EXTERIOR ALGEBRA RESOLUTIONS ARISING FROM HOMOGENEOUS BUNDLES

EXTERIOR ALGEBRA RESOLUTIONS ARISING FROM HOMOGENEOUS BUNDLES

GUNNAR FLØYSTAD

Abstract

We describe resolutions of general maps (resp. general symmetric and skew-symmetric maps) E^a→E(1)^bgiven by linear forms over the exterior algebraE. Via the BGG-correspondence we describe the associated coherent sheaves. We also show how representation theory of algebraic groups enables one to solve these types of problems for much larger classes of maps.

Introduction

WhenRis a commutative ring, the minimal free resolution of a mapR^a →R^b and symmetric and skew-symmetric mapsR^a →R^a, under suitable generality conditions, are well known and have been developed by a series of authors.

See [2, A2.6] for an overview.

In this note we do an analog for the exterior algebra E = ⊕ ∧ⁱ V on a finite dimensional vector spaceV and general graded mapsE^a→E(1)^b, and general graded symmetric and skew-symmetric mapsE^a →E(1)^a. SinceE is both a projective and injectiveE-module, by taking a free (projective) and cofree (injective) resolution of such maps, there is associated an unbounded acyclic complex of freeE-modules, called a Tate resolution, see [3] or [4]. Via the Bernstein-Gel’fand-Gel’fand (BGG) correspondence, this corresponds to a complex of coherent sheaves on the projective spaceP(V^∗). We show that in all the cases above (with the dimension ofV not too low), this complex actually reduces to a coherent sheaf. We describe these coherent sheaves and also describe completely the Tate resolutions.

These descriptions turn out to be simpler to work out than in the corres- ponding commutative case, and, maybe at first surprising, the descriptions are also more geometric.

In fact not only are we able to describe the Tate resolutions and coherent sheaves associated to the maps stated above, but, using the theory of repres- entations of reductive groups, we are able to describe the Tate resolutions and coherent sheaves associated to vast larger classes of natural mapsE^a→E(r)^b,

Received March 3, 2003.

something which would have required considerably more effort for commut- ative rings.

There is only one catch related to all our descriptions. We must assume that the dimension ofV is not too small compared toa andb. For instance for a general mapE^a → E(1)^bwe must assume that the dimension ofV is

≥a+b−1. In case the dimension ofV is smaller than this, the nature of the problem changes, and we do not investigate this case.

1. Tate resolutions and projections

LetV be a finite dimensional vector space over a fieldk. PutW =V^∗and set v=dimP(W). LetE(V )be the graded exterior algebra⊕^v+_i=0¹∧ⁱVwhereVhas degree−1 (because we considerWto have degree 1). LetωE =Hom_k(E, k) be the graded dual, which we consider as a leftE-module. (As suchωE ∼= E(−v−1).) It is the injective hull ofk.

1.1. Tate resolutions

ATate resolutionis an (unbounded) acyclic complexT with components T^p= ⊕i∈ZωE(−i)⊗kV_i^p

of finite rank. (TheV_i^pare finite dimensional vector spaces.) Note that a Tate resolution is completely determined, up to homotopy, by each differentiald^p sinceT^≤pis a projective resolution of the image ofd^pandT^>pis an injective resolution of the image ofd^p.

By [3] or [4], to each coherent sheafF onP(W)there is associated a Tate resolutionT (F)whose terms are

T (F)^p= ⊕^v_i=0ωE(i−p)⊗kHⁱF(p−i).

(In particular we see thatT (F)^p isωE(−p)⊗k H⁰F(p)forp 0.) The maps

ωE(i−p)⊗kHⁱF(p−i)−→ωE(i−p−1)⊗kHⁱF(p+1−i) are determined by the maps in degreep+1−iwhich are the natural maps

W⊗kHⁱF(p−i)−→HⁱF(p+1−i).

Conversely, given a left moduleN = ⊕i∈ZNi overEwe get associated a complexof coherent sheaves

L(N)^∼: · · · →OP(U)(i)⊗kNi →OP(U)(i+1)⊗kNi+1→ · · ·.

In this way we can to each Tate resolutionT associate a complex of coherent sheaves onP(W)by using this construction on kerd^p. (All the L(kerd^p)^∼ become isomorphic in the derived category of coherent sheaves onP(W). If T = T (F)thenL(kerd^p)^∼ is a complex isomorphic toF in this derived category.)

Remark1.1. IfK^◦(E−cF )is the homotopy category of Tate resolutions, andD^b(coh/P(W))is the derived category of coherent sheaves onP(W), the Bernstein-Gel’fand-Gel’fand correspondence says that there is an equivalence of categories

K^◦(E−cF )∼=D^b(coh/P(W)).

See [4] or originally [1].

1.2. Projections

Let U ⊆ W be a subspace and letE(U^∗) = ⊕ ∧ⁱ U^∗. The center of the projectionπ : P(W)P(U)is the linear subspaceP(W/U). Suppose the support ofF is disjoint fromP(W/U). According to [4],

Hom_{E(V )}(E(U^∗), T (F))

is also a Tate resolution (forE(U^∗)) and it is the Tate resolution associated to π∗F. Note that this gives

T (π∗F)^p = ⊕^dim_i=0^P(U)ωE(U^∗)(i−p)⊗kHⁱF(p−i).

IfY → W is a linear map, then let U be the image inW. Thus we get a projection and an embedding

P(W)P(U)−→ⁱ P(Y ).

In this case

Hom_{E(V )}(E(Y^∗), T (F)) is the Tate resolution ofi∗π∗F.

2. Tate resolutions arising from general maps

LetAandBbe finite dimensional vector spaces. Fix a basisx0, . . . , xaofA^∗ andy0, . . . , ybofB^∗Then a map

A^∗−→V ⊗B

corresponds to a(b+1)×(a+1)matrix [vij] where thevijare inV. It comes from the generic map

A^∗−→(A^∗⊗B^∗)⊗B,

corresponding to the generic(b+1)×(a+1)matrix [xjyi], composed with a mapA^∗⊗B^∗→V sendingxjyi tovij. We shall assume this map is general and it gives us a general morphism

ωE⊗A^∗−→ωE(−1)⊗B

which again gives a Tate resolution by taking projective and injective resolu- tions. This again is associated to a complex of coherent sheaves which we now show is a coherent sheaf. Note thata andbare the dimensions ofP(A)and P(B).

Theorem2.1.Given a general morphism

(1) ωE(a)⊗A^∗−→^d ωE(a−1)⊗B

coming from a general surjectionA^∗⊗B^∗→V and supposev≥a+b. Let T be the associated Tate resolution with (1) in components0and1.

a.T is the Tate resolution associated toπ∗L whereL is the line bundle OP(A)×P(B)(−2, a)⊗∧^a+1Aon the Segre embedding of P(A)×P(B)inP(A⊗

B)andπ:P(A⊗B)P(W)is the projection.

b. The Tate resolution has terms T^p=

ωE(−p)⊗S^p−2A⊗S^p+aB⊗ ∧^a+1A, p≥2, ωE(a+b+r)⊗S^b+r+1A^∗⊗S^r−1B^∗⊗ ∧^b+1B^∗, p= −r <0 Proof. We find

H^aL(−a)= ∧^a+1A⊗H^aOP(A)×P(B)(−2−a,0)

= ∧^a+1A⊗A^∗⊗ ∧^a+1A^∗=A^∗ H^aL(−a+1)= ∧^a+1A⊗H^aOP(A)×P(B)(−1−a,1)

= ∧^a+1A⊗ ∧^a+1A^∗⊗B=B

and alsoHⁱL(−i)andHⁱL(−i+1)are zero fori=a. The map (A⊗B)⊗H^aL(−a)→H^aL(−a+1)

is clearly the generic one and hence we get part a. Part b. follows of course by considering the cohomology ofL.

Remark2.2. Consider the generic morphism OP(A⊗B)⊗A^∗→OP(A⊗B)(1)⊗B.

The rank strata of this morphism are the orbits inP(A⊗B)of GL(A)×GL(B). The stratum of lowest rank one, corresponds to the closed orbit of GL(A)× GL(B)which is the Segre embeddingP(A)×P(B) &→P(A⊗B).

3. The Tate resolution of a general symmetric map Given now a symmetric map

A^∗−→V ⊗A.

It corresponds to a symmetric matrix [vij] of sizea+1, i.e. withvij = vji, and comes from the natural map

A^∗→S²A^∗⊗A,

corresponding to the generic symmetric matrix [xixj], composed with the map S²A^∗ →V sendingxixjtovij. We then get a symmetric morphism

ωE⊗A^∗→ωE(−1)⊗A.

Theorem3.1.Given a general symmetric morphism (2) ωE(a)⊗A^∗−→^d ωE(a−1)⊗A

coming from a general surjectionS²A^∗ → V with v ≥ 2a. Let T be the associated Tate resolution with (2) as the components in degree0and1.

a.T is the Tate resolution associated toπ∗L whereL is the line bundle OP(A)×P(A)(−2, a)⊗∧^a+1Aon the Segre embedding ofP(A)×P(A)inP(A⊗A) andπ:P(A⊗A)P(S²A)P(W)is the projection.

b. The Tate resolution has terms T^p=

ωE(−p)⊗S^p−2A⊗S^p+aA⊗ ∧^a+1A, p≥2, ωE(2a+r)⊗S^a+r+1A^∗⊗S^r−1A^∗⊗ ∧^a+1A^∗, p= −r <0 Proof. This follows from the previous theorem by verifying that the Segre embeddingP(A)×P(A) &→ P(A⊗A)does not intersect the center of the projectionP(A⊗A)P(S²A).

In fact the center is the linear subspaceP(∧²A)inP(A⊗A). It consists of all points with coordinates given by non-zero skewsymmetric matrices [kij], i.e.

kij = −kji. On the other hand the image of the Segre embedding inP(A⊗A) are all points given by coordinates [kilj] where(k0, . . . , ka)and(l0, . . . , la)are points inP(A)andP(A). It is easily seen that these sets of points are disjoint.

We now wish to describe the Tate resolution and coherent sheaf associated to a general skew-symmetric morphism. From the preceding one might think that this is quite analogous to what we have done for symmetric maps. However it is quite different due to the fact that the projection center ofP(A⊗A) P(∧²A), which isP(S²A), intersects the Segre embeddingP(A)×P(A) &→ P(A⊗A). In fact the image of the diagonal are the points with coordinates [kikj] and this is a symmetric matrix.

We must therefore take another approach which will give us a much more powerful understanding of what we have just done.

4. Homogeneous bundles on homogeneous spaces

We present here some facts about induced homogeneous bundles on homogen- eous varieties. In particular we consider their cohomology. We base ourselves on [5].

LetGbe a reductive algebraic group over a fieldk of characteristic zero, with Borel groupBand maximal torusT. We letX(T )be the characters of T andR the root system ofG. Corresponding to the choice ofB there is a positive root systemR⁺and letSbe the associated simple roots.

Given a parabolic subgroupPofG, we letP-mod be the category of rational representations ofP. For eachP-module M we get aG-equivariant locally free sheafL(M)of rank dim_kMon the homogeneous varietyG/P [5, I.5.8].

IfN is anotherP-module, thenL(M ⊗kN)=L(M)⊗OG/P L(N). To such a parabolicP is associated a subsetI ⊆Sand the charactersX(P ) are allµ∈ X(T )such thatµ,α =ˇ 0 for allα ∈I. For suchµwe get line bundlesL(µ)onG/P. By [5, II.4.6],L(µ)is ample (in fact very ample) on G/P ifµ,αˇ >0 for allα∈I. Thus we get an embedding

G/P &→P(H⁰(G/P,L(µ))) withOG/P(1)=L(µ).

Let ind^G_P :P-mod→G-mod

be the induction functor [5, I.3.3] and letRⁿind^G_P be its higher derived functors.

ThenHⁿ(G/P,L(M))=Rⁿind^G_PMby [5, I.5.10]. Ifλ∈X(T ), to simplify notation, let Rⁿind^G_Bλ = Hⁿ(λ). This is an irreducible G-module whose

character is determined by the Borel-Bott-Weil theorem [5, II.5.5] and the Weyl character formula [5, II.5.10].

Associated to the inclusionsB ⊆ P ⊆ Gthere is by [5, I.4.5] a spectral sequence

(3) E2^n,m=Rⁿind^G_P◦R^mind^P_BN ⇒R^n+mind^G_BN.

If nowλ∈X(T )is such thatλ,α ≥ˇ 0 for allαinI, thenR^mind^P_Bλ=0 for m >0 by Kempf vanishing [5, II.4.5]. In this case we therefore get by (3) that

Hⁿ(λ)=Rⁿind^G_P(ind^P_Bλ)

=Hⁿ(G/P,L(ind^P_Bλ)).

Considering the G-equivariant induced sheaf L(ind^P_Bλ) on G/P &→ P(H⁰(µ))we are thus able to determine all the cohomology groups

Hⁿ(G/P,L(ind^P_Bλ)(r))=Hⁿ(G/P,L(ind^P_Bλ)⊗L(rµ))

=Hⁿ(G/P,L(ind^P_B(λ+rµ)))=Hⁿ(λ+rµ).

In particular, for the induced homogeneous bundle L = L(ind^P_Bλ)on P(H⁰(µ)), the terms of the Tate resolutionT ofL

T^p = ⊕^dim_i=0^G/PωE(i−p)⊗HⁱL(p−i) may be determined.

Example4.1. LetG=GL(W)and lete0, . . . , evbe a basis forW. IfT is the diagonal matrices we let3i be the character sending diag(t⁰, . . . , tv)toti. We letBbe the lower triangular matrices. The positive roots are then3i−3j

wherei < j and the positive simple roots areαi = 3i−1−3i fori = 1. . . v. LetP be the parabolic subgroup corresponding toα2, . . . , αv, i.e.P consists

of all matrices 





∗ 0 · · · 0

∗ ∗ · · · ∗

∗ ∗ · · · ∗...





.

LettingU =(e1, . . . , ev)there is an exact sequence ofP-modules 0→U →W →(e0)→0.

HereU is the irreducible representation ofP with highest weight31and thus U =ind^P_B31. OnG/P =P(W)there is an exact sequence

0→L(U)→L(W)→L(30)→0.

Since the Picard group of G/P is generated by L(30) we have L(30) = OP(U)(1). Since W is aG-module, L(W) = OP(U)⊗k W. Thus L(U) = 5P(W)(1).

Given now a partition

i:i1≥i2≥ · · · ≥iv

where ij are integers. The Schur bundle Si(5P(W)(1)) is then the induced bundle L(SiU) where Si is the Schur functor, [2, A2.5]. Since SiU is an irreducible representation with highest weighti131+ · · · +iv3vwe getSiU = ind^P_{B j=}ⁿ ₁ij3j

. It then follows that

H^rSⁱ(5P(W)(1))(p−r)=H^r(P(W),L(SⁱU)⊗L((p−r)3⁰))

=H^r

G/P,L

ind^P_B

(p−r)3⁰+ n j=1

ij3j

=H^r

(p−r)3⁰+ n j=1

ij3j

.

By the Borel-Bott-Weil theorem this can be calculated and the answer turns out to be the following.

Leth=h(p)be such thatih > p≥ih+1and leti(p)be the partition i1−1≥ · · · ≥ih−1≥p≥ih+1≥ · · · ≥iv.

It then follows that H^r

(p−r)3⁰+ n j=1

ij3j

=

H^{0 v}_i=0i(p)j3j

, r=h(p)

0, r=h(p).

In particular the Tate resolutionT ofSi(5P(W)(1))has “pure” terms T^p =ωE(h(p)−p)⊗Si(p)W.

Remark4.2. More generally one can show that ifµ=ωαis a fundamental weight corresponding to a short root andPS−{α}is the corresponding parabolic group, then all the induced bundlesL(ind^P_B^S−{α}λ)have "pure" Tate resolutions in the above sense.

5. Tate resolutions arising from general skew-symmetric maps Suppose given a skew-symmetric map

A^∗→V ⊗A.

Such a map comes from the natural map A^∗→ ∧²A^∗⊗A

and a map∧²A^∗→V. It gives rise to a skew-symmetric morphism ωE⊗A^∗→ωE(−1)⊗A.

To describe the Tate resolution and associated coherent sheaf on P(W) consider the Plücker embedding of the Grassmann of lines inP(A)

G(A,2) &→P(∧²A).

Let

0→R→OG⊗A→E →0 be the tautological sequence (where the rank ofE is two).

Theorem5.1.Given a general skew-symmetric morphism (4) ωE(a−1)⊗A^∗→ωE(a−2)⊗A

arising from a general surjection∧²A^∗→V and supposev≥2(a−1). Let T be the associated Tate resolution with components in degree0and1given by (4).

a.T corresponds to the coherent sheafπ∗L whereL is the bundle(S^aE) (−2)⊗ ∧^a+1AonG(A,2) &→ P(∧²A)andπ : P(∧²A) P(W)is the projection.

b. The components ofT are given by

T^p =

ωE(−p)⊗Sa+p−2,p−2A⊗ ∧^a+1A, p≥2

ωE(2a+r−2)⊗Sa+r−1,r−1A^∗⊗ ∧^a+1A^∗, p= −r <0. Proof. Choose a basise0, . . . , eaforAand letBbe the Borel subgroup of lower triangular matrices in GL(A). Let(ei)be the weight space of3i. Then G(A,2)=GL(A)/P whereP is the maximal parabolic subgroup associated to the rootsαi =3i−1−3i,i=2, i.e.P consists of matrices









∗ ∗ 0 · · · 0

∗ ∗ 0 · · · 0

∗ ∗ ∗ · · · ∗

∗ ∗ ∗ · · · ∗...







.

Note that dimG(A,2) = 2(a− 1) and that by the Plücker embedding G(A,2) &→P(∧²A)the canonical bundleOG(1)isL(3⁰+3¹), where3⁰+3¹ is the highest weight of∧²A.

There is a sequence ofP-modules

0→(e2, . . . , ea)→A→(e0, e1)→0 and the tautological sequence onG(A,2)becomes

0→L((e², . . . , ea))→OG⊗A→L((e⁰, e¹))→0. Since(e0, e1)=ind^P_B30asP-modules we get

S^aE =S^aL((e0, e1))=L(S^a(e0, e1))=L(ind^P_B(a30)) and

L :=(S^aE)(−2)⊗ ∧^a+1A=L

ind^P_B

(a−1)30−31+ a

2

3i

.

By the Borel-Bott-Weil theorem we find that H^a−1L(−a+1)=H^a−1

−a31+^a

2

3i

=H⁰(−3a)=A^∗

and

H^a−1L(−a+2)=H^a−1

30−(a−1)31+ a

2

3i

=H⁰(30)=A.

Since we findHⁱL(−i)= 0 andHⁱL(−i+1) = 0 fori = a−1, part a.

follows.

We find that

H⁰L(p)=H⁰

(a+p−1)30+(p−1)31+ a

2

3i

=Sa+p−2,p−2A⊗ ∧^a+1A, p≥2 and by Borel-Bott-Weil that

H^2a−2L(2−2a−r)= ∧^a+1A⊗H^2a−2((−a−r)30+(−2a−r)31)

= ∧^a+1A⊗(∧^a+1A^∗)^⊗2⊗Sa+r−1,r−1A^∗

=Sa+r−1,r−1A^∗⊗ ∧^a+1A^∗, p= −r <0

and also thatHⁱL(p−i)=0 for all otheriandp. Hence b. follows.

6. Some further examples Example6.1. Consider the map

ωE(1)⊗V −→^φ ωE(−1)⊗V^∗

corresponding to the map∧²V^∗⊗V →V^∗. Here we do not get associated a coherent sheaf to the Tate resolutionT (φ)but instead a complex, the truncated Koszul complex

(5) OP(W)(−v)⊗ ∧^vW → · · · →OP(W)(−2)⊗ ∧²W →OP(W)(−1)⊗W In fact, letMbe⊕^v_i=1∧ⁱWwhich is a subquotient ofωE. Then we see that (5) isL(M)^∼.

SinceωE(1)⊗V is the projective cover ofM and ωE(−1)⊗W is the injective hull, we getM =Imφand soT (φ)corresponds to (5).

Example6.2. Supposevis odd. Let∧²W →kbe a non-degenerate sym- plectic form. Then the map∧^v−1W →kgives a morphism

ωE(v−1)−→^φ ωE

equivariant under the symplectic group Sp(W). Since the image ofφ isk⊕ W⊕Imφ2,T (φ)corresponds to a null correlation bundle onP(W)(which is Sp(W)-equivariant).

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2. Eisenbud, D.,Commutative Algebra, Grad. Texts in Math. 150 (1995).

3. Eisenbud, D., Fløystad, G., Schreyer, F.-O.,Sheaf Cohomology and Free Resolutions over Exterior Algebras, Trans. Amer. Math. Soc. 355 (2003), 4397–4426.

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MATEMATISK INSTITUTT JOHS. BRUNSGT. 12 5008 BERGEN NORWAY

E-mail:gunnar@mi.uib.no