grr 1,V Flr 1, r; V, Y Lr = Y r Grr,V Flr 1, r; V.

Hyperplane sections and derived categories 455 Then X Lr is a divisor in X Lr 1, Y Lr 1 is a divisor in Y Lr, and we have the diagrams Y Lr η Y Lr 1 X Lr ξ X Lr 1 ψ Y r φ Y r 1 and ψ X r φ X r 1 where the morphisms ξ, η are divisorial embeddings and φ, ψ are projections. Moreover, the pullbacks E Lr 1 and E Lr of the kernels E r 1 and E r via the projections X Lr 1 Flr 1,r;V Y Lr 1 X r 1 Grr 1,V Y r 1 and X Lr Flr 1,r;V Y Lr X r Grr,V Y r provide us with functors Φ Lr 1 and Φ Lr between the corresponding derived categories. Since all these functors originate from the same object E on QX, Y, we can find certain relations between them; see the details in 4.2. Finally, when steps 1 7 have been performed we can deduce Theorem 1.2. To prove the first claim of Theorem 1.2, we take r = 1 and note that Y 1 = Y. Hence we have a semiorthogonal decomposition D b X 1 = D b Y, A Y, D b PV E 1 1,..., D b PV E 2 i 1 and an isomorphism ω Y/PV = L 1i 1 = O Y i. This immediately implies that ω Y = ωy/pv ω PV = O Y i N. On the other hand, since X 1 is smooth, we deduce that D b Y, A Y is Ext-bounded as a semiorthogonal component of the Ext-bounded category D b X 1. Hence Y, A Y is smooth. The second claim of Theorem 1.2 is deduced from Theorem 2.46 on faithful base change. We apply Theorem 2.46 to the base change Spec k Grr, V corresponding to an admissible r-dimensional subspace L V. The next question we address is how to construct Y starting from X. More precisely, assume that we have data D.1 and D.2 satisfying conditions C.1 C.3. How can we construct data D.3 D.5 such that conditions C.4 C.8 hold? We propose the following approach. Since we must have a morphism φ 1 : W F 1 F 2 on Y, it is natural to construct Y as a subscheme in the moduli space of representations of the quiver Q = W. Condition C.6 suggests that the dimension vector d 1, d 2 of the representations must satisfy θd 1, d 2 = d 2 ranke 2 d 1 ranke 1 = 0. 1.2 We note that this condition determines the dimension vector only up to a multiplicative constant. Let R m denote the representation space of the quiver Q with dimension vector d 1, d 2 satisfying 1.2 and with gcdd 1, d 2 = m, and let R 1, R 2 be the tautological family of representations on R m. Then C.6 suggests considering the GIT-quotient Y m = { ρ, H R m PV supp CokerE 1 R 1ρ e ρ E2 R 2ρ = X H } / χ G,

456 A. G. Kuznetsov where G = GLd 1 GLd 2 /k and χg 1, g 2 = detg 1 ranke1 detg 2 ranke2 is a character of G. To obtain a universal family of quiver representations on Y m, we have to consider a sheaf of Azumaya algebras on Y m. Moreover, if we want C.5 to be satisfied, we have to restrict to the stable locus, thus replacing Y m by Y m = gm 1 PV \ Z m, where gm : Y m PV is the canonical projection and Z m = 2km g ky k. Then we can show under some mild additional assumptions that C.4 C.6 do indeed hold for Y m. It remains to choose m in such a way that C.7 holds. This choice and the verification of C.7 are the most cumbersome parts of the construction of a homologically projectively dual variety. They must be carried out separately for each X. 2. Homological background This section is designed to develop some machinery for working with derived categories in the relative situation. The main results are Theorem 2.46 on faithful base change and Proposition 2.51 a relative version of Bridgeland s trick. Since we need these results to be applicable to the derived categories of sheaves of modules over sheaves of Azumaya algebras, we must work in the corresponding category. We define an Azumaya variety X, A X as an embeddable algebraic variety X of finite type over a field together with a sheaf of Azumaya algebras over X, that is, a sheaf A X of O X -algebras locally isomorphic to a matrix algebra in the étale topology [18]. In particular, A X is a locally free O X -module of finite rank. A morphism f : X, A X Y, A Y of Azumaya varieties consists of a morphism f : X Y of the underlying algebraic varieties and a homomorphism f A : f A Y A X of O X -algebras. Thus we treat Azumaya varieties as ringed spaces with A X playing the role of a sheaf of rings. Azumaya varieties form a category, which contains the category of ordinary embeddable algebraic varieties as a full subcategory the embedding functor takes X to X, O X. In what follows we consider the category of Azumaya varieties. To unburden the notation, we often write X instead of X, A X when it is clear which sheaf of algebras is being used, or when it does not matter. The necessary background on Azumaya varieties is contained in 10, where in particular we define the functors f, f, f!, AX and RHom AX and check that they satisfy all the usual relations. Here we only mention that a morphism f is said to be strict if f A is an isomorphism. Every strict morphism f satisfies f = f and f! = f.! The base field k is assumed to be an algebraically closed field of characteristic zero. 2.1. Kernel functors. We denote by D b qcx, D qcx, D + qcx and D qc X the bounded, upper bounded, lower bounded and unbounded derived categories of quasi-coherent sheaves on an Azumaya variety X. Let DX be the unbounded derived category of quasi-coherent sheaves on X with coherent cohomology. We similarly define D b X, D X and D + X. Finally, D perf X stands for the category of perfect complexes. We write DX, A X and so on whenever we wish to emphasize the role of the Azumaya algebra on X or point it out explicitly. Given an object F DX, we denote by H p F the pth cohomology sheaf of F. For any integers a, b with a b, we denote by D [a,b] X the full subcategory

Hyperplane sections and derived categories 457 of DX formed by those objects F for which H p F = 0 for p [a, b]. We similarly define D a X and D a X. Given a morphism f : X Y, we write f, f and f! for the derived pushforward, derived pullback and twisted pullback functors. Similarly, AX stands for the derived tensor product and RHom AX for the derived local Hom-functor. Lemma 2.1. Suppose that f : X Y is a morphism of Azumaya varieties, F D X, G D + Y and the support suppf is projective over Y. Then f RHom AX F, f! G = RHom AY f F, G. Proof. Let Z be a scheme-theoretical support of F and let i: Z X denote the corresponding closed embedding, so that F = i F, where F D Z. Then both i and f i: Z Y are projective. Using the functoriality of the twisted pullback and the usual duality theorem for i and f i see [12] and Lemma 10.34, we get f RHom AX F, f! G = f RHom AX i F, f! G = f i RHom AX Z F, i! f! G = f i RHom AX Z F, f i! G and we are done. = RHom AY f i F, G = RHom AY f F, G Let X 1, X 2 be Azumaya varieties and let p k : X 1 X 2 X k be the projections. We consider any object K Dqc X1 X 2, A opp A X2 and define the functors Φ K F 1 := p 2 p 1 F 1 AX1 K, Φ! KF 2 := p 1 RHom AX2 K, p! 2 F 2. Then Φ K is an exact functor DqcX 1 DqcX 2 and Φ! K is an exact functor D qcx + 2 D qcx + 1. We call Φ K the kernel functor with kernel K and Φ! K the kernel functor of the second type with kernel K. Lemma 2.2. If X 1 is smooth and K D perf X 1 X 2, then the functor Φ! K is isomorphic to the usual kernel functor with kernel RHom AX2 K, ω X1 A X2 [dim X 1 ]. Proof. This follows from Lemmas 10.7, 10.22 and 10.23 see also 10. The following lemma is obvious. Lemma 2.3. Every morphism φ: K K of kernels induces natural morphisms φ : Φ K Φ K, φ! : Φ! K Φ! K of kernel functors. If φ is an isomorphism, then so are φ and φ!. Lemma 2.4. i If K has coherent cohomology and finite Tor-amplitude over X 1 and if suppk is projective over X 2, then Φ K takes D b X 1 to D b X 2. ii If K has coherent cohomology and finite Ext-amplitude over X 2 and if suppk is projective over X 1, then Φ! K takes Db X 2 to D b X 1. iii If both i and ii hold, then Φ! K is right adjoint to Φ K. Moreover, Φ K takes D perf X 1 to D perf X 2. Proof. i If K has finite Tor-amplitude over X 1, then the object q1 F 1 AX1 K = q1f 1 AX1 A X2 K is bounded for any F 1 D b X 1, and its support is projective over X 2. Therefore Φ K F 1 is bounded and has coherent cohomology. X 1

458 A. G. Kuznetsov ii If K has finite Ext-amplitude over X 2, then we similarly see that the object RHom AX2 K, q! 2 F 2 = RHom A opp X 1 A X2 K, q! 2F 2 is bounded for any F 2 D b X 2, and its support is projective over X 1. Therefore Φ! K F 2 is bounded and has coherent cohomology. iii If both i and ii hold, then we use Lemma 2.1 to deduce that Hom AX1 F1, Φ! KF 2 = Hom AX1 F1, p 1 RHom AX2 K, p! 2 F 2 = Hom AX1 p 1 F 1, RHom AX2 K, p! 2 F 2 = HomAX2 p 1 F 1 AX1 K, p! 2 F 2 = Hom AX2 p2 p 1 F 1 AX1 K, F 2 = HomAX2 ΦK F 1, F 2 for all F 1 D b X 1, F 2 D b X 2. Moreover, the arguments in part ii show that Φ! K has bounded cohomological amplitude, hence the adjointness implies that Φ K takes perfect complexes to complexes of finite Ext-amplitude, that is, to perfect complexes see Corollary 10.45. Lemma 2.5. Suppose that K is a perfect complex, X 2 is smooth and suppk is projective over X 1 and X 2. Let Φ K # : D b X 2 D b X 1 be the kernel functor given by the kernel K # := RHom AX1 K, AX1 ω X2 [dim X 2 ]. Then Φ K # is the left adjoint functor to the kernel functor Φ K : D b X 1 D b X 2. Proof. It follows from Lemmas 2.4 and 2.2 that the right adjoint functor to Φ K # is the kernel functor given by the kernel RHom AX1 K #, A X1 ω X2 [dim X 2 ], which is clearly isomorphic to K. Thus, Φ K is right adjoint to Φ K #. Hence Φ K # is left adjoint to Φ K. Consider the kernels K 12 D X 1 X 2, A opp X 1 A X2, K 23 D X 2 X 3, A opp X 2 A X3. Let p ij : X 1 X 2 X 3 X i X j be the projections. We define the convolution of kernels by K 23 K 12 := p 13 p 12 K 12 AX2 p 23 K 23. Lemma 2.6. If K 12 D X 1 X 2 and K 23 D X 2 X 3, then i Φ K23 Φ K12 = Φ K23 K 12 ; ii Φ! K 12 Φ! K 23 = Φ! K 23 K 12 provided that supp p 12 K 12 AX2 p 23 K 23 is projective over X 1 X 3. Proof. The first assertion is standard. For the second we have Φ! K 12 Φ! K 23 F 3 = p 1 RHom AX2 K12, p! 2 p 2 RHom AX3 K 23, p! 3 F 3 = p 1 RHom AX2 K12, p 12 p! 23 RHom AX3 K 23, p! 3 F 3 = p 1 p 12 RHom AX2 p 12 K 12, RHom AX3 p 23 K 23, p! 23 p! 3 F 3 = p 1 p 13 RHom AX3 p 12 K 12 AX2 p 23 K 23, p! 13 p! 3 F 3 = p 1 RHom AX3 p13 p 12 K 12 AX2 p 23 K 23, p! 3 F 3 = Φ! K 23 K 12 F 3 and we are done.

Hyperplane sections and derived categories 459 Assume that Φ 1, Φ 2, Φ 3 : D D are exact functors between triangulated categories, and α: Φ 1 Φ 2, β : Φ 2 Φ 3, γ : Φ 3 Φ 1 [1] are morphisms of functors. We say that α β γ Φ 1 Φ 2 Φ 3 Φ 1 [1] is an exact triangle of functors if the triangle Φ 1 F αf Φ 2 F βf Φ 3 F γf Φ 1 F [1] is exact in D for any object F D. The following lemma is obvious. Lemma 2.7. If K 1 α K2 β K3 γ K1 [1] is an exact triangle in D X Y, then we have the following exact triangles of functors: Φ K1 α ΦK2 β ΦK3 γ ΦK1 [1], Φ! K 3 β! Φ! K 2 α! Φ! K 1 γ! Φ! K 3 [1]. Lemma 2.8. Let α: X Y be a finite morphism. a If D is a triangulated category and Φ: D DX is a functor such that α Φ = 0, then Φ = 0. b Suppose that X is another variety, K, K D X X are kernels and φ: K K α φ is a morphism such that the morphism α Φ K α Φ K of functors is an isomorphism. Then φ : Φ K Φ K is an isomorphism of functors. Proof. a Since α is finite, we have H i α G = α H i G for any G DX. Therefore α G = 0 implies that α H i G = 0 for all i, whence G = 0 in DX. In particular, α ΦF = 0 implies that ΦF = 0 for all F D. b Let K be the cone of φ: K K in D X X. Then we have an exact triangle of functors Φ K Φ K Φ K, and it follows that α Φ K = 0. Therefore Φ K = 0 by part a. Hence φ is an isomorphism. 2.2. Perfect spanning classes. We will need the following results. Definition 2.9. A class of objects F DX, A X is called a perfect spanning class for X, A X if, for any point x X, there is an object F x F with F x 0 such that 1 F x is a perfect complex; 2 H p F x is supported set-theoretically at x for all p; 3 if p 0 = max { p H p F x 0 }, then H p0 F x = A X OX O x. Lemma 2.10. Any Azumaya variety admits a perfect spanning class. Proof. Choose a closed embedding i: X Y with Y smooth and consider the class consisting of the objects F x := A X OX i i O x for all x X. It is clear that i O x is a perfect complex since Y is smooth and a pullback of a perfect complex is a perfect complex. The following two lemmas are obvious.

460 A. G. Kuznetsov Lemma 2.11. If F is a perfect spanning class for X and G is a perfect spanning class for Y, then F G is a perfect spanning class for X Y. Lemma 2.12. Assume that f : X S and g : Y S are morphisms of algebraic varieties. If F is a perfect spanning class in DX, A X and G is a perfect spanning class in DY, O Y, then p F OX S Y q G is a perfect spanning class in DX S Y, p A X, where p: X S Y X and q : X S Y Y are the projections. Lemma 2.13. Suppose that K D X, A opp X, F is a perfect spanning class for X, A X and we have H X, F AX K = 0 for all F F. Then K = 0. Proof. Assume that K 0 and let s be the greatest integer such that H s K 0. Choose a point x supp H s K and take an object F x F corresponding to this point. Let t be the greatest integer such that H t F x 0. Then it is clear that H p F x AX K is supported at x for all p, H >s+t F x AX K = 0 and H s+t F x AX K = H 0 A X OX O x AX H s K = H 0 H s K OX O x 0. It follows that the hypercohomology spectral sequence E p,q 2 = H q X, H p F x AX K H p+q X, F x AX K of the object F x AX K degenerates at the second term and This is a contradiction. H s+t X, F x AX K = H 0 X, H s+t F x AX K 0. Proposition 2.14. Suppose that K D X Y, A opp X A Y and the functor Φ K : DX, A X DY, A Y is zero on a perfect spanning class F for X, A X. Then K = 0. Proof. Choose a perfect spanning class G for Y, O Y. For all F F and G G we have H X Y, F G AX K = H Y, q p F AX K OX Y q G = H Y, Φ K F OY G = 0. Since F G is a perfect spanning class for X Y, A X, we have K = 0. Corollary 2.15. Suppose that K, K D X Y and let Φ K Φ K be the morphism of kernel functors induced by a morphism φ: K K of kernels. If Φ K Φ K is an isomorphism on a perfect spanning class F for X, then φ is an isomorphism. Proof. Let K be the cone of φ: K K in D X Y. Then we have an exact triangle of functors Φ K Φ K Φ K and it follows that Φ K is zero on the perfect spanning class F. Therefore K = 0. Hence φ is an isomorphism.

Hyperplane sections and derived categories 461 2.3. Koszul complexes. Let X, A X be an Azumaya variety, V a vector bundle on its underlying algebraic variety X and s ΓX, V a section possibly non-regular of V. Then we denote by Kosz X s the Koszul complex of s regarded as an object of the derived category D b X, A X : Kosz X s := { 0 Λ top V OX A X s s V OX A X s AX 0 }, with A X placed in degree 0. Assume that X is Cohen Macaulay. Recall that a section s is regular if the codimension of the zero locus Zs X of s is equal to the rank of V. It is well known that, for regular sections s, we have Kosz X s = i A Zs, where i: Zs X is the embedding and A Zs = A X Zs. Lemma 2.16. Assume that X is a Cohen Macaulay variety. Then we have Kosz X s D [codim X Zs rank V,0] X, A X for any section s ΓX, V. Moreover, Kosz X s is supported scheme-theoretically on an infinitesimal neighbourhood of Zs X and H 0 Kosz X s = i A Zs. Proof. The first part of the claim is local. Locally, we can decompose V = V V in such a way that the first component of s = s, s is regular and rank V = codim X Zs. Let i : Zs X be the embedding. Then we get Kosz X s = Kosz X s AX Kosz X s = i A Zs AX Kosz X s = i Kosz Zs i s, and it remains to note that we have Kosz Zs i s D [ rank V,0] Zs, A Zs by definition, and rank V = rank V rank V = rank V codim X Zs. The second part of the claim is evident since Kosz X s is acyclic on the complement of Zs, and the third part follows from the definition of the sheaf of ideals of Zs as the image of the morphism V s O X. 2.4. Exact cartesian squares. Let f : X, A X S, A S and g: Y, A Y S, A S be morphisms of Azumaya varieties. Assume that either f or g is strict. Then the fibre product X S Y, A X S Y is defined; see Lemma 10.37. Consider the corresponding cartesian square X S Y p X f q g Y S 2.1 Let Γ f : X X S and Γ g : Y S Y denote the graphs of f and g respectively. We put K f = Γ f A X, K g = Γ g A Y, Kf, g = A X S Y and consider the corresponding kernel functors. The following isomorphisms of functors are evident: Φ Kf = f : D X D S, Φ K g = g : D S D Y, Φ Kf,g = q p : D X D Y. Lemma 2.17. We have K f K g D 0 X Y and H 0 K f K g = Kf, g.

462 A. G. Kuznetsov Proof. Let p XS : X S Y X S and p SY : X S Y S Y denote the projections. We consider the object K = p XS K f AS p SY Kg. Since the pullback and the tensor product are left exact, we have K D 0. Furthermore, it is clear that the cohomology of K is supported set-theoretically on the fibre product X S Y = X S S S Y X S Y and H 0 K = A X S Y. Since the projection p XY : X S Y X Y restricted to an infinitesimal neighbourhood of X S Y in X S Y is finite, we deduce that K f K g = p XY K D 0, H 0 K f K g = p XY A X S Y = Kf, g and we are done. The canonical morphism of kernels K f K g H 0 K f K g = Kf, g induces a morphism of functors g f q p. Definition 2.18. A cartesian square 2.1 is said to be exact cartesian if the natural morphism of functors g f q p : D X D Y is an isomorphism. We will denote exact cartesian squares by the symbol EC. Proposition 2.19. The cartesian square 2.1 is exact cartesian if and only if the canonical morphism K f K g H 0 K f K g = Kf, g is an isomorphism. Proof. The only if part follows from Corollary 2.15 and the if part from Lemma 2.3. Corollary 2.15 yields the following assertion. Corollary 2.20. If the natural morphism of functors g f q p is an isomorphism on a perfect spanning class F DX, A X, then the cartesian square 2.1 is exact cartesian. Another consequence of Proposition 2.19 is the following corollary. Corollary 2.21. A cartesian square 2.1 is exact cartesian if and only if the transposed square is exact cartesian. Proof. It suffices to note that the criterion in Proposition 2.19 is symmetric. Lemma 2.22. The cartesian square 2.1 is exact cartesian if and only if the same is true of the underlying square of algebraic varieties. Proof. Note that the fibre product of Azumaya varieties is defined only when one of morphisms f, g is strict. Since the exactness property is symmetric with respect to transposition of the square, we may assume that g is strict. Then p is also strict, and we have q p = q p, and g f = g f. It remains to note that the morphism of functors g f q p is obtained from the morphism of functors g f q p by forgetting the A Y -module structure. Corollary 2.23. If either f or g is flat, then the cartesian square 2.1 is exact cartesian. Proof. Since the exactness property is symmetric with respect to transposition of the square, we may assume that g is flat. Then the underlying square is exact by [12], Chap. II, Proposition 5.12.

Hyperplane sections and derived categories 463 Proposition 2.24. If the diagram 2.1 is exact cartesian, then f! g = p q!. Proof. Note that f! = Φ! K f, g = Φ! K g, p q! = Φ! Kf,g and apply Lemma 2.6, ii. Lemma 2.25. Assume that the square on the right in the following diagram is exact cartesian: X v X v X f EC f S u S u S f and either f or both u and u are strict. Then the ambient square is exact cartesian if and only if the same is true of the square on the left. Proof. Assume that the ambient square is exact and f is strict. We choose perfect spanning classes F D perf X, O X, G D perf S, A S. Then for all F F, G G we have f v v F O f G = f v v F O v f G = f v v F O f u G = f v v F O u G = u u f F O u G = u u f F O G = u f v F O G = u f v F O f G. It remains to note that objects of the type v F O f G form a perfect spanning class in the derived category of X = X S S and to apply Corollary 2.20. Assume that the ambient square is exact and u, u are strict. We choose perfect spanning classes F D perf X, A X, G D perf S, O S. Then for all F F, G G we have f v v F O v f G v v F O f G = f = f v v F O f u G = f v v F O u G = u u f F O u G = u u f F O G = u f v F O G = u f v F O f G. It remains to note that objects of the type v F O f G form a perfect spanning class in the derived category of X = X S S and to apply Corollary 2.20. On the other hand, if the square on the left is exact, then f v v F = f v v F = u f v F = u u f F = u u f F for any F D b X, A X. Hence the ambient square is exact. Lemma 2.26. If g is finite and the canonical morphism f g A Y p q A Y = p A X S Y is an isomorphism, then the square 2.1 is exact cartesian. Proof. Note that g g f F = f F AS g A Y = f F AX f g A Y = f F AX p A X S Y = f p p F = g q p F

464 A. G. Kuznetsov and the resulting isomorphism coincides with the morphism obtained by applying the pushforward functor g to the morphism of functors g f q p. Since the latter is induced by the morphism K f K g Kf, g of kernels, it remains to apply Lemma 2.8, b. Corollary 2.27. Assume that g is a strict closed embedding, Y S is a locally complete intersection and both S and X are Cohen Macaulay. If codim X X S Y = codim S Y, then the square 2.1 is exact cartesian. Proof. By Lemma 2.26, it suffices to check that the canonical homomorphism f g A Y p A X S Y is an isomorphism. The claim is local with respect to S, and so we may assume that Y is the zero locus of a regular section s of a vector bundle V on S with rank V = codim S Y. Then g A Y = KoszS s. Hence, f g A Y = f Kosz S s = Kosz X f s. It is clear that the zero locus of f s on X is the fibre product X S Y, but codim X X S Y = codim S Y = rank V = rank f V, which implies that the section f s is regular and we have an isomorphism Kosz X f s = p A X S Y. Corollary 2.28. Consider the exact cartesian square 2.1 and let U, V and W be vector bundles on S, X and Y respectively, with sections u ΓS, U, v ΓX, V and w ΓY, W. Assume that X, Y, Z =X S Y and S are Cohen Macaulay and the sections p f u + p v + q w ΓX S Y, p f U p V q W, g u + w ΓY, g U W, f u + v ΓX, f U V, u ΓS, U are regular. Denote the corresponding zero loci by S u S, X u,v X, Y u,w Y and Z u,v,w Z. Then the square Z u,v,w X u,v Y v,w S u is exact cartesian. Proof. Consider the diagrams Z u,v Z Y X u,v X S Z u,v,w Z u,v X u,v Y w Y S Z u,v,w Y u,w Y w X u,v S u S In the first diagram, the right-hand square is exact by assumption and the left-hand square is exact by Corollary 2.27. Hence the ambient square is exact by the if

Hyperplane sections and derived categories 465 part of Lemma 2.25. Thus the lower square in the second diagram is exact. On the other hand, its upper square is exact by Corollary 2.27. Hence its ambient square is exact by the if part of Lemma 2.25. Thus the ambient square in the third diagram is exact. On the other hand, its right-hand square is exact by Corollary 2.27. Hence its left-hand square is exact by the only if part of Lemma 2.25. Definition 2.29. A strict morphism φ: T S is said to be faithful with respect to a morphism f : X S if the cartesian square is exact. X S T T Lemma 2.30. A strict and flat base change is faithful with respect to any morphism. Proof. This follows from Corollary 2.23. Corollary 2.31. Let T S be a strict base change faithful with respect to a morphism f : X S. If T S S is a factorization of T S such that T S is strict and finite and S S is strict and smooth, then S S is faithful with respect to f and T S is faithful with respect to f : X S S S. Proof. The smooth base change is faithful by Lemma 2.30 and the finite base change by the only if part of Lemma 2.25. Lemma 2.32. If the square 2.1 is exact cartesian, g has finite Tor-dimension and is strict, and f is projective, then there exists a canonical isomorphism of functors Proof. First of all, we note that φ X S p f! = q! g : D + S D + X S Y. Hom p f! F, q! g F = Hom q p f! F, g F = Hom g f f! F, g F. Hence the canonical adjunction morphism f f! id induces a morphism of functors p f! q! g ; we claim that it is an isomorphism. f1 Decomposing f as X S f 2 S, where f1 is finite and f 2 is strict and smooth, and applying Lemma 2.25, we reduce the proof to two cases, the case of strict and smooth f and the case of finite f, which can be treated separately. If f is strict and smooth, then q is also strict and smooth and f! F = f F OX ω X/S [dim X dim S], q! F = q F OX S Y ω X S Y/Y [dim X S Y dim Y ]. It remains to note that ω X S Y/Y = p ω X/S and dim X S Y = dim X + dim Y dim S since f is smooth. Hence p f! F = p f F OX ω X/S [dim X dim S] = p f F OX S Y p ω X/S [dim X dim S] = q g F OX S Y ω X S Y/Y [dim X S Y dim Y ] = q! g F. f

466 A. G. Kuznetsov Now assume that f is finite. Then q is also finite and, by Lemma 10.26, we have q q! g F = RHomq A X S Y, g F = RHomq p A X, g F = RHomg f A X, g F = g RHomf A X, F = g f f! F = q p f! F. It remains to apply Lemma 2.8, b. 2.5. Semiorthogonal decompositions. We recall the basic definitions and results concerning semiorthogonal decompositions of triangulated categories [2], [4]. A semiorthogonal decomposition of a triangulated category T is a sequence of full triangulated subcategories A 1,..., A n in T such that Hom T A i, A j = 0 for i > j semiorthogonality and for any object T T there exists a sequence of exact triangles T i T i 1 T Ai, where T Ai A i, T 0 = T and T n = 0. A full triangulated subcategory A of a triangulated category T is said to be right admissible if the embedding functor i: A T has a right adjoint functor i! : T A, and left admissible if it has a left adjoint functor i : T A. A subcategory A is said to be admissible if it is both right and left admissible. If A is a full subcategory of a triangulated category T, then the right orthogonal to A in T resp. the left orthogonal to A in T is the full subcategory A resp. A consisting of objects T T such that Hom T A, T = 0 resp. Hom T T, A = 0 for all A A. For any sequence of subcategories A 1,..., A n in T we denote by A 1,..., A n the smallest triangulated subcategory in T containing A 1,..., A n. If T = A 1, A 2 is a semiorthogonal decomposition, then A 1 is left admissible, A 2 is right admissible and A 2 = A 1, A 1 = A 2. Conversely, if the subcategory A T is left resp. right admissible, then T = A, A resp. T = A, A is a semiorthogonal decomposition. More generally, we have the following lemma. Lemma 2.33 [2]. If A 1,..., A n is a semiorthogonal collection in T such that the subcategories A 1,..., A k are left admissible and A k+1,..., A n are right admissible, then A1,..., A k, A 1,..., A k A k+1,..., A n, A k+1,..., A n is a semiorthogonal decomposition of the category T. In particular, a semiorthogonal collection A 1,..., A n of admissible subcategories gives a semiorthogonal decomposition of the category T if and only if A 1,..., A k A k+1,..., A n = 0 for any k. An object E T is said to be exceptional if Hom E, E[t] = 0 for t 0 and HomE, E = k. The triangulated subcategory E T generated by an exceptional object is admissible. A semiorthogonal collection of exceptional objects is called an exceptional collection. 2.6. Derived categories over a base. Consider Azumaya varieties X, A X and Y, A Y over the same smooth algebraic variety S. In other words, we have morphisms f : X, A X S, O S and g : Y, A Y S, O S. A functor Φ: DX, A X DY, A Y is said to be S-linear if, for all F DX, A X and G D b S, O S, there are bifunctorial isomorphisms Φ f G OX F = g G OY ΦF.

Hyperplane sections and derived categories 467 Note that any object G D b S, O S is a perfect complex since S is smooth. Lemma 2.34. If Φ is S-linear and admits a right adjoint functor Φ!, then Φ! is also S-linear. Proof. For all F DX, A X, G DY, A Y and H D b S we have RHom AX F, Φ! g H OY G = RHomAY ΦF, g H OY G Therefore we have = RHom AY g H OY ΦF, G = RHomAY Φ f H OX F, G = RHom AX f H OX F, Φ! G = RHomAX F, f H OX Φ! G. Φ! g H OY G = f H OX Φ! G by Yoneda s lemma, and it is clear that the isomorphism is bifunctorial. Consider a fibre product X S Y of algebraic varieties. Let i: X S Y X Y denote the embedding and let p: X Y X, q : X Y Y denote the projections, so that f p i = g q i. Lemma 2.35. If K Dqc X S Y, A opp X A Y, then the kernel functors ΦiK and are S-linear. Φ! i K Proof. We have f G OX F = q p f G OX F AX i K Φ ik = q p f G OX Y p F AX i K = q p F AX i i p f G OX S Y K = q p F AX i i q g G OX S Y K = q p F AX q g G OX Y i K = g G OY q p F AX i K = g G OY Φ ikf, and similarly for Φ! i K. A strictly full subcategory C DX, A X is said to be S-linear if we have f G OX F C for all F C, G D b S, O S. Lemma 2.36. If C D b X, A X is a strictly full S-linear left resp. right admissible triangulated subcategory, then its left resp. right orthogonal is also S-linear. Proof. Assume for example that C is left admissible and take any F C, F C. Then we have Hom AX F, f G OX F = HomAX f G OX F, F = 0 since f G OX F C. Therefore f G OX F C, and C is S-linear. Lemma 2.37. If D b X, A X = C, C is a semiorthogonal decomposition and C is S-linear, then we have f RHom AX F, F = 0 for all F C, F C. Proof. For all G D b S, O S we have Hom OS G, f RHom AX F, F = HomOX f G, RHom AX F, F = Hom AX f G OX F, F = 0 since f G OX F C. Therefore f RHom AX F, F = 0.