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CANONICAL COMPLEXES ASSOCIATED TO A MATRIX

ANDREW R. KUSTIN

ABSTRACT. Let Φ be anf× g matrix with entries from a commutative Noetherian ringR,with g≤ f. Recall the family of generalized Eagon-Northcott complexes{C i

Φ} associated toΦ. (See, for example, Appendix A2 in “Commutative Algebra with a view toward AlgebraicGeometry” by D. Eisenbud.) For each integeri, C i

Φ is a complex of freeR-modules. Forexample,C 0

Φ is the original “Eagon-Northcott” complex with zero-th homology equal to thering R/Ig(Φ) defined by ideal generated by the maximal order minors ofΦ; andC 1

Φ is the“Buchsbaum-Rim” complex with zero-th homology equal to thecokernel of the transpose ofΦ. If Φ is sufficiently general, then eachC i

Φ, with −1≤ i, is acyclic; and, ifΦ is generic,then these complexes resolve half of the divisor class groupof R/Ig(Φ). The family{C i

Φ}exhibits duality; and, if−1≤ i ≤ f− g+1, then the complexC i

Φ exhibits depth-sensitivitywith respect to the idealIg(Φ) in the sense that the tail ofC i

Φ of length equal to grade(Ig(Φ))

is acyclic. The entries in the differentials ofC iΦ are linear in the entries ofΦ at every position

except at one, where the entries of the differential areg×g minors ofΦ.This paper expands the family{C i

Φ} to a family of complexes{Ci,aΦ } for integersi anda

with 1≤ a≤ g. The entries in the differentials of{Ci,aΦ } are linear in the entries ofΦ at every

position except at two consecutive positions. At one of the exceptional positions the entriesarea×a minors ofΦ, at the other exceptional position the entries are(g−a+1)×(g−a+1)minors ofΦ.

The complexes{C iΦ} are equal to{C

i,1Φ } and{C

i,gΦ }. The complexes{C

i,aΦ } exhibit all

of the properties of{C iΦ}. In particular, if−1≤ i ≤ f− g and 1≤ a≤ g, thenC

i,aΦ exhibits

depth-sensitivity with respect to the idealIg(Φ).

CONTENTS

1. Introduction. 22. Notation, conventions, and preliminary results. 52.A. Ground rules. 52.B. Grade and perfection. 62.C. Multilinear algebra. 73. The classical generalized Eagon-Northcott complexes. 94. Schur and Weyl modules which correspond to hooks. 105. The complexesKΦ, andLΦ associated to a homomorphismΦ. 12

Date: September 16, 2015.AMS 2010Mathematics Subject Classification. Primary 13D02; Secondary 13C40.The author was partially supported by the the Simons Foundation grant number 233597.Keywords: Buchsbaum-Rim complex, depth-sensitivity, determinantal ring, duality, Eagon-Northcott com-

plex, hooks, Koszul complex, maximal Cohen-Macaulay module, perfect module, Schur module, Weyl module.1

http://arxiv.org/abs/1509.04367v1

2 ANDREW R. KUSTIN

6. The complexesKΦ andLΦ whenΦ is a direct sum of homomorphisms. 147. The definition and elementary properties of the complexesC

i,aΦ . 21

8. The acyclicity ofC i,aΦ . 29

References 37

1. INTRODUCTION.

Let R be a commutative Noetherian ring andF andG be freeR-modules of rankf andg,respectively, withg≤ f. Recall that, for eachR-module homomorphism

(1.0.1) Φ : G∗ → F

there is a family of generalized Eagon-Northcott complexes{C iΦ}. (See, for example, Defini-

tion 3.1, [13, Appendix A.2], [3, 2.16], or [20]. A more complete history of these complexesmay be found in the comments on page 26 in [3].) If

(1.0.2) −1≤ i ≤ f− g+1,

thenC iΦ has lengthf−g+1; and, iff−g+1≤ gradeIg(Φ), thenC i

Φ is acyclic fori satisfying(1.0.2). Furthermore, the complexesC i

Φ, for i satisfying (1.0.2), exhibit depth-sensitivity. Inparticular, ifs≤ gradeIg(Φ) for some integers with 0≤ s≤ f −g+1, then Hj(C

iΦ) = 0 for

f− g+2− s≤ j and i satisfying (1.0.2). In the generic situation, the complexes C iΦ, with

i satisfying (1.0.2), resolve the Cohen-Macaulay elements of the divisor class group of thedeterminantal ringR/Ig(Φ). The complexesC i

Φ, with i in the range (1.0.2), exhibit duality:

C iΦ∼= a shift of HomR(C

jΦ,R) in homological degree,

for i + j = f− g. Also, if R is a graded ring, and a matrix representation ofΦ is a matrixof linear forms, then the Betti tables for the complexes{C i

Φ} are pleasing to the eye. Themaps are linear, except at, at most one position where the maps have degreeg. Moreover, theposition of non-linearity slides, along a line of slope 1, from the beginning of the complex tothe end asi varies from 0 tof− g.

We expand the list of canonical complexes which are associated to theR-module homo-morphism (1.0.1). For each pair(i,a) with

(1.0.3) −1≤ i ≤ f− g and 1≤ a≤ g,

we consider a complexC i,aΦ . The classical generalized Eagon-Northcott complexes

(1.0.4) {C iΦ|(1.0.2) holds}

are included in the set

(1.0.5) {Ci,aΦ |(1.0.3) holds}

CANONICAL COMPLEXES ASSOCIATED TO A MATRIX 3

with C iΦ = C

i,1Φ for −1 ≤ i ≤ f− g andC i

Φ = Ci−1,gΦ for 0 ≤ i ≤ f− g+1. The complexes

of (1.0.5) exhibit many of the properties as the listed properties for the generalized Eagon-Northcott complexes (1.0.4). Each complex of (1.0.5) has lengthf− g+1; and,

if f− g+1≤ gradeIg(Φ), then eachC i,aΦ is acyclic.

The complexes of (1.0.5) exhibit depth-sensitivity. In thegeneric situation, the complexC i,aΦ

of (1.0.5)

(1.0.6) resolves a maximal Cohen-Macaulay module over the ring R/Ig(Φ) of rank(

g−1a−1

)

.

The complexes of (1.0.5) exhibit duality:

Ci,aΦ

∼= a shift of HomR(Cj ,b

Φ ,R) in homological degree,

for i + j = f− g−1 anda+b= g+1. Also, if R is a graded ring, andΦ is a map of degree1, then the maps ofCi,a

Φ are linear, except at, at most two adjacent positions where the mapshave degreea and g+ 1− a. Moreover, the position of non-linearity slides, along a lineof slope 1, from the beginning of the complex to the end asi varies from−1 to f− g; seeExample 7.6.

The Eagon-Northcott [11] complexC 0Φ and the Buchsbaum-Rim complex [6, 7, 8]C 1

Φare very important objects in Commutative Algebra and Algebraic Geometry. (For example,[1, 15, 16, 21, 29] is a small sampling of the recent papers about Buchsbaum-Rim multiplicityand its application to equisingularity.) We expect that therest of the family (1.0.5) will proveto be valuable tools in these fields.

The complexesC i,aΦ arise in the study of the homological properties of the primary com-

ponents of the content idealc( f gh) of the product of three generic polynomialsf , g, andh. These components have been identified [9, Thm. 4.2] and all but one of the componentsis known to be Gorenstein [9, Thm. 4.1 and Rem. 4.3]. The complexesC

i,aΦ also arise in

the study of the resolutions of the symmetric algebra Sym(I) and the Rees algebraR (I) ofa grade three Gorenstein idealI = (g1, . . . ,gn) in a polynomial ring over a fieldk; see, forexample, [23] and [24, Cor. 6.3], where the special fiber ringF (I) = k[g1, . . . ,gn] of I isresolved.

The complexesC i,aΦ are straightforward and they are built in a canonical manner. That is,

there are no choices; everything is coordinate-free. The modules inCi,aΦ are Schur modules

and Weyl modules corresponding to hooks. In other words, themodules inC i,aΦ all are kernels

of Koszul complex maps or Eagon-Northcott complex maps; see4.1 and 4.3. The complexC

i,aΦ is obtained by concatenating three finite complexes:

K→∧

→ L,

whereK andL are standard complexes of Weyl and Schur modules, respectively, and∧

consists of a single exterior power concentrated in one position; see Definition 7.2 for thedetails. The complexesK andL are introduced in Section 5.

4 ANDREW R. KUSTIN

The main result of this paper is Theorem 8.4 which states thatif Φ is sufficiently gen-eral,−1≤ i, and 1≤ a≤ g, thenC

i,aΦ is an acyclic complex of freeR-modules and H0(C

i,aΦ )

is a torsion-freeR/Ig(Φ)-module of rank(

g−1a−1

)

. The most important applications occur

when i also satisfiesi ≤ f− g. Indeed, in this situation,C i,aΦ has lengthf− g+ 1 and,

if f− g+ 1 ≤ gradeIg(Φ), then H0(Ci,aΦ ) is a perfectR-module of projective dimension

f− g+ 1 resolved byCi,aΦ and Extf−g+1

R (H0(Ci,aΦ ),R) is a perfectR-module resolved by

Cf−g−i−1,g+1−aΦ ; furthermore, even if gradeIg(Φ)< f−g+1, the complexC i,a

Φ exhibits depth-sensitivity with respect to the idealIg(Φ) in the sense that

(1.0.7) Hj(Ci,aΦ ) = 0 for f− g+2−gradeIg(Φ)≤ j, when−1≤ i ≤ f− g and 1≤ a≤ g.

The depth-sensitivity (1.0.7) allows one to use truncations of variousC i,aΦ as acyclic strands

in resolutions even whenIg(Φ) is known to be less thanf− g+1.There is a basic similarity between the present paper and andthe paper [25], which pro-

duces a family of complexes{Dqρ} for each almost alternating homomorphismρ. The fam-

ily {Dqρ} shares many properties with the family of generalized Eagon-Northcott complexes

{C iΦ}. The main difference between [25] and the present paper is that the homomorphism

ρ of [25] is special, in the sense that it is an almost alternating homomorphism; whereas,the complexes{C

i,aΦ } of the present paper and the generalized Eagon-Northcott complexes

{C iΦ} are both constructed from an arbitrary homomorphismΦ. Nonetheless, the statement

of Theorem 8.4 and some steps in its proof are modeled on [25, Thm. 8.3], although [25]does not contain any analogue to Lemma 8.1, which is the key calculation in the presentpaper. In place of a result like Lemma 8.1, [25] first treats the generic case and then special-izes to the non-generic case. In the present paper, Lemma 8.1shows that if the image ofΦcontains a basis element ofF , then

Ci,aΦ and C

i,aΦ′ ⊕C

i,a−1Φ′

have isomorphic homology for some (“smaller”)R-module homomorphismΦ′. To proveTheorem 8.4, we iterate Lemma 8.1 and apply the acyclicity lemma. The representationtheory that is used in the proof of Lemma 8.1 is begun in (6.3.4) and (6.3.5) and carried outin Proposition 6.4.

The complexesC i,aΦ are defined in 7.2; examples are given in 7.6 and 7.7; the duality is

treated in 7.9; and the zero-th homology

H0(Ci,aΦ ) =

∧f−g+a F

im(∧a Φ)∧

∧f−g F

, if i =−1,

coker(∧

g−a+1 Φ∗), if i = 0, and

Lg−ai+1 G

Φ∗(F)·Lg−ai G

, if 1 ≤ i.

is calculated in 7.11. The Schur moduleLpqG is described in 4.1 and 4.3.


2. NOTATION, CONVENTIONS, AND PRELIMINARY RESULTS.

There are three subsections: Ground rules, Grade and perfection, and Multilinear algebra.

2.A. Ground rules.

2.1. Unless otherwise noted,R is a commutative Noetherian ring and all functors are functorsof R-modules; that is,⊗, Hom,( )∗, Symi, Di , and

∧i mean⊗R, HomR, HomR( ,R), SymRi ,

DRi , and

∧iR, respectively.

2.2. A complexC : · · · → C2 → C1 → C0 → 0 of R-modules is calledacyclic if H j(C ) = 0for 1≤ j.

2.3. If a complexC has the form

0→ Cℓ → Cℓ−1 → ·· · → C1 → C0 → 0,

with C0 6= 0 andCℓ 6= 0, then we say that thelength ofC is ℓ and we write length(C ) = ℓ.

2.4. If A is a complex andn is an integer, thenA[n] is a new complex with[

A[n]]

j = An+ j .

2.5. If (A,a)θ−→ (B,b) is a map of complexes, then thetotal complex(or mapping cone)

of θ, denoted Tot(θ), is the complex(T, t) with T = A[−1]⊕B as a graded module. Thedifferential inT is given by

T j =A j−1⊕B j

t j−−−−→

A j−2⊕

B j−1

= T j−1,

with

t j =

[

a j−1 0θ j−1 −b j

]

.

2.6. If Φ is a matrix (or a homomorphism of freeR-modules), thenIr(Φ) is the ideal gener-ated by ther × r minors ofΦ (or any matrix representation ofΦ).

2.7. If S is a statement then

χ(S) =

{

1, if S is true,

0, if S is false.

6 ANDREW R. KUSTIN

2.B. Grade and perfection.

2.8. Thegradeof a proper idealI in a Noetherian ringR is the length of a maximalR-regularsequence inI . The unit idealR of R is regarded as an ideal of infinite grade.

2.9. Let M be a non-zero finitely generated module over a Noetherian ring Rand let ann(M)

be the annihilator ofM and pdRM be the projective dimension ofM. It is well-known that

gradeann(M) = min{ j | Ext jR(M,R) 6= 0};

therefore, it follows that

(2.9.1) gradeann(M)≤ pdRM.

If equality holds in (2.9.1), thenM is called aperfect R-module. Recall, for example, thatif R is Cohen-Macaulay andM is perfect, thenM is Cohen-Macaulay. (This is not the fullstory. For more information, see, for example, [3, Prop. 16.19] or [2, Thm. 2.1.5].)

The idealI in R is called aperfect idealif R/I is a perfectR-module.

2.10. Let M be a finitely generatedR-module. TheR-moduleM hasrank r if Mp is a freeRp-module of rankr for all associated primesp of AssR. If every non-zero-divisor inR isregular onM, thenM is calledtorsion-free. A proof of the following result may be found in[25, Prop. 1.25].

Proposition 2.10.1.Let M be a non-zero finitely generatedR-module with finite projectivedimension. Suppose thatI is a perfect ideal ofR with IM = 0. For each integerw, with1≤ w≤ pdRM, let Fw be the ideal ofR generated by

{x∈ R | pdRxMx < w}.

If w+ 1 ≤ gradeFw for all w with gradeI + 1 ≤ w ≤ pdRM, thenM is a torsion-freeR/Imodule. In particular, if pdRM ≤ gradeI , thenM is a torsion-free(R/I)-module. �

2.11. The following statement is well-known; see, for example, [19, Cor. 6.10]. It followsfrom the fact that the ifM is a perfect module, then the idealsFw for M (in the sense 2.10.1)and the annihilator ofM all have the same radical.

Proposition 2.11.1.LetA→Rbe a homomorphism of Noetherian rings andM be a non-zerofinitely generated perfectA-module. IfM⊗A R 6= 0, then

grade((annM)R) = pdAM−max{i | TorAi (M,R) 6= 0}. �

In our favorite applications of 2.11.1, we focus on the complex F⊗A R, whereF is aresolution ofM by projectiveA-modules. We are satisfied with an inequality; and therefore,there is no need for us to assume thatM ⊗A R 6= 0. Furthermore, in practice,A is usuallya polynomial ring over the ring of integers. We refer to the following statement as depth-sensitivity. Of course, these ideas were first worked out in [12, 27, 28, 17], and especially[18, Cor. 3.1].


Proposition 2.11.2.Let A→ R be a homomorphism of Noetherian rings,M be a non-zerofinitely generated perfectA-module, andF be a resolution ofM by projectiveA-moduleswith the length ofF equal to the projective dimension ofM. Then

H j(F⊗A R) = 0 for pdAM−grade((annM)R)+1≤ j. �

It is worth observing that if annM is replaced by any sub-ideal, then the inequality of2.11.2 continues to hold.

2.C. Multilinear algebra.

2.12. Our complexes are described in a coordinate-free manner. Let V be a free module offinite rankd overR. We make much use of the symmetric algebra Sym•V, the divided poweralgebraD•(V∗), and the exterior algebras

∧•V and∧•(V∗). We use the fact thatD•(V∗) is

a module over Sym•V and the fact that∧•V and

∧•(V∗) are modules over one another. Wealso use the fact that these module actions give rise to natural perfect pairings

ev : Symi V ⊗Di(V∗)→ R and ev :

∧i V ⊗∧i(V∗)→ R,

for each integeri. The duals

ev∗ : R→ Di(V∗)⊗Symi V and ev∗ : R→

∧i(V∗)⊗∧i V

of the above evaluation maps are completely independent of coordinates. It follows that, if{mℓ} is a basis for Symi V (or

∧i V) and{m∗ℓ} is the corresponding dual basis forDi(V∗) (or

∧i(V∗)), then the element

(2.12.1) ev∗(1) = ∑ℓ

m∗ℓ ⊗mℓ ∈ Di(V

∗)⊗Symi V (or∧i(V∗)⊗

∧i V)

is completely independent of coordinates. These elements will also be used extensively inour calculations.

2.12.2. We emphasize a special case of (2.12.1). IfωV∗ is a basis for∧d(V∗) andωV is

the corresponding dual basis for∧dV, then the elementωV∗ ⊗ωV is a canonical element of

∧d(V∗)⊗∧dV. This element is also used in our calculations.

The following facts about the interaction of the module structures of∧•V on

∧•(V∗) and∧•(V∗) on

∧•V are well known; see [4, section 1], [5, Appendix], and [22, section 1].

Proposition 2.13. Let V be a free module of rank d over a commutative Noetherian ring Rand let br ∈

∧r V, cp ∈∧pV, andαq ∈

∧q(V∗).

(a) If r = 1, then(br(αq))(cp) = br ∧ (αq(cp))+(−1)1+qαq(br ∧cp).

(b) If q = d, then(br(αq))(cp) = (−1)(d−r)(d−p)(cp(αq))(br).

(c) If p = d, then[br(αq)](cp) = br ∧αq(cp).

8 ANDREW R. KUSTIN

(d) If Ψ : V →V ′ is a homomorphism of free R-modules andδs+r ∈∧s+r(V ′∗), then

(∧sΨ∗)[((

∧r Ψ)(br))(δs+r)] = br [(∧s+r Ψ∗)(δs+r)]. �

The next result is an application of Proposition 2.13 and theideas of 2.12. The result isused in the proof of Observation 7.12.

Proposition 2.14. Let V be a free module of rank d over a commutative Noetherian ring Rand let br ∈

∧r V, cp ∈∧pV, andαq ∈

∧q(V∗). Then br ∧ (αq(cp)) is equal to a sum ofelements of the form b′r ′ ∧ (α′

q′(cp)) where

b′r ′ ∈∧r ′V, α′

q′ ∈∧q′(V∗), r ′−q′ = r −q, and q′ ≤ q+d− r − p.

Remark. The elementcp has not been changed; but an upper bound has been imposed onthe degree of the

∧•(V∗) contribution to the expression. Of course, the assertion isonlyinteresting whend < r + p

Proof. Let ev∗(1) = ∑ℓm∗ℓ ⊗mℓ ∈

∧d−p(V∗)⊗∧d−pV, as described in (2.12.1). Notice that

(2.14.1) ∑ℓ

m∗ℓ(mℓ∧cp) = cp.

To establish (2.14.1), it suffices to test the proposed equation after multiplying both sides onthe left by an arbitrary elementxd−p of

∧d−pF. The left side becomes

∑ℓ

xd−p∧m∗ℓ(mℓ∧cp) = ∑

ℓ

[xd−p(m∗ℓ)](mℓ∧cp) by 2.13.c

=(

∑ℓ

[xd−p(m∗ℓ)] ·mℓ

)

∧cp

= xd−p∧cp,

as desired. Apply (2.14.1) and 2.13.c to see that

br ∧αq(cp) = ∑ℓ

br ∧ (αq∧m∗ℓ)(mℓ∧cp) = ∑

ℓ

[br(αq∧m∗ℓ)](mℓ∧cp).

Eachmℓ is a sum of elements of the formv1∧· · ·∧vd−pwith vi ∈V. Apply 2.13.a numeroustimes to write

[br(αq∧m∗ℓ)](v1∧· · ·∧vd−p∧cp) =

{

±v1∧ [br(αq∧m∗ℓ)](v2∧· · ·∧vd−p∧cp)

±[(v1∧br)(αq∧m∗ℓ)](v2∧· · ·∧vd−p∧cp)

=

±v2∧v1∧ [br(αq∧m∗ℓ)](v3∧· · ·∧vd−p∧cp)

±v1∧ [(v2∧br)(αq∧m∗ℓ)](v3∧· · ·∧vd−p∧cp)

±v2∧ [(v1∧br)(αq∧m∗ℓ)](v3∧· · ·∧vd−p∧cp)

±[(v2∧v1∧br)(αq∧m∗ℓ)](v3∧· · ·∧vd−p∧cp)

= · · · .


We continue is this manner and expressbr ∧αq(cp) as a sum of elements of the form

x∧ [(x′∧br)(αq∧m∗ℓ)](cp),

wherex andx′ are homogeneous elements of∧•F and degx+degx′ = d− p. The proof is

complete because

deg[(x′∧br)(αq∧m∗ℓ)] = q+d− p−degx′− r ≤ q+d− p− r. �

3. THE CLASSICAL GENERALIZED EAGON-NORTHCOTT COMPLEXES.

We recall the classical generalized Eagon-Northcott complexes{C iΦ | i ∈ Z} which were

introduced at (1.0.1).

Definition 3.1. Let Rbe a commutative Noetherian ring,F andG be freeR-modules of rankf andg, respectively, withg ≤ f, Φ : G∗ → F be anR-module homomorphism, andi be aninteger.

(a) The complexC iΦ is

· · ·ηΦ−→

∧f−g−i−2F ⊗D2(G∗)

ηΦ−→

∧f−g−i−1F ⊗D1(G∗)

ηΦ−→

∧f−g−i F ⊗D0(G∗)(3.1.1)

∧g Φ

−−−→∧

f−i F ⊗∧

g G⊗Sym0GKosΦ−−−→

∧f−i+1F ⊗

∧g G⊗Sym1G

KosΦ−−−→ ·· · ,

with∧

f−g−i F ⊗D0(G∗) in positioni+1. In particular, if j is an integer, then the module(C i

Φ) j is

(3.1.2) (C iΦ) j =

{∧f− j F ⊗

∧g G⊗Symi− j G, if j ≤ i, and

∧f−g− j+1F ⊗D j−i−1(G∗), if i +1≤ j.

(b) The mapsηΦ and KosΦ may be found in Definition 5.2.a.

(c) The map∧aF ⊗D0(G∗)

∧g Φ

−−−→∧a+g F ⊗

∧g G⊗Sym0G is

fa 7→ fa∧ (∧

g Φ)(ωG∗)⊗ωG,

whereωG∗ ⊗ωG is the canonical element of∧

g(G∗)⊗∧

g G from (2.12.2).

Example 3.2. Adopt the notation of 3.1. We record the classical generalized Eagon North-cott complexesC i

Φ which have the form:

0→ (C iΦ)f−g+1 → (C i

Φ)f−g → ·· · → (C iΦ)2 → (C i

Φ)1 → (C iΦ)0 → 0.

Of course, these complexes have lengthf− g+1. The corresponding indicesi are given in(1.0.4):

C−1Φ : 0→

∧0F ⊗Df−g+1(G∗)ηΦ−→

∧1F ⊗Df−g(G∗)ηΦ−→ . . .

ηΦ−→

∧f−g+1F ⊗D0(G∗)

→ 0,

10 ANDREW R. KUSTIN

C 0Φ : 0→

∧0F ⊗Df−g(G∗)ηΦ−→

∧1F ⊗Df−g−1(G∗)ηΦ−→ . . .

ηΦ−→∧

f−g F ⊗D0(G∗)∧

g Φ−−−→

∧f F ⊗

∧g G⊗Sym0G→ 0,

C 1Φ : 0→

∧0F ⊗Df−g−1(G∗)ηΦ−→

∧1F ⊗Df−g−2(G∗)ηΦ−→ . . .

ηΦ−→∧

f−g−1F ⊗D0(G∗)∧

g Φ−−−→

∧f−1F ⊗

∧g G⊗Sym0G

KosΦ−−−→∧

f F ⊗∧

g G⊗Sym1G→ 0,

C 2Φ : 0→

∧0F ⊗Df−g−2(G∗)ηΦ−→

∧1F ⊗Df−g−3(G∗)ηΦ−→ . . .

ηΦ−→

∧f−g−2F ⊗D0(G∗)

∧g Φ

−−−→∧

f−2F ⊗∧


∧f−1F ⊗

∧g G⊗Sym1G


f F ⊗∧

g G⊗Sym2G→ 0,...

Cf−g−1Φ : 0→

∧0F ⊗D1(G∗)ηΦ−→

∧1F ⊗D0(G∗)∧gΦ−−−→

∧g+1F ⊗

∧g G⊗Sym0G


f−g−2F ⊗∧

g G⊗Sym1GKosΦ−−−→ . . .


f F ⊗∧

g G⊗Symf−g−1G→ 0,

Cf−g

Φ : 0→∧0(F∗)⊗D0(G∗)

∧g Φ

−−−→∧

g F ⊗∧


∧g+1F ⊗

∧g G⊗Sym1G

KosΦ−−−→ . . .KosΦ−−−→

∧f F ⊗

∧g G⊗Symf−g G→ 0, and

Cf−g+1Φ : 0→

∧g−1F ⊗

∧g G⊗Sym0G


g F ⊗∧

g G⊗Sym1GKosΦ−−−→ . . .


f F ⊗∧

g G⊗Symf−g+1G→ 0.

The mapsηΦ and KosΦ may be found in Definition 5.2.a.

4. SCHUR AND WEYL MODULES WHICH CORRESPOND TO HOOKS.

The modules which comprise the complexes{Ci,aΦ } are Schur and Weyl modules which

correspond to hooks. We recall some of the elementary properties of these modules.

4.1. LetV be a non-zero free module of rankd over the commutative Noetherian ringRandlet a andb be integers. Define theR-module homomorphisms

κab :

∧aV ⊗SymbV →∧a−1V ⊗Symb+1V and

ηab :

∧aV ⊗DbV →∧a+1V ⊗Db−1V


to be the compositions

∧aV ⊗SymbVev∗(1)−−−→(V∗⊗V)⊗

∧aV ⊗SymbV(4.1.1)rearrange−−−−−→(V∗⊗

∧aV)⊗ (V ⊗SymbV)

ModAct⊗mult−−−−−−−−→

∧a−1V ⊗Symb+1V and

∧aV ⊗DbVev∗(1)−−−→

∧aV ⊗ (V ⊗V∗)⊗DbV(4.1.2)regroup−−−−→(

∧aV ⊗V)⊗ (V∗⊗DbV)

mult⊗ModAct−−−−−−−−→

∧a+1V ⊗Db−1V,

respectively. The map ev∗(1) is discussed in (2.12.1), “mult” is multiplication in the sym-metric algebra or the exterior algebra, and “ModAct” is the module action of

∧•(V∗) on∧•V or Sym•(V

∗) on D•V. Define theR-modules

LabV = kerκa

b and KabV = kerηa

b.

In the future, we will often writeκ andη in place ofκab andηa

b.

4.2. The R-modulesLabV andKa

bV have been used by many authors in many contexts. Inparticular, they are studied extensively in [4]; although our indexing conventions are differentthan the conventions of [4]; that is,

the module we callLabV is calledLb

a+1V in [4].

4.3. The modulesLabV andKa

bV may also be thought of as the Schur modulesLλV and WeylmodulesKλV which correspond to certain hooksλ. We use the notation of Examples 2.1.3.hand 2.1.17.h in Weyman [30] to see that the module we callLa

bV is also the Schur moduleL(a+1,1b−1)V and the module we callKa

bV is also the Weyl moduleK(b+1,1a−1)V.

4.4. The complex

0→∧dV ⊗Symc−dV

κ−→

∧d−1V ⊗Symc−d+1Vκ−→(4.4.1)

· · ·κ−→

∧1V ⊗Symc−1Vκ−→

∧0V ⊗SymcV → 0,

which is a homogeneous strand of an acyclic Koszul complex, is split exact for all non-zerointegersc; hence,La

bV is a projectiveR-module. In fact,LabV is a freeR-module of rank

rankLabV =

(

d+b−1a+b

)(

a+b−1a

)

;

see [4, Prop. 2.5]. Similarly,KabV is a freeR-module of rank

rankKabV =

(

d+ba+b

)(

a+b−1b

)

.

12 ANDREW R. KUSTIN

The perfect pairing(∧aV ⊗SymbV

)

⊗(∧a(V∗)⊗Db(V

∗))

→ R,

induces a perfect pairing

(4.4.2) LabV ⊗Ka+1

b−1(V∗)→ R, provided(a,b) 6= (0,0) or (−1,1).

Each assertion in Observation 4.5 is obvious; but it is very convenient to have all of thesefacts gathered in one place.

Observation 4.5.Let R be a commutative Noetherian ring, V be a free R-module ofrank d,andℓ be an integer. Then the following modules are canonically isomorphic:

(a) L0ℓV = SymℓV,

(b) Ld−1ℓ V ∼=

∧dV ⊗Symℓ−1V, providedℓ+d 6= 1,(c) Ld

ℓV = 0, providedℓ+d 6= 0,(d) Lℓ

0V = 0, providedℓ 6= 0,(e) Lℓ

1V∼=

∧ℓ+1V, providedℓ 6=−1,(f) K0

ℓV = 0, providedℓ 6= 0,(g) K1

ℓV ∼= Dℓ+1V, providedℓ 6=−1,(h) Kd

ℓ V =∧dV ⊗DℓV, and

(i) Kℓ0V =

∧ℓV.

Proof. Assertions (a), (h), and (i) follow from the definitions, andassertions (b), (c), (d), and(e) are immediate consequences of split exact complex (4.4.1). If c is non-zero, then we mayapply Hom(−,R) to the split exact complex (4.4.1) to obtain the split exact complex

0→∧0(V∗)⊗Dc(V∗)

η−→

∧1(V∗)⊗Dc−1(V∗)η−→ . . .

η−→

∧d(V∗)⊗Dc−d(V∗)→ 0.

ReplaceV∗ with V in order to see that

(4.5.1) 0→∧0V ⊗DcV

η−→

∧1V ⊗Dc−1Vη−→ . . .

η−→

∧dV ⊗Dc−dV → 0

is also split exact. Assertions (f) and (g) are consequence of (4.5.1). �

5. THE COMPLEXESKΦ, AND LΦ ASSOCIATED TO A HOMOMORPHISMΦ.

The complexesK andL contain Weyl modulesKab(G

∗) and Schur modulesLabG, respec-

tively. The complexC i,aΦ , which is the focal point of the present paper, is obtained bycon-

catenating the complexesK→∧→ L; see Definition 7.2.

Data 5.1. Let Rbe a commutative Noetherian ring,F andG be freeR-modules of finite rankf andg, respectively, andΦ : G∗ → F be anR-module homomorphism.


Definition 5.2. Adopt Data 5.1.

(a) If r andq are integers, then define theR-module homomorphisms

ηΦ :∧r F ⊗Dq(G∗)→

∧r+1F ⊗Dq−1(G∗) and

KosΦ :∧r F ⊗SymqG→

∧r+1F ⊗Symq+1G

to be the compositions

∧r F ⊗Dq(G∗)1⊗ev∗(1)⊗1−−−−−−−→

∧r F ⊗G∗⊗G⊗Dq(G∗)

1⊗Φ⊗ModAct−−−−−−−−→

∧r F ⊗∧1F ⊗Dq−1(G∗)

mult⊗1−−−−→

∧r+1F ⊗Dq−1(G∗)

and

∧r F ⊗SymqG1⊗ev∗(1)⊗1−−−−−−−→

∧r F ⊗G∗⊗G⊗SymqG

1⊗Φ⊗mult−−−−−−→

∧r F ⊗∧1F ⊗Symq+1G

mult⊗1−−−−→

∧r+1F ⊗Symq+1G,

respectively, where ev∗(1) is described in (2.12.1), ModAct is the module action ofSym•G on D•(G∗) and mult is multiplication in the exterior algebra

∧•F or the sym-metric algebra Sym•G.

(b) If N andp are integers, then defineKN,pΦ to be the maps and modules

KN,pΦ : 0→

∧0F ⊗KpN(G

∗)ηΦ−→

∧1F ⊗KpN−1(G

∗)ηΦ−→ . . .

ηΦ−→∧N F ⊗Kp

0 (G∗)→ 0

with [KN,pΦ ] j =

∧N− j F ⊗Kpj (G

∗); and defineLN,pΦ to be the maps and modules

0→∧N+1F ⊗Lp

1GKosΦ−−−→

∧N+2F ⊗Lp2G

KosΦ−−−→ ·· ·KosΦ−−−→

∧f F ⊗Lp

f−NG→ 0,

with

(5.2.1) [LN,pΦ ] j =

{

0 if (p, j) = (0, f−N), and∧

f− j F ⊗Lpf−N− jG, otherwise,

where the maps are induced by the homomorphismsηΦ and KosΦ of (a) and the modulesKp

q (G∗) andLpqG are defined in (4.1.2) and (4.1.1).

Remarks 5.3. (a) The map

ηab :

∧a(G∗)⊗Db(G∗)→∧a+1(G∗)⊗Db−1(G∗)

of (4.1.2) is equal to the map

ηidG∗ :∧a(G∗)⊗Db(G∗)→

∧a+1(G∗)⊗Db−1(G∗)

of 5.2.a.

14 ANDREW R. KUSTIN

(b) The map

κab :

∧aG⊗SymaG→∧a−1G⊗Symb+1G

of (4.1.1) is related to the map KosidG∗ of 5.2.a by way of the following commutativediagram:

∧g−aG∗⊗SymbG⊗

∧g G

ModAct ∼=��

(−1)g−aKosidG∗//∧

g−a+1G∗⊗Symb+1G⊗∧

g G

ModAct∼=��∧aG⊗SymbG

κab

//∧a−1G⊗Symb+1G.

(c) The maps and modules ofKN,pΦ andLN,p

Φ form complexes because the diagrams

0→∧r F ⊗Kp

q (G∗)� � //

ηΦ��✤

✤

✤

∧r F ⊗∧p(G∗)⊗Dq(G∗)

ηidG∗//

ηΦ��

∧r F ⊗∧p+1(G∗)⊗Dq−1(G∗)

ηΦ��

0→∧r+1F ⊗Kp

q−1(G∗) �

�//∧r+1 F ⊗

∧p(G∗)⊗Dq−1(G∗)ηidG∗

//∧r+1F ⊗

∧p+1(G∗)⊗Dq−2(G∗)

and

0→∧r F ⊗Lp

qG� � //

KosΦ��✤

✤

✤

∧r F ⊗∧pG⊗SymqG

κ//

KosΦ��

∧r F ⊗∧p−1G⊗Symq+1G

KosΦ��

0→∧r+1F ⊗Lp

q+1G� � //

∧r+1F ⊗∧pG⊗Symq+1G κ

//∧r+1 F ⊗

∧p−1G⊗Symq+2G

commute.

6. THE COMPLEXESKΦ AND LΦ WHEN Φ IS A DIRECT SUM OF HOMOMORPHISMS.

In this section we assume that the homomorphismΦ : G∗ → F is a direct sum of homo-morphisms. In Proposition 6.4 we relate the complexesK

N,pΦ andLN,p

Φ of Definition 5.2.b tosimilar complexes built from smaller data. This result plays a prominent role in the proof ofthe acyclicity of the complexesC i,a

Φ ; see Theorem 8.4, which is the main result of the paper.

Data 6.1. Let Rbe a commutative Noetherian ring,F andG be freeR-modules of rankf andg, respectively, andΦ : G∗ → F be anR-module homomorphism. DecomposeF andG as

F = F ′⊕F ′′ and G= G′⊕G′′,

whereF ′, F ′′, G′ andG′′ are freeR-modules and rankF ′′ = rankG′′ = 1 and let

F∗ = F ′∗⊕F ′′∗ and G∗ = G′∗⊕G′′∗


be the corresponding decompositions ofF∗ andG∗. Assume that

Φ =

[

Φ′ 00 Φ′′

]

,

whereΦ′ : G′∗ → F ′ andΦ′′ : G′′∗ → F ′′ areR-module homomorphisms.

Notation 6.2. Adopt Data 6.1.

(a) LetA(Φ′′) andB(Φ′′) represent the complexes

A(Φ′′) : 0→ RKosΦ′′−−−→ F ′′⊗G′′ → 0

and

B(Φ′′) : 0→ G′′∗ Φ′′

−→ F ′′ → 0.

(b) Letπ′ : F →F ′ be the projection map which corresponds to the direct sum decompositionF = F ′⊕F ′′.

(c) If a, b, c, d, andeare integers, then let

incl† :∧aF ′⊗

∧bG⊗SymcG⊗∧d F ′′⊗SymeG′′ →

∧a+d F ⊗∧bG⊗Symc+eG

be theR-module homomorphism given by

(multiplication in∧•F)⊗ (the identity map in

∧•G)⊗ (multiplication inSym•G).

(d) If a, b, c, d, andeare integers, then let

quot† :∧a+d F ⊗

∧b+eG⊗SymcG→∧aF ′⊗

∧bG′⊗SymcG′⊗∧d F ′′⊗

∧eG′′


the projection map∧a+d F →∧aF ′⊗

∧d F ′′

induced byF = F ′⊕F ′′

⊗

the projection map∧b+eG→∧bG′⊗

∧eG′′

induced byG= G′⊕G′′

⊗

the quotient mapSym•G→ Sym•G′

induced byG/G′′ = G′

.

(e) If a, b, c, andd are integers, then let

incl‡ :∧aF ′⊗

∧b(G′∗)⊗Dc(G′∗)⊗∧d(G′′∗)→

∧aF ⊗∧b+d(G∗)⊗Dc(G∗)


(inclusion)⊗ (multiplication)⊗ (inclusion).

(f) If a, b, c, d, andeare integers, then let

quot‡ :∧a+d F ⊗

∧b(G∗)⊗Dc+e(G∗)→∧aF ′⊗

∧b(G′∗)⊗Dc(G∗)⊗∧d F ′′⊗De(G′′∗)

16 ANDREW R. KUSTIN


the projection map∧a+d F →∧a F ′⊗

∧d F ′′

induced byF = F ′⊕F ′′

⊗

(

the identity mapon

∧•(G∗)

)

⊗

Dc+e(G∗)ev∗(1)⊗1−−−−−→

De(G′′∗)⊗SymeG′′⊗Dc+e(G∗)

1⊗ModAct−−−−−−→ De(G′′∗)⊗Dc(G∗)

exchange−−−−−→ Dc(G∗)⊗De(G′′∗)

,

where ev∗(1) is described in (2.12.1) and ModAct is the module action of Sym•G onD•(G∗).

Proposition 6.4 asserts that (6.4.1) and (6.4.3) are short exact sequences of complexes. Ob-servation 6.3 considers the modules in (6.4.1) and (6.4.3) in the arbitrary positionj. Recallthe functionχ from 2.7.

Observation 6.3.Adopt Data6.1and Notation6.2.(a) If p,q, r are integers, then

0→

∧r F ′⊗LpqG

⊕∧r−1 F ′⊗Lpq−1G⊗F ′′⊗G′′

incl†−−→

∧rF⊗LpqG

quot†−−−→

∧r−1F ′⊗LpqG′⊗F ′′

⊕

χ((p,q) 6= (1,0))(∧r−1F ′⊗Lp−1

q G′⊗F ′′⊗G′′)

→ 0

is an exact sequence of R-modules.(b) If p,q, r are integers, then

0→

∧r F ′⊗Kpq (G′∗)

⊕∧r F ′⊗Kp−1

q (G′∗)⊗G′′∗

incl‡−−→

∧r F ⊗Kpq (G∗)

quot‡−−−→

χ(

(p,q) 6= (0,1))(∧r F ′⊗Kp

q−1(G∗)⊗G′′∗

)

⊕∧r−1F ′⊗Kpq (G∗)⊗F ′′

→ 0

is an exact sequence of R-modules.

Remark 6.3.3. If r = 1 andF ′ = 0, thenF = F ′′ has rank one and no harm occurs if we setF = F ′′ equal toR (that is, apply−⊗F∗). In this case, 6.3.a is

(6.3.4) 0→ Lpq−1G⊗G′′ incl†

−−→ LpqG

quot†−−−→ Lp

qG′⊕χ(

(p,q) 6= (1,0))(

Lp−1q G′⊗G′′

)

→ 0.

If r = 0, then 6.3.b is

(6.3.5) 0→Kp

q (G′∗)⊕

(

Kp−1q (G′∗)⊗G′′∗

)

incl‡−−→ Kp

q (G∗)quot‡−−−→ χ

(

(p,q) 6= (0,1))(

Kpq−1(G

∗)⊗G′′∗)

→ 0.

The exact sequences (6.3.4) and (6.3.5) are results from representation theory; see, for example, [30,2.3.1] or [14, Exercise 6.11]. The versions we give are stated explicitly, require no assumptions aboutcharacteristic, and are precisely the results that we use inthe proof of Lemma 8.1. In fact, our proofof Proposition 6.4 was created by starting with proofs of (6.3.4) and (6.3.5).

Proof. (a). If p< 0 orq< 0, then all of the modules in 6.3.a are zero. Ifq= 0, then 6.3.a is

0→ χ(p= 0)∧r F ′ incl†

−−→ χ(p= 0)∧rF

quot†−−−→ χ(p= 0)

∧r−1F ′⊗F ′′ → 0,


0

��

0

��

0

��

0→

∧r F ′⊗LpqG

⊕∧r−1 F ′⊗Lpq−1G⊗F ′′⊗G′′

incl†//

� _

��

∧r F ⊗LpqG

quot†//

� _

��

∧r−1 F ′⊗LpqG′⊗F ′′

⊕∧r−1 F ′⊗Lp−1

q G′⊗F ′′⊗G′′

→ 0

� _

��

0→

∧r F ′⊗∧p G⊗SqG⊕∧r−1 F ′⊗

∧p G⊗Sq−1G⊗F ′′⊗G′′

incl†//

κ

��

∧r F ⊗∧p G⊗SqG

quot†//

κ

��

∧r−1 F ′⊗∧p G′⊗SqG′⊗F ′′

⊕∧r−1 F ′⊗∧p−1G′⊗SqG′⊗F ′′⊗G′′

→ 0

κ

��

0→

∧r F ′⊗∧p−1G⊗Sq+1G

⊕∧r−1 F ′⊗∧p−1G⊗SqG⊗F ′′⊗G′′

incl†//

κ��

∧r F ⊗∧p−1 G⊗Sq+1G

quot†//

κ

��

∧r−1 F ′⊗∧p−1G′⊗Sq+1G′⊗F ′′

⊕∧r−1 F ′⊗∧p−2 G′⊗Sq+1G′⊗F ′′⊗G′′

→ 0

κ��

0→

∧r F ′⊗Lp−2q+2G

⊕∧r−1 F ′⊗Lp−2

q+1G⊗F ′′⊗G′′

incl†//

��

∧r F ⊗Lp−2q+2G

quot†//

��

∧r−1 F ′⊗Lp−2q+2G′⊗F ′′

⊕∧r−1 F ′⊗Lp−3

q+2G′⊗F ′′⊗G′′

→ 0

��

0 0 0

FIGURE 6.3.1. This picture is a commutative diagram which is used intheproof of Observation 6.3.a. The middle two rows are exact. The columns areexact provided 2≤ p+q. We use “S” as an abbreviation for “Sym”.

which is exact for allp. (Notice that we used the factorχ((p,q) 6= (0,0)) in our proof that6.3.a is exact for allp whenq= 0.) If p= 0, then 6.3.a is

0→

∧r F ′⊗Symq G⊕∧r−1F ′⊗Symq−1 G⊗F′′⊗G′′

incl†−−→

∧r F ⊗SymqGquot†−−−→

∧r−1F ′⊗SymqG′⊗F ′′ → 0,

which is exact for allq. Henceforth, we assume 2≤ p+q. Observe that Figure 6.3.1 is acommutative diagram; each column is split exact; and the middle two rows are split exact.The assertion follows from the snake lemma and the fact that the bottom incl† in Figure 6.3.1is automatically an injection.

(b). If p< 0 orq< 0, then all of the modules in 6.3.b are zero. Ifq= 0, then 6.3.b is

0→

∧r F ′⊗∧p(G′∗)⊕∧r F ′⊗

∧p−1(G′∗)⊗G′′∗

incl‡−−→

∧r F ⊗∧p(G∗)

quot‡−−−→

∧r−1F ′⊗∧p(G∗)⊗F ′′ → 0,

18 ANDREW R. KUSTIN

0

��

0

��

0

��

0→

∧r F ′⊗Kpq (G′∗)

⊕∧r F ′⊗Kp−1

q (G′∗)⊗G′′∗

incl‡//

� _

��


quot‡//

� _

��

∧r F ′⊗Kpq−1(G

∗)⊗G′′∗

⊕∧r−1 F ′⊗Kpq (G∗)⊗F ′′

→ 0

� _

��

0→

∧r F ′⊗∧p(G′∗)⊗Dq(G′∗)

⊕∧r F ′⊗∧p−1(G′∗)⊗Dq(G′∗)⊗G′′∗

incl‡//

ηidG∗

��

∧r F ⊗∧p(G∗)⊗Dq(G∗)

quot‡//

ηidG∗

��

∧r F ′⊗∧p(G∗)⊗Dq−1(G∗)⊗G′′∗

⊕∧r−1 F ′⊗∧p(G∗)⊗Dq(G∗)⊗F ′′

→ 0

ηidG∗

��

0→

∧r F ′⊗∧p+1(G′∗)⊗Dq−1(G′∗)

⊕∧r F ′⊗∧p(G′∗)⊗Dq−1(G′∗)⊗G′′∗

incl‡//

ηidG∗

��

∧r F ⊗∧p+1(G∗)⊗Dq−1(G∗)

quot‡//

ηidG∗

��

∧r F ′⊗∧p+1(G∗)⊗Dq−2(G∗)⊗G′′∗

⊕∧r−1 F ′⊗∧p+1(G∗)⊗Dq−1(G∗)⊗F ′′

→ 0

ηidG∗

��

0→

∧r F ′⊗Kp+2q−2 (G

′∗)

⊕∧r F ′⊗Kp+1

q−2 (G′∗)⊗G′′∗

incl‡//

��

∧r F ⊗Kp+2q−2 (G

∗)quot‡

//

��

∧r F ′⊗Kp+2q−3 (G

∗)⊗G′′∗

⊕∧r−1 F ′⊗Kp+2

q−2 (G∗)⊗F ′′

→ 0

��

0 0 0

FIGURE 6.3.2. This picture is a commutative diagram which is used intheproof of Observation 6.3.b. The middle two rows are exact. The columns areexact provided 2≤ p+q.

which is exact for all integersp. If p= 0, then 6.3.b is

0→ χ(q= 0)∧r F ′ incl‡

−−→ χ(q= 0)∧r F

quot‡−−−→ χ(q= 0)

∧r−1F ′⊗F ′′ → 0,

which is exact for all integersq. Henceforth, we assume 2≤ p+q. Observe that (6.3.2) isa commutative diagram; each column is split exact; and the middle two rows are split exact.Once again, the assertion follows from the snake lemma and the fact that the bottom incl‡ inFigure 6.3.2 is automatically an injection. �

The complexesLN,pΦ andKN,p

Φ may be found in Definition 5.2.b; the complexesA(Φ′′)

andB(Φ′′) are defined in Notation 6.2; and some comments about the totalcomplex (ormapping cone) of a map of complexes is in 2.5.

Proposition 6.4. Adopt Data6.1and Notation6.2. Let p and N be integers. The followingstatements hold.


(a) Assume that p6= 0. Then there is a canonical short exact sequence of complexes

0→ Tot(

LN,pπ′◦Φ ⊗A(Φ′′)

)

incl†−−→ L

N,pΦ

quot†−−−→

LN−1,pΦ′ ⊗F ′′

⊕

LN−1,p−1Φ′ ⊗F ′′⊗G′′

→ 0.(6.4.1)

In particular, if Φ′′ is an isomorphism, thenquot† induces an isomorphism

(6.4.2) Hj(LN−1,pΦ′ )⊕H j(L

N−1,p−1Φ′ )∼= H j(L

N,pΦ ),

for all j.(b) Assume that p6= 0. Then there is a canonical short exact sequence of complexes

0→K

N,pΦ′

⊕

(KN,p−1Φ′ ⊗G′′∗)

incl‡−−→K

N,pΦ

quot‡−−−→ Tot

(

KN−1,pπ′◦Φ ⊗B(Φ′′)

)

→ 0.(6.4.3)

In particular, if Φ′′ is an isomorphism, thenincl‡ induces an isomorphism

(6.4.4) Hj(KN,pΦ′ )⊕H j(K

N,p−1Φ′ )∼= H j(K

N,pΦ ),

for all j.

Remark 6.4.5.The conclusion of Proposition 6.4.b does not hold whenp= 0 and 1≤N≤ f.In this case,KN,p

Φ′ andKN,pΦ are 0→

∧N F ′ → 0 and 0→∧N F → 0, respectively, with each

non-zero module in position zero, andKN,p−1Φ′ is the zero complex. Thus, (6.4.4) does not

hold.The conclusion of Proposition 6.4.a also does not hold whenp= 0. In particular,LN,0

Φ isthe complex

0→∧N+1F ⊗Sym1G

KosΦ−−−→∧N+2F ⊗Sym2G

KosΦ−−−→ . . .KosΦ−−−→

∧f F ⊗Symf−N G→ 0.

In the typical situation

Hf−N−1(LN,0Φ ) =

∧N F, Hf−N−1(LN−1,0Φ′ ) =

∧N−1F ′,

and (6.4.2) fails to hold.

Proof. (a). There is nothing to prove ifp≤ 0; so we assume that 1≤ p. Notice that (6.4.1)in position j is

0→

∧f− j F ′⊗Lpf− j−NG

⊕∧f− j−1 F ′⊗Lp

f− j−N−1G⊗F ′′⊗G′′

incl†−−−→

∧f− jF ⊗Lpf− j−N

Gquot†−−−→

∧f− j−1 F ′⊗Lpf− j−NG′⊗F ′′

⊕

χ((p, j) 6= (1,f−N))∧f− j−1 F ′⊗Lp−1

f− j−NG′⊗F ′′⊗G′′

→ 0,

which is 6.3.a withr replaced byf− j andq replaced byf− j −N; hence, each row of (6.4.1)is a short exact sequence. To see that that (6.4.1) is a map of complexes, one verifies thatFigure 6.4.6 is a commutative diagram, and this is straightforward. Thus, (6.4.1) is a shortexact sequence of complexes.

20 ANDREW R. KUSTIN

∧r F ′⊗LpqG

⊕∧r−1F ′⊗Lpq−1G⊗F ′′⊗G′′

incl†//

[

Kosπ′◦Φ 0KosΦ′′ −Kosπ′◦Φ

]

��

∧r F ⊗LpqG

quot†//

KosΦ

��

∧r−1 F ′⊗LpqG′⊗F ′′

⊕∧r−1F ′⊗Lp−1

q G′⊗F ′′⊗G′′

[

−KosΦ′ 00 −KosΦ′

]

��∧r+1 F ′⊗Lpq+1G

⊕∧r F ′⊗LpqG⊗F ′′⊗G′′

incl†//∧r+1F ⊗Lp

q+1Gquot†

//

∧r F ′⊗Lpq+1G

′⊗F ′′

⊕∧r F ′⊗Lp−1

q+1G′⊗F ′′⊗G′′

FIGURE 6.4.6. In the proof of Proposition 6.4.a one verifies that this diagram(with p andq both positive) commutes in order to see that that (6.4.1) is amapof complexes.

∧r F ′⊗Kpq (G′∗)

⊕∧r F ′⊗Kp−1

q (G′∗)⊗G′′∗

incl‡//

[

ηπ′◦Φ 00 ηπ′◦Φ

]

��


quot‡//

ηΦ

��

∧r F ′⊗Kpq−1(G

∗)⊗G′′∗

⊕∧r−1 F ′⊗Kpq (G∗)⊗F ′′

[ηΦ′ 0Φ′′ −ηΦ′

]

��∧r+1 F ′⊗Kpq−1(G

∗)

⊕∧r+1F ′⊗Kp−1

q−1 (G∗)⊗F ′′⊗G′′∗

incl‡//∧r+1F ⊗Kp

q−1(G∗)

quot‡//

∧r+1F ′⊗Kpq−2(G

∗)⊗G′′∗

⊕∧r F ′⊗Kpq−1(G

∗)⊗F ′′

FIGURE 6.4.7. In the proof of Proposition 6.4.b one verifies that this diagramcommutes in order to see that that (6.4.3) is a map of complexes.

The assertion (6.4.2) is now obvious. Indeed, the hypotheses thatΦ′′ is an isomorphism

ensures that the complex Tot(

LN,pπ′◦Φ ⊗A(Φ′′)

)

on the left side of (6.4.1) has homology zero

and the long exact sequence of homology associated to the short exact sequence of complexes(6.4.1) yields (6.4.2).

(b). Assume that 1≤ p. Notice that (6.4.3) in positionj is

0→

∧N− j F ′⊗Kpj (G

′∗)

⊕∧N− j F ′⊗Kp−1

j (G′∗)⊗G′′∗

incl‡−−→

∧N− j F ⊗Kpj (G

∗)quot‡−−−→

∧N− j F ′⊗Kpj−1(G

∗)⊗G′′∗

⊕∧N− j−1F ′⊗Kpj (G

∗)⊗F ′′→ 0,

which is 6.3.b withr replaced byN− j andq replaced byj. The parameterp is is not zeroby hypothesis; hence, Observation 6.3.b ensures that each row of (6.4.3) is a short exactsequence. To see that that (6.4.3) is a map of complexes, one verifies that Figure 6.4.7 is acommutative diagram, and this is straightforward. Thus, (6.4.3) is a short exact sequence ofcomplexes. The assertion (6.4.4) is now obvious, as in the proof of (6.4.2). �


7. THE DEFINITION AND ELEMENTARY PROPERTIES OF THE COMPLEXESCi,aΦ .

The maps and modules ofCi,aΦ are introduced in 7.2. It is shown in 7.4 that eachC

i,aΦ

is a complex. Examples are given in 7.6 and 7.7. The relationship between the classicalgeneralized Eagon-Northcott complexes{C i

Φ} and the complexes{Ci,aΦ } is examined in 7.8.

The duality in the family{Ci,aΦ } is studied in 7.9. Information about the length ofC

i,aΦ is

recorded in 7.10. The zero-th homology ofCi,aΦ , for −1≤ i, may be found in 7.11. The fact

thatIg(Φ) annihilates H0(Ci,aΦ ), for −1≤ i, is established in 7.12.

Data 7.1. Let Rbe a commutative Noetherian ring,F andG be freeR-modules of rankf andg, respectively, withg≤ f, Φ : G∗ → F be anR-module homomorphism.

Definition 7.2. Adopt Data 7.1. Recall the complexesKN,pΦ andLN,p

Φ of Definition 5.2.b andRemark 5.3.c and the homomorphismκ of 4.1. Leti anda be integers with 1≤ a≤ g. Definethe maps and modules(C i,a

Φ ,d) to be

(7.2.1) 0→Kf−g−i−1,aΦ [−i −2]

d−→

∧f−g+a−i−1F

d−→ L

f−i−1,g−aΦ → 0,

with [C i,aΦ ]i+1 =

∧ f−g+a−i−1 F. The differentials

[Ci,aΦ ]i+2

di+2−−→ [C

i,aΦ ]i+1

di+i−−→ [C

i,aΦ ]i

are

[Ci,aΦ ]i+2 = [K

f−g−i−1,aΦ ]0 =

∧f−g−i−1F ⊗Ka

0(G∗) =

∧f−g−i−1F ⊗

∧a(G∗)

1⊗∧aΦ

−−−−→∧

f−g−i−1F ⊗∧aF

mult−−→

∧f−g+a−i−1F = [C

i,aΦ ]i+1

and

[Ci,aΦ ]i+1 =

∧f−g+a−i−1F

1⊗ev∗(1)−−−−−→

∧f−g+a−i−1F ⊗

∧g+1−a(G∗)⊗

∧g+1−a G

1⊗∧

g+1−a Φ⊗κ−−−−−−−−−→

∧f−g+a−i−1F ⊗

∧g+1−aF ⊗Lg−a

1 G

mult⊗1−−−−→

∧f−i F ⊗Lg−a

1 G= [Lf−i−1,g−aΦ ]i = [C i,a

Φ ]i .

Remark 7.3. We give two other descriptions of the modules inCi,aΦ :

(7.3.1) [Ci,aΦ ] j =

∧f− j F ⊗Lg−a

i+1− jG, if j ≤ i,∧

f−g+a−i−1F, if j = i +1, and∧

f−g+1− j F ⊗Kaj−i−2(G

∗), if i +2≤ j,

andCi,aΦ looks like

· · ·ηΦ−→


1(G∗)

ηΦ−→


0(G∗)

∧a Φ−−−→

∧f−g+a−i−1F(7.3.2)

∧g+1−a Φ

−−−−−→∧

f−i F ⊗Lg−a1 G


f−i+1F ⊗Lg−a2 G

KosΦ−−−→ ·· · ,

22 ANDREW R. KUSTIN

with∧

f−g+a−i−1F in positioni +1.

Observation 7.4.Adopt Data7.1. Let and i and a be integers with1≤ a≤ g. Then the mapsand modules(C i,a

Φ ,d) of Definition7.2 form a complex.

Proof. One obtainsC i,aΦ by pasting together two well-known complexes; hence it suffices to

show that

(C i,aΦ )i+3

di+3−−→ (C i,a

Φ )i+2di+2−−→ (C i,a

Φ )i+1di+1−−→ (C i,a

Φ )idi−→ (C i,a

Φ )i−1

is a complex; furthermore, by the duality of Observation 7.9, it suffices to show that

(C i,aΦ )i+2

di+2−−→ (C i,a

Φ )i+1di+1−−→ (C i,a

Φ )idi−→ (C i,a

Φ )i−1

is a complex. Takef ∈∧

f−g+a−i−1F = (Ci,aΦ )i+1. We compute

(di ◦di+1)( f ) = di

(

∑ℓ f ∧ (∧

g+1−a Φ)(m∗ℓ)⊗κ(mℓ) ∈

∧f−i F ⊗Lg−a

1 G= (Ci,aΦ )i

)

= ∑ℓ∑ℓ′ f ∧ (∧

g+1−a Φ)(m∗ℓ)∧Φ(n∗ℓ′)⊗κ(mℓ) · (1⊗nℓ′)

∈∧

f−i+1F ⊗Lg−a2 G= (C

i,aΦ )i−1

= ∑ℓ∑ℓ′ ∑ℓ′′ f ∧ (∧

g+2−a Φ)(m∗ℓ ∧n∗ℓ′)⊗n∗ℓ′′(mℓ)⊗nℓ′′ ·nℓ′

= ∑L ∑ℓ′ ∑ℓ′′ f ∧ (∧

g+2−a Φ)(M∗L)⊗n∗ℓ′′(n

∗ℓ′(M))⊗nℓ′′ ·nℓ′

= 0,

where

ev∗(1) = ∑ℓm∗ℓ ⊗mℓ ∈

∧g+1−aG∗⊗

∧g+1−aG,

ev∗(1) = ∑ℓ′ n∗ℓ′ ⊗nℓ′ = ∑ℓ′′ n

∗ℓ′′ ⊗nℓ′′ ∈ G∗⊗G, and

ev∗(1) = ∑L M∗L ⊗ML ∈

∧g+2−aG∗⊗

∧g+2−aG

are the canonical elements of (2.12.1). One easily verifies that

∑ℓ∑ℓ′ m∗ℓ ∧n∗ℓ′ ⊗mℓ⊗nℓ′ = ∑L ∑ℓ′ M

∗L⊗n∗ℓ′(ML)⊗nℓ′ ∈

∧g+2−aG∗⊗

∧g+1−aG⊗G.

(Merely evaluate both sides at a typical element of∧

g+2−aG⊗∧

g+1−aG∗⊗G∗.) It is obvi-ous that

∑ℓ′ ∑ℓ′′ n∗ℓ′ ∧n∗ℓ′′ ⊗nℓ′ ·nℓ′′ = 0∈

∧2G∗⊗Sym2G.

Take f ∈∧

f−g−i−1F andγ ∈∧a(G∗) = Ka

0(G∗), see 4.5.i. So,f ⊗ γ is in (C i,a

Φ )i+2. Wecompute

(di+1◦di+2)( f ⊗ γ) = di+1

(

f ∧ (∧a Φ)(γ) ∈

∧f−g+a−i−1F = (C i,a

Φ )i+1

)

= ∑ℓ f ∧ (∧aΦ)(γ)∧ (

∧g−a+1 Φ)(m∗

ℓ)⊗κ(mℓ)

∈∧

f−i F ⊗Lg−a1 G= (C

i,aΦ )i

= ∑ℓ f ∧ (∧

g+1Φ)(γ∧m∗ℓ)⊗κ(mℓ)


= 0. �

Remark 7.5. If a is equal to 0 org+1, then it is possible to construct a complexCi,aΦ using

the recipe of Remark 7.3. These complexes are

Ci,0Φ : 0→

∧f−g−i−1F

identity map−−−−−−−→

∧f−g−i−1F → 0,

with the non-zero modules appearing in positionsi +2 andi +1; and

Ci,g+1Φ : 0→

∧f−i F → 0,

with the non-zero module appearing in positioni +1.

Example 7.6. Adopt Data 7.1 with(g, f) = (5,9). We record the complexesC i,aΦ of Defini-

tion 7.2 and Observation 7.4 which have the form:

0→ (Ci,aΦ )5 → (C

i,aΦ )4 → (C

i,aΦ )3 → (C

i,aΦ )2 → (C

i,aΦ )1 → (C

i,aΦ )0 → 0.

Of course, these complexes have lengthf− g+1= 5.

C−1,1Φ : 0→

∧0F⊗K14(G

∗)ηΦ−→∧1F⊗K1

3(G∗)

ηΦ−→∧2F⊗K12(G

∗)ηΦ−→∧3F⊗K1

1(G∗)

ηΦ−→∧4F⊗K10(G

∗)

∧1Φ−−−→∧5F→0

C0,1Φ : 0→

∧0F⊗K13(G

∗)ηΦ−→∧1F⊗K1

2(G∗)

ηΦ−→∧2F⊗K11(G

∗)ηΦ−→∧3F⊗K1

0(G∗)

∧1Φ−−−→∧4F

∧5Φ−−−→∧9F⊗L4

1G→0

C1,1Φ : 0→

∧0F⊗K12(G

∗)ηΦ−→∧1F⊗K1

1(G∗)

ηΦ−→∧2F⊗K10(G

∗)

∧1 Φ−−−→∧3F

∧5 Φ−−−→∧8F⊗L4

1GKosΦ−−−→∧9F⊗L4

2G→0

C2,1Φ : 0→

∧0F⊗K11(G

∗)ηΦ−→∧1F⊗K1

0(G∗)

∧1 Φ−−−→∧2F

∧5 Φ−−−→∧7F⊗L4



3G→0

C3,1Φ : 0→

∧0F⊗K10(G

∗)

∧1 Φ−−−→∧1F

∧5 Φ−−−→∧6F⊗L4




4G→0

C4,1Φ : 0→

∧0F

∧5Φ−−−→∧5F⊗L4





5G→0

C5,1Φ : 0→

∧4F⊗L41G

KosΦ−−−→∧5F⊗L42G




KosΦ−−−→∧9F⊗L46G→0

C−1,2Φ : 0→

∧0F⊗K24(G

∗)ηΦ−→∧1F⊗K2

3(G∗)

ηΦ−→∧2F⊗K22(G

∗)ηΦ−→∧3F⊗K2

1(G∗)

ηΦ−→∧4F⊗K20(G

∗)

∧2Φ−−−→∧6F→0

C0,2Φ : 0→

∧0F⊗K23(G

∗)ηΦ−→∧1F⊗K2

2(G∗)

ηΦ−→∧2F⊗K21(G

∗)ηΦ−→∧3F⊗K2

0(G∗)

∧2Φ−−−→∧5F

∧4Φ−−−→∧9F⊗L3

1G→0

C1,2Φ : 0→

∧0F⊗K22(G

∗)ηΦ−→∧1F⊗K2

1(G∗)

ηΦ−→∧2F⊗K20(G

∗)

∧2 Φ−−−→∧4F

∧4 Φ−−−→∧8F⊗L3


2G→0

C2,2Φ : 0→

∧0F⊗K21(G

∗)ηΦ−→∧1F⊗K2

0(G∗)

∧2 Φ−−−→∧3F

∧4 Φ−−−→∧7F⊗L3



3G→0

C3,2Φ : 0→

∧0F⊗K20(G

∗)

∧2 Φ−−−→∧2F

∧4 Φ−−−→∧6F⊗L3




4G→0

C4,2Φ : 0→

∧1F

∧4Φ−−−→∧5F⊗L3





5G→0

C−1,3Φ : 0→

∧0F⊗K34(G

∗)ηΦ−→∧1F⊗K3

3(G∗)

ηΦ−→∧2F⊗K32(G

∗)ηΦ−→∧3F⊗K3

1(G∗)

ηΦ−→∧4F⊗K30(G

∗)

∧3Φ−−−→∧7F→0

C0,3Φ : 0→

∧0F⊗K33(G

∗)ηΦ−→∧1F⊗K3

2(G∗)

ηΦ−→∧2F⊗K31(G

∗)ηΦ−→∧3F⊗K3

0(G∗)

∧3Φ−−−→∧6F

∧3Φ−−−→∧9F⊗L2

1G→0

C1,3Φ : 0→

∧0F⊗K32(G

∗)ηΦ−→∧1F⊗K3

1(G∗)

ηΦ−→∧2F⊗K30(G

∗)

∧3 Φ−−−→∧5F

∧3 Φ−−−→∧8F⊗L2


2G→0

24 ANDREW R. KUSTIN

C2,3Φ : 0→

∧0F⊗K31(G

∗)ηΦ−→∧1F⊗K3

0(G∗)

∧3 Φ−−−→∧4F

∧3 Φ−−−→∧7F⊗L2



3G→0

C3,3Φ : 0→

∧0F⊗K30(G

∗)

∧3 Φ−−−→∧3F

∧3 Φ−−−→∧6F⊗L2




4G→0

C4,3Φ : 0→

∧2F

∧3Φ−−−→∧5F⊗L2





5G→0

C−1,4Φ : 0→

∧0F⊗K44(G

∗)ηΦ−→∧1F⊗K4

3(G∗)

ηΦ−→∧2F⊗K42(G

∗)ηΦ−→∧3F⊗K4

1(G∗)

ηΦ−→∧4F⊗K40(G

∗)

∧4Φ−−−→∧8F→0

C0,4Φ : 0→

∧0F⊗K43(G

∗)ηΦ−→∧1F⊗K4

2(G∗)

ηΦ−→∧2F⊗K41(G

∗)ηΦ−→∧3F⊗K4

0(G∗)

∧4Φ−−−→∧7F

∧2Φ−−−→∧9F⊗L1

1G→0

C1,4Φ : 0→

∧0F⊗K42(G

∗)ηΦ−→∧1F⊗K4

1(G∗)

ηΦ−→∧2F⊗K40(G

∗)

∧4 Φ−−−→∧6F

∧2 Φ−−−→∧8F⊗L1


2G→0

C2,4Φ : 0→

∧0F⊗K41(G

∗)ηΦ−→∧1F⊗K4

0(G∗)

∧4 Φ−−−→∧5F

∧2 Φ−−−→∧7F⊗L1



3G→0

C3,4Φ : 0→

∧0F⊗K40(G

∗)

∧4 Φ−−−→∧4F

∧2 Φ−−−→∧6F⊗L1




4G→0

C4,4Φ : 0→

∧3F

∧2Φ−−−→∧5F⊗L1





5G→0

C−2,5Φ : 0→

∧1F⊗K55(G

∗)ηΦ−→∧2F⊗K5

4(G∗)

ηΦ−→∧3F⊗K53(G

∗)ηΦ−→∧4F⊗K5

2(G∗)

ηΦ−→∧5F⊗K51(G

∗)ηΦ−→∧6F⊗K5

0(G∗)→0

C−1,5Φ : 0→

∧0F⊗K54(G

∗)ηΦ−→∧1F⊗K5

3(G∗)

ηΦ−→∧2F⊗K52(G

∗)ηΦ−→∧3F⊗K5

1(G∗)

ηΦ−→∧4F⊗K50(G

∗)

∧5Φ−−−→∧9F→0

C0,5Φ : 0→

∧0F⊗K53(G

∗)ηΦ−→∧1F⊗K5

2(G∗)

ηΦ−→∧2F⊗K51(G

∗)ηΦ−→∧3F⊗K5

0(G∗)

∧5Φ−−−→∧8F

∧1Φ−−−→∧9F⊗L0

1G→0

C1,5Φ : 0→

∧0F⊗K52(G

∗)ηΦ−→∧1F⊗K5

1(G∗)

ηΦ−→∧2F⊗K50(G

∗)

∧5 Φ−−−→∧7F

∧1 Φ−−−→∧8F⊗L0


2G→0

C2,5Φ : 0→

∧0F⊗K51(G

∗)ηΦ−→∧1F⊗K5

0(G∗)

∧5 Φ−−−→∧6F

∧1 Φ−−−→∧7F⊗L0



3G→0

C3,5Φ : 0→

∧0F⊗K50(G

∗)

∧5 Φ−−−→∧5F

∧1 Φ−−−→∧6F⊗L0




4G→0

C4,5Φ : 0→

∧4F

∧1Φ−−−→∧5F⊗L0





5G→0

Example 7.7. If f= g+1 andi = 0, thenC0,aΦ ⊗

∧f F∗ is

0→∧0F ⊗

∧f F∗⊗Ka

0(G∗)

∧aΦ−−−→

∧aF ⊗∧

f F∗∧

g+1−a Φ−−−−−→

∧f F ⊗

∧f F∗⊗Lg−a

1 G→ 0,

which is naturally isomorphic to

(7.7.1) 0→∧

f F∗⊗∧a(G∗)

d2−→∧

f−aF∗ d1−→∧

f−aG→ 0,

with d2(ωF∗ ⊗ γa) = [(∧a Φ)(γa)](ωF∗) andd1(φf−a) = (

∧f−aΦ∗)(φf−a), for ωF∗ ∈

∧f F∗,

γa ∈∧a(G∗), andφf−a ∈

∧f−aF. In particular, iff= 4, g= 3, a= 2, andΦ = (Φi, j) is given

by a 4×3 matrix, then (7.7.1) is

0→ R3 d2−→ R6 d1−→ R3 → 0

with

d1 =

∆(1,2;1,2) ∆(1,3;1,2) ∆(1,4;1,2) ∆(3,4;1,2) ∆(2,4;1,2) ∆(2,3;1,2)∆(1,2;1,3) ∆(1,3;1,3) ∆(1,4;1,3) ∆(3,4;1,3) ∆(2,4;1,3) ∆(2,3;1,3)∆(1,2;2,3) ∆(1,3;2,3) ∆(1,4;2,3) ∆(3,4;2,3) ∆(2,4;2,3) ∆(2,3;2,3)


and

d2 =

∆(3,4;1,2) ∆(3,4;1,3) ∆(3,4;2,3)−∆(2,4;1,2) −∆(2,4;1,3) −∆(2,4;2,3)∆(2,3;1,2) ∆(2,3;1,3) ∆(2,3;2,3)∆(1,2;1,2) ∆(1,2;1,3) ∆(1,2;2,3)−∆(1,3;1,2) −∆(1,3;1,3) −∆(1,3;2,3)∆(1,4;1,2) ∆(1,4;1,3) ∆(1,4;2,3)

,

where

∆(i, j;k, ℓ) = det

[

Φi,k Φi,ℓΦ j ,k Φ j ,ℓ

]

.

Observation 7.8.Adopt Data7.1. Recall the complexes{C iΦ} of Definition3.1and the com-

plexes{Ci,aΦ } of Definition7.2and Observation7.4. Then, for each integer i, the complexes

C iΦ, C

i,1Φ , and C

i−1,gΦ ⊗

∧g G

are canonically isomorphic.

Proof. Use the formulas (3.1.2) and (7.3.1), Observation 4.5, and the fact that∧

g G⊗∧

g G∗

is canonically isomorphic toR to see that the modules

(C iΦ) j , (C i,1

Φ ) j , and (C i−1,gΦ ) j ⊗

∧g G

are canonically isomorphic for allj. These canonical isomorphisms induce the requiredcanonical isomorphisms of complexes. �

Observation 7.9. Adopt Data7.1. Let and i and a be integers with1 ≤ a ≤ g. Then thecomplexes

Ci,aΦ ⊗

∧f(F∗) and (Cf−g−i−1,g+1−a

Φ )∗[−(f− g+1)]

of Definition7.2and Observation7.4are canonically isomorphic.

Remark. The symbol “[−(f− g+1)]” refers to a shift in homological degree, see 2.4. Inparticular, for each integerj,

((Cf−g−i−1,g+1−aΦ )∗[−(f− g+1)]) j = ((Cf−g−i−1,g+1−a

Φ )f−g+1− j)∗.

Proof. Use the formula (7.3.1), Observation 4.5, and the fact that∧ℓF∗ is canonically iso-

morphic to∧

f−ℓF ⊗∧

f(F∗), for all integersℓ to see that the modules

(Ci,aΦ ) j ⊗

∧f(F∗) and ((Cf−g−i−1,g+1−a

Φ )f−g+1− j)∗

are canonically isomorphic for allj. Indeed,

((Cf−g−i−1,g+1−aΦ )f−g+1− j)

∗

=

(∧

g−1+ j F ⊗La−1j−i−1G)∗ if i +2≤ j,

(∧

g+i+1−aF)∗, if i +1= j, and

(∧ j F ⊗Kg+1−a

i− j (G∗))∗, if j ≤ i

26 ANDREW R. KUSTIN

∼=

∧g−1+ j(F∗)⊗Ka

j−i−2(G∗), if i +2≤ j,

∧g+i+1−a(F∗), if i +1= j, and

∧ j(F∗)⊗Lg−ai− j+1G, if j ≤ i

by (4.4.2)

∼=

∧f−g+1− j F ⊗Ka

j−i−2(G∗)⊗

∧f(F∗), if i +2≤ j,

∧f−g−i−1+aF ⊗

∧f(F∗), if i +1= j, and

∧f− j F ⊗Lg−a

i− j+1G⊗∧

f(F∗), if j ≤ i

=(C i,aΦ ) j ⊗

∧f(F∗).

The complexC i,aΦ is obtained by patching together two well-known complexes.The duality

among the pieces is well understood. We focus on the duality at the patch:

(C i,aΦ )i+2⊗

∧f(F∗)d

//

��

(C i,aΦ )i+1⊗

∧f(F∗)d

//

��

(C i,aΦ )i ⊗

∧f(F∗)

��

((C f−g−i−1,g+1−aΦ )f−g−1−i)

∗ d∗// ((C f−g−i−1,g+1−a

Φ )f−g−i)∗ d∗

// ((C f−g−i−1,g+1−aΦ )f−g+1−i)

∗,

which is the same as

∧f−g−i−1 F⊗Ka0(G

∗)⊗∧f(F∗)

d//

��

∧f−g+a−i−1 F⊗∧f(F∗)

d//

��

∧f−i F⊗Lg−a1 G⊗

∧f(F∗)

��

(∧g+1+i F⊗La−1

1 G)∗d∗

// (∧g+i+1−a F)∗

d∗// (∧i F⊗Kg+1−a

0 (G∗))∗.

The vertical maps are comprised of the canonical isomorphisms∧ℓF ⊗

∧f F∗ ∼=∧f−ℓF∗ and the

isomorphisms induced by the perfect pairing of 4.4.2. Thereis no difficulty in checking that thediagram commutes up to sign. �

Our conventions concerning the length of a complex are givenin 2.3.

Observation 7.10.Adopt Data7.1. Let and i and a be integers with1≤ a≤ g. Recall thecomplexes{C

i,aΦ } of Definition7.2and Observation7.4.

(a) If −1≤ i, then(C i,aΦ ) j = 0 for j ≤−1 and length(C i,a

Φ )≤ f.

(b) If i ≤ f− g andf− g+2≤ j, then(C i,aΦ ) j = 0.

(c) Assume

−1≤ i ≤ f− g, or (i,a) = (f− g+1,1), or (i,a) = (−2,g).

Then(C i,aΦ ) j = 0 for j ≤ −1 and for f− g+2 ≤ j; in particular, C

i,aΦ is a complex of

length at mostf− g+1.

Proof. Use (7.3.1). Ifj ≤−1≤ i, then the the∧•F contribution to(C i,a

Φ ) j is∧

f− j F = 0. Ifi ≤ f−g andf−g+2≤ j, then the

∧•F contribution to(C i,aΦ ) j is

∧f−g+1− j F = 0. The other

assertions are checked in a similar manner. �



i,aΦ } of Definition7.2and Observation7.4. Then

H0(Ci,aΦ ) =

∧f−g+a F

im(∧a Φ)∧

∧f−g F

, if i =−1,

coker(∧

g−a+1 Φ∗), if i = 0, and

Lg−ai+1 G

Φ∗(F)·Lg−ai G

, if 1≤ i.

Proof. Fix a basis elementωF∗ of∧

f F∗ and, for eachj, let σ :∧ j F →

∧f− j F∗ be the

non-canonical isomorphism which sendsf j to f j(ωF∗).For i = 0, consider the commutative square

(7.11.1) (C0,aΦ )1 =

∧f−g+a−1F

d1//

∼= σ��

(C0,aΦ )0 =

∧f F ⊗Lg−a

1 G

∼= σ⊗(4.5.e)��

// H0(C0,aΦ ) // 0

∧g+1−aF∗

∧g+1−a Φ∗

//∧

g−a+1G.

For 1≤ i, consider the commutative square

(7.11.2) (Ci,aΦ )1 =

∧f−1F ⊗Lg−a

i Gd1

//

∼= σ1⊗1��

(Ci,aΦ )0 =

∧f F ⊗Lg−a

i+1 G

∼= σ0⊗1��

// H0(Ci,aΦ ) // 0

F∗⊗Lg−ai G

δ// Lg−a

i+1 G,

whereδ(φ⊗∑ℓAℓ⊗Bℓ) = ∑ℓAℓ⊗Φ∗(φ) ·Bℓ for φ ∈ F∗, Aℓ ∈∧

g−aG, Bℓ ∈ Symi G and

∑ℓAℓ⊗Bℓ ∈ Lg−ai G⊆

∧g−aG⊗Symi G.

In both diagrams, (7.11.1) and (7.11.2), the top line is exact by 7.10.a. Fori = −1, thesequence

(C−1,aΦ )1 =

∧f−g F ⊗

∧aG∗ d1−→ (C−1,aΦ )0 =

∧f−g+aF → H0(C

−1,aΦ )→ 0

is exact andd1 sendsff−g⊗ γa to ff−g∧ (∧a Φ)(γa). �


i,aΦ } of Definition7.2and Observation7.4. Then Ig(Φ) annihilatesH0(C

i,aΦ ) for

all i with −1≤ i.

Proof. We show thatIg(Φ) ·(C i,aΦ )0⊆ imd1 for each relevanti. Consider the arbitrary element

r = [(∧

g Φ)(ωG∗)](φg) of Ig(Φ), whereωG∗ ∈∧

g G∗ andφg ∈∧

g F∗.We prove the assertion fori =−1 by showing thatr ·

∧f−g+aF ⊆ im(

∧a Φ)∧∧

f−g F . Letff−g+a ∈

∧f−g+aF. Apply Proposition 2.14 in order to writer · ff−g+a, which is equal to

ff−g+a∧φg

(

(∧

gΦ)(ωG∗))

,

28 ANDREW R. KUSTIN

as a sum of elements of the form

f ∧φ(

(∧

gΦ)(ωG∗))

,

for homogeneous elementsf ∈∧•F andφ ∈

∧•F with

degφ ≤ g+ f− (f− g+a)− g= g−a.

Apply Proposition 2.13.d to see that

f ∧φ(

(∧

gΦ)(ωG∗))

= f ∧ (∧

g−degφΦ)([

(∧degφΦ∗)(φ)

]

(ωG∗))

.

This completes the proof wheni =−1 becausea≤ g−degφ.Consideri = 0. In light of (7.11.1), it suffices to show that

(7.12.1) r(∧

g+1−a G)⊆ (∧

g+1−a Φ∗)(∧

g+1−aF∗).

Let gg+1−a be an element of∧

g+1−aG. Observe that[

(∧a−1 Φ)[gg+1−a(ωG∗)]

]

(φg)

is an element of∧

g+1−aF∗ and∧

g+1−a Φ∗ carries this element to

(∧

g+1−a Φ∗)(

[

(∧a−1 Φ)[gg+1−a(ωG∗)]

]

(φg))

=[gg+1−a(ωG∗)](

(∧

g Φ∗)(φg))

by 2.13.d

=gg+1−a∧ωG∗

(

(∧

g Φ∗)(φg))

by 2.13.c

=r ·gg+1−a.

The claim (7.12.1) has been established. This completes theproof wheni = 0.One further consequence of (7.12.1) is thatrG ⊆ Φ∗(F∗). (Takea to beg to obtain this

conclusion.)We prove the assertion for 1≤ i by showing that

rLg−ai+1 G⊆ δ(F∗⊗Lg−a

i G),

in the notation of (7.11.2). Observe that the exact sequence(4.4.1) yields

Lg−ai+1 G= κ(

∧g−a+1G⊗Symi G)

since 0< g−a+ i +1. On the other hand,

κ(∧

g−a+1G⊗Symi−1G)⊆ Lg−ai G.

It follows that

rLg−ai+1 G⊆ rG ·

(

κ(∧

g−a+1G⊗Symi−1G))

⊆ rG ·(

Lg−ai G

)

⊆ Φ∗(F∗) ·Lg−ai G= δ(F∗⊗Lg−a

i G),

where the multiplication “·” means multiply into the the symmetric algebra factor. The proofis complete for 1≤ i. �


8. THE ACYCLICITY OF Ci,aΦ .

Theorem 8.4 is the main result of the paper. It asserts ifΦ is sufficiently general, thenC i,aΦ

is a resolution of H0(Ci,aΦ ) and H0(C

i,aΦ ) is a torsion-free(R/Ig(Φ))-module, for−1≤ i and

1≤ a≤ g. The depth-sensitivity assertion, Corollary 8.5, was promised in (1.0.7), and is infact our main motivation for writing the paper. Corollary 8.6 records the fact, promised in(1.0.6), that in the generic situation, with−1≤ i ≤ f− g, then H0(C

i,aΦ ) is a maximal Cohen-

Macaulay module of rank(

g−1a−1

)

over the determinantal ringR/Ig(Φ). Recall, from 7.11,that

H0(Ci,aΦ ) =

∧f−g+a F

im(∧a Φ)∧

∧f−g F

, if i =−1,

coker(∧

g−a+1 Φ∗), if i = 0, and

Lg−ai+1 G

Φ∗(F)·Lg−ai G

, if 1 ≤ i.

Theorem 8.4 follows readily from Lemma 8.1 by way of the acyclicity lemma. If I1(Φ) = R,then it is shown in Lemma 8.1 that

Ci,aΦ and C

i,aΦ′ ⊕C

i,a−1Φ′

have isomorphic homology for some smallerR-module homomorphismΦ′.

Lemma 8.1.Adopt Data6.1and Notation6.2with g≤ f andΦ′′ an isomorphism. Let a and ibe integers with1≤ a≤ g−1. Recall the complexC i,a

Φ of Definition7.2and Observation7.4.Then there there exists a canonical complexD of free R-modules and canonical short exactsequences

(8.1.1) 0→ Tot(

Lf−i−1,g−aπ′◦Φ ⊗A(Φ′′)

)

incl†−−→ C

i,aΦ

quot†−−−→ D → 0

and

(8.1.2) 0→ Ci,aΦ′ ⊕

(

Ci,a−1Φ′ ⊗G′′∗

) incl‡−−→ D


(

Kf−g−i−2,aπ′◦Φ [−i −2]⊗B(Φ′′)

)

→ 0.

In particular, there are canonical isomorphisms

(8.1.3) Hj(Ci,aΦ )∼= H j(C

i,aΦ′ )⊕

(

H j(Ci,a−1Φ′ )⊗G′′∗

)

for all integers j.

Remarks 8.1.4. (a) Lemma 8.1 does apply whena= 1 provided one interpretsC i,a−1Φ′ using

Remark 7.5. The complexC i,0Φ′ of Remark 7.5 is split exact; so the ultimate conclusion

of Lemma 8.1 whena= 1 is

H j(Ci,1Φ )∼= H j(C

i,1Φ′ ),

for all integersj. In light of Observation 7.8, this conclusion is well-known.

30 ANDREW R. KUSTIN

(b) Lemma 8.1 is false whena= g, even if one interpretsC i,gΦ′ using Remark 7.5. The correct

statement is that

H j(Ci,gΦ )∼= H j(C

i,g−1Φ′ ) and Hj(C

i,gΦ ) 6∼= H j(C

i,gΦ′ )⊕H j(C

i,g−1Φ′ ).

Indeed,

H j(Ci,gΦ )∼= H j(C

i+1,1Φ ) by Observation 7.8

∼= H j(Ci+1,1Φ′ ) by (a)

∼= H j(Ci,g−1Φ′ ) by Observation 7.8.

On the other hand, the complexCi,gΦ′ has non-zero homology because the complex is

non-zero in exactly one position; see Remark 7.5.

Proof of Lemma 8.1. We know from Definition 7.2 thatC i,aΦ is the complex

0→Kf−g−i−1,aΦ [−i −2]

d−→

∧f−g+a−i−1F

d−→ L

f−i−1,g−aΦ → 0,

with [C i,aΦ ]i+1 =

∧ f−g+a−i−1F. The parameterg−a is positive; so, we know from Proposi-tion 6.4.a that

0→ Tot(


)

incl†−−→ L

f−i−1,g−aΦ

quot†−−−→

Lf−i−2,g−aΦ′ ⊗F ′′

⊕

Lf−i−2,g−a−1Φ′ ⊗F ′′⊗G′′

→ 0

is a short exact sequence of complexes. It follows that

(8.1.5) Tot(


)

incl†−−→ C

i,aΦ

is an injection of complexes. LetD be the cokernel of (8.1.5). Observe that (8.1.1) is a shortexact sequence of complexes and thatD is isomorphic to

0→Kf−g−i−1,aΦ [−i −2]

d−→

∧f−g+a−i−1F

quot† ◦d−−−−→

Lf−i−2,g−aΦ′ ⊗F ′′

⊕

Lf−i−2,g−a−1Φ′ ⊗F ′′⊗G′′

→ 0.

The parametera is positive; and therefore, we also know from Proposition 6.4.b that

0→K

f−g−i−1,aΦ′

⊕

(Kf−g−i−1,a−1Φ′ ⊗G′′∗)

incl‡−−→K

f−g−i−1,aΦ


(

Kf−g−i−2,aπ′◦Φ ⊗B(Φ′′)

)

→ 0

is a short exact sequence of complexes. It follows that

Dquot‡−−−→ Tot

(


)

[−i −2]


is a surjection of complexes with kernel isomorphic to the complex:

0→K

f−g−i−1,aΦ′ [−i −2]

⊕

Kf−g−i−1,a−1Φ′ [−i −2]⊗G′′∗

d◦incl‡−−−−→

∧f−g+a−i−1Fquot† ◦d−−−−→

Lf−i−2,g−aΦ′ ⊗F ′′

⊕

Lf−i−2,g−a−1Φ′ ⊗F ′′⊗G′′

→ 0.(8.1.6)

Thus,

(8.1.7) 0→ (8.1.6)incl‡−−→ D


(


)

[−i −2]→ 0

is a short exact sequence of complexes. Observe that

(8.1.8) Ci,aΦ′ ⊕ (C

i,a−1Φ′ ⊗G′′∗)

��

(8.1.6),

given by

0→K

f−g−i−1,aΦ′ [−i −2]

⊕

Kf−g−i−1,a−1Φ′ [−i −2]⊗G′′∗

[

d 00 d

]

//

[

id 00 id

]

��

∧f−g+a−i−1F ′

⊕∧f−g+a−i−2F ′⊗G′′∗

[

d 00 d

]

//

[ incl incl† ◦(1⊗Φ′′)]

��

Lf−i−2,g−a−1Φ′

⊕

Lf−i−2,g−aΦ′ ⊗G′′∗

→ 0

[

0 1⊗Φ′′

1⊗KosΦ′′ 0

]

��

0→K

f−g−i−1,aΦ′ [−i −2]

⊕

Kf−g−i−1,a−1Φ′ [−i −2]⊗G′′∗

d◦incl‡//∧f−g+a−i−1F

quot† ◦d//

Lf−i−2,g−aΦ′ ⊗F ′′

⊕

Lf−i−2,g−a−1Φ′ ⊗F ′′⊗G′′

→ 0,

is an isomorphism of complexes. (The complexesCi,aΦ′ and C

i,a−1Φ′ have been read from

Definition 7.2.) Combine (8.1.7) and (8.1.8) in order to see that (8.1.2) is a short exactsequence of complexes. The isomorphisms of (8.1.3) follow immediately from the shortexact sequences (8.1.1) and (8.1.2) because the two total complexes have all homology equalto zero sinceΦ′′ is an isomorphism. �

In Lemma 8.3 we iterate Lemma 8.1. The basic set-up is similarto, but not the same as,the set-up of Data 6.1.

Data 8.2. Let Rbe a commutative Noetherian ring,F andG be freeR-modules of rankf andg, respectively, withg≤ f, andΦ : G∗ → F be anR-module homomorphism. DecomposeFandG as

F = F ′⊕F ′′ and G= G′⊕G′′,

whereF ′, F ′′, G′ andG′′ are freeR-modules and rankF ′′ = rankG′′ = r for some integerrwith 1≤ r ≤ g−1 and and let

F∗ = F ′∗⊕F ′′∗ and G∗ = G′∗⊕G′′∗

32 ANDREW R. KUSTIN

be the corresponding decompositions ofF∗ andG∗. Assume that

(8.2.1) Φ =

[

Φ′r 0

0 Φ′′r

]

,

whereΦ′r : G′∗ → F ′ is anR-module homomorphism andΦ′′

r : G′′∗ → F ′′ is anR-moduleisomorphism.

Lemma 8.3. Adopt Data8.2. Let i and a be integers with1≤ a≤ g. Recall the complexC i,aΦ

of Definition7.2and Observation7.4.

(a) If 1≤ r ≤ g−1, then

(8.3.1) Hj(Ci,aΦ )∼=

g−r⊕

β=1

H j(Ci,βΦ′

r)(

ra−β).

(b) If r = g−1, then the following statements hold:

(i) H j(Ci,aΦ )∼= H j(C

i,1Φ′

g−1)(

g−1a−1),

(ii) if Ig(Φ) = R, then the complexC i,aΦ is split exact; and,

(iii) if Ig(Φ) is a proper ideal of grade at leastf− g+ 1, thenCi,aΦ is a resolution of

(R/Ig(Φ))(g−1a−1).

Proof. (a). We are given

Φ =

[

Φ′r 0

0 Φ′′r

]

,

whereΦ′′r is an isomorphism of free modules of rankr. We may rearrange the data so that

Φ =

Φ′r 0 0

0 Φ′′′ 00 0 Φ′′

,

whereΦ′′ is an isomorphism of free modules of rank one andΦ′′′ is an isomorphism of freemodules of rankr −1. Let

Φ′ =

[

Φ′r 0

0 Φ′′′

]

.

Apply Lemma 8.1 and Remarks 8.1.4.a and 8.1.4.b to obtain

(8.3.2) Hj(Ci,aΦ )∼= χ(a≤ g−1)H j(C

i,aΦ′ )⊕χ(2≤ a)H j(C

i,a−1Φ′ ),

whereχ is described in 2.7. Notice that (8.3.2) agrees with (8.3.1)whenr = 1 because

(

1a−β

)

=

1, if β = a and 1≤ a≤ g−1,

1, if β = a−1 and 2≤ a≤ g,

0, if β 6∈ {a−1,a} and 1≤ β ≤ g−1.


Induction onr applied to (8.3.2) now yields

H j(Ci,aΦ )∼= χ(a≤ g−1)

(g−1)−(r−1)⊕

β=1

H j(Ci,βΦ′

r)(

r−1a−β)⊕χ(2≤ a)

(g−1)−(r−1)⊕

β=1

H j(Ci,βΦ′

r)(

r−1a−1−β).

The constraintsχ(a≤ g−1) andχ(2≤ a) are redundant. Indeed, ifg≤ a, then

β ≤ g− r =⇒ r ≤ g−β ≤ a−β =⇒(r−1

a−β)

= 0,

and ifa≤ 1, then

1≤ β =⇒ a−1−β ≤−1 =⇒( r−1

a−1−β)

= 0.

Thus,

H j(Ci,aΦ )∼=

g−r⊕

β=1

H j(Ci,βΦ′

r)(

r−1a−β)⊕

g−r⊕

β=1

H j(Ci,βΦ′

r)(

r−1a−1−β)

∼=g−r⊕

β=1

H j(Ci,βΦ′

r)(

ra−β).

(b). Assertion (bi) is a special case of (a). Recall thatΦ′g−1 : G′∗ → F ′ is a homomorphism,

G′ is a free module of rank one, andF ′ is a free module of rankf− g+ 1. The complexC

i,1Φ′

g−1is the Koszul complex on a generating set forI1(Φ′

g−1) = Ig(Φ). If Ig(Φ) = R, then

Ci,1Φ′

g−1is split exact. This is (bii). Otherwise, the hypothesisf− g+1≤ gradeIg(Φ) ensures

that gradeIg(Φ) is generated by a regular sequence and thereforeCi,1Φ′

g−1is a resolution of

H0(Ci,1Φ′

g−1) = R/Ig(Φ). This is (biii). �

Theorem 8.4 is the main result of the paper.

Theorem 8.4. Adopt Data7.1. Let i and a be integers with1≤ a≤ g. Recall the complexC

i,aΦ from Definition7.2and Observation7.4. Assume that Ig(Φ) is a proper ideal of R with

f− g+1≤ gradeIg(Φ).

(a) If length(C i,aΦ ) = f−g+1 and(C i,a

Φ ) j = 0 for j ≤−1, then the following statements hold.

(i) The complexC i,aΦ is acyclic.

(ii) The R-moduleH0(Ci,aΦ ) is perfect of projective dimensionf− g+1.

(iii) The(R/Ig(Φ))-moduleH0(Ci,aΦ ) is torsion-free.

(iv) If f− g+2≤ gradeIg−1(Φ), then the(R/Ig(Φ))-moduleH0(Ci,aΦ ) has rank

(

g−1a−1

)

.

(b) If −1≤ i and f− t +1≤ gradeIt(Φ) for all t with f+1− length(C i,aΦ )≤ t ≤ g−1, then

(i) the complexC i,aΦ is acyclic,

(ii) H j(Cf−g−i−1,g+1−aΦ ) = Extf−g+1− j

R (H0(Ci,aΦ ),R) for all j, and

(iii) H j(Cf−g−i−1,g+1−aΦ ) = 0 for 1≤ j.

34 ANDREW R. KUSTIN

(c) If −1≤ i, f− g+2≤ length(C i,aΦ ), and

(8.4.1) f− t +2≤ gradeIt(Φ), for all t with f+1− length(C i,aΦ )≤ t ≤ g−1,

thenH0(Ci,aΦ ) is a torsion-free(R/Ig(Φ))-module of rank

(

g−1a−1

)

.

Remarks 8.4.2. (a) Recall from Lemma 8.3.bii that ifIg(Φ) = R, thenCi,aΦ is split exact.

(b) Observation 7.10 contains elementary facts about the length of the complexesC i,aΦ . In

particular, the hypotheses of (a) are satisfied when

−1≤ i ≤ f− g, or (i,a) = (f− g+1,1), or (i,a) = (−2,g).

(c) In the generic case (whenΦ can be represented by a matrix of variables) all of the gradehypotheses of Theorem 8.4 are automatically satisfied because

f− t +2≤ (g− t+1)(f− t+1), whenever 1≤ t ≤ g−1.

(d) The modules H0(Ci,aΦ ) are recorded in Observation 7.11.

Proof. Throughout this proof, letC representC i,aΦ andℓ and represent length(C ).

(ai). The complexC has the form

0→ Cf−g+1 → ·· · → C1 → C0 → 0;

see Observation 7.10. Letp be a prime ideal ofR with gradep≤ f− g. Observe that

gradep≤ f− g< f− g+1≤ gradeIg(Φ)≤ gradeIg−1(Φ).

Thus, there is a(g− 1)× (g− 1) minor of Φ which is a unit inRp and we may applyLemma 8.3.biii in order to conclude thatCp is acyclic. The acyclicity criterion (see, forexample, [2, 1.4.13]) now guarantees thatC is acyclic.

(aii). We know from Observation 7.12 thatIg(Φ)⊆ ann(H0(C )); and therefore,

pdRH0(C )≤ f− g+1 by ai

≤ gradeIg(Φ) by hypothesis

≤ gradeann(H0(C ))

≤ pdRH0(C ) by (2.9.1).

The proof of (aii) is complete; see 2.9, if necessary.

Assertion (aiii) is a consequence of Proposition 2.10.1.

(aiv). TheR-moduleR/Ig(Φ) is perfect of projective dimensionf− g+1. (See, for example,[10, Cor. 5.2], [26, Thm. 1]. or [3, 2.7].) Letp ∈ SpecR be an associated prime ofR/Ig(Φ).It follows that gradep= f−g+1 andIg−1(Φ) 6⊆ p. Thus, Corollary 8.3.biii may be applied to

Φp in order to conclude that H0(Ci,aΦ )p ∼= H0(R/Ig(Φ))

(g−1a−1)

p . The proof of (aiv) is complete;see 2.10.


(bi). We induct onℓ. The base case,ℓ = f− g+1, is established in (ai). We now study thecase withf− g+2 ≤ ℓ. As in the proof of (ai) we apply the acyclicity criterion andprovethatCp is acyclic for all prime idealsp of R with gradep< ℓ. Fix such ap. The hypothesesof (b) with t = f+1− ℓ now ensure that

gradep< ℓ≤ gradeIf+1−ℓ(Φ).

Thus, there is an(f+ 1− ℓ)× (f+ 1− ℓ) minor of Φ which is a unit in inRp and, afterrearrangement,

Φp =

[

Φ′f+1−ℓ 00 Φ′′

f+1−ℓ

]

,

as described in (8.2.1). Apply Corollary 8.3.a in order to conclude that

(8.4.3)H j(C )p is a direct of suitably many copies of modules from the set

{H j(Ci,βΦ′

f+1−ℓ) | 1≤ β ≤ g− (f+1− ℓ)}.

Observe that

f− t +1≤ gradeIt(Φ)≤ gradeIt(Φp) = gradeIt−(f+1−ℓ)(Φ′f+1−ℓ)

for all t with f+1− ℓ≤ t ≤ g−1. Let

f′ = f− (f+1− ℓ), g

′ = g− (f+1− ℓ), and t ′ = t − (f+1− ℓ).

We have shown that

f′− t ′+1≤ gradeIt ′(Φ′

f+1−ℓ) for all t ′ with 0≤ t ′ ≤ g′−1.

It follows that the hypotheses of (b) apply toCi,βΦ′

f+1−ℓfor eachβ with 1≤ β ≤ g′. On the other

hand,

length(C i,βΦ′

f+1−ℓ)≤ rank(the target ofΦ′

f+1−ℓ) = f′ = f− (f+1− ℓ) = ℓ−1.

By induction onℓ, eachCi,βΦ′

f+1−ℓis acyclic; and thereforeCp is also acyclic by (8.4.3).

(bii) and (biii). Now that we know thatC is a resolution of H0(C ), assertion (bii) can be readfrom Observation 7.9; and (biii) follows from (bii) becausethe gradef− g+1 idealIg(Φ) iscontained in the annihilator of H0(C ).

(c). We already know from (bi) that

(8.4.4) C is a free resolution of H0(C ) of lengthℓ.

For each integerw, let Fw be the ideal inR generated by:

{x∈ R | pdRxH0(C

i,aΦ )x < w}.

Claim 8.4.5. If f+1− ℓ≤ t ≤ g−1, thenIt(Φ)⊆ Ff−t+1.

36 ANDREW R. KUSTIN

Proof of Claim 8.4.5. If∆ is at × t minor of Φ, then one can arrange the data so that

Φ∆ =

[

Φ′t 0

0 Φ′′t

]

,

whereΦ′′t is an isomorphism of freeR∆-modules of rankt as is described in Data 8.2. Apply

Corollary 8.3.a to see that

(8.4.6) Hj(C )∆ ∼=g−t⊕

β=1

H j(Ci,βΦ′

t)(

ta−β), for all j.

Apply (8.4.4) to see that eachC i,βΦ′

twhich actually appears in (8.4.6) (that is, with

( ta−β

)

6= 0)is also acyclic. Thus,

pdR∆H0(C )∆ ≤ max{length(C i,β

Φ′t) | 1≤ β ≤ g− t} ≤ rank(the target ofΦ′

t) = f− t.

This completes the proof of Claim 8.4.5.

Combine hypothesis (8.4.1) and Claim 8.4.5 to see that

f+1− ℓ≤ t ≤ g−1 =⇒ (f− t+1)+1= f− t +2≤ gradeIt(Φ)≤ gradeFf−t+1.

Let w= f− t +1. We have shown that

f− g+2≤ w≤ ℓ =⇒ w+1≤ gradeFw.

Apply Proposition 2.10.1 to conclude that theR/Ig(Φ) module H0(C ) is torsion-free.We re-use the rank calculation of (aiv). Ifp ∈ AssR(R/Ig(Φ)), then

gradep= f− g+1< f− g+3≤ gradeIg−1(Φ);

hence, Corollary 8.3.biii may be applied toΦp, as was done in (aiv), to conclude thatrankH0(C ) =

(

g−1a−1

)

. �

The next result was promised in (1.0.7), and is our main motivation for writing the paper.

Corollary 8.5. Adopt Data7.1. Let and i and a be integers with1 ≤ a ≤ g. Recall thecomplexC i,a

Φ from Definition7.2and Observation7.4. If −1≤ i ≤ f− g, then

H j(Ci,aΦ ) = 0 for f− g+2−gradeIg(Φ)≤ j.

Proof. Assertion 8.4.a may be applied in the generic case with the ring equal to the polyno-mial ringZ[{xi, j}] and the homomorphism given by a matrix of indeterminates. The presentassertion is a consequence of Proposition 2.11.2. �

The next result was promised in (1.0.6).

Corollary 8.6. If k is a field,g≤ f are positive integers, R is the polynomial ring

R= k[{xi, j | 1≤ j ≤ g,1≤ i ≤ f}],


Φ : Rg → Rf is the generic map given by

x1,1 . . . x1,g...

...xf,1 . . . xf,g

,

and M= H0(Ci,aΦ ) for some i and a with−1≤ i ≤ f−g and1≤ a≤ g, thenannR(M) = Ig(Φ)

and M is a maximal Cohen-Macaulay(R/Ig(Φ))-module of rank(

g−1a−1

)

.

Proof. The ringR is Cohen-Macaulay and theR-moduleM is perfect (by Theorem 8.4.a);henceM is a Cohen-MacaulayR-module. Ifp is in AssRM, thenIg(Φ) ⊆ ann(M) ⊆ p (seeObservation 7.12) andf− g+ 1 = gradep (by [2, 1.4.15]). On the other hand,Ig(Φ) isalready a prime ideal ofR of gradef− g+1. It follows that Ig(Φ) = p, Ig(Φ) = annRM,SuppR(M) = SuppR(R/Ig(Φ)), and AssRM = {Ig(Φ)}. �

Remark 8.7. A module over a local Artinian ring has rank only if it is free.Example 7.7shows that the hypothesisf− g+2≤ gradeIg−1(Φ) is needed in Theorem 8.4.aiv. Indeed, ifR= k[x,y], for some fieldk, and

Φ =

x 0 0y x 00 y x0 0 y

,

then H0(C0,2Φ ) is anR/I3(Φ)-module which does not have any rank. It is easy to see that the

length of H0(C0,2Φ ) is twice the length ofR/I3(Φ) = k[x,y]/(x,y)3; however, H0(C

0,2Φ ) has

the wrong Betti numbers, as a module overR, to be a freeR/I3(Φ)-module.

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DEPARTMENT OFMATHEMATICS, UNIVERSITY OF SOUTH CAROLINA , COLUMBIA , SC 29208E-mail address: [email protected]

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