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Horrocks correspondence

Prabhakar Rao

Department of Mathematics and Computer ScienceUniversity of Missouri-St.Louis

May, 2016

Prabhakar Rao (Department of Mathematics and Computer Science University of Missouri-St.Louis)Horrocks correspondence May, 2016 1 / 23

Pn over a field k . For any sheaf B on Pn,

H i∗(B) := ⊕k∈ZH

i (X ,B(k)),

a graded module over the polynomial ring S .

H0∗(F), then

F∨ = H0∗ (F∨).

Free resolution for F∨:

0→ Pn−1∨ · · · → P3∨ → P2∨ → P1∨ → P0∨ → F∨ → 0. (1)


Dualize this: get a complex, with cohomology modules of finite length

0→ F → P0 → P1 → P2 → · · · → Pn−2 → Pn−1 → 0. (2)

or sheafified: get an exact sequence of bundles

0→ F → P0 → P1 → P2 → · · · → Pn−2 → Pn−1 → 0. (3)

(Notation: P i is a free module, P i is a free bundle/sheaf.)From this it becomes clear that

H i∗(F) = H i (P ), 0 ≤ i ≤ n − 1

Next resolve F and splice it to P .


Get enlarged P ,

0→ P−n → · · · → P0 → · · · → Pn−1 → 0

Properties:

H i (P ) = 0 if i /∈ 1, . . . , n − 1,H i (P ) = finite length module if i ∈ 1, . . . , n − 1

Horrocks’ correspondence is to match a bundle F to such a complex offree modules. Horrocks calls it a Z-complex.

Given another such complex Q quasi-isomorphic to P ,let G = ker(Q0 → Q1), let G be the sheafification of the module G . ThenG and F are stably isomorphic: ie.There exist free bundles R, S such that

F ⊕R ∼= G ⊕ S.


Call such a Z-complex a Horrocks data complex since it contains as dataall the intermediate cohomology of a bundle as well as data of a bundle Fwith that cohomology.

Horrocks data complexes modulo quasi-isomorphismsl

vector bundles modulo stable equivalence

Fact: There are non-isomorphic bundles F1,F2 on P3 with identicalintermediate cohomology modules: for null-correlation bundles,H1(Fi (−1) = k ,H2(Fi (−3) = k are the only non-zero cohomologies.


Horrocks Correspondence Theorem:

Let F ,G be two vector bundles on Pn with no free summands. If

φ : F → G induces isomorphisms

H i∗(φ) : H i

∗(F)→ H i∗(G), for all 1 ≤ i ≤ n − 1,

then φ is an isomorphism.

Horrocks Splitting Criterion:

If F is a vector bundle on Pn and for all 1 ≤ i ≤ n − 1, H i∗(F) = 0, then

F is free (F =⊕

j OPn(aj)).


ACM varieties

X n a smooth ACM variety with respect to the polarization OX (1). Thisimplies that H i

∗(OX ) = 0 for all 1 ≤ i ≤ n − 1.A = H0

∗ (OX ) is a graded noetherian ring.

E a vector bundle on X . H i∗(E) is an A-module of finite length for any

1 ≤ i ≤ n − 1.A vector bundle B on X is called an ACM bundle if H i

∗(B) = 0 for all1 ≤ i ≤ n − 1.Examples: OX (a), ωX (b), free bundles

⊕j OX (aj).

E bundle on X . E := H0∗ (E) an A-module. E∨ = H0

∗ (E∨).Repeat Horrocks’ steps. (Joint work with F. Malaspina)


Free resolution for E∨: but stop after Cn−2∨

0→ K → Cn−2∨ · · · → C 3∨ → C 2∨ → C 1∨ → C 0∨ → E∨ → 0. (4)

K sheafified gives K an ACM bundle.Dualize.

0→ E → C 0 → C 1 → · · · → Cn−2 → K∨ → 0. (5)

Sheafify:0→ E → C0 → C1 → · · · → Cn−2 → K∨ → 0. (6)

H i∗(E) = H i (C ) for all i ≤ n − 1.

Next resolve E (maybe infinite free resolution) and splice it to C .


Get enlarged C

. . .C−m → C−m+1 → · · · → C 0 → C 1 → · · · → Cn−2 → K∨ → 0

with cohomology supported in degrees 1, 2, . . . , n − 1.

Choose a free resolution P of C .

P has the same cohomology, supported in degrees 1, 2, . . . , n − 1.Call P a Horrocks data complex on X .

Let F equal kernel P0 → P1. F is F sheafified.Call F a Horrocks data bundle on X .

We get a diagram


0→ F → P0δ1P−−→ P1

δ2P−−→ . . .

δn−2P−−−→ Pn−2 → Pn−1 → 0

↓ β ↓ ↓ ↓ ↓

0→ E → C0δ1C −−→ C1

δ2C −−→ . . .

δn−2C −−−→ Cn−2 → K∨ → 0

and β : F → E induces H i∗(F) ∼= H i

∗(E).

Properties of FF is determined by E up to direct sum with a free bundle.

The dual gives a finite free resolution for F∨.

F has no summands of the form B or B∨, where B is a non-free ACMbundle.

The complex can be replaced by a minimal complex givingβ : Fm → E , where Fm has no free summand.

For any two minimal Horrocks data bundles F ,F ′, if φ : F → F ′induces an isomorphism on intermediate cohomology modules, then φis an isomorphism.


History: Old ideas going back to Auslander.β∨ : E∨ → F∨ can be modified to

0→ N → E∨ ⊕ L→ F∨ → 0

where N is a maximal Cohen Macaulay module when X is arithmeticallyGorenstein and where L is free.Auslander’s approximation theorem.Buchweitz extends this to non-commutative rings.


Constructing (F , β) given E will be called finding Horrocks data for E .

Choosing a minimal complex 0→ Fm → P0m → P1

m . . . gives minimalHorrocks data (Fm, β) for E .

Push out the short exact sequence 0→ Fm → P0m → Gm → 0, using β,

getting

0→ E → A(E)→ Gm → 0.

A(E) is an ACM bundle determined by E .


Correspondence Theorem I

Let σ : E → E ′ where E and E ′ have no ACM summands and H i∗σ is an

isomorphism for 1 ≤ i ≤ n − 1. Suppose that one of the following holds:

H1∗ (E ⊗ A(E ′)∨) ∼= H1

∗ (E ′ ⊗A(E ′)∨).

H1∗ (E ⊗ B∨) ∼= H1

∗ (E ′ ⊗ B∨) for all irreducible ACM bundles B.

Then σ is an isomorphism.

Correspondence Theorem II

Given bundles E , E ′ with no ACM summands and suppose they haveHorrocks data (Fm, β), (Fm, β

′) with the same Fm. For any ACM bundleB, let VB denote the kernel of H1

∗ (Fm ⊗ B∨)→ H1∗ (E ⊗ B∨). Likewise

define V ′B for E ′.If one of the following holds:

VB = V ′B for each irreducible non-free summand B appearing in A(E)or A(E ′).

VB = V ′B for all irreducible non-free ACM bundles BThen E ∼= E ′.


Let B be an ACM bundle,⊕

j OX (aj)→ B a surjection onto globalsections. Fm gives rise to

H1∗ (Fm ⊗ B∨)→ H1

∗ (Fm ⊗⊕j

OX (−aj)).

H1∗ (Fm ⊗ B∨)soc will denote the kernel of this map and will be called the

sub-module of H1∗ (Fm ⊗ B∨) consisting of B-socle elements.

Given E , Horrocks data (Fm, β) and an ACM bundle B,

VB ⊆ H1∗ (Fm ⊗ B∨)soc.

Reason:H1∗ (Fm ⊗ B∨) → H1

∗ (Fm ⊗⊕

j OX (−aj))

↓ ↓H1∗ (E ⊗ B∨) → H1

∗ (E ⊗⊕

j OX (−aj))


Conversely, given a minimal Horrocks data bundle Fm, any elementα ∈ H1(Fm(t)⊗ B∨)soc gives rise the the extension

0→ Fmβ−→ D → B(−t)→ 0.

Pullback by any section OX (a)→ B is split hence (Fm, β) gives Horrocksdata for D.

Question

Given Fm, what are all the submodules VB ⊆ H1∗ (Fm ⊗ B∨)soc that arise

from bundles E with Fm as Horrocks data bundle?


Splitting Criterion?

Mainly a tautology: Requirements for E to be free of the form⊕

j OX (aj):

H i∗(E) = 0, 1 ≤ i ≤ n − 1.

For all ACM bundles B, H1∗ (E∨ ⊗ B) = 0.

But if E is a non-free ACM bundle, then a short resolution of E

0→ C →⊕j

OX (ej)→ E → 0

yields C is ACM and H1(E∨ ⊗ C) 6= 0.


Example 1

Qn quadric hypersurface of odd dimension. Knorrer proved Q has (up totwists) a unique non-free irreducible ACM bundle: Σ the spinor bundle.Knowing this

the splitting criterion becomes Ottaviani’s theorem.

Correspondence Theorem I simplifies considerably.

Correspondence Theorem II can be pursued further.


H1∗ (Fm ⊗ Σ∨)soc as an A-module is annihilated by the ideal

(X0,X1, . . . ,Xn+2), hence is just a graded vector space.

Theorem

For any graded subspace V of H1∗ (Fm ⊗ Σ∨)soc, there exists a bundle E

with invariants (Fm,V ).

Proof: A basis of V yields an element in H1∗ (Fm ⊗ V∨) where

V = V ⊗k Σ. Hence get E where

0→ Fm → E → V → 0

More work with Σ is needed to see that now VΣ is not larger than V .

Conclusion

Up to sums of ACM bundles,E⇔

(Fm,V )

.


Example 2

X equals Veronese surface in P5 , OX (1) the restriction of OP5(1).

The only non-free indecomposable ACM bundles (up to twists) areL = OP2(1) and U = Ω1

X ⊗ L.

H1∗ (Fm ⊗ L∨)soc) and H1

∗ (Fm ⊗ U∨)soc) again have A-module structure ofa graded vector space.Given E with Horrocks data (Fm, β), get kernels VL,VU .

The natural map 3L(−1)→ U gives H1∗ (Fm⊗U∨)

ψFm−−−→ 3H1∗ (Fm⊗L∨(1).

Extra requirement: ψFm(VU ) ⊆ 3VL(1).

Conclusion

Up to sums of ACM bundles,E⇔

(Fm,V ,W ) | ψFm(V ) ⊆ 3W (1).


Example 3: in progress

X is P1 × P1 embedded in P5 by OX (1) = O(1, 2). Non-trivial ACM linebundles up to twist are

L = O(0,−1),M = O(−1, 1),L∨ = O(0, 1) and O(−1, 0).

Ext1(M,L) = H1(O(1,−2) = k ⊕ khence there is a 1-parameter family of ACM bundles Dλ

0→ L → Dλ →M→ 0.

Faenzi & Malaspina show that every irreducible ACM bundle D aftertwists, appears in a short exact sequence

0→ L⊕a → D →M⊕b → 0.

So the splitting criterion reduces to requiring H1∗ (E∨ ⊗ B) = 0 for

B = L,M,L∨,O(−1, 0).


For Correspondence Theorem I, assume E , E ′ have no ACM bundlesummands, and φ : E → E ′ satisfies

H1∗ (φ) is an isomorphism,

H1∗ (E ⊗ B∨) ∼= H1

∗ (E ′ ⊗ B∨) are isomorphisms forB = L,M,L∨,O(−1, 0).

Does this imply that H1∗ (E ⊗ D∨) ∼= H1

∗ (E ′ ⊗D∨) for any ACM bundle D?


Possible framework for future: Costa,Miro-Roig have a definition ofCohen-Macaulay exceptional block collections on a smooth projective X n.

A sheaf/bounded complex of sheaves F is exceptional ifExt0(F ,F) = k,Exti (F ,F) = 0, i 6= 0.

(F1,F2, . . . ,Fm) is an exceptional collection if each Fi is exceptionaland for j < k , Ext(Fk ,Fj) = 0.

(F1,F2, . . . ,Fm) is a block of length m if each Fi is exceptional andfor j 6= k , Ext(Fk ,Fj) = 0.

An m-block collection of type (α0, α1, . . . , αm) is a sequence(F0,F1, . . . ,Fm) of blocks Fi of length αi such that the totality ofmembers in the blocks (in that order) forms an exceptional collection.

An m-block collection is full if the totality of members in the blocksgenerates the derived category of X .

On (X n,OX (1)), an n-block collection of sheaves is Cohen-Macaulayif for each Fn

p ∈ Fn and Fkq ∈ Fk , k < n, Fn

p ⊗Fkq is ACM.


This definition of a full Cohen-Macaulay n-block collection on (X n,OX (1))includes the cases of quadrics, Grassmannians. In many of these instances,Fn will consist of a single sheaf OX . So all Fk

q will be ACM bundles on X .In this context, their theorem is

Costa, Miro-Roig

Let X be a smooth ACM variety of dimension n with a fullCohen-Macaulay n-block collection, with Fn = OX. Let E be a vectorbundle such that E ⊗ Fk

q is ACM for all k , q. Then E is free.

Question

A similar correspondence theorem?


prabhakar rao - nd.educonf/mcaag/talk_slides/rao_mcaagslides.pdf · prabhakar rao department of...

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