COHOMOLOGY OVER GRADED COMPLETE INTERSECTIONS

OF CODIMENSION TWO

LUCHEZAR L. AVRAMOV AND ZHENG YANG

Abstract. Standard graded complete intersections of codimension 2 that lie

on a quadric are characterized by the property that all their finitely generatedgraded modules have eventually arithmetic sequences of Betti numbers.

Contents

Introduction 11. Cohomology over local rings 32. Multiplicative structures in (co)homology 53. Graded exterior algebras of rank two 84. Local complete intersections of codimension two 105. Bigraded exterior algebras of rank two 146. Graded complete intersections of codimension two 157. Betti numbers over graded rings 20References 22

Introduction

This paper is concerned with free resolutions of finitely generated graded modulesover a standard graded commutative algebra R =

⊕j∈NRj over a field k; that is, for

R of the form P/I, where P is polynomial ring over k with indeterminates of degree1 and I is a homogeneous ideal contained in

⊕i>2 Pi. When I is principal, R is

called a hypersurface; more generally, if I can be generated by a P -regular sequenceof c elements, then R is said to be a complete intersection of codimension c.

Every finitely generated graded R-module N admits a resolution where the mod-ules are free, the differentials preserve degrees, and the entries of their matrices liein m =

⊕j>1Rj . Such a minimal resolution is unique up to isomorphism. In

particular, the rank of the module at the ith step of a minimal resolution is aninvariant of N , known as its ith Betti number and denoted by βRi (N).

We study how patterns of Betti sequences reflect and affect the structure of R.To describe antecedents, set d = depthR. For i > d − depthN the Hilbert

Syzygy Theorem and the Auslander-Buchsbaum Equality yield βRi (N) = 0 when R

Date: November 11, 2014.2010 Mathematics Subject Classification. Primary: 13D02, 13H10. Secondary: 13D40, 14M10,

16E45.Key words and phrases. Graded rings, complete intersections, free resolutions, Betti numbers.Research partly supported by NSF grant DMS-1103176.

1

2 L. L. AVRAMOV AND Z. YANG

is a polynomial rings, and Eisenbud [9] shows that βRi+1(N) = βRi (N) holds when R

is a hypersurface. Strong converses hold: If βRd+1(R/m) = 0, then R is a polynomial

ring by Serre’s characterization of regularity; if βRd+1(R/m) = βRd (R/m), then R is

a singular hypersurface, as otherwise (βRi (N))i strictly increases, by Gulliksen [11].Here we identify those rings over which the next simplest pattern occurs.

Theorem A. For a standard graded k-algebra R the following are equivalent:

(i) R is a complete intersection of codimension 2 that lies on a quadric.(ii) (βRi (R/m2))i>d is arithmetic and unbounded.

(iii) (βRi (N))i0 is arithmetic for every finitely generated graded R-module Nand is unbounded for some graded R-module.

As the first result of Section 1, we deduce from theorems in [3, 4, 13] that theanalogs of the implications (i) ⇐⇒ (ii) ⇐= (iii) obtained by replacing “arithmeticsequence” with “polynomial of degree at most c − 1” hold in any codimension c.The implication (i) =⇒ (iii) answers in codimension two a question raised in [3]:

If R is a codimension c complete intersection lying on c− 1 quadrics,is then every sequence (βRi (N))i0 given by some polynomial in i?

Positive answers are known when c = 1 (noted above) and for any c when all thedefining relations are quadratic (proved in [3] by using Koszul duality). For c = 2we obtain such an answer from a sharper, more general statement described below.

For the rest of the introduction we assume R = Q/(g1, . . . , gc), where Q is astandard graded k-algebra and g1, . . . , gc a Q-regular sequence of forms. Whenproj dimQN is finite Gulliksen [12] proved that the sequences (βR2j(N))j0 and

(βR2j+1(N))j0 are each given by some polynomial in j of degree less then c, andAvramov [2] showed that these polynomials have equal degrees and leading terms.

In particular, when c = 2 there exist rational numbers a, b0, and b1 such thatβR2j+r = aj + br holds for r = 0, 1 and j 0. This result provides the context forthe next theorem. It is recovered, in more precise form, in the course of the proofof the theorem, which takes up Sections 2 through 7.

Theorem B. Let R = Q/(g1, g2) for a Q-regular sequence g1, g2 with deg g1 = 2.Let N be a finitely generated graded R-module; set m = d− depthRN and

l = m+ max2βQm(N) , 2βQm+1(N)− 2βQm(N)− 1If proj dimQN is finite, then for some a ∈ Z the following equalities hold:

βRi+l(N) = ai+ βRl (N) for i ≥ 0

Appropriate forms of all the results mentioned above—with the exception ofTheorems A and B—are known to hold in the more general context of finitelygenerated modules over local rings and concern complete intersections of arbitrarycodimension. We proceed to discuss, in this order, the roles played in our argumentsby codimension two, the quadric in I, and the gradings of R and N .

Codimension two appears twice, as distinct avatars. The first one is homo-logical and forces the formality of certain Koszul complexes. This allows us tomove between an abstract complete intersection, R, and the very concrete exte-rior algebra on two generators, Λ. The R-module N corresponds to the Λ-moduleN = TorQ(N, k); if proj dimQN = 2, then βRi (N) = βΛ

i (N) holds for all i.The second entrance of codimension two is through representation theory, in the

guise of Kronecker’s description of the indecomposable Λ-modules. That theory is

GRADED COMPLETE INTERSECTIONS 3

reviewed in Section 3, along with the cohomology of the indecomposable Λ-modules,computed in [6]. These data show that (βΛ

i (N))i is eventually arithmetic if and onlyif the multiplicities of the indecomposable components of N satisfy a ‘key relation’.

Ever since the work in [12, 9, 2] the main tool for studying resolutions overcomplete intersections has been provided by commutative polynomial rings of co-homology operators of degree 2 attached to regular sequences defining R. New tothis paper is the realization that quadratic equations carry additional information:They give rise to cohomology operators that are squares of (non-central) operatorsof degree 1. This general homological fact is proved in Section 2 and is applied inSection 4 to derive new restrictions on the Λ-module structure of N.

In Sections 1 through 4 we work entirely in a local context.The utilization of internal gradings is a second novelty in our approach. Their

presence has often been ignored when investigating resolutions over complete inter-sections because general linear combinations of the defining equations usually failto be homogeneous. We exploit the additional rigidity imposed by internal gradingsto analyze the bigraded representation theory of Λ in Section 5, then the Hilbertseries of N and the module structures of N and ExtR(N, k) Section 6.

These results impose further restrictions on the multiplicities of indecomposablecomponents of N. They provide the last pieces of information needed to verify the‘key relation’, and this is done in Section 7, the final one. Although gradings playan essential role in the proof of Theorem B, we know of no case when the localanalog fails. We end the paper with remarks about the status of the local question.

This article may be viewed as a continuation of the work of Avramov and Buch-weitz in [6], where representation theory was first applied to the study of resolu-tions over complete intersections. Two results from that paper are indispensablehere as well—the comparison of ExtR(N, k) and ExtΛ(N, k) and the computationof ExtΛ(X, k) for the indecomposable Λ-modules X. From this arguments proceedin opposite directions: In [6] the Λ-structure of ExtQ(N, k) is used to study themodule structure of ExtR(N, k) over a polynomial subring of ExtR(k, k). Here we

get information on TorQ(N, k) by analyzing the action of ExtR(k, k) on ExtR(N, k).

1. Cohomology over local rings

In this section (R,m, k) is a local ring; that is, R is a commutative noetherianring with unique maximal ideal m, and k is the residue field R/m.

We start by recalling some notions used to describe minimal free resolutions. Forbackground material on resolutions we routinely refer to [5].

1.1. The ith Betti number βRi (N) of a finitely generated R-module N is defined tobe the rank of the ith module in a minimal free resolution of N over R. It can becomputed by using either torsion or extension functors, since

rankk TorRi (N, k) = βRi (N) = rankk ExtiR(N, k)

When N is non-zero, its complexity is the number

cxRN = infd ∈ N ∪ 0 | βRi (N) ≤ aid−1 for some a ∈ N and all i 0

and we set cxR 0 = −1. The Poincare series of N is the formal power series

PRN =∑i>0

βRi (N)zi

4 L. L. AVRAMOV AND Z. YANG

1.2. Let R be the m-adic completion of R. Cohen’s Structure Theorem yields a

regular local ring (P, n, k), an ideal I ⊆ n2 and an isomorphism P/I ∼= R. We callany such isomorphism a minimal Cohen presentation of R.

The following numbers are invariants of R:

(1.2.1) ε1(R) = rankk n/n2 , ε2(R) = rankk I/nI , and q(R) = rankk I/(I ∩ n3) .

This is well-known for the first two, which record the minimal number of gener-ators of m and the minimal number of relations of R; see [5, 7.1.5]. The third onecounts the number of “quadratic” relations of R. For it the exact sequence

(1.2.2) 0→ I/(I ∩ n3)→ n2/n3 → n2/(I + n3)→ 0

and the isomorphism n2/(I +n3) ∼= m2/m3 yield the following invariant expression:

(1.2.3) q(R) = rankk n2/n3 − rankk m

2/m3 =

(ε1(R) + 1

2

)− rankk m

2/m3 .

Recall that R is said to be complete intersection (of codimension c) if in some Co-hen presentation 1.2 the defining ideal I can be generated by a P -regular sequence(of length c); this is independent of the choice of presentation; see [5, 7.3.3].

1.3. Theorem. Assume ε1(R) ≥ 2 and let c be a positive integer.The first two conditions below are equivalent and are implied by the third one.

(i) R is complete intersection of codimension c and c ≤ q(R) + 1.(ii) βRi (R/m2) = β(i) for i ≥ depthR and some β ∈ Q[x] with deg β = c− 1.

(iii) For each R-module N there exists βN ∈ Q[x] with deg βN < c, such that

βRi (N) = βN (i) for i 0

and deg βM = c− 1 for some R-module M .

Proof. We set e = ε1(R) and s = q(R) and recall some general results:

(1) If m2 6= 0, then cxR(R/m2) = cxR k by [4, Corollary 5]; see also [5, 10.3.8].(2) If cxR k = c, then R is complete intersection of codimension c by [13, 2.3]; see

also [5, 8.1.2].(3) If R is complete intersection of codimension c, then [3, 2.1] gives

(1.3.1) PRR/m2 =1

z

((1 + z)s+1−c

(1− z)c−s+

(1 + z)e−c(ez − 1)

(1− z)c

)We will use a basic fact concerning generating functions; see, e.g. [19, 4.3.1].

(4) The condition βRi (N) = βN (i) for i 0 and some βN ∈ Q[x] with deg βN < d isequivalent to PRN = hN/(1−z)d for some hN ∈ Z[z]; in addition, deg βN = d−1if and only if hN (1) 6= 0, and then βRi (N) = βN (i) holds for i ≥ deg hN .

(i) =⇒ (ii). We have s ≤ c by definition, c ≤ e becauseR is complete intersection,and c ≤ s + 1 by hypothesis. Thus, the right-hand side of (1.3.1) has the formh/(1− z)c with h ∈ Z[z] and deg h = e− c = depthR. Since e ≥ 2 holds, the sameformula gives h(1) 6= 0, so the desired polynomial β exists by (4).

(ii) =⇒ (i). Since cxR(R/m2) = deg β + 1 = c ≥ 1, we have m2 6= 0. Now (1)and (2) show that R is complete intersection of codimension c, so formula (1.3.1)holds. By (4) its right-hand has no pole at z = −1, and this forces c ≤ s+ 1.

(iii) =⇒ (i). Every R-module N satisfies βRi (N) ≤ aNβRi (k) for i 0 and someaN ∈ N; see [5, 4.1.9]. This yields the first inequality in the string of relations

c = deg βM + 1 = cxRM ≤ cxR k = deg βk + 1 ≤ c

GRADED COMPLETE INTERSECTIONS 5

Thus, R is complete intersection of codimension c, by (2). The hypothesis e ≥ 2implies m2 6= 0, so (1) gives cxR(R/m2) = c. Since βRi (R/m2) is eventually equalto a polynomial of degree c− 1 ≥ 0, from (4) and (3) we get c ≤ s+ 1.

1.4. Remark. For c = 0 or c = 1 condition (i) of the theorem holds for anyvalue of q(R), and its equivalence with (iii) is part of standard characterizations ofregular local rings and local hypersurfaces, respectively. In those cases (ii) is usuallyreplaced with βRd+1(k) = 0 or βRd+1(k) = βRd (k), respectively, where d = depthR.

A well-known result of Tate gives PRk = (1+ t)e−c/(1− t)c for any local completeintersection ring of codimension c, see [20, Theorem 6] or [5, 7.1.1 and 7.3.3]. Thus,the Betti sequence of k does not detect quadratic relations. It is an interestingopen question whether (ii) can be replaced with a condition that involves only aspecified finite segment of the Betti sequence of R/m2.

2. Multiplicative structures in (co)homology

In this section (R,m, k) denotes a local ring and N a finitely generated Q-module.As usual, ΩlR(N) stands for the cokernel of the differential ∂l+1 in some minimalfree resolution of N over R; in particular, Ω0

R(N) = N .We discuss and study various multiplicative structures carried by different

(co)homological functors. Their interaction is at the heart of our investigation.For any R-module we set M∗ = HomR(M,k). We let ΣpM denote the graded

R-module defined by (ΣpM)p = M and (ΣpM)i = M for i 6= p; in case uppergradings are used, this means (ΣpM)−p = M and (ΣpM)i = M for i 6= −p.

2.1. Let N be a finitely generated R-module and form the graded k-vector spaces

E = ExtR(k, k) and N = ExtR(N, k)

For all i, j ∈ Z composition products yield k-linear maps E i ⊗k N i → N i+j . Theyturn E into graded algebra and N into a left graded E-module; see [17, §1].

2.2. The homotopy Lie algebra of R is a graded subspace of E that contains thecommutators [ϑ, θ] = ϑθ − (−1)ijθϑ of all ϑ ∈ πiR and θ ∈ πjR and the squares ϑ2

of all ϑ ∈ πiR when i is odd. We briefly recall the construction given in [18], using

canonical isomorphisms as equalities; for instance, ExtiR(k, k) = TorR(k, k)∗.

2.2.1. By [14, 2.3.4], TorR(k, k) is naturally a graded k-algebra with divided powers.

Set πR = (T R)∗, where T R is the quotient of TorRi (k, k) by the subspace spannedby the products of elements of positive degrees and the divided powers of degree 2or higher of elements of even degree; in particular, π1

R = E1 = m∗.

The assignment R 7→ πR defines a contravariant functor from the category oflocal rings with residue field k and homomorphisms of rings that induce the identityon k to the category of graded Lie algebras over k and their homomorphisms.

2.2.2. A minimal Cohen presentation R ∼= P/I, see 1.2, gives π2R = I∗, an equality

πR = πR of graded Lie algebras, and there is an exact sequence of such algebras

0→ π>2R → πR → πP → 0

Indeed, TorP (k, k) is the exterior algebra on Σ(n/n2). This gives T R = Σ(n/n2),

hence π>2R = 0, and thus exactness in degrees different from 1. Exactness in degree 1

means that m∗ → n∗ is bijective, see 2.2.1, and it is because I is contained in n2.

6 L. L. AVRAMOV AND Z. YANG

2.2.3. If (Q, q, k) is a local ring, J an ideal generated by Q-regular sequence in q2,and Q = Q/J , then there is a natural exact sequence of graded Lie algebras

0→ Σ−2(J∗)ζ−→ πQ

πψ−−→ πQ → 0

where the image of Σ−2J∗ is contained in the center of πQ.

Indeed, in the proof of [14, 3.4.1], Gulliksen shows that the map ψ : Q → Q

induces isomorphisms ψi : T Qi ∼= TQi for i ≥ 3 and an exact sequence

0→ T Q2ψ2−−→ T Q2 → J/qJ → T Q1

ψ1−−→ T Q1 → 0

The condition J ⊆ q2 means that ψ1 is bijective. These properties yield the desiredexact sequence, because πi = (ψi)

∗ for each i.On the other hand, [7, 2.7, p. 700] shows that Im(ζ) is in the center of E .

The main result of this section concerns the structure of πR in low degrees.

2.3. Theorem. Assume R ∼= Q/J , where (Q, q, k) is a complete local ring and Jan ideal generated by Q-regular sequence contained in q2, and set

e = rankk(q/q2) , c = rankk(J/qJ) , and b = rankk J/(q3 ∩ J)

If rankk q2/q3 =

(e+12

), then E = ExtR(k, k) contains elements τ1, . . . , τb in π1

R

and linearly independent elements χ1, . . . , χc in π2R with the following properties:

χr = τ2r for 1 ≤ r ≤ b(2.3.1)

χrξ = ξχr for 1 ≤ r ≤ c and all ξ ∈ E(2.3.2)

Proof. From the exact sequence of k-vector spaces

0→ (q3 ∩ J)/qJ → J/qJ → J/(q3 ∩ J)→ 0

we get rankk(q3∩J)/qJ = c−b. Choose first elements g1, . . . , gb of J whose residueclasses form a basis of J/(q3∩J), then elements gb+1, . . . , gc of q3∩J whose classesform a basis of (q3 ∩ J)/qJ . The exact sequence shows that the set g1, . . . , gcmaps to a basis of J/qJ , and so minimally generates J ; thus, g1, . . . , gc is Q-regular.

Form the local ring Q′ = Q/(gb+1, . . . , gc) with maximal ideal q′ = qQ′, and notethat the ideal J ′ = JQ′ is generated by the Q′-regular sequence gb+1, . . . , gc.

Let Q = P/I be a minimal Cohen presentation with regular local ring (P, n, k).Due to the inclusion J ⊆ q2 we have J ′ ⊆ q′2, so the surjective ring homomorphisms

(2.3.3) P Q Q′ R induce n/n2 ∼= q/q2 ∼= q′/q′2 ∼= m/m2

Thus, setting I = Ker(P → R) we get a minimal Cohen presentation R ∼= P/I.Since J = IQ, we have canonical surjections I/mI → J/qJ → J ′/q′J ′. Thus,

we may pick elements f1, . . . , fb in I so that fi maps to gi for i = 1, . . . , b andelements fc+1, . . . , fc in n3 ∩ I so that fr maps to gr for r = b + 1, . . . , c. Nowchoose fb+1, . . . , fn in Ker(P → Q) ∩ I so that f1, . . . , fn minimally generates I.

Next we show that the images of f1, . . . , fb form a k-basis of I/(n3∩I). They arelinearly independent because their images in J/(q3 ∩ J) form a basis, so it sufficesto show that fr is in m3 for r ≥ b+ 1. This is how fr was chosen for b+ 1 ≤ r ≤ c.On the other hand, (2.3.3) gives rankk(n/n2) = e, whence rankk(n2/n3) =

(e+12

)because P is regular. Due to the hypothesis on rankk(q2/q3) the surjective mapn2/n3 q2/q3 is an isomorphism. This gives fr ∈ n3 for b+ 1 ≤ r ≤ c, as desired.

GRADED COMPLETE INTERSECTIONS 7

Choose t1, . . . , te to minimally generate n. As f1, . . . , fn are in n2, we have

(2.3.4) fr =∑

16i6j6e

arijtitj for r = 1, . . . , n

with arij ∈ P . Let a denote the image in k of a ∈ R. We then have

arij = 0 for b+ 1 ≤ r ≤ n(2.3.5)

rankk(arij)16r6b16i6j6e = b(2.3.6)

Indeed, the set titj16i6j6e yields a basis of n2/n3, so the inclusions fr ∈ n3 give(2.3.6), while the linear independence of the images of f1, . . . , fb in I/(n3 ∩ I) andthe isomorphisms I/(n3 ∩ I) ∼= (I + n3)/n3) ⊆ n2/n3 imply (2.3.6).

The ring homomorphisms in (2.3.3) induce a commutative diagram

0 // Σ−2(J ′∗) //

πR // πQ′ //

0

0 // Σ−2(J∗)ζ

//

πR // πQ //

0

0 // π>2R

// πR // πP // 0

of graded Lie algebras; see 2.2. The rows are exact, by 2.2.3 for the top two and by2.2.2 for the bottom one. Due to the commutativity of the diagram, the maps inthe right-hand column are surjective and those in the left-hand one are injective.

Let φ1, . . . , φc be the basis of π2R, dual to the basis of I/nI defined by

f1, . . . , fn; see 2.2.2. In cohomological degree 2 the maps in the left-hand columnare the k-duals of the canonical maps I/mI → J/qJ → J ′/q′J ′. Thus, φ1, . . . , φband φ1, . . . , φc are bases for the images of Σ−2(J ′∗) and of Σ−2(J∗), respectively.

Let ϑ1, . . . , ϑe be the basis π1R, dual to the basis of n/n2 defined by t1, . . . , te;

see 2.2.1. Sjodin [17, Theorem 4], see also [5, 10.2.2], gives equalities

[ϑi, ϑj ] =b∑r=1

arijφr for 1 ≤ i < j ≤ e

ϑ2i =

b∑r=1

ariiφr for 1 ≤ i ≤ e

where summations go up only to b, due to (2.3.5).Let [π1

R, π1R] denote the subspace pf π2

R, spanned by the commutators and thesquares of the elements in π1

R. The formulas displayed above give an inclusion[π1R, π

1R] ⊆ (φ1, . . . , φb). Equality holds, because by (2.3.6) we can select b equations

with linearly independent right-hand sides and solve them for φ1, . . . , φb. Any twoelements ϑ, ξ in π1

R satisfy the relation [ϑ, ξ] = (ϑ+ξ)2−ϑ2−ξ2, so the space [π1R, π

1R]

is spanned by the squares of the elements in π1R. Thus, there exist τ1 . . . , τb in π1

R

such that τ21 . . . , τ2b is a basis of the image of Σ−2(J ′∗).With χr = τ2r for r = 1, . . . , b and χr = φr for r = b+ 1, . . . , c we get a basis of

the subspace Im(ζ) of π2R that satisfies (2.3.1) by choice and (2.3.2) due to 2.2.3.

8 L. L. AVRAMOV AND Z. YANG

2.4. Assume R ∼= Q/J , where (Q, q, k) is a complete local ring and J an ideal

generated by Q-regular sequence of contained in q2. Set N = R⊗R N and

Λ = TorQ(R, k) and N = TorQ(N , k)

Homology products give k-linear maps Λi ⊗k Nj → Ni+j that turn Λ into a graded-commutative k-algebra with Λ0 = k and N into a left graded module; see [6, §1].

Let E be the Koszul complex on some minimal set of generators of J . It is a

minimal Q-free resolution of R, which gives isomorphisms of graded k-algebras

(2.4.1) TorQ(R, k) = H(E ⊗R k) = E ⊗R k

Thus, Λ is the exterior algebra on Σ(J/qJ), where we have used TorQ1 (R, k) = J/qJ .

When L is a graded Λ-module ExthΛ(L, k) is graded by total cohomological degree;see [6, 3.2]. Composition products turn S = ExtΛ(k, k) into a graded k-algebra andExtΛ(L, k) into a graded S-module, and S is the symmetric algebra on Σ−2((Λ1)∗).

We recall the definitions of shifts for graded modules over Λ and E .

2.5. Let p be an integer, L a graded Λ-module, and L a graded S-module.A graded Λ-module ΣpL is defined by setting (ΣpL)j = Lj−p and letting the

product of x ∈ Λi and y ∈ ΣpLj be the element (−1)ipxy ∈ Lj+i−p = (ΣpL)j+1.A graded E-module ΣpL is defined by setting (ΣpL)j = Lj+p and letting the

product of of χ ∈ E i and υ ∈ ΣpLj be the element (−1)ipχυ ∈ Li+j+p = (ΣpL)i+j .There are natural isomorphisms of graded modules over E and S, respectively:

ExtΛ(ΣpL, k) ∼= Σ−p ExtΛ(L, k)(2.5.1)

ExtR(ΩlR(N), k) ∼= Σl Ext>lR (ΩlR(N), k)(2.5.2)

The following reduction technique plays a crucial role in the rest of the paper.

2.6. The image of the k-linear map (Λ1)∗ = J∗ζ−→ π2

R, see 2.2.3, lies in the cen-ter of E , so by 2.4 it extends to a canonical homomorphism of graded k-algebrasExtΛ(k, k) = S → E . In particular, ExtR(N, k) is canonically a graded S-module.

If some minimal Q-free resolution of N admits a structure of DG E-module, then

N = TorQ(N , k) yields for l ≥ 0 isomorphisms of graded left S-modules

(2.6.1) ExtR(ΩlR(N), k) ∼= Σl Ext>lΛ (N, k)

This comes from [6, 3.7] for l = 0, and (2.5.2) extends it to l ≥ 0.

3. Graded exterior algebras of rank two

In this section k denotes an algebraically closed field.Representation theory and homological algebra over graded exterior algebras

with two generators are the essential tools in the rest of paper. We start by recallingbasic facts concerning the representation theory of graded k-algebras.

3.1. Let G be a graded k-algebra with G0 = k and Gi = 0 for i < 0 or i > 0.We say that G has the graded Krull-Remak-Schmidt property if every finitely

generated graded G-module is a direct sum of indecomposable graded G-modulesand the summands in such sums are defined uniquely up to isomorphism. A catalogof graded indecomposables for G is a set that contains a single representative fromevery isomorphism class of indecomposable graded modules.

GRADED COMPLETE INTERSECTIONS 9

If rankkG is finite, or if G is finitely generated and commutative, then G has thegraded Krull-Remak-Schmidt property, with a catalog of graded indecomposablesgiven by the modules ΣpX with p ∈ Z and X ranging over the gradable modulesin some catalog of indecomposables for G; see Gordon and Green [10, 3.2, 4.1] orAuslander and Reiten [1, Propositions 8, 9], respectively.

Assume that G has the graded Krull-Remak-Schmidt property and N is a finitelygenerated graded G-module. The multiplicity of an indecomposable G-module Xin N is the largest s ∈ N for which N contains a direct summand isomorphic to Xs.We write µNX(p) for the multiplicity of ΣpX in N ; if (X [n])n∈N is a family of graded

indecomposables, then µNX(n, p) denotes the multiplicity of ΣpX [n] in N .

3.2. Let Λ be a graded exterior algebra k〈x1, x2〉, with x1, x2 ∈ Λ1

For each n ∈ N let X[n] denote any one of the Λ-modules described below.

3.2.1. The vector space A[n] with basis u1, . . . , un, v1, . . . , vn+1, where ui ∈ A[n]0

and vj ∈ A[n]1 , and Λ acts by the formulas x1vi = 0 = x2vi and

x1ui = vi for 1 ≤ i ≤ n and x2ui = −vi+1 for 1 ≤ i ≤ n

3.2.2. The vector space B[n] with basis u1, . . . , un, v1, . . . , vn−1, where ui ∈ B[n]0

and vj ∈ B[n]1 , and Λ acts by the formulas x1vi = 0 = x2vi and

x1ui =

vi for 1 ≤ i ≤ n− 1

0 for i = nand x2ui =

0 for i = 1

−vi−1 for 2 ≤ i ≤ n

3.2.3. The vector space C[n] with basis u1, . . . , un, v1, . . . , vn, where ui ∈ C[n]0 and

vj ∈ C[n]1 , and Λ acts by the formulas x1vi = 0 = x2vi and

x1ui =

vi+1 for 1 ≤ i ≤ n− 1

0 for i = nand x2ui = −vi for 1 ≤ i ≤ n

3.2.4. The vector spaces λD[n] for λ ∈ k, with bases u1, . . . , un, v1, . . . , vn, where

ui ∈ λD[n]0 and vj ∈ λD

[n]1 , and Λ acts by the formulas x1vi = 0 = x2vi and

x1ui = vi for 1 ≤ i ≤ n and x2ui =

λvi − vi+1 for 1 ≤ i ≤ n− 1

λvn for i = n

Notice that the subset u1, . . . , un ⊂ X[n]0 minimally generates X[n], and that

B[1] = k

3.3. The modules ΣpX[n] with (n, p) ∈ N × Z for X ∈ A,B,C, λD, and ΣpΛ withp ∈ Z form a catalog of graded indecomposables for Λ.

Indeed, the modules listed above form a catalog of non-graded indecomposablesfor Λ by a celebrated result of Kronecker; see Dieudonne [8] for a modern proof. In3.2 we showed that every indecomposable is gradable, so 3.1 gives the assertion.

3.4. Let S be the polynomial ring k[χ1, χ2], with χ1, χ2 ∈ S2.For each n ∈ N, let X[n] denote any one of the S-modules described below:

A[n] = Σ2−2n Homk

(S

(χ1, χ2)n, k

)B[n] = Σ2n−2 (χ1, χ2)n−1

C[n] = Σ2n (χ1, χ2)n

(χ2)nλD[n] = Σ2n (χ1, χ2)n

(χ1 + λχ2)nfor λ ∈ k

10 L. L. AVRAMOV AND Z. YANG

Notice that X[n] is minimally generated by n elements in X 0[n], and that

A[1] = k and B[1] = S

Next we describe the cohomology of the indecomposable graded Λ-modules.

3.5. For each n ≥ 1 there are isomorphisms of graded S-modules

ExtΛ(A[n], k) ∼= A[n] ⊕ Σ−(2n+1)S ExtΛ(B[n], k) ∼= B[n]ExtΛ(C[n], k) ∼= C[n] ExtΛ(λD[n], k) ∼= λD[n]

ExtΛ(Λ, k) ∼= k

They are proved in [6, §4], where the indecomposable Λ-modules carry differentnames and cohomological notation is used. The translation, with n ≥ 1, follows:

L(n) = Σ−1A[n] , L(1− n) = B[n] ,

M(n,∞) = C[n] , M(n, λ) = λD[n] for λ ∈ k

Furthermore, by [6, 3.12] the graded S-module ExtΛ(X[n], k) is equal to the totalcohomology of a complex of free graded S-modules constructed from X[n]. Theone associated to L(n) is denoted by L•(−n) for n ∈ Z, and the one associated toM(n, λ) is denoted by M•(n, λ) for n ∈ N and λ ∈ k ∪ ∞. The cohomology ofthese complexes is computed in [6, 4.5.2, 4.5.3], giving the results recorded above.

Recall that when N is an S-module AssS(N ) denotes the set of prime ideals ofS associated to N , and when N is graded its associated primes are homogeneous.

3.6. Lemma. The S-modules X[n] in 3.4 are indecomposable and satisfy

AssS(A[n]) = p with p = (χ1, χ2) and (A[n])p ∼= Homk

(Sp/pnSp, k

)AssS(B[n]) = p with p = (0) and (B[n])p ∼= SpAssS(C[n]) = p with p = (χ2) and (C[n])p ∼= Sp/pnSp

AssS(λD[n]) = p with p = (χ1 + λχ2) and (λD[n])p ∼= Sp/pnSp

Proof. The formulas for A[n] hold because (χ1, χ2)nA[n] = 0, and for B[n] because

its shift is an ideal. Setting Ji = (χi1χn−i2 , . . . , χn2 ) we get inclusions of S-modules

0 ⊂ J1/(χn2 ) ⊂ · · · ⊂ Jn/(χn2 ) = C[n]with subquotients isomorphic to S/(χ2); the formulas for C[n] follows. A similarargument, where χ2 is replaced with χ1 + λχ2, proves the formulas for λD[n].

As AssS(X[n]) = p, every non-zero submodule Y of X[n] has AssS(Y) = p;in particular, Yp is non-zero. Thus, a non-trivial decomposition of X[n] over Swould yield a non-trivial decomposition for (X[n])p over the local ring Sp withmaximal ideal pSp. However, (X[n])p is indecomposable: For X 6= A this holdsbecause (X[n])p is cyclic. The module (A[n])p has finite k-rank, so it is isomorphic

to its double dual over k. Thus, it is enough to show that Homk

((A[n])p, k

)is

indecomposable over Sp; to see this, just note that it is isomorphic to Sp/pnSp.

4. Local complete intersections of codimension two

Let (Q, q, k) be a local ring with k algebraically closed and g1, g2 a Q-regularsequence in q2. Set R = Q/J where J = (g1, g2). Using Theorem 2.3, we identify

• TorQ(R, k) and the graded exterior algebra Λ = k〈x1, x2〉 with xi ∈ Λ1.

GRADED COMPLETE INTERSECTIONS 11

• ExtΛ(k, k) and the graded polynomial ring S = k[χ1, χ2] with χi ∈ S2.• S and a central graded subalgebra of E = ExtR(k, k).

Let N 6= 0 denote a finitely generated R-module and set N = TorQ(N, k).Let µN

X(n, p) for X ∈ A,B,C, λD and µNΛ (p) denote the multiplicities in N of the

indecomposable graded Λ-modules ΣpX[n] and ΣpΛ, respectively; see 3.3.Our goal is to find restrictions on the graded Λ-module structure of N. The first

one is read off a minimal Q-free resolution of N ; it was noted in [6, 4.9].

4.1. Proposition. The graded Λ-module N = TorQ(N, k) satisfies the equality

(4.1.1)∑n,p

(−1)pµNA(n, p) =

∑n,p

(−1)pµNB(n, p)

Proof. For each Λ-module N, set ε(N) =∑j(−1)j rankk Nj . We have

ε(ΣpA[n]) = (−1)p+1 and ε(ΣpB[n]) = (−1)p

and ε(ΣpC[n]) = ε(ΣpλD[n]) = ε(ΣpΛ) = 0 for (n, p) ∈ N× Z; see 3.2. Thus, we get

ε(N) =∑

X,n,p

µNX(n, p)ε(ΣpX[n]) =

∑n,p

(−1)p+1µNA(n, p) + (−1)p

∑n,p

µNB(n, p)

Recall now that ε(N) = 0 holds, as the Q-module N has a non-zero annihilator.

4.2. Tracking projective dimensions in a minimal R-free resolution of N gives

(4.2.1) proj dimQ ΩlR(N) = sup2,proj dimQN − l

When proj dimQN ≤ 2 Iyengar [15, 2.1] shows that any minimal free resolutionof N over Q has a structure of DG module over the Koszul complex E on g1, g2;see also [5, 2.2.5]. In view of 2.6, there are isomorphisms of graded S-modules

(4.2.2) ExtR(ΩlR(N), k) ∼= Σl Ext>lΛ (N, k)

The finer points in the proof of the next result involve tracking the degrees ofthe basis elements of free direct summands of the S-module ExtR(N, k).

4.3. Theorem. Assume proj dimQN = 2 holds, and for p ∈ Z set

(4.3.1) aNp = supn | µN

A(n, p) 6= 0

When L = ΩlR(N) for some l ≥ 0 the Λ-module L = TorQ(L, k) satisfies

µLA(n, p) = 0 for n ≥ 1 if p 6= 0, 1 or l ≥ 2aN

p + p− 1 .(4.3.2)

µLB(1, p) = 0 if p 6= 0, 1, 2 .(4.3.3)

µLB(1, 2) = 0 if l ≥ max2aN

p + pp=0,1 .(4.3.4)

µLB(n, p) = 0 for n ≥ 2 if p 6= 0, 1.(4.3.5)

µLC(n, p) = 0 for n ≥ 1 if p 6= 0, 1.(4.3.6)

µ LλD(n, p) = 0 for n ≥ 1 if p 6= 0, 1.(4.3.7)

µLΛ(p) = 0 if p 6= 0 or l ≥ 1 .(4.3.8)

We single out a case with easily verified hypotheses and uniform conclusion. Inview of 3.5 the corollary provides a more precise version of [6, 4.1].

12 L. L. AVRAMOV AND Z. YANG

4.4. Corollary. Set βQi = βQi (N) and let l be an integer satisfying

l ≥ max2βQ0 , 2(βQ1 − βQ0 )− 1

For all n ≥ 1 and λ ∈ k the following equalities hold:

µLA(n, p) = µL

Λ(p) = 0 for p ∈ Z(4.4.1)

µLB(n, p) = µL

C(n, p) = µ LλD(n, p) = 0 for p 6= 0, 1(4.4.2)

Proof. For brevity, set ap = aNp . The definitions yield the inequalities below

a0 ≤ rankk N0 = βQ0

a1 + 1 = rankk(ΣA[a1])2 ≤ rankk N2 = βQ2 = βQ1 − βQ0

Thus, the hypothesis on l implies the bounds in (4.3.2), (4.3.4), and (4.3.8).

Proof of Theorem 4.3. Since proj dimQ L ≤ 2, see 4.2, every indecomposable directsummand X of L has Xi = 0 for i 6= 0, 1, 2. By consulting the catalog from 3.3we see that X must be isomorphic to Λ, to ΣpX[n] for n ≥ 1 and p = 0, 1 withX ∈ A,B,C, λD, or to Σ2B[1]. This proves the assertions that do not depend on l.

Next we prove (4.3.8). Assume, by way of contradiction, that L has a directsummand N isomorphic to Λ. The hypothesis l ≥ 1 provides an exact sequence

(4.4.3) 0→ L→ F →M → 0

of R-modules, where M = Ωl−1R (N) and F →M is a projective cover. It induces a

homomorphism of graded Λ-modules ι : L→ F, where F = TorQ(F, k). As the map

F ⊗Q k →M ⊗Q k is bijective, the homology exact sequence of TorQ(?, k) definedby (4.4.3) implies ι0(L0) = 0, hence ι(L) = 0, and thus Ker(ι2) ⊇ L2 6= 0. On the

other hand, we have TorQ3 (M,k) = 0, so the same homology exact sequence showsthat ι2 is injective. This is a contradiction, which proves µL

Λ(p) = 0.Assume that we have µL

B(1, 2) 6= 0 or µLA(n, p) 6= 0 with p = 0, 1. From 3.5 we see

that the S-module ExtR(L, k) contains a direct summand isomorphic to Σ−rS withr = 2 or r = 2n+ 1 + p, respectively. From (2.6.1) we see that ExtΛ(N, k) containsa direct summand isomorphic to Σ−r−lS. As we already know, N is a direct sumof indecomposable Λ-modules X with Xi = 0 for i 6= 0, 1, 2. In view of 3.5, those Xthat contribute to ExtΛ(N, k) a direct summand of the desired form are isomorphicto ΣqA[m] with 2m+ 1 + q = 2 + l or 2m+ 1 + q = r + l, respectively. Both Λ andS have the graded Krull-Remak-Schmidt property, see 3.1, so N has to contain asummand of one of those types. Accordingly, we get

l = 2m+ 1 + q − 2 = 2m+ q − 1 ≤ 2aNq + q − 1 or

l = 2m+ 1 + q − r ≤ 2m+ q − 2 ≤ 2aNq + q − 2 ,

respectively. Thus, l ≥ max2aNp + pp=0,1 implies µL

B(1, 2) = 0, and l ≥ 2ap+ p− 1

implies µLA(n, p) = 0. This finishes the proofs of (4.3.4) and (4.3.2).

Now we turn to results that depends on the existence of quadratic relations.

4.5. Let N be a graded E-module. The vector space underlying N decomposes asN even ⊕ N odd, where (N even)i = N i for even i and (N even)i = 0 for odd i, andN odd is defined accordingly. These are S-submodules, because Sodd = 0.

Since S is in the center of E , left multiplication with τ1 induces S-linear maps

(4.5.1) α : N even → N odd and β : N odd → N even

GRADED COMPLETE INTERSECTIONS 13

of degree one. Their compositions yield S-linear maps

(4.5.2) βα : N even → N even and αβ : N odd → N odd

of degree two that are given by left multiplication with χ1.Recall that when J is a homogeneous ideal of S the elements of N annihilated

by some power of J form a graded submodule, denoted by H0J (N ).

The proof of the next theorem hinges on utilizing the action on ExtR(N, k) of asubalgebra of E that strictly contains S.

4.6. Theorem. Assume J/(q3 ∩ J) 6= 0 and pick χ1 = τ21 for some τ1 ∈ E1; seeTheorem 2.3. Assume also proj dimQN = 2 and let L = ΩlR(N) for some l ≥ 0.

The numbers aNp from (4.3.1) satisfy the inequalities

(4.6.1) aN1 − 1 ≤ aN

0 ≤ aN1 + 1

and the following equalities hold for the graded Λ-module L = TorQ(L, k):

µLC(n, 0) = µL

C(n, 1) for n ≥ 1.(4.6.2)

µ LλD(n, 0) = µ L

λD(n, 1) for n ≥ 1 if λ 6= 0.(4.6.3)

µL0D(n, 0) = µL

0D(n, 1) for n ≥ 1 if rankk J/(q3 ∩ J) = 2.(4.6.4)

Proof. Set N = ExtR(N, k).Let first p denote the maximal ideal (χ1, χ2) of S. Due to the isomorphisms in

3.5, Lemma 3.6 and Theorem 4.3 yield isomorphisms of graded S-modules

H0p(N even) ∼=

⊕n>1

(A[n]

)µNA(n,0) and H0

p(N odd) ∼=⊕n>1

(Σ−1A[n]

)µNA(n,1)

that we use to identify the parties involved. In particular, we get

maxi | (H0p(N even))i 6= 0 = maxi | Ai[n] 6= 0µN

A(n,0)6=0 = 2aN0 − 2

maxi | (H0p(N odd))i 6= 0 = maxi | Ai[n] 6= 0µN

A(n,1)6=0 = 2aN1 − 1

From 4.5 we get, by functoriality, degree one homomorphisms

α′ : H0p(N even)→ H0

p(N odd) and β′ : H0p(N odd)→ H0

p(N even)

of graded A-modules, such that α′β′ and β′α′ are given by multiplication with χ1.It is evident that the first inequality in (4.6.1) holds when aN

1 ≤ 1, and that thesecond inequality holds when aN

0 ≤ 1. When aN1 ≥ 2 the inclusions

τ1(H0p(N even)2n−4) ⊇ χ1((Σ−1A[n])

2n−3) = (Σ−1A[n])2n−1 = A2n−2

[n] 6= 0

give H0p(N even)2n−4 6= 0, hence 2aN

1 − 4 ≤ 2aN0 − 2. When aN

0 = n ≥ 2, from

τ1(H0p(N odd)2n−3) ⊇ χ1(A2n−4

[n] ) = A2n−2[n] 6= 0

we get H0p(N odd)2n−3 6= 0, hence 2aN

0 − 3 ≤ 2aN1 − 1. Now (4.6.1) is proved.

As proj dimQ L = 2, see 4.2, for the rest of the proof we may assume L = N .

Let (X[n], p) stand for (C[n], (χ2)) or for (λD[n], (χ1 + λχ2)). As above, we get

(N even)p ∼=⊕n>1

(Sp/pnSp

)µNX(n,0) ⊕

(Sp)∑

n>1(µNA(n,1)+µ

NB(n,0)+µ

NB(n,2))

(N odd)p ∼=⊕n>1

(Sp/pnSp

)µNX(n,1) ⊕

(Sp)∑

n>1(µNA(n,0)+µ

NB(n,1))

14 L. L. AVRAMOV AND Z. YANG

as Sp-modules, and homomorphisms of Sp-modules

αp : (N even)p → (N odd)p and βp : (N odd)p → (N even)p

such that αpβp and βpαp are given by multiplication with χ1. This element isinvertible in Sp when λ 6= 0, so (N even)p and (N odd)p are isomorphic Sp-modules.Since Sp is a discrete valuation ring with maximal ideal pSp, the structure theoremfor finitely generated modules over PIDs implies µN

X(n, 0) = µNX(n, 1) for n ≥ 1.

When rankk J/(q3 ∩ J) = 2 Theorem 2.3 yields also χ2 = τ22 for some τ2 ∈ E1.

The preceding argument, with τ1 replaced by τ2, applies to the pair (0D[n], (χ1)).

5. Bigraded exterior algebras of rank two

In this section k denotes an algebraically closed field.Here we refine the representation theory and homological algebra over graded

exterior algebras with two generators in the presence of an additional grading.

5.1. Fix integers d1 and d2 that satisfy 1 ≤ d1 ≤ d2 and set d = d2 − d1.Let Λ be the exterior algebra k〈x1, x2〉, bigraded by xr ∈ Λ1,dr for r = 1, 2.A bigraded Λ-module is a graded Λ-module equipped with a direct sum decom-

position Ni =⊕

i∈Z Ni,j satisfying xrNi,j ⊆ Ni+1,j+dr .

For (p, q) ∈ Z2 the bigraded Λ-module ΣpN(q) has ΣpN(q)i,j = Ni−p,j+q, and theproduct of xr and y ∈ ΣpN(q)i,j is (−1)pxry ∈ Ni+1−p,j+q+dr = ΣpN(q)i+1,j+dr .

Indecomposability of bigraded modules and the bigraded Krull-Remak-Schmidtproperty of algebras are defined by the obvious extensions of the notions from 3.1.The Gordon-Green results recalled there also extend; see [16, 9.6.6, 9.6.7].

5.2. Lemma. A catalog of bigraded indecomposables is provided by the modules

ΣpX[n](q) with (n, p, q) ∈ N× Z2 for X ∈ A,B,C

ΣpλD[n](q) with (n, p, q) ∈ N× Z2 and

λ ∈ k if d1 = d2

λ = 0 if d1 < d2(5.2.1)

ΣpΛ(q) with (p, q) ∈ Z2

obtained from the eponymous modules in 3.2 by bigrading them as follows:

ui ∈ A[n]0,d(i−1) for 1 ≤ i ≤ n and vi ∈ A

[n]1,d(i−1)+d1 for 1 ≤ i ≤ n+ 1

ui ∈ B[n]0,d(n−i) for 1 ≤ i ≤ n and vi ∈ B

[n]1,d(n−i)+d1 for 1 ≤ i ≤ n− 1

ui ∈ C[n]0,d(n−i) for 1 ≤ i ≤ n and vi ∈ C

[n]1,d(n−i)+d2 for 1 ≤ i ≤ n

ui ∈ λD[n]0,d(i−1) for 1 ≤ i ≤ n and vi ∈ λD

[n]1,d(i−1)+d1 for 1 ≤ i ≤ n

Proof. It is easy to verify that for the listed modules the assigned bidegrees agreewith the multiplications tables in 3.2. The only modules D from the catalog in 3.3that are not list above are isomorphic to Σp

λD[n](q) for some λ 6= 0, when d1 < d2.Thus, in order to finish the proof we need to show that D cannot be bigraded.

Assume the contrary and set j = minj′ ∈ Z | Dp,j′ 6= 0. Multiplication by x1yields a bijective k-linear map Dp,∗ → Dp+1,∗, so minj′ ∈ Z | Dp+1,j′ 6= 0 = d1+j.As λ 6= 0 multiplication by x2 also is bijective, so minj′ ∈ Z | Dp+1,j′ 6= 0 = d2+j.This is impossible because d1 6= d2, so we have the desired contradiction.

Next we review the bigradings induced on cohomology.

GRADED COMPLETE INTERSECTIONS 15

5.3. Let N be a bigraded Λ-module, such that rankk N is finite.For each i ∈ Z the k-space N j = ExtjΛ(N, k) decomposes as N j =

⊕j∈ZN i,j ;

the index j is induced by the second gradings of the modules in the resolution.Composition products preserve bigradings: Si,jN i′,j′ ⊆ N i+i′,j+j′ . They turn S

into a bigraded k-algebra and N into a bigraded S-module. As in Theorem 2.3(3),S is the polynomial ring k[χ1, χ2] with χr ∈ S2,dr for r = 1, 2.

For (p, q) ∈ Z2 the bigraded S-module ΣpN (q) has ΣpN (q)i,j = N i+p,j+q, and

the product of χ ∈ Si,j and υ ∈ (ΣpN (q))i′,j′ equals χυ. The analog of (2.5.1) is

the existence of natural isomorphisms of bigraded S-modules

(5.3.1) ExtΛ(ΣpN(q), k) ∼= Σ−p ExtΛ(N, k)(−q)

5.4. For n ∈ N and admissible λ, see (5.2.1), define bigraded S-modules

A[n] = Σ2−2n Homk

(S

(χ1, χ2)n, k

)(nd2 − d2) B[n] = Σ2n−2 (χ1, χ2)n−1(nd1 − d1)

C[n] = Σ2n (χ1, χ2)n

(χ2)n(nd1) λD[n] = Σ2n (χ1, χ2)n

(χ1 + λχ2)n(nd1 + d)

by endowing them with the bigradings induced from the bigrading of S.The bigraded S-modules X[n] listed above have the following properties:

(5.4.1) X 0,0[n]∼= k and X 0,j

[n] = 0 for j < 0

They are minimally generated by n elements in X 0,∗[n] , and satisfy

A[1] = k and B[1] = S

5.5. For each n ≥ 1 and every admissible λ, see (5.2.1), there are isomorphisms

ExtΛ(A[n], k) ∼= A[n] ⊕ Σ−(2n+1)S(nd1 + d) ExtΛ(B[n], k) ∼= B[n]ExtΛ(C[n], k) ∼= C[n] ExtΛ(λD[n], k) ∼= λD[n]

ExtΛ(Λ, k) ∼= k

of bigraded S-modules. Indeed, 3.5 shows that both sides of each formula areisomorphic up to some shift of internal degrees; that shift is identified from (5.4.1),

since Ext0,jΛ (X[n], k) is non-zero for j = 0 and is zero for j < 0.

6. Graded complete intersections of codimension two

Throughout this section k denotes an algebraically closed field, Q = k[t1, . . . , te]a standard graded polynomial ring, q the ideal (t1, . . . , te), and g1, g2 a regularsequence of forms. We set dr = deg gr for r = 1, 2 and assume d2 ≥ d1 ≥ 2.

Let R be the graded ring Q/(g1, g2) and N a finitely generated graded R-module.

6.1. Set Λi,j = grTorQi (R, k)j and Ni,j = grTorQi (N, k)j , where the modules on theright come from resolutions of R and N consisting of graded free Q-modules andhomomorphisms of degree zero. They yield natural direct sum decompositions

TorQi (R, k) =⊕j∈Z

Λi,j and TorQi (N, k) =⊕j∈Z

Ni,j

that are compatible with homology products. The homology products of 2.4 arecompatible with the bigradings and turns N = TorQ(N, k) into a bigraded module

16 L. L. AVRAMOV AND Z. YANG

over Λ-module, where Λ = TorQ(R, k). The graded analog of Theorem 2.3(2)identifies Λ and the bigraded exterior algebra k〈x1, x2〉 from 5.1, with xr ∈ Λ1,dr .

Let µNX(n, p; q) for X ∈ A,B,C, λD and µN

Λ (p; q) denote the multiplicities in Nof the indecomposable bigraded Λ-modules ΣpX(q) and ΣpΛ(q), respectively; seeLemma 5.2. We form the following aggregated numerical invariants:

µNX =

∑n,p,q

(−1)pµNX(n, p; q) and µN

Λ =∑p,q

(−1)pµNΛ (p; q)(6.1.1)

νNX =

∑n,p,q

(−1)pµNX(n, p; q)n and νN

Λ =∑p,q

(−1)pµNΛ (p; q)n(6.1.2)

κNX =

∑n,p,q

(−1)pµNX(n, p; q)q and κN

Λ =∑p,q

(−1)pµNΛ (p; q)q(6.1.3)

Our main results show that these invariants are related in non-trivial ways.The first formula in the next theorem is a formal consequence of (4.1.1), see

(6.3.1) below, but the second one does not even make sense unless N is bigraded.

6.2. Theorem. The bigraded Λ-module N = TorQ(N, k) satisfies the equalities

µNB = µN

A(6.2.1)

κNB = κN

A + d1

(νN

B +∑λ

ν NλD

)+ d2

(νN

A + νNC

)(6.2.2)

Proof. The argument uses computations with Hilbert series of graded vector spacesV with rankk Vj finite for j ∈ Z and Vj = 0 for j 0; for such V we set

HV (y) =∑j∈Z

(rankk Vj)yj ∈ Z[[y]][y−1]

If, in addition, V is bigraded and Vi,j = 0 for i 0, then we set

HV (y, z) =∑

(i,j)∈Z2

(rankk Vi,j)yjzi ∈ Z[[y, z]][(yz)−1]

With dim denoting Krull dimension, the Hilbert-Serre Theorem gives an equality

HN (y) =hN (y)

(1− y)dimNfor some hN (y) ∈ Z[y] with hN (1) 6= 0

and a similar one for Q. A minimal Q-free resolution of N gives an exact sequence

0→ Q⊗k Ns,∗ → · · · → Q⊗k N1,∗ → Q⊗k N0,∗ → N → 0

of graded Q-modules. It yields the first equality in the string

HN (y) =

s∑i=0

(−1)iHQ(y)

(∑j∈Z

rankk Ni,jyj

)=

hQ(y)

(1− y)dimQHN(y,−1)

Setting m = dimQ− dimN we obtain in Z[y] an equality

HN(y,−1)hQ(y) = (1− y)mhN (y)

As hQ(1) 6= 0 we see that HN(y,−1) is divisible by (1− y)m, and hence by (1− y)2

because (g1, g2)N = 0 implies m ≥ 2. Thus, writing f ′ for ∂f/∂y we get

(6.2.3) HN(1,−1) = 0 and H ′N(1,−1) = 0 ,

GRADED COMPLETE INTERSECTIONS 17

By expressing N as a direct sum of indecomposable we get an equality

(6.2.4) HN(y, z) = A(y, z) +B(y, z) + C(y, z) +∑λ

λD(y, z) + E(y, z)

where the summands have the following values:

(6.2.5) X(y, z) =

∑n,p,q

µNX(n, p; q) yqHX[n](y, z)zp for X = A,B,C, λD∑

p,q

µNΛ (p; q) yqHΛ(y, z)zp for X = E

In view of Lemma 5.2, the Hilbert series in (6.2.5) are given by the formulas

(6.2.6)

HA[n](y, z) = gn(y) + yd1gn+1(y)z HB[n](y, z) = gn(y) + yd2gn−1(y)z

HC[n](y, z) = gn(y)(1 + yd2z) HλD[n](y, z) = gn(y)(1 + yd1z)

HΛ(y, z) = (1 + yd1z)(1 + yd2z)

where the right hand sides are expressed in terms of the polynomials

gn(y) = 1 + yd2−d1 + · · ·+ y(d2−d1)(n−1)

The values at 1 of gn(y) and of its derivative g′n(y) are given by

gn(1) = n and g′n(1) = (d2 − d1)n(n− 1)/2

By using first formulas (6.2.4), (6.2.5) and (6.2.6), then (6.1.1) we obtain

HN(1,−1) =∑n,p,q

(−1)p+1µNA(n, p; q) +

∑n,p,q

(−1)pµNB(n, p; q) = µN

B − µNA

Since HN(1,−1) = 0 holds, by (6.2.3), this implies the validity of (6.2.1).Differentiating the equalities (6.2.6) and evaluating the results at (1,−1) we get

(6.2.7)

H ′ΣpA[n](q)(1,−1) = (−1)p+1(nd2 + d1 + q)

H ′ΣpB[n](q)(1,−1) = (−1)p+1(nd1 − d1 − q)

H ′ΣpC[n](q)(1,−1) = (−1)p+1nd2

H ′ΣpλD[n](q)(1,−1) = (−1)p+1nd1

H ′ΣpΛ(q)(1,−1) = 0

By using first (6.2.4), (6.2.5) and (6.2.7), then (6.1.1), (6.1.2) and (6.1.3), we obtain

−H ′N(1,−1) =∑n,p,q

(−1)pµNA(n, p; q)(nd2 + d1 + q)

+∑n,p,q

(−1)pµNB(n, p; q)(nd1 − d1 − q)

+∑n,p,q

(−1)pµNC(n, p; q)(nd2)

+∑λ

∑n,p,q

(−1)pµNλD(n, p; q)(nd1)

= d1

(µN

A − µNB + νN

B +∑λ

ν NλD

)+ d2

(νN

A + νNC

)+ κN

A − κNB

We have µNA = µN

B by (6.2.1) and H ′N(1,−1) = 0 by (6.2.3), so (6.2.2) follows.

18 L. L. AVRAMOV AND Z. YANG

Results transfer smoothly from the local context to the graded one. Validationsof this remark for homology and cohomology are presented in separate discussions.

6.3. Let Q and R be the completions of Q and R at the ideal Q = (t1, . . . , te).

As k = Q/q = Q/qQ and the canonical maps Q → Q and R → R are flat, the

R-module N = R⊗R N has proj dimQ N = proj dimQN and βRi (N) = βRi (N).

When l is a non-negative integer ΩlR(N) denotes the cokernel of the differential∂l+1 in some minimal graded free resolution of N over R. Here flatness yields

canonical isomorphisms R⊗R ΩlR(N) ∼= ΩlR

(N), and for each i isomorphisms⊕j∈Z

Λi,j ∼= TorQi (R, k) and⊕j∈Z

Ni,j ∼= TorQi (N , k)

that are compatible with homology products. We use them to identify TorQ(R, k)

and TorQ(N , k) and the graded objects underlying Λ and N, respectively.In particular, the multiplicities in Sections 6 and 4 are linked by the formulas∑

q

µNX(n, p; q) = µN

X(n, p) and∑q

µNΛ (p; q) = µN

Λ (p)(6.3.1)

for X ∈ A,B,C, λD; as in (5.2.1), the parameter λ may take the following values:

(6.3.2) λ ∈ k when d1 = d2 , respectively, λ = 0 when d1 < d2

6.4. Let E denote the graded k-algebra ExtR(k, k) and N its graded moduleExtR(N, k); see 2.1. As in 6.1, resolutions of k and N by graded free R-modulesand homomorphisms of degree zero define k-vector spaces E i,j = grExtiR(k, k)j andN i,j = grExtiR(N, k)j , respectively. They yield natural decompositions

(6.4.1) E i =⊕j∈ZE i,j and N i =

⊕j∈ZN i,j

that are compatible with composition products.Next we review some properties of E . It is clear that E0,∗ = k and that E1,∗

has a basis τ1, . . . , τe of elements in E1,1. To go further, note that analogs ofthe arguments for Theorem 2.3 can be performed in the graded setting by usingonly by homogeneous operations. More precisely, the relations g1, . . . , gc can bepreordered so that deg gr ≤ deg gr+1 holds, then only relations of degree 2 need tobe manipulated, and all the changes affecting them can be made with coefficientsin k. Thus, Theorem 2.3 has graded analogs.

Under the hypotheses in force in this section they show that E contains a centralbigraded polynomial algebra k[χ1, χ2] with χr ∈ E2,dr for r = 1, 2, that when d1 = 2we may assume τ1 = χ2

1, and when d1 = 2 = d2 we may assume τr = χ2r for r = 1, 2.

They also identify k[χ1, χ2] and the bigraded algebra S = ExtΛ(k, k), so thatthe following graded analog of 4.2.2 holds: For each (l, n) ∈ N0 × Z the bigraded

Λ-module N = TorQ(N, k) yields an isomorphism of bigraded S-modules

(6.4.2) ExtR(Ωl(N)(−q), k) ∼= Σl Ext>lΛ (N, k)(q)

The first two formulas in the next theorem directly concern the bigraded struc-ture of L, and so they cannot be obtained from results about local rings.

6.5. Theorem. Assume proj dimQN = 2 and let l be an integer satisfying

l ≥ max2βQ0 (N) , 2(βQ1 (N)− βQ0 (N))− 1

GRADED COMPLETE INTERSECTIONS 19

If d1 = 2 and L = Ωl(N), then there is an isomorphism

(6.5.1)⊕n,p,q

B[n](q)µLB(n,p;q) ∼=

s⊕j=1

B[nj ](qj)⊕s⊕i=1

ΣB[n′i](qi + 1) .

of bigraded Λ-modules, and the following equalities hold:

−κLB =

∑n,q

µLB(n, 0; q) = s(6.5.2)

µLB =

∑n,q

µLB(n, 0; q)−

∑n,q

µLB(n, 1; q) = 0(6.5.3)

Proof. Set s =∑n,q µ

LB(n, 0; q). In view of (6.3.1), we get µL

B = s−∑n,q µ

LB(n, 1; q)

from (4.4.2), and µLA =

∑n,q µ

LA(n, 0; q) −

∑n,q µ

LA(n, 1; q) = 0 from (4.4.1). Now

(6.2.1) gives µLB = µL

A = 0, so (6.5.3) holds.Since Λ has the bigraded Krull-Remak-Schmidt property, (6.5.3) yields

(6.5.4)⊕n,p,q

B[n](q)µLB(n,p;q) ∼=

s⊕j=1

B[nj ](qj)⊕s⊕i=1

ΣB[n′i](q′i) .

with nj 6= 0 6= n′i for 1 ≤ i, j ≤ s. We assume, as we may, that the following hold:

q1 ≤ · · · ≤ qs and q′1 ≤ · · · ≤ q′sWe will prove (6.5.1) by showing that q′i = qi + 1 holds for i = 1, . . . , s.

Choose χ1 and χ2 with χ1 = τ21 for some τ1 ∈ E1,1, see 6.4, and set

B = L/(

H0(χ2)(L) +

∑λ

H0(χ1+λχ2)(L)

)The local cohomology modules in the preceding formulas are bigraded submodulesof L. In view of the isomorphisms in 5.5, 5.4, and 5.3, formula (6.5.4) gives

B ∼=s⊕j=1

B[nj ](−qj)⊕s⊕i=1

Σ−1B[n′i](−q′i) .

Thus, we make the following identifications of the bigraded S-modules:

B even,∗ =

s⊕j=1

B[nj ](−qj) and B odd,∗ =

s⊕i=1

Σ−1B[n′i](−q′i)

As in 4.5, multiplication with τ1 induces S-linear maps

B even,∗ α−−→ B odd,∗ β−−→ B even,∗ α−−→ B odd,∗

of bidegree (1, 1). Both βα and αβ are given by multiplication with χ1, and so areinjective because B[n] is torsion-free; thus, α and β are injective.

Next we prove by induction that for each integer r with 0 ≤ r < s we have

q′1 = q1 + 1, . . . , q′r = qr + 1

The hypothesis is vacuous for r = 0, so we may suppose that the assertion has beenproved for some r ≥ 0. Assume, by way of contradiction, that qr+1 + 1 < q′r+1

holds. For 1 ≤ j ≤ r + 1 and r + 1 ≤ i ≤ s we then have inequalities

qj + 1 ≤ qr+1 + 1 < q′r+1 ≤ q′i

20 L. L. AVRAMOV AND Z. YANG

As (B[nj ](−qj))0,qj 6= 0 = (Σ−1B[n′i](−q′i))

1,qj+1, see (5.4.1), the composed map

αi,j : B[nj ](−qj) → Beven α−−→ B odd Σ−1B[n′i](−q

′i)

of bigraded S-modules satisfies Ker(αi,j)0,qj 6= 0.

Let F denote the field of fractions of S. From 5.4 we have inclusions

Σ2−2nB[n](2− 2n) = (χ1, χ2)n−1 ⊆ S

They induce canonical isomorphisms F ⊗S B[n] ∼= F . Treating these as identifica-

tions leads to equalities F ⊗S B even = F s = F ⊗S B odd. Accordingly, F ⊗S α isidentified with the s× s matrix (aij) over F , and so F ⊗S αi,j : F → F is given bymultiplication with ai,j ∈ F . Thus, Ker(F ⊗S αi,j) 6= 0 implies ai,j = 0, hence

F ⊗S α =

a1,1 · · · a1,r+1 a1,r+2 · · · a1,s...

. . ....

.... . .

...ar,1 · · · ar,r+1 ar,r+2 · · · ar,s0 · · · 0 ar+1,r+2 · · · ar+1,s

.... . .

......

. . ....

0 · · · 0 as,r+2 · · · as,s

The first r+ 1 columns of F ⊗S α are linearly dependent, so the F -rank of F ⊗S αis strictly smaller than s. This is impossible because α is injective, and hence so isF ⊗S α. The contradiction obtained shows that q′r+1 ≤ qr+1 + 1 holds.

Set L′ = Ωl+1R (N), and form the bigraded S-modules L′ = ExtR(L′, k) and

B′ = L′/(

H0(χ2) L

′ +∑λ

H0(χ1+λχ2) L

′)In view of (6.4.2), we have isomorphisms of bigraded S-modules

B′ even,∗ ∼=s⊕i=1

B′[n′i](−q′i) and B′ odd,∗ ∼=

s⊕j=1

Σ−1B′[nj+1](−qj − 2)

The conditions of the theorem apply to L′ as well as to L. The induction hypothesiscan be rewritten in the form

q1 + 2 = q′1 + 1, . . . , qr + 2 = q′r + 1

so the preceding stretch of the argument gives qr+1 + 2 ≤ q′r+1 + 1. It follows thatqr+1 + 1 ≤ q′r+1 holds, so we have proved the equality q′r+1 = qr+1 + 1.

The induction step is now complete, so q′i = qi + 1 holds for i = 1, . . . , s. Thisyields the isomorphism (6.5.1), which gives the second equality in the string

κLB =

∑n,p,q

(−1)pµLB(n, p; q)q =

s∑i=1

qi −s∑i=1

(qi + 1) = −s

This establishes (6.5.2), and so finishes the proof of the theorem.

7. Betti numbers over graded rings

We prove the results announced in the introduction, starting with Theorem B.

GRADED COMPLETE INTERSECTIONS 21

7.1. Theorem. Assume R = Q/(g1, g2), where Q is a standard graded k-algebraand g1, g2 is a Q-regular sequence with deg g1 = 2, and set d = depthR.

Let N be a finitely generated graded R-module, and set m = d− depthRN and

l = m+ max2βQm(N) , 2βQm+1(N)− 2βQm(N)− 1

If proj dimQN is finite, then for some a ∈ Z the following equalities hold:

βRi+l(N) = ai+ βRl (N) for i ≥ 0

Proof. Let k denote an algebraic closure of k. For each i ∈ Z there is a natu-

ral isomorphism TorR⊗kki (N ⊗k k, k) ∼= TorRi (N, k) ⊗k k, which gives an equality

βRi (N) = βR⊗kki (N ⊗k k). Thus, we may assume that k is algebraically closed.The discussions in 6.3 and 6.4, in particular the equalities (6.3.1), allow us to

use in the graded context results proved over local rings in Section 4.Set L = Ωl(N) and L = TorR(L, k). Dimension shifting and (6.4.2) give

βRi+l(N) = βRi+l−m(ΩmR (N)) = βRi (L) = βΛi (L) for i ≥ 0

We write βi for βΛi (L) proceed to show that the sequence (βi)i>0 is arithmetic.

From the Auslander-Buchsbaum Equality and (4.3.1) we get

proj dimQN = depthQ− depthQN = depthR+ 2− depthRN = m+ 2

Thus, Corollary 4.4 shows that L is a direct sum of Λ-modules isomorphic to ΣpX[n]

with X ∈ B,C, λD and p = 0, 1. Their multiplicities satisfy the following relations:

νLC =

∑n,q

µLC(n, 0; q)n−

∑n,q

µLC(n, 1; q)n = 0(7.1.1)

∑λ

ν LλD =

∑λ

∑n,q

µ LλD(n, 0; q)n−

∑λ

∑n,q

µ LλD(n, 1; q)n = ν L

0D(7.1.2)

κLB = 2

(νL

B + ν L0D

)(7.1.3)

Indeed, from formula (4.6.2) we obtain

νLC =

∑n,q

µLC(n, 0; q)n−

∑n,q

µLC(n, 1; q)n

=∑n

(∑q

µLC(n, 0; q)−

∑q

µLC(n, 1; q)

)n

= 0

A similar computation with (4.6.3) yields (7.1.2). Now (6.2.2) simplifies to (7.1.3).For each u ≥ 0, the isomorphisms in 3.5 provide equalities:

βΛ2u(B[n]) = u+ n βΛ

2u(C[n]) = n βΛ2u(λD[n]) = n

βΛ2u+1(B[n]) = 0 βΛ

2u+1(C[n]) = 0 βΛ2u+1(λD[n]) = 0

Referring again to Corollary 4.4, we obtain

β2u =∑n,q

µLB(n, 0; q)(u+ n) +

∑n,q

µLC(n, 0; q)n+

∑λ

∑n,q

µ LλD(n, 0; q)n

β2u+1 =∑n,q

µLB(n, 1; q)(u+ n) +

∑n,q

µLC(n, 1; q)n+

∑λ

∑n,q

µ LλD(n, 1; q)n

22 L. L. AVRAMOV AND Z. YANG

From these expressions and formulas (6.5.3), (7.1.1), and (7.1.2) we get

β2u+1 − β2u = −µLBu− νL

B − νLC −

∑λ

ν LλD

= −(νL

B + ν L0D

)The same references as above give the first two equalities in the string

β2u+2 − β2u+1 = µLBu+

∑n,q

µLB(n, 0; q) + νL

B + νLC +

∑λ

ν LλD

=∑n,q

µLB(n, 0; q) + νL

B + ν L0D

= −κLB + νL

B + ν L0D

= −(νL

B + ν L0D

)The last two equalities come from (6.5.2) and (7.1.3), respectively.

Thus, βRi (N) = −(νL

B + νL0D

)i+ βRl (N) holds for i ≥ 0.

The final result of the paper was stated in the introduction as Theorem A.

7.2. Theorem. For a standard graded ring R of depth d the following are equivalent.

(i) R ∼= P/(f1, f2), where P is a standard graded polynomial ring and f1, f2 isa P -regular sequence of forms with deg f1 = 2 ≤ deg f2.

(ii) (βRi (R/m2))i>d is an unbounded arithmetic sequence.(iii) (βRi (N))i0 is arithmetic for every finitely generated graded R-module N

and is unbounded for some graded R-module.

Proof. Theorem 1.3 gives the implications (iii) =⇒ (ii) =⇒ (i), due to the equalities

βRi (N) = βRi (N) in 6.3; the implication (i) =⇒ (iii) comes from Theorem 7.1.

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Luchezar L. Avramov, Department of Mathematics, University of Nebraska, Lincoln,

NE 68588, U.S.A.E-mail address: avramov@math.unl.edu

Zheng Yang, Department of Mathematics, University of Nebraska, Lincoln, NE

68588, U.S.A.E-mail address: s-zyang8@math.unl.edu