# Vector bundles on elliptic curves over a discrete valuation ring

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Journal of Algebra 255 (2002) 288–296 www.academicpress.com

Vector bundles on elliptic curves over a discrete valuation ring ✩

E. Ballico

Department of Mathematics, University of Trento, 38050 Povo (TN), Italy Received 28 February 2001

Communicated by Michel Broué

Abstract Here we study vector bundles on elliptic curves over a DVR. In particular, we classify

the vector bundles whose restriction to the special ﬁber is stable. For singular genus one curves over a DVR, we consider the same problem for ﬂat sheaves whose restriction to the special ﬁber is torsion free and obtained taking iterated extensions of a non-locally free sheaf with rank one and degree 0. 2002 Elsevier Science (USA). All rights reserved. Keywords: Vector bundle; Elliptic curve; DVR; Discrete valuation ring; Elliptic curve over a discrete valuation ring; Vector bundles on elliptic curves; Indecomposable vector bundle; Semistable vector bundle; Stable vector bundle; Torsion free sheaf

1. Introduction

Let D be a DVR with ﬁeld of fractions K, maximal ideal m and containing a ﬁeld k such that the induced map k → R/m is an isomorphism. The last condition implies that k is integrally closed in K. Let k be the algebraic closure of k. Fix a smooth geometrically connected curve T of genus g over k with T (k) smooth and T (k) = ∅. Set C = TD = T ×Spec(k) Spec(D). Let π : C → T and f : C → Spec(D) be the projections. For any vector bundle E on C let Ek be

✩ The author was partially supported by MURST and GNSAGA of INdAM (Italy). E-mail address: [email protected]

0021-8693/02/$ – see front matter 2002 Elsevier Science (USA). All rights reserved. PII: S 0 0 2 1 - 8 6 9 3 ( 0 2 ) 0 0 1 5 9 - X

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its restriction to the special ﬁber of f . Since T (k) = ∅, the proper map f has a section and it is universally connected, i.e. f∗(OC ) ∼= OSpec(D) and the same is true after any base change. By [3, Proposition 4 at p. 204], Pic(C) is an extension of the relative Picard functor Pic(C/ Spec(D)) by Pic(Spec(D)). Since D is a principal domain, Spec(D) is trivial and we obtain the following observation.

Remark 1. The natural map π∗ : Pic(T ) → Pic(C) induced by π is an isomorphism.

Deﬁnition 1. A vector bundle E on C will be called cohomologically ﬂat if for every A ∈ Pic(C) the ﬁnitely generated D-module H 1(C, E ⊗ A) has no torsion, i.e. it is free.

A quite complete picture of all vector bundles on the projective line over Spec(D) (the case g = 0) is given in [7]. In particular Hüble and Sun proved that for g = 0 the direct sums of line bundles are the only cohomologically ﬂat vector bundles [7, Theorem 1.4], but that even for g = 0 there are many other vector bundles; they worked even in mixed characteristic. It is a natural question to give conditions on a vector bundle E on C which assures that E ∼= π∗(F ) for some vector bundle F on T . By Lemma 1 below for any vector bundle F on T the vector bundle π∗(F ) is cohomologically ﬂat. Given a bundle G on T it is a natural question to classify all bundles E on C with Ek ∼= G. In general, the last question seems to be hopeless, as shown for g = 0 and G arbitrary direct sum of line bundles in [7]. Here we study both questions in the case g = 1 using Atiyah’s classiﬁcation of vector bundles on an elliptic curve over k (see [2, Part II], or [11]). For the second question we consider the case in which G is geometrically indecomposable and with non-integral slope. From now on (except in Section 4) we assume g = 1.

Fix P ∈ T (k). By [2, Theorem 7], for all integers r, d with r > 0 and every L ∈ Pic(T (k)) there is a vector bundle Fr,L on T (k) with rank(Fr,L) = r, det(Fr,L) ∼= L, and Fr,L indecomposable; such vector bundle Fr,L is unique if and only if r and deg(L) are coprime; in the general case there are ﬁnitely many such bundles, all obtained from one of them by twisting with a torsion line bundle [2, Theorem 7 and Corollary at p. 437]; Fr,L will denote any of them. Any such bundle Fr,L is semistable; Fr,L is stable if and only if r and deg(L) are coprime. If L ∈ Pic(T (k)), then Fr,L is deﬁned over k, i.e. it comes from a vector bundle on T . If L ∼= OC (dP ), we will write F (r, d) instead of Fr,L.

Deﬁnition 2. For all integers r, d with r > 0 set E(r, d) = π∗(F (r, d)).

Remark 2. By Deﬁnition 2 we have E(r, d)k ∼= F (r, d). By Lemma 1 every bundle E(r, d) is cohomologically ﬂat. We have E(r, d)∗ ∼= E(r, −d) (with the usual ambiguity if r and d are not coprime) and in particular E(r, 0)∗ ∼= E(r, 0).

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Since π∗ commutes with tensor products and direct sums, every vector bundle on C obtained from ﬁnitely many E(ri , di)’s taking tensor products and direct sums is cohomologically ﬂat.

Here is a sample of our results.

Theorem 1. Let E be a cohomologically ﬂat vector bundle on C. Assume the existence of integers s, ri , di with s > 0, 0 di < ri , and Mi ∈ Pic(T ), 1 i s, such that Ek ∼= F (r1, d1) ⊗ M1 ⊕ · · · ⊕ F (rs , ds) ⊗ Ms . We have E ∼= E(r1, d1) ⊗ π ∗(M1) ⊕ · · · ⊕ E(rs, ds) ⊗ π ∗(Ms ) if and only if the bundles Hom(E(ri, di), E), 1 i s, are cohomologically ﬂat.

Theorem 2. Let E be a vector bundle on C such that there are integers t, s, ri , 1 i s, di, 1 i s, with s 1, ri 2, and t < di/ri < t + 1, and Mi ∈ Pic0(T ), 1 i s, with Ek ∼= F (r1, d1) ⊗ M1 ⊕ · · · ⊕ F (rs , ds) ⊗ Ms . Then E is cohomologically ﬂat.

Notice that Theorem 2 covers all cases in which Ek is stable. In Section 3 we consider the case in which T is a singular geometrically integral curve with pa(T ) = 1. Most of our results proved in the smooth case are true with the same proofs, but here we also study the case in which E is not locally free but only ﬂat over Spec(D) and with Ek obtained taking iterated extensions of a non-locally free torsion free sheaf with rank one and degree 0 (see Proposition 3). In Section 4 we brieﬂy consider the case g 2.

2. Proofs of Theorems 1 and 2

In this section we prove Theorems 1 and 2. At the end of the section we consider the case in which Ek is irreducible but not geometrically indecomposable.

Let F be a coherent sheaf on C. We have Ri f∗(F ) = H i(C, F )∼, where for every D-module M, M or M∼ denotes the coherent sheaf on Spec(D) associated with M. Let p be the closed point of Spec(D). It follows that the natural map

Ri f∗(F ) ⊗ k(p) → H i(T , Fk) can be identiﬁed with the natural map

φi : H i(C, F ) ⊗ (D/m) → H i(T , Fk). In this set-up Theorem II.12.11 of [7] can be translated as follows.

Theorem 3. (1) φ0 is injective. (2) φ1 is an isomorphism. (3) The two following statements are equivalent:

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(i) φ0 is surjective; (ii) H 1(C, F ) is a free D-module.

It follows that F is cohomologically ﬂat if and only if for every L ∈ Pic(T ) the natural map

H 0 C, F ⊗ π ∗(L) ⊗ (D/m) → H 0(T , Fk ⊗ L) is surjective, and in this case H 0(C, F ⊗ π∗(L)) is a free D-module.

Lemma 1. Assume only that T is a projective and geometrically irreducible curve over k with Treg = ∅. Let F , G be vector bundles on T . We have π ∗(F )k ∼= F . If π∗(F ) ∼= π∗(G) then F ∼= G. The vector bundle π∗(F ) is cohomologically ﬂat.

Proof. The isomorphism π∗(F )k ∼= F is obvious. This isomorphism applied to the bundle Hom(F, G) gives easily the second assertion (or use the projection formula). Fix A ∈ Pic(C). Hence there is M ∈ Pic(T ) with A ∼= π∗(M). By the projection formula the canonical map

ρ : H 0 C, π ∗(F ) ⊗ A ⊗ (D/m) ∼= H 0 C, π ∗(F ⊗ M) → H 0(T , F ⊗ M) is surjective. Hence H 1(C, π∗(F ) ⊗ A) is free (Theorem 3 or [10, Proposition 4(ii)]). ✷

From now on again T is a smooth curve of genus 1 over Spec(k). As an immediate consequence of Theorem 3 and Lemma 1 we obtain the following observation.

Remark 3. Take L ∈ Pic(T ). The sheaf H 1(C, π∗(L)) is always free. If deg(L) < 0, then H 1(C, π ∗(L)) ∼= D⊕−deg(L). If deg(L) = 0 and L is not trivial, then H 1(C, π ∗(L)) = 0. If L is trivial, then H 1(C, π ∗(L)) ∼= H 0(C, π ∗(L)) ∼= D. If deg(L) > 0, then H 1(C, π ∗(L)) = 0 and H 0(C, π ∗(L)) ∼= D⊕ deg(L).

Proposition 1. Let E be a cohomologically ﬂat vector bundle on C such that Ek ∼= L1 ⊕ · · · ⊕ Lr with Li ∈ Pic(T ). Then E ∼= π ∗(L1) ⊕ · · · ⊕ π ∗(Lr ).

Proof. We ﬁx an isomorphism u = (u1, . . . , ur ) : L1 ⊕ · · · ⊕ Lr → Ek. Since E is cohomologically ﬂat, we may lift each morphism ui to a morphism vi : π ∗(Li ) → Ek. We obtain a morphism v = (v1, . . . , vr ) : π ∗(L1) ⊕ · · · ⊕ π∗(Lr ) → E lifting u. By Nakayama’s Lemma, v is a surjection between vector bundles with the same rank and hence it is an isomorphism. ✷

Proof of Theorem 1. The only if part follows from Lemma 1. Assume that all bundles Hom(E(ri , di), E), 1 i s, are cohomologically ﬂat. Hence we may lift the sections ui ∈ H 0(T , Hom(F (ri , di), Ek)), 1 i s, with

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(u1, . . . , us ) : F (r1, di) ⊗ M1 ⊕ · · · ⊗ F (rs .ds) ⊗ Ms → Ek inducing an isomorphism to a morphism v : E(r1, d1) ⊗ π ∗(M1) ⊕ · · · ⊕ E(rs , ds) ⊗ π ∗(Ms ) → E. By Nakayama’s Lemma, v is an isomorphism. ✷

By the classiﬁcation of indecomposable vector bundles on T ×Spec(k) Spec(k) [2, Part II], our convention on the bundles Fr,L and the isomorphism of Pic(T ) and Pic(C), the following corollaries are obvious consequences of Theorem 1.

Corollary 1. Assume k algebraically closed. Let E be a cohomologically ﬂat vector bundle on C. We have E ∼= E(r1, d1) ⊗ L1 ⊕ · · · ⊕ E(rs, ds) ⊗ Ls for some Li ∈ Pic(C) and some integers ri , di with 0 di < ri , 1 i s, if and only if for all integers r, d with 0 d < r the vector bundle Hom(E(r, d), E) is cohomologically ﬂat.

Corollary 2. Assume k algebraically closed. Let E be a vector bundle on C. We have E ∼= π∗(F ) for some vector bundle F on T if and only if for all integers r, d with 0 d < r the vector bundle Hom(E(r, d), E) is cohomologically ﬂat.

Proof of Theorem 2. Take integers r, d with r > 0 and t < d/r < t + 1. By [2, Lemma 15], we have h1(T , F (r, d) ⊗ M) = 0 for every M ∈ Pic(T ) with deg(M) −t and h0(T , F (r, d) ⊗ M) = 0 for every M ∈ Pic(T ) with deg(M) −t − 1. Hence h1(T , Ek ⊗ M) = 0 for every M ∈ Pic(T ) with deg(M) −t and h0(T , Ek ⊗ M) = 0 for every M ∈ Pic(T ) with deg(M) −t − 1. Hence the cohomological ﬂatness of E follows from Theorem 3 or [11, Proposition 4]. ✷

For any vector bundle G on a geometrically integral projective curve over a ﬁeld, set µ(G) = deg(G)/ rank(G) (the slope of G). Let µ+(G) be the maximal slope of a non-zero subsheaf of G. Set µ−(G) = −µ+(G∗). G is semistable if and only if µ+(G) = µ(G). G is semistable if and only if µ+(G) = µ−(G). By the Atiyah’s classiﬁcation of indecomposable vector bundles on an elliptic curve (use [2, Lemma 15], and Serre duality) we obtain the following lemma.

Lemma 2. Let F be a vector bundle on T . Assume the existence of an integer t such that t < µ−(F ) µ+(F ) < t + 1. For every M ∈ Pic(T ) with deg(M) −t we have H 1(T , F ⊗ M) = 0. For every M ∈ Pic(T ) with deg(M) −t − 1 we have H 0(T , F ⊗ M) = 0.

By Lemma 2 and Theorem 3 or [10, Proposition 4], Theorem 2 may be rephrased in the following way.

Corollary 3. Let E be a vector bundle on C. Assume the existence of an integer t such that t < µ−(Ek) µ+(Ek) < t + 1. Then E is cohomologically ﬂat.

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Now we will see some new features arising when k is not algebraically closed. Here ⊗ will be a tensor product ⊗k over k. Recall that k is integrally closed in K. For any algebraic extension L/k, set TL = T ⊗Spec(k) Spec(L), CL⊗D = C ⊗Spec(D) Spec(L ⊗ D) and call π : CL⊗D → Spec(L ⊗ D) the corresponding projection. For any vector bundle E on C, let EL⊗D be the associated vector bundle on CL⊗D. We will say that E is geometrically cohomologically ﬂat if for every algebraic extension L/k, L a ﬁeld, and every M ∈ Pic(TL), the L ⊗ Dmodule H 1(CL⊗D, EL⊗D ⊗ π ∗(M)) is a free L ⊗ D-module.

Theorem 4. Let E be a vector bundle on C such that Ek is indecomposable, but Ek ⊗ k splits as a direct sum of line bundles. Then E and π∗(Ek) are cohomologically ﬂat. E is geometrically cohomologically ﬂat if and only if E ∼= π ∗(Ek).

Proof. There is a ﬁnite extension L/k such that EL := (Ek) ⊗ L ∼= L1 ⊕ · · · ⊕ Lr with Li ∈ Pic(TL). Now we will check that Ek is semistable. Assume Ek not semistable. Hence the ﬁrst step of the Harder–Narasimhan ﬁltration of Ek is a saturated subbundle F of Ek with 1 rank(F ) < rank(Ek) and µ(F ) > µ(J ) for every indecomposable factor J of Ek/F . Since the Harder–Narasimhan ﬁltration is invariant for extensions of the base ﬁeld [9, Proposition 3] we have µ(E ⊗ k) > µ+(A) for every indecomposable factor A of (Ek/F ) ⊗ k. Hence H 1(Tk, Hom(Ek/F, F ) ⊗ k) = 0. Since the extension k/k is ﬂat, we obtain H 1(T , Hom(Ek/F, F )) = 0 by ﬂat base change [5, III.9.3]. Thus Ek ∼= F ⊕ (Ek/F ), contradicting the indecomposability of Ek. Thus Ek is semistable. By [9, Proposition 3], EL is semistable. Hence deg(Li) = deg(L1) for every i.

Claim. None of the line bundles Li , 1 i r, is deﬁned over k.

Proof. Assume for instance L1 deﬁned over k, say L1 ∼= BL with B ∈ Pic(T )(k). By ﬂat base change [5, III.9.3], we have H 0(T , Hom(B, Ek)) ⊗ L ∼= H 0(TL, Hom(L1, L1 ⊕ · · ·⊕ Lr )) = 0. Set s = dimk(H 0(T , Hom(B, Ek))). Since Ek is semistable and deg(B) = deg(L1), for every u ∈ H 0(T , Hom(B, Ek)) the subsheaf u(B) of Ek is a maximal degree saturated subbundle and s is the maximal integer such that Ek has a subbundle isomorphic to B⊕s . Since Ek is indecomposable, we have 1 s < r. Since s = dimL(H 0(TL, Hom(L1, L1 ⊕ · · · ⊕ Lr ))), the Krull–Schmidt uniqueness of the direct sum decomposition of EL [1, Theorem 3], implies that s is the number of indices i with 1 i r and Li ∼= L1. Just to ﬁx the notation assume Li ∼= L1 if and only if 1 i s. Hence (Ek/B⊕s )L ∼= Ls+1 ⊕ · · · ⊕ Lr . Since H 1(TL, Hom(Ls+1 ⊕ · · · ⊕ Lr , L⊕1 s )) = 0, by ﬂat base change we obtain H 1(T , Hom(Ek/B⊕s , B⊕s )) = 0. Hence Ek ∼= B⊕s ⊕ (Ek/B⊕s ), contradiction. The contradiction concludes the proof of the claim. ✷

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Fix A ∈ Pic(T ). If deg(A) < − deg(L1), we have H 0(TL, Li ⊗ A) = 0 for every i. If deg(A) > − deg(L1) we have H 1(TL, Li ⊗ A) = 0 for every i. If deg(A) = − deg(L1) none of the line bundles Li ⊗ A are trivial by the claim. Hence by Theorem 3 we obtain that E is cohomologically ﬂat. Now assume E geometrically cohomologically ﬂat. Since Ek ⊗ k is a direct sum of line bundles, H 1(Ck, Hom(π ∗(Ek), E) ⊗ k) is free. By Theorem 3 the natural map

φ0 : H 1 Ck, Hom π ∗(Ek), E ⊗ k → H 0 Tk, Hom(Ek, Ek)

is an isomorphism. Thus we may lift the identity Ek ⊗ k → Ek ⊗ k to an isomorphism u : π∗(Ek) ⊗ k → Ek ⊗ k. By ﬂat base change [5, III.9.3] we easily see the existence of such isomorphism u deﬁned over k. ✷

3. Singular curves of genus 1

In this section we consider the case of singular integral curves of arithmetic genus 1. Here we assume k algebraically closed. Fix an integral projective curve T over k with pa(T ) = 1. In Proposition 3 for simplicity we will assume char(k) = 2, 3 so that T has a Weierstrass equation and we may quote an elementary part of [4]. Assume T singular. Set C = TD = T ×Spec(k) Spec(D). Let π : C → T and f : C → Spec(D) be the projections. Since Treg(k) = ∅, the proper map f has a section and it is universally connected, i.e. f∗(OC) ∼= OSpec(D) and the same is true after any base change. By [3, Proposition 4 at p. 204], Pic(T ) is an extension of PicC/ Spec(D) Spec(D) by Pic(Spec(D)). Since D is a principal domain, Pic(Spec(D)) is trivial and we obtain the following observation.

Remark 4. As in the smooth case the natural map f ∗ : Pic(T ) → Pic(C) induced by f is an isomorphism.

Remark 5. Take L ∈ Pic(T ). The sheaf H 1(C, π∗(L)) is always free. If deg(L) < 0, then H 1(C, π ∗(L)) ∼= D⊕−deg(L). If deg(L) = 0 and L is not trivial, then H 1(C, π ∗(L)) = 0. If L is trivial, then H 1(C, π ∗(L)) ∼= H 0(C, π ∗(L)) ∼= D. If deg(L) > 0, then H 1(C, π ∗(L)) = 0 and H 0(C, π ∗(L)) ∼= D⊕ deg(L).

Proposition 2. Let E be a vector bundle on C. Assume the existence of an integer t such that t < µ−(Ek) µ+(Ek) < t + 1. Then E is cohomologically ﬂat.

Proof. Fix A ∈ Pic(C) and set m = deg(Ak). We have µ+((E ⊗ A)k) = µ+(Ek ⊗ Ak) = µ(Ek) + m and µ−((E ⊗ A)k) = µ−(Ek ⊗ Ak) = µ−(Ek) + m. By the very deﬁnition of µ+ we have h0(Ck, Ek ⊗ Ak) = 0 if µ+(Ek ⊗ Ak) < 0, i.e. if m −t − 1. By Serre duality and the deﬁnition of µ− we have h1(Ck, Ek ⊗ Ak) = 0 if µ−(Ek ⊗ Ak) > 0, i.e. if m −t. Hence E is cohomologically ﬂat by Theorem 3 or [10, Proposition 4]. ✷

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From now on we study the same problem for non locally free sheaves. Let E be a coherent sheaf on C ﬂat over Spec(D) and such that Ek := E ⊗D (D/m) is torsion free on T . Again H 0(C, E ⊗ A) is torsion free (and hence free) for every A ∈ Pic(C). We will say that E is cohomologically ﬂat if H 1(C, E ⊗ A) is torsion free (and hence free) for every A ∈ Pic(C). Assume char(k) = 2, 3. By [4, Lemma 0.2], there is a unique rank one torsion free sheaf F on T with deg(F ) = 0 and F not locally free. For every M ∈ Pic(T ) with deg(M) 0 we have h0(T , F ⊗ M) = 0. For every M ∈ Pic(T ) with deg(M) 0 we have h1(T , F ⊗ M) = 0. Hence the usual proof given quoting Theorem 3 or [10, Proposition 4], gives the following result.

Proposition 3. Assume char(k) = 2, 3. Let E be a coherent sheaf on C ﬂat over Spec(D) and equipped with an increasing ﬁltration {Ei}0 i r with E0 = 0, Er = E , Ei+1/Ei torsion free and ﬂat over Spec(D) with (Ei+1/Ei )k ∼= F for 0 i < r. Then E is cohomologically ﬂat.

4. Genus at least two

In this section we brieﬂy consider the case of smooth curves of genus g 2. We assume k algebraically closed and char(k) = 0. Let X be a smooth, connected projective curve. Set Y = X ×Spec(k) Spec(D). Let π : Y → X and f : Y → Spec(D) be the projections.

Remark 6. Since k is algebraically closed, X(k) = ∅ and hence f has a section. By [3, Proposition 4 at p. 204], the map π∗ : Pic(X) → Pic(Y ) is bijective. Hence by Lemma 1 for every A ∈ Pic(Y ) the ﬁnitely generated D-module H 1(Y, A) is free.

For all integers r, d with r > 0 let M(X; r, d) be the moduli scheme of all rank r stable vector bundles on X with degree d. It is known [13] that M(X; r, d) is a smooth, irreducible variety of dimension (r2 − 1)(g − 1) + g. For any rank r vector bundle F on X set s1(F ) = deg(F ) − deg(A), where A is a maximal degree line subbundle of E.

Theorem 5. Fix integers s > 0, ri , 1 i s, di, 1 i s, and t with ri > 0 and t < di/ri < t + 1 for every i. Fix general Fi ∈ M(X; ri, di), 1 i s, and let E be a vector bundle on Y with Ek ∼= F1 ⊕ · · · ⊕ Fs . Then E is cohomologically ﬂat.

Proof. Fix A ∈ Pic(Y ). We have (E ⊗ A)k ∼= Ek ⊗ Ak ∼= (F1 ⊗ Ak) ⊕ · · · ⊕ (Fs ⊗ Ak). For a general F ∈ M(X; r, d), s1(F ) is the least integer u such that u (r − 1)(g − 1 + d)/r (see [6, Section 4], or [8, Remark 3.14], or [12, Theorem 1.2] for a published proof). Furthermore, ωX ⊗ F ∗ may be considered

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as a general element of M(X; r, r(2g − 2) − d). Hence if deg(Ak) g − t − 1, we have h1(X, Ek ⊗ Ak) = 0, while if deg(Ak) g − 2 + t, we have h0(X, Ek ⊗

Ak) = 0. We obtain the cohomological ﬂatness of E quoting Theorem 3 or [10,

Proposition 4] in the usual way. ✷

References

[1] M.F. Atiyah, On the Krull–Schmidt theorem with applications to sheaves, Bull. Soc. Math. France 84 (1957) 307–317; reprinted in: Michael Atiyah Collected Works, Vol. 1, Oxford Science Publications, 1988, pp. 81–93.

[2] M.F. Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc. (3) 7 (1957) 414– 472; reprinted in: Michael Atiyah Collected Works, Vol. 1, Oxford Science Publications, 1988, pp. 105–143.

[3] S. Bosch, W. Lütkebohmert, M. Raynaud, Neron Models, in: Ergeb. Math. Grenzeb. (3), Band 21, Springer-Verlag, 1990.

[4] R. Friedman, J.W. Morgan, E. Witten, Vector bundles over elliptic ﬁbrations, J. Algebraic Geom. 8 (1999) 279–401.

[5] R. Hartshorne, Algebraic Geometry, in: Grad. Texts in Math., Vol. 52, Springer-Verlag, 1977. [6] A. Hirschowitz, Problémes de Brill–Noether en rang supérieur, Prépubl. Math. n. 91, Nice, 1986. [7] R. Hüble, X. Sun, Vector bundles on the projective line over a discrete valuation ring and the

cohomology of canonical sheaves, Comm. Algebra 27 (1999) 3513–3529. [8] H. Lange, Some geometrical aspects of vector bundles on curves, Aportaciones Mat. Notas

Investigación 5 (1992) 53–74. [9] S. Langton, Valuative criteria for families of vector bundles on algebraic varieties, Ann. of

Math. 101 (1975) 88–110. [10] D. Mumford, K. Suominen, Introduction to the theory of moduli, in: F. Oort (Ed.), Algebraic

Geometry, Oslo, 1970, Wolters-Noordhoff, 1972, pp. 171–222. [11] T. Oda, Vector bundles on an elliptic curve, Nagoya Math. J. 43 (1971) 41–72. [12] B. Russo, M. Teixidor i Bigas, On a conjecture of Lange, J. Algebraic Geom. 8 (1999) 483–496. [13] C.S. Seshadri, Fibrés vectoriels sur les courbes algébriques, Astérisque 96, 1982.

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Journal of Algebra 255 (2002) 288–296 www.academicpress.com

Vector bundles on elliptic curves over a discrete valuation ring ✩

E. Ballico

Department of Mathematics, University of Trento, 38050 Povo (TN), Italy Received 28 February 2001

Communicated by Michel Broué

Abstract Here we study vector bundles on elliptic curves over a DVR. In particular, we classify

the vector bundles whose restriction to the special ﬁber is stable. For singular genus one curves over a DVR, we consider the same problem for ﬂat sheaves whose restriction to the special ﬁber is torsion free and obtained taking iterated extensions of a non-locally free sheaf with rank one and degree 0. 2002 Elsevier Science (USA). All rights reserved. Keywords: Vector bundle; Elliptic curve; DVR; Discrete valuation ring; Elliptic curve over a discrete valuation ring; Vector bundles on elliptic curves; Indecomposable vector bundle; Semistable vector bundle; Stable vector bundle; Torsion free sheaf

1. Introduction

Let D be a DVR with ﬁeld of fractions K, maximal ideal m and containing a ﬁeld k such that the induced map k → R/m is an isomorphism. The last condition implies that k is integrally closed in K. Let k be the algebraic closure of k. Fix a smooth geometrically connected curve T of genus g over k with T (k) smooth and T (k) = ∅. Set C = TD = T ×Spec(k) Spec(D). Let π : C → T and f : C → Spec(D) be the projections. For any vector bundle E on C let Ek be

✩ The author was partially supported by MURST and GNSAGA of INdAM (Italy). E-mail address: [email protected]

0021-8693/02/$ – see front matter 2002 Elsevier Science (USA). All rights reserved. PII: S 0 0 2 1 - 8 6 9 3 ( 0 2 ) 0 0 1 5 9 - X

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its restriction to the special ﬁber of f . Since T (k) = ∅, the proper map f has a section and it is universally connected, i.e. f∗(OC ) ∼= OSpec(D) and the same is true after any base change. By [3, Proposition 4 at p. 204], Pic(C) is an extension of the relative Picard functor Pic(C/ Spec(D)) by Pic(Spec(D)). Since D is a principal domain, Spec(D) is trivial and we obtain the following observation.

Remark 1. The natural map π∗ : Pic(T ) → Pic(C) induced by π is an isomorphism.

Deﬁnition 1. A vector bundle E on C will be called cohomologically ﬂat if for every A ∈ Pic(C) the ﬁnitely generated D-module H 1(C, E ⊗ A) has no torsion, i.e. it is free.

A quite complete picture of all vector bundles on the projective line over Spec(D) (the case g = 0) is given in [7]. In particular Hüble and Sun proved that for g = 0 the direct sums of line bundles are the only cohomologically ﬂat vector bundles [7, Theorem 1.4], but that even for g = 0 there are many other vector bundles; they worked even in mixed characteristic. It is a natural question to give conditions on a vector bundle E on C which assures that E ∼= π∗(F ) for some vector bundle F on T . By Lemma 1 below for any vector bundle F on T the vector bundle π∗(F ) is cohomologically ﬂat. Given a bundle G on T it is a natural question to classify all bundles E on C with Ek ∼= G. In general, the last question seems to be hopeless, as shown for g = 0 and G arbitrary direct sum of line bundles in [7]. Here we study both questions in the case g = 1 using Atiyah’s classiﬁcation of vector bundles on an elliptic curve over k (see [2, Part II], or [11]). For the second question we consider the case in which G is geometrically indecomposable and with non-integral slope. From now on (except in Section 4) we assume g = 1.

Fix P ∈ T (k). By [2, Theorem 7], for all integers r, d with r > 0 and every L ∈ Pic(T (k)) there is a vector bundle Fr,L on T (k) with rank(Fr,L) = r, det(Fr,L) ∼= L, and Fr,L indecomposable; such vector bundle Fr,L is unique if and only if r and deg(L) are coprime; in the general case there are ﬁnitely many such bundles, all obtained from one of them by twisting with a torsion line bundle [2, Theorem 7 and Corollary at p. 437]; Fr,L will denote any of them. Any such bundle Fr,L is semistable; Fr,L is stable if and only if r and deg(L) are coprime. If L ∈ Pic(T (k)), then Fr,L is deﬁned over k, i.e. it comes from a vector bundle on T . If L ∼= OC (dP ), we will write F (r, d) instead of Fr,L.

Deﬁnition 2. For all integers r, d with r > 0 set E(r, d) = π∗(F (r, d)).

Remark 2. By Deﬁnition 2 we have E(r, d)k ∼= F (r, d). By Lemma 1 every bundle E(r, d) is cohomologically ﬂat. We have E(r, d)∗ ∼= E(r, −d) (with the usual ambiguity if r and d are not coprime) and in particular E(r, 0)∗ ∼= E(r, 0).

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Since π∗ commutes with tensor products and direct sums, every vector bundle on C obtained from ﬁnitely many E(ri , di)’s taking tensor products and direct sums is cohomologically ﬂat.

Here is a sample of our results.

Theorem 1. Let E be a cohomologically ﬂat vector bundle on C. Assume the existence of integers s, ri , di with s > 0, 0 di < ri , and Mi ∈ Pic(T ), 1 i s, such that Ek ∼= F (r1, d1) ⊗ M1 ⊕ · · · ⊕ F (rs , ds) ⊗ Ms . We have E ∼= E(r1, d1) ⊗ π ∗(M1) ⊕ · · · ⊕ E(rs, ds) ⊗ π ∗(Ms ) if and only if the bundles Hom(E(ri, di), E), 1 i s, are cohomologically ﬂat.

Theorem 2. Let E be a vector bundle on C such that there are integers t, s, ri , 1 i s, di, 1 i s, with s 1, ri 2, and t < di/ri < t + 1, and Mi ∈ Pic0(T ), 1 i s, with Ek ∼= F (r1, d1) ⊗ M1 ⊕ · · · ⊕ F (rs , ds) ⊗ Ms . Then E is cohomologically ﬂat.

Notice that Theorem 2 covers all cases in which Ek is stable. In Section 3 we consider the case in which T is a singular geometrically integral curve with pa(T ) = 1. Most of our results proved in the smooth case are true with the same proofs, but here we also study the case in which E is not locally free but only ﬂat over Spec(D) and with Ek obtained taking iterated extensions of a non-locally free torsion free sheaf with rank one and degree 0 (see Proposition 3). In Section 4 we brieﬂy consider the case g 2.

2. Proofs of Theorems 1 and 2

In this section we prove Theorems 1 and 2. At the end of the section we consider the case in which Ek is irreducible but not geometrically indecomposable.

Let F be a coherent sheaf on C. We have Ri f∗(F ) = H i(C, F )∼, where for every D-module M, M or M∼ denotes the coherent sheaf on Spec(D) associated with M. Let p be the closed point of Spec(D). It follows that the natural map

Ri f∗(F ) ⊗ k(p) → H i(T , Fk) can be identiﬁed with the natural map

φi : H i(C, F ) ⊗ (D/m) → H i(T , Fk). In this set-up Theorem II.12.11 of [7] can be translated as follows.

Theorem 3. (1) φ0 is injective. (2) φ1 is an isomorphism. (3) The two following statements are equivalent:

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(i) φ0 is surjective; (ii) H 1(C, F ) is a free D-module.

It follows that F is cohomologically ﬂat if and only if for every L ∈ Pic(T ) the natural map

H 0 C, F ⊗ π ∗(L) ⊗ (D/m) → H 0(T , Fk ⊗ L) is surjective, and in this case H 0(C, F ⊗ π∗(L)) is a free D-module.

Lemma 1. Assume only that T is a projective and geometrically irreducible curve over k with Treg = ∅. Let F , G be vector bundles on T . We have π ∗(F )k ∼= F . If π∗(F ) ∼= π∗(G) then F ∼= G. The vector bundle π∗(F ) is cohomologically ﬂat.

Proof. The isomorphism π∗(F )k ∼= F is obvious. This isomorphism applied to the bundle Hom(F, G) gives easily the second assertion (or use the projection formula). Fix A ∈ Pic(C). Hence there is M ∈ Pic(T ) with A ∼= π∗(M). By the projection formula the canonical map

ρ : H 0 C, π ∗(F ) ⊗ A ⊗ (D/m) ∼= H 0 C, π ∗(F ⊗ M) → H 0(T , F ⊗ M) is surjective. Hence H 1(C, π∗(F ) ⊗ A) is free (Theorem 3 or [10, Proposition 4(ii)]). ✷

From now on again T is a smooth curve of genus 1 over Spec(k). As an immediate consequence of Theorem 3 and Lemma 1 we obtain the following observation.

Remark 3. Take L ∈ Pic(T ). The sheaf H 1(C, π∗(L)) is always free. If deg(L) < 0, then H 1(C, π ∗(L)) ∼= D⊕−deg(L). If deg(L) = 0 and L is not trivial, then H 1(C, π ∗(L)) = 0. If L is trivial, then H 1(C, π ∗(L)) ∼= H 0(C, π ∗(L)) ∼= D. If deg(L) > 0, then H 1(C, π ∗(L)) = 0 and H 0(C, π ∗(L)) ∼= D⊕ deg(L).

Proposition 1. Let E be a cohomologically ﬂat vector bundle on C such that Ek ∼= L1 ⊕ · · · ⊕ Lr with Li ∈ Pic(T ). Then E ∼= π ∗(L1) ⊕ · · · ⊕ π ∗(Lr ).

Proof. We ﬁx an isomorphism u = (u1, . . . , ur ) : L1 ⊕ · · · ⊕ Lr → Ek. Since E is cohomologically ﬂat, we may lift each morphism ui to a morphism vi : π ∗(Li ) → Ek. We obtain a morphism v = (v1, . . . , vr ) : π ∗(L1) ⊕ · · · ⊕ π∗(Lr ) → E lifting u. By Nakayama’s Lemma, v is a surjection between vector bundles with the same rank and hence it is an isomorphism. ✷

Proof of Theorem 1. The only if part follows from Lemma 1. Assume that all bundles Hom(E(ri , di), E), 1 i s, are cohomologically ﬂat. Hence we may lift the sections ui ∈ H 0(T , Hom(F (ri , di), Ek)), 1 i s, with

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(u1, . . . , us ) : F (r1, di) ⊗ M1 ⊕ · · · ⊗ F (rs .ds) ⊗ Ms → Ek inducing an isomorphism to a morphism v : E(r1, d1) ⊗ π ∗(M1) ⊕ · · · ⊕ E(rs , ds) ⊗ π ∗(Ms ) → E. By Nakayama’s Lemma, v is an isomorphism. ✷

By the classiﬁcation of indecomposable vector bundles on T ×Spec(k) Spec(k) [2, Part II], our convention on the bundles Fr,L and the isomorphism of Pic(T ) and Pic(C), the following corollaries are obvious consequences of Theorem 1.

Corollary 1. Assume k algebraically closed. Let E be a cohomologically ﬂat vector bundle on C. We have E ∼= E(r1, d1) ⊗ L1 ⊕ · · · ⊕ E(rs, ds) ⊗ Ls for some Li ∈ Pic(C) and some integers ri , di with 0 di < ri , 1 i s, if and only if for all integers r, d with 0 d < r the vector bundle Hom(E(r, d), E) is cohomologically ﬂat.

Corollary 2. Assume k algebraically closed. Let E be a vector bundle on C. We have E ∼= π∗(F ) for some vector bundle F on T if and only if for all integers r, d with 0 d < r the vector bundle Hom(E(r, d), E) is cohomologically ﬂat.

Proof of Theorem 2. Take integers r, d with r > 0 and t < d/r < t + 1. By [2, Lemma 15], we have h1(T , F (r, d) ⊗ M) = 0 for every M ∈ Pic(T ) with deg(M) −t and h0(T , F (r, d) ⊗ M) = 0 for every M ∈ Pic(T ) with deg(M) −t − 1. Hence h1(T , Ek ⊗ M) = 0 for every M ∈ Pic(T ) with deg(M) −t and h0(T , Ek ⊗ M) = 0 for every M ∈ Pic(T ) with deg(M) −t − 1. Hence the cohomological ﬂatness of E follows from Theorem 3 or [11, Proposition 4]. ✷

For any vector bundle G on a geometrically integral projective curve over a ﬁeld, set µ(G) = deg(G)/ rank(G) (the slope of G). Let µ+(G) be the maximal slope of a non-zero subsheaf of G. Set µ−(G) = −µ+(G∗). G is semistable if and only if µ+(G) = µ(G). G is semistable if and only if µ+(G) = µ−(G). By the Atiyah’s classiﬁcation of indecomposable vector bundles on an elliptic curve (use [2, Lemma 15], and Serre duality) we obtain the following lemma.

Lemma 2. Let F be a vector bundle on T . Assume the existence of an integer t such that t < µ−(F ) µ+(F ) < t + 1. For every M ∈ Pic(T ) with deg(M) −t we have H 1(T , F ⊗ M) = 0. For every M ∈ Pic(T ) with deg(M) −t − 1 we have H 0(T , F ⊗ M) = 0.

By Lemma 2 and Theorem 3 or [10, Proposition 4], Theorem 2 may be rephrased in the following way.

Corollary 3. Let E be a vector bundle on C. Assume the existence of an integer t such that t < µ−(Ek) µ+(Ek) < t + 1. Then E is cohomologically ﬂat.

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Now we will see some new features arising when k is not algebraically closed. Here ⊗ will be a tensor product ⊗k over k. Recall that k is integrally closed in K. For any algebraic extension L/k, set TL = T ⊗Spec(k) Spec(L), CL⊗D = C ⊗Spec(D) Spec(L ⊗ D) and call π : CL⊗D → Spec(L ⊗ D) the corresponding projection. For any vector bundle E on C, let EL⊗D be the associated vector bundle on CL⊗D. We will say that E is geometrically cohomologically ﬂat if for every algebraic extension L/k, L a ﬁeld, and every M ∈ Pic(TL), the L ⊗ Dmodule H 1(CL⊗D, EL⊗D ⊗ π ∗(M)) is a free L ⊗ D-module.

Theorem 4. Let E be a vector bundle on C such that Ek is indecomposable, but Ek ⊗ k splits as a direct sum of line bundles. Then E and π∗(Ek) are cohomologically ﬂat. E is geometrically cohomologically ﬂat if and only if E ∼= π ∗(Ek).

Proof. There is a ﬁnite extension L/k such that EL := (Ek) ⊗ L ∼= L1 ⊕ · · · ⊕ Lr with Li ∈ Pic(TL). Now we will check that Ek is semistable. Assume Ek not semistable. Hence the ﬁrst step of the Harder–Narasimhan ﬁltration of Ek is a saturated subbundle F of Ek with 1 rank(F ) < rank(Ek) and µ(F ) > µ(J ) for every indecomposable factor J of Ek/F . Since the Harder–Narasimhan ﬁltration is invariant for extensions of the base ﬁeld [9, Proposition 3] we have µ(E ⊗ k) > µ+(A) for every indecomposable factor A of (Ek/F ) ⊗ k. Hence H 1(Tk, Hom(Ek/F, F ) ⊗ k) = 0. Since the extension k/k is ﬂat, we obtain H 1(T , Hom(Ek/F, F )) = 0 by ﬂat base change [5, III.9.3]. Thus Ek ∼= F ⊕ (Ek/F ), contradicting the indecomposability of Ek. Thus Ek is semistable. By [9, Proposition 3], EL is semistable. Hence deg(Li) = deg(L1) for every i.

Claim. None of the line bundles Li , 1 i r, is deﬁned over k.

Proof. Assume for instance L1 deﬁned over k, say L1 ∼= BL with B ∈ Pic(T )(k). By ﬂat base change [5, III.9.3], we have H 0(T , Hom(B, Ek)) ⊗ L ∼= H 0(TL, Hom(L1, L1 ⊕ · · ·⊕ Lr )) = 0. Set s = dimk(H 0(T , Hom(B, Ek))). Since Ek is semistable and deg(B) = deg(L1), for every u ∈ H 0(T , Hom(B, Ek)) the subsheaf u(B) of Ek is a maximal degree saturated subbundle and s is the maximal integer such that Ek has a subbundle isomorphic to B⊕s . Since Ek is indecomposable, we have 1 s < r. Since s = dimL(H 0(TL, Hom(L1, L1 ⊕ · · · ⊕ Lr ))), the Krull–Schmidt uniqueness of the direct sum decomposition of EL [1, Theorem 3], implies that s is the number of indices i with 1 i r and Li ∼= L1. Just to ﬁx the notation assume Li ∼= L1 if and only if 1 i s. Hence (Ek/B⊕s )L ∼= Ls+1 ⊕ · · · ⊕ Lr . Since H 1(TL, Hom(Ls+1 ⊕ · · · ⊕ Lr , L⊕1 s )) = 0, by ﬂat base change we obtain H 1(T , Hom(Ek/B⊕s , B⊕s )) = 0. Hence Ek ∼= B⊕s ⊕ (Ek/B⊕s ), contradiction. The contradiction concludes the proof of the claim. ✷

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Fix A ∈ Pic(T ). If deg(A) < − deg(L1), we have H 0(TL, Li ⊗ A) = 0 for every i. If deg(A) > − deg(L1) we have H 1(TL, Li ⊗ A) = 0 for every i. If deg(A) = − deg(L1) none of the line bundles Li ⊗ A are trivial by the claim. Hence by Theorem 3 we obtain that E is cohomologically ﬂat. Now assume E geometrically cohomologically ﬂat. Since Ek ⊗ k is a direct sum of line bundles, H 1(Ck, Hom(π ∗(Ek), E) ⊗ k) is free. By Theorem 3 the natural map

φ0 : H 1 Ck, Hom π ∗(Ek), E ⊗ k → H 0 Tk, Hom(Ek, Ek)

is an isomorphism. Thus we may lift the identity Ek ⊗ k → Ek ⊗ k to an isomorphism u : π∗(Ek) ⊗ k → Ek ⊗ k. By ﬂat base change [5, III.9.3] we easily see the existence of such isomorphism u deﬁned over k. ✷

3. Singular curves of genus 1

In this section we consider the case of singular integral curves of arithmetic genus 1. Here we assume k algebraically closed. Fix an integral projective curve T over k with pa(T ) = 1. In Proposition 3 for simplicity we will assume char(k) = 2, 3 so that T has a Weierstrass equation and we may quote an elementary part of [4]. Assume T singular. Set C = TD = T ×Spec(k) Spec(D). Let π : C → T and f : C → Spec(D) be the projections. Since Treg(k) = ∅, the proper map f has a section and it is universally connected, i.e. f∗(OC) ∼= OSpec(D) and the same is true after any base change. By [3, Proposition 4 at p. 204], Pic(T ) is an extension of PicC/ Spec(D) Spec(D) by Pic(Spec(D)). Since D is a principal domain, Pic(Spec(D)) is trivial and we obtain the following observation.

Remark 4. As in the smooth case the natural map f ∗ : Pic(T ) → Pic(C) induced by f is an isomorphism.

Remark 5. Take L ∈ Pic(T ). The sheaf H 1(C, π∗(L)) is always free. If deg(L) < 0, then H 1(C, π ∗(L)) ∼= D⊕−deg(L). If deg(L) = 0 and L is not trivial, then H 1(C, π ∗(L)) = 0. If L is trivial, then H 1(C, π ∗(L)) ∼= H 0(C, π ∗(L)) ∼= D. If deg(L) > 0, then H 1(C, π ∗(L)) = 0 and H 0(C, π ∗(L)) ∼= D⊕ deg(L).

Proposition 2. Let E be a vector bundle on C. Assume the existence of an integer t such that t < µ−(Ek) µ+(Ek) < t + 1. Then E is cohomologically ﬂat.

Proof. Fix A ∈ Pic(C) and set m = deg(Ak). We have µ+((E ⊗ A)k) = µ+(Ek ⊗ Ak) = µ(Ek) + m and µ−((E ⊗ A)k) = µ−(Ek ⊗ Ak) = µ−(Ek) + m. By the very deﬁnition of µ+ we have h0(Ck, Ek ⊗ Ak) = 0 if µ+(Ek ⊗ Ak) < 0, i.e. if m −t − 1. By Serre duality and the deﬁnition of µ− we have h1(Ck, Ek ⊗ Ak) = 0 if µ−(Ek ⊗ Ak) > 0, i.e. if m −t. Hence E is cohomologically ﬂat by Theorem 3 or [10, Proposition 4]. ✷

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From now on we study the same problem for non locally free sheaves. Let E be a coherent sheaf on C ﬂat over Spec(D) and such that Ek := E ⊗D (D/m) is torsion free on T . Again H 0(C, E ⊗ A) is torsion free (and hence free) for every A ∈ Pic(C). We will say that E is cohomologically ﬂat if H 1(C, E ⊗ A) is torsion free (and hence free) for every A ∈ Pic(C). Assume char(k) = 2, 3. By [4, Lemma 0.2], there is a unique rank one torsion free sheaf F on T with deg(F ) = 0 and F not locally free. For every M ∈ Pic(T ) with deg(M) 0 we have h0(T , F ⊗ M) = 0. For every M ∈ Pic(T ) with deg(M) 0 we have h1(T , F ⊗ M) = 0. Hence the usual proof given quoting Theorem 3 or [10, Proposition 4], gives the following result.

Proposition 3. Assume char(k) = 2, 3. Let E be a coherent sheaf on C ﬂat over Spec(D) and equipped with an increasing ﬁltration {Ei}0 i r with E0 = 0, Er = E , Ei+1/Ei torsion free and ﬂat over Spec(D) with (Ei+1/Ei )k ∼= F for 0 i < r. Then E is cohomologically ﬂat.

4. Genus at least two

In this section we brieﬂy consider the case of smooth curves of genus g 2. We assume k algebraically closed and char(k) = 0. Let X be a smooth, connected projective curve. Set Y = X ×Spec(k) Spec(D). Let π : Y → X and f : Y → Spec(D) be the projections.

Remark 6. Since k is algebraically closed, X(k) = ∅ and hence f has a section. By [3, Proposition 4 at p. 204], the map π∗ : Pic(X) → Pic(Y ) is bijective. Hence by Lemma 1 for every A ∈ Pic(Y ) the ﬁnitely generated D-module H 1(Y, A) is free.

For all integers r, d with r > 0 let M(X; r, d) be the moduli scheme of all rank r stable vector bundles on X with degree d. It is known [13] that M(X; r, d) is a smooth, irreducible variety of dimension (r2 − 1)(g − 1) + g. For any rank r vector bundle F on X set s1(F ) = deg(F ) − deg(A), where A is a maximal degree line subbundle of E.

Theorem 5. Fix integers s > 0, ri , 1 i s, di, 1 i s, and t with ri > 0 and t < di/ri < t + 1 for every i. Fix general Fi ∈ M(X; ri, di), 1 i s, and let E be a vector bundle on Y with Ek ∼= F1 ⊕ · · · ⊕ Fs . Then E is cohomologically ﬂat.

Proof. Fix A ∈ Pic(Y ). We have (E ⊗ A)k ∼= Ek ⊗ Ak ∼= (F1 ⊗ Ak) ⊕ · · · ⊕ (Fs ⊗ Ak). For a general F ∈ M(X; r, d), s1(F ) is the least integer u such that u (r − 1)(g − 1 + d)/r (see [6, Section 4], or [8, Remark 3.14], or [12, Theorem 1.2] for a published proof). Furthermore, ωX ⊗ F ∗ may be considered

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as a general element of M(X; r, r(2g − 2) − d). Hence if deg(Ak) g − t − 1, we have h1(X, Ek ⊗ Ak) = 0, while if deg(Ak) g − 2 + t, we have h0(X, Ek ⊗

Ak) = 0. We obtain the cohomological ﬂatness of E quoting Theorem 3 or [10,

Proposition 4] in the usual way. ✷

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