Vector Bundles onG(1, 4) without Intermediate Cohomology

Journal of Algebra 214, 128᎐142 Ž1999. Article ID jabr.1998.7700, available online at http:rrwww.idealibrary.com on

Vector Bundles on GŽ 1, 4. without Intermediate Cohomology E. Arrondo* and B. Grana ˜ Departamento de Algebra, Facultad de Ciencias Matematicas, Uni¨ ersidad Complutense ´ de Madrid, 28040 Madrid, Spain E-mail: [email protected], [email protected] Communicated by Corrado de Concini Received February 18, 1998

INTRODUCTION A known result by Horrocks Žsee wHx. characterizes the line bundles on a projective space as the only indecomposable vector bundles without intermediate cohomology. This result has been generalized by Ottaviani Žsee wO1, O2x. to quadrics and Grassmannians. More precisely, he characterizes direct sums of line bundles as those vector bundles without intermediate cohomology and satisfying other cohomological conditions. Žsee wKx. has proved for any quadric that the More generally, Knorrer ¨ line bundles and spinor bundles Žand their twists by line bundles. are characterized by the property of being indecomposable and not having intermediate cohomology Žthere is an unpublished independent proof of this fact by I. Sols, which has been the starting point of the present work.. A proof of this characterization in the particular case of the four-dimensional quadric can be found in wAS, Proposition 1.3x. Buchweitz, Greuel, and Schreyer Žsee wBGSx. proved a ‘‘converse’’ of such results: hyperplanes and quadrics are the only hypersurfaces in projective space for which there are, up to a twist, a finite number of indecomposable vector bundles without intermediate cohomology. * Partially supported by DGICYT Grant PB96-0659. We acknowledge the tremendous help derived from the Maple package Schubert, created by S. A. Strømme and S. Katz wKSx. Its use has significantly contributed to an efficient computation of the cohomology of vector bundles in GŽ1, 4.. 128 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.

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Extensions of Horrocks’ theorem to some other particular varieties have been studied by several authors. For instance, C. Madonna has given a cohomological criterion for the splitting of rank-two vector bundles on smooth hypersurfaces in ⺠ 4 Žsee wMx., and L. Costa and the first author studied vector bundles without intermediate cohomology on Fano threefolds of index two Žsee wACx.. The goal of this paper is to generalize Horrocks’ result to the Grassmann variety GŽ1, 4. of lines in ⺠ 4 , in the sense of characterizing those vector bundles on it without intermediate cohomology. The paper is distributed in three sections. In the first one, we give the preliminaries on vector bundles on GŽ1, 4. that will be needed for the sequel. In the second section, we characterize the universal bundles on GŽ1, 4. as those without intermediate cohomology an verifying other cohomology vanishings. Finally, in the last section we prove our main result, in whichᎏaccording to the mentioned result in wBGSx ᎏwe obtain big families of vector bundles without intermediate cohomology.

1. PRELIMINARIES Let G s GŽ1, 4. denote the Grassmann variety of lines in ⺠ 4 s ⺠Ž V ., the projective space of hyperplanes of V. We will assume the ground field to have characteristic zero, although all our results are likely to hold in any characteristic different from two. Consider the universal exact sequence on G defining the universal vector bundles Q and S of respective ranks two and three, 0 ª Sˇª V m OG ª Q ª 0

Ž 1.1.

Ža check means a dual vector bundle.. The second symmetric power of the above epimorphism induces a long exact sequence S 2 V m OG

6 6

S2Q

6

Sˇm V

6

S Ž y1 .

6

0

0

6

M

6

Ž 1.2. 6

0

0

where the rank-twelve vector bundle M is defined to be the corresponding kernel, and we made the identification n2 Sˇ( S Žy1. Žas usual we write OG Ž1. ( n3 S ( n2 Q ..

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On the other hand, taking the second exterior power in the dual universal sequence we have the following natural long exact sequence Ždefining the rank-seven vector bundle K as a kernel.: 66

H2 S

6

H2 V * m OG

6

Qˇ m V *

6

S 2 Qˇ

6

0

0

6

K

6

Ž 1.3. 6

0

0

It is easy to see that Ext S , K . s V *, so that, for any natural numbers i, j there are non-trivial extensions 1Ž

0 ª K [i ª G ª S [ j ª 0.

Ž 1.4.

DEFINITION. An indecomposable direct summand of a vector bundle G as in Ž1.4. will be called a ¨ ector bundle of type ŽI.. EXAMPLE 1.1. Consider the following commutative diagram of exact sequences coming from Ž1.3. and the dual of Ž1.2., which defines P as a pull-back:

ˇ M 6

ˇ M 6

0

0

0

6

S m V* 6

6

P 6

0

6

SˇŽ1. 6

6

H 2 V * m OG 6

6

K

0 6

6

0

6

6

K

0

0 6

ˇ . s 0, it follows that the middle vertical exact sequence Since Ext 1 Ž OG , M ˇ is a vector bundle of type ŽI.. In fact, the middle splits. This shows that M horizontal exact sequence is an element in Ext 1 Ž S m V *, K . ( HomŽ V, V ., which is represented by the identity map on V. ˇ K . s V. This means that, in Similarly, one can observe that Ext 1 Ž K, general, for a vector bundle G appearing in an exact sequence Žsplit or not. as in Ž1.4. there are non-trivial extensions 0 ª G ª G ⬘ ª Kˇ[l ª 0.

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ˇ S . s 0, it follows that such a G ⬘ appears in an exact Since Ext 1 Ž K, sequence 0 ª K [i ª G ⬘ ª S [ j [ Kˇ[l ª 0.

Ž 1.5.

DEFINITION. An indecomposable direct summand of a vector bundle G ⬘ as in Ž1.5. will be called a ¨ ector bundle of type ŽII.. Finally, the left side of the exact sequence Ž1.3. shows that K Ž1. is generated by its global sections. This yields the following exact sequence defining EŽ1. as a kernel, 0 ª E Ž 1 . ª V m V * m OG ª K Ž 1 . ª 0,

Ž 1.6.

where E is a vector bundle of rank 18. The vector bundle EŽ1. has the following non-zero extensions groups: Ext 1 Ž K Ž1., EŽ1.. Žgenerated by the ˇ EŽ1.., and Ext 1 Ž S , EŽ1... We give extension Ž1.6.., Ext 1 Ž Q, EŽ1.., Ext 1 Ž K, a general definition containing in particular vector bundles of type ŽII. and their duals: DEFINITION. A vector bundle of type ŽIII. will be an indecomposable direct summand of a Žmaybe trivial. extension 0 ª E Ž 1.

[i

[ K [ j [ Sˇ[k ª G ª S [l [ Kˇ[m [ Q [n ª 0.

EXAMPLE 1.2. From Ž1.6. and the first short exact sequence in Ž1.3. we obtain the following commutative diagram of exact sequences defining P as a pull-back:

S 2 Qˇ 6

S 2 Qˇ 6

0

0

0

6

Qˇ m V * 6

6

P 6

0

6

K 6

6

V m V * m OG Žy1. 6

6

E

0 6

6

0

6

6

E

0

0 6

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Since Ext 1 Ž OG Žy1., S 2 Qˇ. s 0, it follows that the middle vertical exact sequence splits, and hence S 2 Qˇ is of type ŽIII.. DEFINITION. A vector bundle F on G is said not to ha¨ e intermediate cohomology if H i Ž G, F Ž l .. s 0 for all l g ⺪ and i s 1, . . . , 5 Žsince all cohomology groups are taken on G, we will for short write H i Ž F ...

ˇ E are Remark. It is easy to see that the vector bundles Q , S , Sˇ, K, K, simple Ži.e., their only endomorphisms are multiplications by constants ., indecomposable, and have no intermediate cohomology. This implies in particular that all vector bundles of type ŽIII. have no intermediate cohomology Žin fact, all the vector bundles appearing in this section are, maybe up to a twist, of type ŽIII., as we have remarked.. The goal of this paper is to prove that any vector bundle on G without intermediate cohomology is obtained in this way. Table 1.3. For the reader’s convenience, we list here the only non-zero intermediate cohomology of the above five vector bundles when tensored with Q and Sˇ: h1 Ž Q m Sˇ Žy1.. s h5 Ž S m Q Žy5.. s h1 Ž Sˇm Q Žy1.. s h 2 Ž Sˇm Sˇ Žy1.. s h 2 Ž K m Q Žy2.. s h3 Ž K m Sˇ Žy2. s h4 Ž Kˇm Q Žy4.. s h5 Ž Kˇ m Sˇ Žy4.. s h 3 Ž E m Q Žy2.. s h4 Ž E m Sˇ Žy2. s 1. h1 Ž K m Sˇ. s h1 Ž E m Q . s h 2 Ž E m Sˇ. s h1 Ž E m Sˇ Ž1.. s 5. Most of the above equalities can be derive from the others by using the universal exact sequence Ž1.1. or the Serre duality, taking into account that the canonical line bundle on G is ␻G s OG Žy5.. 2. CHARACTERIZATION OF THE UNIVERSAL BUNDLES We start by recalling Ottaviani’s characterization of direct sums of line bundles, when particularized to GŽ1, 4.. THEOREM 2.1 ŽOttaviani wO1, O2x.. Let F be a ¨ ector bundle on GŽ1, 4.. Then F is a direct sum of line bundles if and only if the following conditions hold: Ža. Žb. Žc.

F has no intermediate cohomology H i Ž F m Q Ž l .. s 0 for any i s 1, . . . , 5 and l g ⺪ H 1 Ž F m SˇŽ l .. s 0 for any l g ⺪.

Remark. Ottaviani’s original statement is not as we gave it. Instead of conditions Žb. and Žc., his conditions are Žsee wO1, Theorem 1Žc. for k s 1, n s 4.: Žb⬘. Žc⬘.

H i Ž F m S Ž l .. s 0 for any i s 3, 4, 5 and l g ⺪ H i Ž F m SˇŽ l .. s 0 for any i s 1, 2, 3 and l g ⺪.

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These are clearly equivalent to Žb. and Žc. in our statement by taking cohomology in the universal exact sequence Ž1.1. and its dual tensored with F Ž l . and using the assumption that F has no intermediate cohomology. The idea for proving the main theorem is to successively remove the six extra cohomological conditions appearing in Žb. and Žc. to eventually characterize those vector bundles without intermediate cohomology. Each time we remove a condition, a new family of vector bundles will appear. Table 1.3 indicates which vector bundles must appear each time. We will characterize in this section the universal vector bundles, by removingᎏone by oneᎏthe conditions in Theorem 2.1 that they do not satisfy. The first condition we remove will be Žc., which is the only condition that Q does not verify. Hence, Q should be characterized by conditions Ža. and Žb. in Ottaviani’s theorem. Notice that then we obtain a result completely analogous to Horrocks’ theorem, the role of line bundles being played now by line bundles and their tensor products with Q. We will prove this result in detail, the others being sketched as long as they are similar Žin fact, this proof will have a difficulty at the beginning not appearing in the remaining proofs.. The precise statement is: THEOREM 2.2. Let F be an indecomposable ¨ ector bundle on GŽ1, 4.. Then F is, up to a twist with a line bundle, either the tri¨ ial line bundle or Q if and only if the following conditions hold: Ža. Žb.

F has no intermediate cohomology H i Ž F m Q Ž l .. s 0 for any i s 1, . . . , 5 and l g ⺪.

Proof. Let F be a vector bundle satisfying Ža. and Žb.. We will prove our result by induction on Ý l h1 Ž F m SˇŽ l ... If this sum is zero, then we are in the hypotheses of Ottaviani’s Theorem 2.1, so that F is a direct sum of line bundles. So assume that h1 Ž F m SˇŽ l .. / 0 for some l g ⺪. By changing if necessary F with a twist, we can assume l s 0. Then we have a non-zero element in Ext 1 Ž S , F ., which yields a non-trivial extension 0 ª F ª P ª S ª 0.

Ž 2.1.

We first claim that P also verifies conditions Ža. and Žb.. Indeed the only vanishing to check is that of H 5 Ž P m Q Žy5.., since h 5 Ž S m Q Žy5.. s 1. To prove this vanishing, we first dualize and suitably twist the exact sequences Ž1.1., Ž1.2., Ž1.3., and using the natural isomorphisms n2 S (

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SˇŽ1. and S 2 Qˇ ( S 2 Q Žy2. we get a long exact sequence 0 ª Q Ž y5 . ª OG Ž y4 .

[5

ª OG Ž y3 .

[10

ª Q Ž y3 .

[5

ª OG Ž y1 .

[15

ˇ Ž y1. ª 0. ªM The fact that F and S Žand hence also P . satisfy conditions Ža. and, for i s 2, 3, also Žb. easily implies that there is a commutative diagram

ˇŽy1.., H 1Ž S m M

6

ˇŽy1.. H 1Ž P m M

H 5 Ž S m Q Žy5.. 6

6

H 5 Ž P m Q Žy5.. 6

where the vertical maps are isomorphisms from the long exact sequence above. Since we just need to prove that the map on the top is zero, it suffices to prove the same for the map in the bottom. By looking at the dual of the first short exact sequence in Ž1.2., that map appears in a commutative diagram 6

H 0 Ž P m Sˇ.

H 0 Ž S m Sˇ.

6

6

ˇŽy1... H 1Ž S m M

6

ˇŽy1.. H 1Ž P m M

The claim follows by observing that the vertical map on the left is an epimorphism Žits cokernel lies in H 1 Ž P m S Žy1. m V *. s H 2 Ž P m QˇŽy1. m V *. s 0., while the map on the top is zero since the extension Ž2.1. was non-trivial and S is simple. It is also immediate to check that h1 Ž P m SˇŽ l .. s h1 Ž F m SˇŽ l .. for any l except l s 0, for which h1 Ž P m Sˇ. s h1 Ž F m Sˇ. y 1. The latter follows from the exact sequence H 0 Ž P m Sˇ. ª H 0 Ž S m Sˇ. ª H 1 Ž F m Sˇ. ª H 1 Ž P m Sˇ. ª H 1 Ž S m Sˇ. s 0, in which, as we observed, the first map is zero and h 0 Ž S m Sˇ. s 1. We can therefore apply the induction hypothesis to P and conclude that it decomposes as a direct sum of summands of the type OG Ž l . and Q Ž l ..

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We next consider the following commutative diagram defining P⬘ as a pull-back Žand in which the right vertical map is the dual of Ž1.1..: 0 6

S 6

Qˇ 6

Qˇ 6

0

0

0

6

6

OG[5 6

0

6

6

P⬘ 6

6

F

P 6

6

0

6

6

F

0

0 6

The middle horizontal exact sequence splits since F has not intermediate cohomology. The middle vertical exact sequence also splits, since Ext 1 Ž P, Qˇ. ( H 5 Ž P m Q Žy5..* s 0. Hence P [ Qˇ ( F [ OG[5, from which the theorem follows. THEOREM 2.3. Let F be an indecomposable ¨ ector bundle on GŽ1, 4.. Then F is, up to a twist with a line bundle, either the tri¨ ial line bundle, or Q , or S if and only if the following conditions hold: Ža. Žb.

F has no intermediate cohomology H i Ž F m Q Ž l .. s 0 for any i s 1, 2, 3, 4 and l g ⺪.

Proof. We prove it by induction on Ý l h 5 Ž F m Q Ž l .., the zero case being Theorem 2.2. Hence we assume h5 Ž F m Q Ž l .. / 0 for some l, and we can suppose without loss of generality that l s y5. Hence, by Serre duality, there is a non-zero element in Ext 1 Ž Q , Fˇ.. This produces a non-trivial extension 0 ª Fˇ ª P ª Q ª 0.

Ž 2.2.

It is now immediate Žin contrast with the proof of Theorem 2.2. to prove that Pˇ still satisfies conditions Ža. and Žb. Žthis is because Q m Q has no intermediate cohomology.. Also, it holds that h5 Ž Pˇ m Q Ž l .. s h5 Ž F m Q Ž l .. for any l, except for l s y5 for which h5 Ž Pˇ m Q Žy5.. s h5 Ž F m Q Žy5.. y 1 Žthe proof being, by using Serre duality, as in Theorem 2.2.. Therefore, by induction hypothesis, Pˇ decomposes as a direct sum of summands of the type OG Ž l ., Q Ž l ., and S Ž l ..

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We consider now the following commutative diagram defining P⬘ as a pull-back: 0 6

Sˇ 6

Sˇ 6

0

0

0

6

OG[5 6

0

6

Q 6

6

6

P⬘ 6

6

P 6

6

Fˇ

6

0

Fˇ

6

0

0 6

As in the proof of Theorem 2.2, the middle horizontal exact sequence splits. Therefore, our result will follow if the middle vertical exact sequence also splits Žthis is the only difficulty that did not appear in Theorem 2.2 and that will appear in the rest of the proofs.. To see this, we study the direct summands of Ext 1 Ž P, Sˇ. ( H 1 Ž Pˇ m Sˇ. corresponding to the decomposition of P into direct summands. Only a summand of the type Q ; P Žif it exists. produces a non-zero summand Ext 1 Ž Q , Sˇ. ; Ext 1 Ž P, Sˇ.. But then the corresponding component of the element of ␰ g Ext 1 Ž P, Sˇ. defined by the vertical extension must be zero. Indeed, ␰ is the image of the universal extension Ž1.1. under the map ␲ * : Ext 1 Ž Q , Sˇ. ª Ext 1 Ž P, Sˇ. induced by the projection ␲ : P ª Q in Ž2.2.. Since Ž2.2. is non-split and the only endomorphisms of Q are the multiplications by a constant, it follows that the restriction of ␲ to any direct summand Q ; P is zero. Finally, we prove a theorem characterizing the universal vector bundles on GŽ1, 4. by means of their cohomology vanishings. THEOREM 2.4. Let F be an indecomposable ¨ ector bundle on GŽ1, 4.. Then F is, up to a twist with a line bundle, either the tri¨ ial line bundle, or Q , or S , or Sˇ if and only if the following conditions hold: Ža. Žb.

F has no intermediate cohomology H i Ž F m Q Ž l .. s 0 for any i s 2, 3, 4 and l g ⺪.

Proof. We use now induction on Ý l h1 Ž F m Q Ž l .., the zero case being now Theorem 2.3. Again we can assume h1 Ž F m Q Žy1.. / 0. Therefore,

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there is a non-zero element in Ext 1 Ž Q , F ., which yields a non-trivial extension 0 ª F ª P ª Q ª 0. As before, P still satisfies conditions Ža. and Žb. and h1 Ž P m Q Ž l .. s h1 Ž F m Q Ž l .. for any l, except for l s y1 for which h1 Ž P m Q Žy1.. s h1 Ž F m Q Žy1.. y 1. Hence, by induction hypothesis, P decomposes as a direct sum of summands of the type OG Ž l ., Q Ž l ., S Ž l ., and SˇŽ l .. Finally, we consider the following commutative diagram defining P⬘ as a pull-back: 0 6

Q 6

Sˇ 6

Sˇ 6

0

0

0

6

6

OG[5 6

0

6

6

P⬘ 6

6

F

P 6

6

0

6

6

F

0

0 6

The middle horizontal exact sequence splits as usual, and the splitting of the middle vertical exact sequence is proved as in Theorem 2.3. This proves the theorem.

3. VECTOR BUNDLES WITHOUT INTERMEDIATE COHOMOLOGY We continue here the strategy of the previous section. The difference now is that we will not obtain a finite number of vector bundles when removing any of the conditions in Theorem 2.4Žb.. THEOREM 3.1. Let F be an indecomposable ¨ ector bundle on GŽ1, 4.. Then F is, up to a twist with a line bundle, either the tri¨ ial line bundle or Q , S , Sˇ or a ¨ ector bundle of type ŽI. if and only if the folloiwng conditions hold: Ža. Žb.

F has no intermediate cohomology H i Ž F m Q Ž l .. s 0 for any i s 3, 4 and l g ⺪.

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Proof. We use induction on Ý l h2 Ž F m Q Ž l .., the zero case being Theorem 2.4. We can assume h2 Ž F m Q Žy2.. / 0. Hence, by using the universal exact sequence Ž1.1. there is a non-zero element in Ext 1 Ž SˇŽ1., F ., which yields a non-trivial extension 0 ª F ª P ª SˇŽ 1 . ª 0. As in the proofs of the preceding section, P still satisfies conditions Ža. and Žb. and h2 Ž P m Q Ž l .. s h2 Ž F m Q Ž l .. for any l, except for l s y2 for which h2 Ž P m Q Žy2.. s h2 Ž F m Q Žy2.. y 1. By the induction hypothesis, P decomposes as a direct sum of summands of the type OG Ž l ., Q Ž l ., S Ž l ., SˇŽ l . and twists of vector bundles of type ŽI.. Now we consider the following commutative diagram defining P⬘ as a pull-back, and in which the right column is the second short exact sequence in Ž1.3.:

SˇŽ1. 6

6

OG[10 6

0

K 6

K 6

0

0

0

6

6

P⬘ 6

6

P 6

6

F

0 6

6

0

6

6

F

0

0 6

The middle horizontal exact sequence splits as usual, because F has not intermediate cohomology. As for the middle vertical exact sequence, the main difference now with the proofs in the preceding sections is that the element Ext 1 Ž P, K . defining the extension can have non-zero components different from those corresponding to possible summands SˇŽ1. ; P Žwhich as usual we know to produce a zero coordinate.. Indeed, any direct summand of type ŽI. Žwe include here possible direct summands S ; P . will yield a non-zero summand of Ext 1 Ž P, K .. We decompose P s P1 [ P2 , where P1 is a sum of vector bundles of type ŽI. and P2 does not have any summand of type ŽI.. Since our extension lives in Ext 1 Ž P, K . s Ext 1 Ž P1 , K . [ Ext 1 Ž P2 , K . and the second coordinate is zero, it follows that P⬘ ( P1X [ P2 , where P1X appears in an exact sequence 0 ª K ª P1X ª P1 ª 0.

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But from this exact sequence it is immediate to see that P1X is a direct sum of vector bundles of type ŽI. Žsince P1 is too.. This completes the proof. THEOREM 3.2. Let F be an indecomposable ¨ ector bundle on GŽ1, 4.. Then F is, up to a twist with a line bundle, either the tri¨ ial line bundle, or Q , or S , or Sˇ, or a ¨ ector bundle of type ŽII., or the dual of a ¨ ector bundle of type ŽII. if and only if the following conditions hold: Ža. Žb.

F has no intermediate cohomology H 3 Ž F m Q Ž l .. s 0 for any l g ⺪.

Proof. We use induction on Ý l h 4 Ž F m Q Ž l .., the zero case being Theorem 3.1. We can assume h 2 Ž Fˇ m QˇŽy1.. s h 4 Ž F m Q Žy4.. / 0. Hence, by using the universal exact sequence Ž1.1. there is a non-zero element in Ext 1 Ž SˇŽ1., Fˇ., which yields a non-trivial extension 0 ª Fˇ ª P ª SˇŽ 1 . ª 0. ˇ As usual, P still satisfies conditions Ža. and Žb. and h 4 Ž Pˇ m Q Ž l .. s h 4 Ž F

m Q Ž l .. for any l, except for l s y4 for which h 4 Ž P m Q Žy4.. s h 4 Ž F m Q Žy4.. y 1. By induction hypothesis, P decomposes as a direct sum of summands of the type OG Ž l ., Q Ž l ., S Ž l ., SˇŽ l . and twists of vector bundles of type ŽII. or their duals. Now we consider the following commutative diagram defining P⬘ as a pull-back:

P 6

SˇŽ1. 6

6

OG[10 6

0

K 6

K 6

0

0

0

6

6

6

P⬘ 6

6

0 6

6

Fˇ

6

0

Fˇ

6

0

0 6

Once more, the middle horizontal exact sequence splits, and the only non-zero components Žcorresponding to direct summands of P . in the element Ext 1 Ž P, K . defining the middle vertical extension can be those coming from possible summands of type ŽII. of P Žas in the preceding proofs, the coordinates corresponding to possible factors SˇŽ1. are zero.. As in the proof of Theorem 3.1, we also consider S and Kˇ as vector

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bundles of type ŽII.. We decompose P s P1 [ P2 , where P1 is a sum of vector bundles of type ŽII. and P2 does not have any summand of type ŽII.. Hence, it follows that P⬘ ( P1X [ P2 , where P1X appears in an exact sequence 0 ª K ª P1X ª P1 ª 0. But, again as in the proof of Theorem 3.1, P1X is a direct sum of vector bundles of type ŽII., which completes the proof. We can finally state and prove our result characterizing vector bundles on GŽ1, 4. without intermediate cohomology. THEOREM 3.3. An indecomposable ¨ ector bundle on GŽ1, 4. without intermediate cohomology is, up to a twist with a line bundle, a ¨ ector bundle of type ŽIII.. Proof. Let F be a vector bundle on GŽ1, 4. without intermediate cohomology. We use induction on Ý l h3 Ž F m Q Ž l .., the zero case being Theorem 3.2. Without loss of generality, we assume h3 Ž F m Q Žy3.. / 0. Hence, by using the dual of the exact sequences Ž1.1. and Ž1.3. and the identification n2 S ( SˇŽ1., there is a non-zero element in Ext 1 Ž K Ž1., F ., which yields a non-trivial extension 0 ª F ª P ª K Ž 1 . ª 0.

Ž 3.1.

As usual, Pˇ has no intermediate cohomology and h Pˇ m Q Ž l .. s h F m Q Ž l .. for any l, except for l s y3 for which h3 Ž P m Q Žy3.. s h3 Ž F m Q Žy3.. y 1. Hence, by induction hypothesis, P decomposes as a direct sum of summands of twists of vector bundles of type ŽIII.. Now we consider the following commutative diagram defining P⬘ as a pull-back, and in which the right column is the exact sequence Ž1.6.: 3Ž

P 6

K Ž1. 6

6

OG[25 6

0

EŽ1. 6

EŽ1. 6

0

0

0

6

6

P⬘ 6

6

0 6

6

F

6

0

6

6

F

0

0 6

3Ž

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GŽ1, 4.

141

The middle horizontal exact sequences splits once more because F has not intermediate cohomology. As for the middle vertical exact sequence, we decompose P s P1 [ P2 , where P1 is a sum of vector bundles of type ŽIII. and the summands of P2 are not of type ŽIII., but twists of vector bundles of type ŽIII. with a non-trivial line bundle. As in the other proofs in this section, it suffices to prove that the element ␰ g Ext 1 Ž P, EŽ1.. s Ext 1 Ž P1 , EŽ1.. [ Ext 1 Ž P2 , EŽ1.. corresponding to the middle vertical extension has zero as its second component in this decomposition. The extra difficulty now is that, if a summand H ; P2 produces a non-zero summand Ext 1 Ž H, EŽ1.. ; Ext 1 Ž P2 , EŽ1.., it is not necessarily H s K Ž1., but it could happen, more generally, that H Žy1. is a vector bundle of type ŽIII.. In this case, H is a direct summand of a vector bundle G Ž1. fitting in an exact sequence 0 ª E Ž 2.

[i

[ K Ž 1.

[l

[ KˇŽ 1 .

ª S Ž 1.

[j

[ SˇŽ 1 .

[m

[k

[ Q Ž 1.

ª G Ž 1.

[n

ª 0.

By contradiction, assume that the component of ␰ in Ext 1 Ž H, EŽ1.. is not zero. Then, at least one summand K Ž1. of that exact sequence must produce a non-zero map K Ž 1. ; E Ž 2.

[i

[ K Ž 1.

[j

[ SˇŽ 1 .

[k

ª G Ž 1. ª H ; P ª K Ž 1.

Žthe projection P ª K Ž1. being that of Ž3.1... Since the only endomorphisms of K are multiplications by constants, this implies that the projection P ª K Ž1. has a section, so that the extension Ž3.1. must be trivial, which is a contradiction.

REFERENCES wACx

E. Arrondo and L. Costa, Vector bundles on Fano 3-folds without intermediate cohomology, preprint, 1998. wASx E. Arrondo and I. Sols, On congruences of lines in the projective space, Mem. ´ Soc. Math. France 50 Ž1992.. wBGSx R. O. Buchweitz, G. M. Greuel, and F. O. Schreyer, Cohen᎐Macaulay modules on hypersurface singularities, II, In¨ ent. Math. 88 Ž1987., 165᎐182. wHx G. Horrocks, Vector bundles on the punctured spectrum of a ring, Proc. London Math. Soc. Ž 3 . 14 Ž1964., 689᎐713. wKSx S. Katz and S. A. Strømme, Schubert, a Maple package for intersection theory, available at http:rrwww.math.okstaste.edur ;;katzrschubert.html or by anonymous ftp from ftp.math.okstate.edu or linus.mi.uib.no, cd pubrschubert. wKx H. Knorrer, Cohen᎐Macaulay modules on hypersurface singularities, I, In¨ ent. Math. ¨ 88 Ž1987., 153᎐164.

142 wMx wO1x wO2x

ARRONDO AND GRANA ˜

C. Madonna, A splitting criterion for rank 2 vector bundles on hypersurfaces in ⺠ 4 , Rend. Sem. Torino, in press. G. Ottaviani, Criteres ` de scindage pour les fibres ´ vectoriels sur les grassmannianes et les quadriques, C. R. Acad. Sci. Paris Ser. ´ I 305 Ž1987., 257᎐260. G. Ottaviani, Some extensions of Horrocks criterion to vector bundles on Grassmannians and quadrics, Ann. Mat. Pura Appl. Ž 4 . 155 Ž1989., 317᎐341.