Some Remarks on Rigid Sheaves, Helices and Exceptional Vector Bundles on Fano Varieties Over Arbitrary Fields

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In the special case where #Bi = 1, we obtain the following result. Theorem (Theorem 6.14). Let X be a Fano variety and a rigid vector bundle. Let s E s k k = 1 ... r the Krull–Schmidt decomposition on X k k . Denote by Bi = E ⊗ A the⊕ orbit⊕ A of under the action of s . If ⊗B , then is the i1 , ..., iri i Gal(k k) # i = 1 {Bdirect sumB of} weak exceptionalA sheaves. | E Related to Theorem 6.12 is the problem whether, for instance, any separable exceptional sheaf on a del Pezzo surface (or more general on a Fano variety) can be included into a full separable exceptional collection. Analogously to Theorem 6.12, we prove the following criterion for an separable exceptional collection to be part of a full separable exceptional collection: Theorem (Theorem 6.13). Let X be a Fano variety over k and a separable exceptional s E s vector bundle. Let k k = 1 ... r the Krull–Schmidt decomposition on X k k . E⊗ A ⊕ ⊕ A s ⊗ Denote by Bi = i1 , ..., i the orbit of i under the action of Gal(k k). Then can {B B ri } A | E be included into a full separable exceptional collection on X if and only if the Bi can be s included into a full exceptional collection on X k k consisting of Galois invariant split semisimple blocks. ⊗ In view of the fact that on del Pezzo surfaces over an algebraically closed ﬁeld any rigid vector bundle is the direct sum of exceptional ones and that furthermore any exceptional collection can be included into a full exceptional collection (see [11]), the above criteria show what to prove to produce examples which give evidence for Rudakov’s conjecture to be true.

Conventions. Throughout this work k denotes an arbitrary ground ﬁeld and ks and k¯ a separable respectively algebraic closure. Furthermore, any locally free sheaf is assumed to be of ﬁnite rank and will be called vector bundle.

2. Fano varieties and del Pezzo surfaces over arbitrary fields By a variety over k we mean a separated scheme of finite type over k. In this paper we will consider only smooth projective and geometrically integral varieties. A Fano variety is by definition a variety over k with ample anti-canonical sheaf. A del Pezzo surface is a Fano variety of dimension two. If X is a del Pezzo surface we write KX for the class of ωX in Pic(X). The intersection number d = (KX , KX ) is called the degree of X. Riemann–Roch theorem and Casteln- s s uovo’s criterion show that X k k is rational. Moreover, X k k is isomorphic to either 1 1 ⊗ 2 ⊗ P P , or to the blow up of P at r 8 distinct closed points. The precise statement is × ≤ the content of the following well-known theorem (see [21], Theorem 1.6). Theorem 2.1. Let X be a del Pezzo surface over a separably closed field k of degree d. Then either X is isomorphic to the blow up of P2 at 9 d points in general position in k − P2 (k), or d = 8 and X is isomorphic to P1 P1 . k k × k 3

To give some non-trivial examples of Fano varieties, we recall some basic facts on Brauer–Severi varieties. For details we refer to [2] and [8]. Recall that a finite-dimensional k-algebra A is called central simple if it is an associative k-algebra that has no two-sided ideals other than 0 and A and if its center equals k. If the algebra A is a division algebra it is called central division algebra. Note that A is a central simple k-algebra if and only if there is a finite field extension k L, such that ⊂ A k L Mn(L). This is also equivalent to A k k¯ Mn(k¯). An extension k L such ⊗ ≃ ⊗ ≃ ⊂ that A k L Mn(L) is called splitting field for A. ⊗ ≃ The degree of a central simple algebra A is defined to be deg(A) := √dimkA. According to the Wedderburn Theorem, for any central simple k-algebra A there is an unique integer n> 0 and a division k-algebra D such that A Mn(D). The division algebra D is unique ≃ up to isomorphism and its degree is called the index of A and is denoted by ind(A). n A Brauer–Severi variety of dimension n is a variety X such that X k L P for a finite ⊗n ≃ field extension k L. A field extension k L for which X k L P is called splitting ⊂ s ⊂ ⊗ ≃ field of X. Clearly, k and k¯ are splitting fields for any Brauer–Severi variety. In fact, every Brauer–Severi variety always splits over a finite separable field extension of k. By embedding the finite separable splitting field into its Galois closure, a Brauer–Severi variety therefore always splits over a finite Galois extension. It follows from descent theory that X is projective, integral and smooth over k. Via Galois cohomology, n-dimensional Brauer– Severi varieties are in one-to-one correspondence with central simple algebras of degree n + 1. Note that for a n-dimensional Brauer–Severi variety X one has ωX = X ( n 1). O − − For details and proofs on all mentioned facts we refer to [2] and [8]. To a central simple k-algebra A one can also associate twisted forms of Grassmannians. Let A be of degree n and 1 d n. Consider the subset of Grassk(d n, A) consisting of ≤ ≤ · those subspaces of A that are left ideals I of dimension d n. This subset can be given the · structure of a projective variety which turns out to be a generalized Brauer–Severi variety. It is denoted by BS(d,A). After base change to some splitting field E of A the variety BS(d,A) becomes isomorphic to GrassE (d, n). For d = 1 the generalized Brauer–Severi variety is the Brauer–Severi variety associated to A. Note that BS(d,A) is a Fano variety. For details and properties of generalized Brauer–Severi varieties see [4]. Note that the finite product X of (generalized) Brauer–Severi varieties is a Fano variety. In particular, if X is two-dimensional, it is a del Pezzo surface.

3. Tilting objects and exceptional collections Let be a triangulated category and a triangulated subcategory. The subcategory D C is called thick if it is closed under isomorphisms and direct summands. For a subset C A of objects of we denote by A the smallest full thick subcategory of containing D h i ⊥ D the elements of A. Furthermore, we define A to be the subcategory of consisting of D all objects M such that HomD(E[i], M) = 0 for all i Z and all elements E of A. We ∈ say that A generates if A⊥ = 0. Now assume admits arbitrary direct sums. An D D c object B is called compact if HomD(B, ) commutes with direct sums. Denoting by − D the subcategory of compact objects we say that is compactly generated if the objects of c D generate . One has the following important theorem (see [5], Theorem 2.1.2). D D Theorem 3.1. Let be a compactly generated triangulated category. Then a set of objects c D c A generates if and only if A = . ⊂ D D h i D For a smooth projective variety X over k, we denote by D(Qcoh(X)) the derived category of quasicoherent sheaves on X. The bounded derived category of coherent sheaves is denoted by Db(X). Note that D(Qcoh(X)) is compactly generated with compact objects being all of Db(X). For details on generating see [5]. Definition 3.2. Let k be a field and X a smooth projective variety over k. An object D(Qcoh(X)) is called tilting object on X if the following hold: T ∈ 4

(i) Ext vanishing: Hom( , [i]) = 0 for i = 0. T T 6 (ii) Generation: If D(Qcoh(X)) satisfies RHom( , ) = 0, then = 0. N ∈ T N N (iii) Compactness: Hom( , ) commutes with direct sums. T − Below we state the well-known tilting correspondence which is a direct application of a more general result on triangulated categories (see [10], Theorem 8.5). We denote by Mod(A) the category of right A-modules and by Db(A) the bounded derived category of finitely generated right A-modules. Furthermore, perf(A) D(Mod(A)) denotes the ⊂ full triangulated subcategory of perfect complexes, those quasi-isomorphic to a bounded complexes of finitely generated projective right A-modules. Theorem 3.3. Let X be a smooth projective variety over k. Suppose we are given a tilting object on X and let A = End( ). Then the following hold: T T (i) The functor RHom( , ): D(Qcoh(X)) D(Mod(A)) is an equivalence. T − → ∼ (ii) If Db(X), this equivalence restricts to an equivalence Db(X) Db(A). T ∈ → Definition 3.4. Let A be a division algebra over k, not necessarily central. An object Db(X) is called weak exceptional if End( ) = A and Hom( , [r]) = 0 for r = 0. If E ∈ E E E 6 A = k the object is called exceptional. If A is a separable k-algebra, the object is called E separable exceptional.

Definition 3.5. A totally ordered set 1, ..., n of weak exceptional (resp. separable ex- {E E } ceptional) objects on X is called a weak exceptional collection (resp. separable exceptional collection) if Hom( i, j [r]) = 0 for all integers r whenever i > j. A weak exceptional E E b (resp. separable exceptional) collection is full if 1, ..., n = D (X) and strong if h{E E }i Hom( i, j [r]) = 0 whenever r = 0. If the set 1, ..., n consists of exceptional objects E E 6 {E E } it is called exceptional collection. Notice that the direct sum of objects forming a full strong weak exceptional (resp. separable exceptional) collection is a tilting object in the sense of Definition 3.2. Remark 3.6. If the ring A in Definition 3.4 is required to be a semisimple algebra, the object is also called semi-exceptional object in the literature (see [18]). Consequently, one can also define (full) semi-exceptional collections. Example 3.7. Let Pn be the projective space and consider the ordered collection of invertible sheaves Pn , Pn (1), ..., Pn (n) . In [3] Beilinson showed that this is a full {O O O } strong exceptional collection.

1 1 Example 3.8. Let X = P P . Then X , X (1, 0), X (0, 1), X (1, 1) is a full strong × {O O O O } exceptional collection on X. Here we write X (i, j) for (i) ⊠ (j). O O O Definition 3.9. Let X be a smooth projective variety over k. A collection 1, ..., n of b {E E } objects in D (X) is called semi-exceptional block if Hom( i, j [l]) = 0 for any i, j whenever n E E l = 0. If furthermore End( i) is the product of matrix algebras over k, the collection 6 i=1 E is called split semisimple exceptional block. L A generalization of the notion of a full weak exceptional collection is that of a semiorthogonal decomposition of Db(X). Recall that a full triangulated subcategory of Db(X) is D .called admissible if the inclusion ֒ Db(X) has a left and right adjoint functor D → Definition 3.10. Let X be a smooth projective variety over k. A sequence 1, ..., n of b D b D full triangulated subcategories of D (X) is called semiorthogonal if all i D (X) are ⊥ b D ⊂ admissible and j i = D (X) Hom( , ) = 0, i for i > j. Such D ⊂ D {F ∈ | G F b ∀G∈D } a sequence defines a semiorthogonal decomposition of D (X) if the smallest full thick b subcategory containing all i equals D (X). D b For a semiorthogonal decomposition we write D (X)= 1, ..., n . hD D i 5

Remark 3.11. Let 1, ..., n be a full weak exceptional collection on X. It is easy to verify E E b that by setting i = i one gets a semiorthogonal decomposition D (X)= 1, ..., n . D hE i hD D i For a wonderful and comprehensive overview of the theory on semiorthogonal decompositions and its relevance in algebraic geometry we refer to [12].

4. w-helices on Fano varieties We slightly extend the deﬁnition of a helix which is given in [7].

Deﬁnition 4.1. As sequence of objects H = ( i)i∈Z on a Fano variety X is called a V w-helix of type (n, d) if there are positive integers n, d with d 2 such that ≥ (i) for each l Z the corresponding thread ( l+1, ..., l+n) is a full w-exceptional ∈ V V collection, (ii) for each l Z one has l−n =( l ωX )[dim(X) + 1 d]. ∈ V V ⊗ − A w-helix H =( i)i∈Z of type (n, d) is said to be geometric if Hom( i, j [r]) = 0 for all V V V i < j unless r = 0. Moreover, it is called strong if each thread is a full strong w-exceptional collection. Remark 4.2. If the corresponding thread in Deﬁnition 4.1 is a full exceptional collection, the w-helix is called helix in the literature (see [6], [7] or [11]). Instead of using full weak exceptional collections in the threads one can also use full semi-exceptional collections.

d−1 Example 4.3. Take X = P . Note that ωX = X ( d). Then H = ( X (i))i∈Z is a O − O geometric helix of type (d,d).

1 1 Example 4.4. Take X = P P . The canonical sheaf is ωX = X ( 2, 2). Then × O − − H =(..., X , X (1, 0), X (0, 1), X (1, 1), X (2, 2), ...) is a geometric helix of type (4, 3). O O O O O One can give a geometric interpretation of a w-helix via the so called rolled-up w-helix algebras. For this, let H =( i)i∈Z be a w-helix of type (n, d) and deﬁne the w-helix algebra V as

A(H)= Hom( i, j ). V V l≥ j−i l M0 Y= This is a graded algebra which has a Z-action induced by the Serre functor

( ωX )[1 d]: Hom( i, j ) Hom( i−n, j−n). −⊗ − V V −→ V V The rolled-up w-helix algebra B(H) is deﬁned to be the subalgebra of A(H) of invariant elements. Obviously, the algebra B(H) is graded, too. To both algebras A(H) and B(H) one can associate a quiver. The quiver underlying A(H) has vertices labeled by the elements of Z and ai,j arrows connecting vertex i with vertex j. Here ai,j denotes the dimension of the cokernel map

Hom( i, l) Hom( l, j ) Hom( , j ). V V ⊗ V V −→ V V i

nij = ai,j+pn p∈Z X arrows from vertex i to vertex j. The geometric interpretation of the w-helix H in terms of the rolled-up w-helix algebra B(H) is stated in the corollary below. We ﬁrst ﬁx some notation. For a smooth projective variety X over k let A( ) := pec(S•( )), where S•( ) E S E E is the symmetric algebra of the vector bundle on X. The associated structure morphism E is π : A( ) X. Recall the following theorem (see [14], Theorem 5.1). E → 6

Theorem 4.5. Let X be a smooth projective variety over k and a vector bundle. Suppose i ∨ l E is a tilting bundle on X. If H (X, S ( ))=0 for all i = 0 and all l> 0, then T∗ T ⊗T ⊗ E 6 π is a tilting bundle on A( ). T E ∨ Now let X be a Fano variety and set Y := A(ωX ). Furthermore, let n = rk(K0(X)) and d = dim(Y ). Now Theorem 4.5 has the following consequence. Corollary 4.6. Let X be a Fano variety and B(H) the rolled up w-helix algebra of a given geometric w-helix H = ( i)i∈Z of type (n, d). Then B(H) is a graded algebra and for a V given thread 1, ..., n there is an equivalence V V ∗ b b RHom(π ( i), ): D (Y ) D (B(H)). V − −→ i M i ∨ l ∨ Proof. According to Theorem 4.5 we only have to verify that H (X, S (ωX ))=0 n l ∨T ⊗T− ⊗l for all i = 0 and all l > 0, where = j . Note that S (ω ) = ω so that the 6 T j=1 V X X vanishing of the desired cohomology follows from the fact that H =( i)i∈Z is a geometric L V w-helix of type (n, d). It is easy to see that End( ) B(H). The rest is (ii) of Theorem T ≃ 3.3.

5. Examples of w-helices on Fano-varieties In this section we consider the Fano varieties from Example 2.2 and provide two examples of w-helices. Recall from [13] the following deﬁnition. Deﬁnition 5.1. Let X be a variety over k. A vector bundle on X is called absolutely E split if it splits as a direct sum of invertible sheaves on X k k¯. For an absolutely split ⊗ vector bundle we shortly write AS-bundle. In [13] we classify all AS-bundles on proper k-schemes. Among others, we study in detail the AS-bundles on Brauer–Severi varieties. So in Section 6 of loc.cit. it is proved that on an arbitrary Brauer–Severi variety X the indecomposable AS-bundles are vector ⊕ind(A⊗i) bundles i, i Z, satisfying i k k¯ (i) , where A is the central simple W ∈ W ⊗ ≃ O algebra corresponding to X. These i are unique up to isomorphism and one has 0 W W ≃ X . Furthermore, the vector bundles i satisfy a symmetry and periodicity relation O W which are stated in the following lemma. Lemma 5.2. Let X be a n-dimensional Brauer–Severi variety of period p. Then the following hold: ∨ (i) −i, Wi ≃W (ii) i+rp i X (rp). W ≃W ⊗O Proof. This follows from [13], Proposition 5.3.

Proposition 5.3. Let X be a n-dimensional Brauer–Severi variety and i the inde- W composable AS-bundles from above. Then H = ( i)i∈Z is a geometric w-helix of type W (n + 1, n + 1).

⊕ind(A⊗i) Proof. As i k k¯ (i) we get End( i) k k¯ M ⊗i (k¯) and hence W ⊗ ≃ O W ⊗ ≃ ind(A ) End( i) is a central simple k-algebra. In fact End( i) is a central division algebra since W W i is indecomposable by construction. That H = ( i)i∈Z is a geometric w-helix of type W W (n + 1, n + 1) now follows from base change to some splitting ﬁeld and (ii) of Lemma 5.2 as the period p divides n + 1.

Let X be the del Pezzo surface C1 C2, where C1 and C2 are Brauer–Severi curves × corresponding to quaternion algebras (a,b) and (c, d) with a = c and b = d. The indecom- 6 6 posable AS-bundles on C1 and C2 are denoted by i and j respectively. V W 7

Proposition 5.4. Let X be as above. Then H = (..., X , 1 ⊠ C2 , C1 ⊠ 1, 1 ⊠ O V O O W V 1, C1 (2) ⊠ C2 (2), ...) is a geometric w-helix of type (4, 3). W O O Proof. Applying the K¨unneth formula (see [22], p.86), we ﬁnd End( i ⊠ j ) End( i) V W ≃ V ⊗ End( j ). Now the assumption on the quaternion algebras ensures that End( i ⊠ j ) is W V W again a quaternion algebra and hence a central division algebra over k. This follows from ⊗i the fact that End( i) (resp. End( j )) is by construction Brauer-equivalent to (a,b) ⊗j V W (resp. (c, d) ) and from [8], Theorem 1.5.5. The rest of the proof is left to the reader.

Remark 5.5. The assumption on the quaternion algebras in Proposition 5.4 is of techni- cally nature and ensures that any thread is a full weak exceptional collection. If we would deal with full semi-exceptional collections in each thread, this assumption can be omitted.

6. Rigid sheaves and exceptional vector bundles Definition 6.1. Let X be a Fano variety over a field k. A coherent sheaf is called rigid 1 F if Ext ( , ) = 0. F F Recall the following theorems of Kuleshov and Orlov (see [11], Theorems 5.2 and 6.11). Theorem 6.2. An arbitrary rigid bundle on a del Pezzo surface over an algebraically closed field splits into a direct sum of exceptional bundles. Theorem 6.3. On an arbitrary del Pezzo surface over an algebraically closed field each exceptional collection is part of a full exceptional collection. In [20] Rudakov formulated several conjectures. We summarize two of them in the following: Conjecture 6.4. Any rigid sheaf on a del Pezzo surface over an arbitrary field k is the direct sum of weak exceptional ones and any weak exceptional sheaf can be included into a full w-exceptional collection. Moreover, we modify the conjecture of Rudakov and formulate: Question 6.5. Is any rigid vector bundle on a Fano variety over an arbitrary field k the direct sum of separable exceptional vector bundles and is it possible to include any separable exceptional vector bundle into a full separable exceptional collection? To prove our main results from the introduction, we recall some facts on classical descent theory for vector bundles on proper k-schemes. For details see [1] and [13]. For a vector bundle on a proper k-scheme X we set A( ) := End( )/rad(End( )), E E E E where rad(End( )) is the Jacobson radical of the endomorphism ring. Furthermore, Z( ) E E denotes the center of A( ). The assumption that X is a proper k-scheme ensures that E vector bundles or more generally coherent sheaves enjoy a Krull–Schmidt decomposition. Thus A( ) is a semisimple k-algebra. If = m ⊕di is the Krull–Schmidt decompo- E E i=1 Ei sition, then A( ) is the product of the matrix algebras Md (A( i)). In particular, is E i E E indecomposable if and only if A( ) is a divisionL algebra over k. If k L is a separable E ⊂ extension, we have A( ) k L = A( k L). E ⊗ E⊗ Lemma 6.6. Let be an indecomposable vector bundle on X and k L a normal exten- E ⊂ sion containing Z( ) and splitting A( ). Write m = [Z( ) : k]sep and d = degZ(E)(A( )). E E E m ⊕d E Then k L has a Krull–Schmidt decomposition of the form ( i) , where i are E ⊗ i=1 E E indecomposable vector bundles on X k L with A( i)= L. ⊗ E L Proof. This is Lemma 1.1 of [1].

Proposition 6.7. Let X be a proper variety over k and and coherent sheaves. If F G k k¯ k k¯, then is isomorphic to . F ⊗ ≃G⊗ F G 8

Proof. See [13], Proposition 3.3.

Remark 6.8. The proof of Proposition 6.7 shows that the statement also holds for ks instead of k¯.

Proposition 6.9. Let X be a proper variety over k. If is an indecomposable Gal(ks k)- s E | invariant vector bundle on X k k , then there exists an up to isomorphism unique inde- ⊗ s ⊕m composable vector bundle on X such that k k . V V⊗ ≃E Proof. This is [1], Proposition 3.4. We reproduce the proof as its idea will be used for the proof of Proposition 6.10 below. So let k M be a finite Galois extension inside of s s ⊂ k such that M k for some vector bundle on X k M. Then let π∗ be the E≃N⊗ N ⊗ s N sheaf on X obtained by the projection π : X k M X. As the Gal(k k)-conjugates of s ∗ ⊗ →⊕[M:k] | M k are all isomorphic to , we have π π∗ . Applying the Krull–Schmidt N⊗ E N ≃E Theorem we can consider a direct summand of π∗ . Since is indecomposable, the s ⊕d M N E vector bundle satisfies k k for a suitable positive integer d > 0. To M M ⊗ ≃ E ′ prove the uniqueness, we assume that there is another indecomposable vector bundle ′ s ⊕d′ ⊕d′ s ′⊕d s M satisfying k k . Then ( ) k k ( ) k k , and Proposition 6.7 in M ⊗ ≃E M ′⊗ ≃ M ⊗ combination with Remark 6.8 implies ⊕d ′⊕d. The Krull–Schmidt Theorem yields ′ M ≃M . M≃M To continue, we need the following modification of Proposition 6.9.

Proposition 6.10. Let X be a smooth projective variety over k. Let be an indecompos- s s E able vector bundle on X k k and suppose 1, ..., r is the Gal(k k)-orbit of . Then ⊗ {E E } | E there is an up to isomorphism unique indecomposable vector bundle on X such that s r ⊕d F k k for a unique positive integer d> 0. F ⊗ ≃ i=1 Ei L Proof. Note that the vector bundle := 1 2 ... r is by assumption Galois V E ⊕ E ⊕ ⊕ E invariant. Since is indecomposable it follows that any i is indecomposable, too. To get E E our assertion, we proceed as in the proof of Proposition 6.9 to obtain a vector bundle s ⊕m r ⊕m M on X such that k k = i=1 i for a suitable positive integer m> 0. Take M⊗ ≃ V E s ⊕r any direct summand of and observe that k k for some positive integer W M L W⊗ ≃ V s r m. In fact this follows from the assumption that 1, ..., r is the Gal(k k)-orbit of ≤ r ⊕m {E E } | the indecomposable bundle and since i=1 i is the Krull–Schmidt decomposition of s E E k k . Now choose among the direct summands of a bundle with the smallest M⊗ M F rank and denote this rank by d. Finally,L one proceeds as in the proof of Proposition 6.9 to conclude with the Krull–Schmidt Theorem, Proposition 6.7 and Remark 6.8 that is F unique up to isomorphism.

Deﬁnition 6.11. Let X be a smooth projective variety over k. Let be an indecom- s E s posable vector bundle on X k k and suppose E := 1, ..., r is the Gal(k k)-orbit of ⊗ {E E } | . Let d> 0 be the unique smallest positive integer for which there is an indecomposable E s r ⊕d ⊕d ⊕d ⊕d vector bundle on X with k k . We call the set E := , ..., V V ⊗ ≃ i=1 Ei {E1 Er } the minimal descent-orbit of E. L The next two theorems give us a necessary and suﬃcient condition for Question 6.5 to be true.

Theorem 6.12. Let X be a Fano variety over k and a rigid vector bundle on X. Let s E s k k = 1 ... r be the Krull–Schmidt decomposition on X k k . Denote by BE ⊗ A ⊕ the⊕ A orbit of under the action of s . Then⊗ is the direct i = i1 , ..., iri i Gal(k k) sum of{B separableB exceptional} vectorA bundles if and only if the minimal| descent-orbitsE of all s Bi are split semisimple exceptional blocks over k . 9

′ s ′ s Proof. Since is rigid on X, we see that := k k is rigid on X := X k k and ′ ′ E ′ E E ⊗ ⊗′ ′ ˜ = ks k¯ on X ks k¯. Now consider the Krull–Schmidt decomposition of on X E E ⊗ ⊗ E ′ = 1 2 ... r. E A ⊕ A ⊕ ⊕ A ′ Note that all j are indecomposable vector bundles on X . Proposition 3.1 of [19] implies A ′ that all ˜j = j ks k¯ remain indecomposable vector bundles after base change to X ks k¯. A A ⊗ ′ ′ ⊗ Now Theorem 6.2 yields that ˜ = ks k¯ decomposes as the direct sum of exceptional ′ E E ⊗ vector bundles on X ks k¯. Let ⊗ ′ (1) ˜ = 1 2 ... s E G ⊕G ⊕ ⊕G be such a decomposition. Since ˜j are indecomposable, we get a second decomposition of A ˜′ into indecomposable vector bundles which is given as E ′ ′ (2) ˜ = ks k¯ = ˜1 ˜2 ... ˜r. E E ⊗ A ⊕ A ⊕ ⊕ A The Krull–Schmidt Theorem however implies r = s and that decompositions (1) and (2) s are up to permutation the same. In particular we have End( j ) = k . By Proposition A ri ⊕di 6.10 there are positive integers di > 0 such that the vector bundles descent to j=1 Bij B⊕di (indecomposable) vector bundles i on X. From the assumption thatL any i is a split V l semisimple exceptional block, we conclude Ext ( i, i) = 0 for l = 0. Moreover, we have V V 6 ri s ⊕di End( i) k k End( ) V ⊗ ≃ Bij j=1 Ms s s Mm1 (k ) Mm2 (k ) ... Mm (k ), ≃ × × × t and hence End( i) is a separable algebra over k. We set d to be the least common multiple V of all di. By the deﬁnition of d there are positive integers ni Z such that ni di = d. ∈ · We then have r ri ⊕d ( ( ⊕d)) E ≃ Bij i=1 j=1 M M r ri ( ( ⊕(di·ni))) ≃ Bij i=1 j=1 M M ⊕n1 ... ⊕nr . ≃ V1 ⊕ ⊕ Vr From the Krull–Schmidt Theorem and the fact that 1, ..., r are separable-exceptional, V V we conclude that is the direct sum of separable exceptional vector bundles on X. E For the other implication assume the vector bundle is the direct sum of separable E exceptional vector bundles, say

1 ... q. E≃C ⊕ ⊕C mi ⊕dj For any i, let i = be its Krull–Schmidt decomposition. The i are inde- C C j=1 Dij D j composable and therefore End( ij ) are division algebras over k. In particular, End( ij ) L D D s are separable algebras over k. From Lemma 6.6 we get that any i decomposes over k D j as si,j s ⊕bi i k k ( i ) , D j ⊗ ≃ D j l l M=1 g s with unique bi and ( ij )l indecomposable. Therefore, End(( ij )l)= k . The assumption r D s D r Ext ( i, i) = 0 for r > 0 implies after base change to k that Ext (( ij )l, ( ij )l) = 0 C C D s D for r > 0. Now sinceg1, ..., q are separable exceptional, weg see that k k decomposes C C ′ E ⊗ as the direct sum of the exceptional vector bundles ( ij )l on X . Moreover,g g since the s D sets ( ij )1, ..., ( ij )si,j are by construction Gal(k k)-invariant, they decompose as the { D D } | g g g 10

s disjoint union of the Gal(k k)-orbits. Let us denote these orbits by Bi1 , ..., Bi . By con- | qi ⊕bi struction, the minimal descent orbits B , j = 1, ..., qi, form split semisimple exceptional ij s blocks. We see that in the Krull–Schmidt decomposition of k k the Galois orbits of the E⊗ direct summands can be rearranged in such a way that the obtained decomposition give rise to minimal descent orbits forming split semisimple exceptional blocks. This completes the proof.

Theorem 6.13. Let X be a Fano variety over k and a separable-exceptional vector s E s bundle. Let k k = 1 ... r the Krull–Schmidt decomposition on X k k . Denote E⊗ A ⊕ ⊕ A s ⊗ by Bi = i1 , ..., i the orbit of i under the action of Gal(k k). Then can be {B B ri } A | E included into a full separable exceptional collection on X if and only if the Bi can be s included into a full exceptional collection on X k k consisting of Galois invariant split semisimple blocks. ⊗

Proof. Assume can be included into a full separable exceptional collection 1, ..., m . E {E E } Without loss of generality = 1. Consider the Krull–Schmidt decompositions of the s E E i k k , given by E ⊗ s i k k = i1 i2 ... i . E ⊗ F ⊕F ⊕ ⊕F si l Since all i are separable-exceptional, we obtain that all ij satisfy Ext ( ij , ij ) = 0 E s F F F for l > 0. Moreover, End( ij ) = k and hence all ij are exceptional vector bundles on ′ s F s F s X := X k k . The Galois group Gal(k k) acts on the sets i1 , ..., i since i k k ⊗ | {F F si } E ⊗ is Galois invariant. We then obtain the following block decomposition

11 21 m1 F F F 12 22 m2 b ′ D (X )=  F.  ,  F.  , ...,  F.  . * . . . +              1s1   2s2   msm  F  F  F  Note that the Bi are part of the first block and can therefore be included into a block decomposition with each block being Galois invariant. Obviously, all blocks occurring in the block decomposition are split semisimple by construction. For the other implication assume the Galois orbits B1, ..., Br of can be included into a full exceptional collection on X′ consisting of Galois invariant split-semisimple blocks. Let us denote this block decomposition by b ′ D (X )= B1, ..., Br, Br+1, ..., Br+m . h i Note that it is no restriction if we put B1, ..., Br at the first components of our block decomposition. We denote by l1 , ..., l the vector bundles in the block Bl, i.e Bl = ql E E ql l1 , ..., l for l = 1, ..., r + m. Since End( l ) is the product of matrix algebras ql i=1 i {E s E } s E over k , we conclude End( li )= k and hence all li are indecomposable vector bundles on ′ E L E X . From Proposition 6.10 we know that for the orbits B1, ..., Br there exist unique vector s ql ⊕dl bundles 1, ..., r on X, such that l k k for suitable positive integers dl V V V ⊗ ≃ i=1 Eli and l = 1, ..., r. Denote by d the least common multiple of d1, ..., dr. By definition there L are positive integers nl such that d = dl nl. Then we have · ⊕d s ⊕n1 ⊕nr k k ... . E ⊗ ≃ V1 ⊕ ⊕ Vr In the same way one can show that there are vector bundles r+1, ..., r+m on X such that ⊕dj R R s for suitable dj the bundles , for j = 1, ..., m, are after base change to k isomorphic Rr+j ⊕p1 ⊕pmi to l1 ... l where the l1 , ..., lq are the vector bundles occurring in the block E ⊕ ⊕E ql E E l Br+j, for j = 1, ..., m. We then get a semiorthogonal decomposition Db(X) ⊕d, ⊕d1 , ..., ⊕dr ≃ hE Rr+1 Rr+mi , r+1, ..., r+m ≃ hE R R i 11 and observe that can be included into a full separable exceptional collection on X. E s Theorem 6.14. Let X be a Fano variety over k and a rigid vector bundle. Let k k = the Krull–Schmidt decomposition on E s. Denote by B E⊗ 1 ... r X k k i = i1 , ..., iri A ⊕ ⊕ A s ⊗ {B B } the orbit of i under the action of Gal(k k). If #Bi = 1, then is the direct sum of weak exceptionalA sheaves. | E

′ s ′ s Proof. Since is rigid on X, we see that := k k is rigid on X := S k k and ′ ′ E ′ E E ⊗ ⊗′ ′ ˜ = ks k¯ on X ks k¯. Now consider the Krull–Schmidt decomposition of on X E E ⊗ ⊗ E ′ = 1 2 ... r, E A ⊕ A ⊕ ⊕ A ′ where j are indecomposable vector bundles on X . Proposition 3.1 of [19] implies that A ′ all ˜j = j ks k¯ remain indecomposable vector bundles after base change to X ks k¯. A A ⊗ ′ ′ ⊗ Now Theorem 6.2 yields that ˜ = ks k¯ decomposes as the direct sum of exceptional ′ E E ⊗ vector bundles on X ks k¯. Let ⊗ ′ (3) ˜ = 1 2 ... s E G ⊕G ⊕ ⊕G be such a decomposition. Since ˜j are indecomposable, we get a second decomposition of A ˜′ into indecomposable vector bundles given as E ′ ′ (4) ˜ = ks k¯ = ˜1 ˜2 ... ˜r. E E ⊗ A ⊕ A ⊕ ⊕ A The Krull–Schmidt Theorem however implies that r = s and therefore decompositions (3) s and (4) are up to permutation the same. In particular we have End( j ) = k . It is also A′ easy to see that all j are actually exceptional vector bundles on X . Moreover, from A s the assumption #Bi = 1 we obtain that all j are Gal(k k)-invariant. So we can apply A | s Proposition 6.9 to get indecomposable vector bundles j on X such that ( j ) k k ⊕m V V ⊗ ≃ j . Now observe that Aj s ⊕mj End( j ) k k End( j ) V ⊗ ≃ A s Mm (k ), ≃ j and hence End( j ) is a central simple algebra over k. We recall that j is indecomposable V V so that End( j ) is indeed a central division algebra. Moreover, it follows easily from s V⊕mj l ( j ) k k that Ext ( j , j ) = 0 for l> 0. Thus all j are weak exceptional vector V ⊗ ≃ Aj V V V bundles. Now consider the bundles j for j = 1, ..., r and set d := lcm(m1,m2, ..., mr) to V be the least common multiple. By the deﬁnition of the least common multiple there are ⊕d positive integers nj Z such that nj mj = d. We now consider the vector bundle ∈ · E and ﬁnd r r ⊕d s ⊕d ⊕mj ·nj ( ) k k ( j ) . E ⊗ ≃ A ≃ Aj j=1 i=j M M ⊕mj Since the vector bundles descent to j , we obtain Aj V r ⊕d s ⊕nj s ( ) k k ( ) k k . E ⊗ ≃ Vj ⊗ j=1 M From Proposition 6.7 we conclude r ⊕n ⊕d j . E ≃ Vj j=1 M The Krull–Schmidt Theorem now implies that has to be the direct sum of some of the E j . Hence it is the direct sum of weak exceptional sheaves. This completes the proof. V 12

Example 6.15. Let C be a Brauer–Severi curve over k and a rigid vector bundle. s s s 1 E After base change to k the bundle k k on C k k Pks is according to a theorem of E⊗ ⊗ ≃ s Grothendieck the direct sum of P1 (i). Since any P1 (i) is Gal(k k) invariant, Theorem O ks O ks | 6.13 implies that is the direct sum of w-exceptional vector bundles. Note that this fact E also follows from the classiﬁcation of vector bundles on C. In [13] it is proved that any vector bundle is the direct sum of indecomposable AS-bundles. The indecomposable E AS-bundles however are weak exceptional vector bundles so that in fact any bundle on C is the direct sum of weak exceptional ones.

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MATHEMATISCHES INSTITUT, HEINRICH–HEINE–UNIVERSITAT¨ 40225 DUSSELDORF,¨ GERMANY E-mail adress: [email protected]