LAURENT COEFFICIENTS AND EXT

OF FINITE GRADED MODULES

Luchezar L. Avramov Ragnar-Olaf Buchweitz Judith D. Sally

Let R = ⨁_{n⩾0} R_{n} be a graded commutative ring generated over a field K = R_{0} by homogeneous elements x_{1}, …, x_{e} of positive degrees d_{1}, …, d_{e}. The Hilbert-Serre Theorem shows that for each finite graded R–module M = ⨁_{n∈ℤ} M_{n} the Hilbert series ∑_{n∈ℤ}(rank_{K} M_{n})t^{n} is the Laurent expansion around 0 of a rational function

H_{M}(t) = \frac{q_{M}(t)}{∏_{i=1}^{e}(1 − t^{d_{i}})}

with q_{M}(t) ∈ ℤ[t, t^{−1}]. We demonstrate that Laurent expansions [M]_{z} of H_{M}(t) around other points z of the extended complex plane \overline{ℂ} also carry important structural information.

When R is generated in degree one and z = 1 such an approach has been used implicitly, since the coefficients of the principal part of [M]_{1} are then also coefficients of the Hilbert polynomial of M. Without assumptions on the generation of R, coefficients of [M]_{1} have been sighted occasionally in invariant theory. In that context Benson and Crawley-Boevey [3] have discovered recently that for all finite graded modules over a graded normal domain the coefficient ψ of 1/(1 − t)^{dim R−1} satisfies an equality

(1)

ψ(Hom_{R}(M, N)) − ψ(Ext_{R}^{1}(M, N)) =
rank_{R}(M) rank_{R}(N)ψ(R) + rank_{R}(M)ψ(N) − rank_{R}(N)ψ(M) .

The present investigation started as an attempt to “explain” this intriguing formula, and to find out whether analogous equalities exist for subsequent coefficients of Laurent expansions around 1. Fairly complete answers are presented in Sections 3 and 4. They depend on embedding (1) into an equality of Laurent series, which “predicts” an infinite sequence of similar relations among higher Laurent coefficients. The results are proved in Section 1, and are used in Section 2 to obtain non-trivial lower bounds for the Bass numbers of certain graded modules of finite injective dimension.

Typeset by AMS-TEX

1980 Mathematics Subject Classification (1985 Revision). 13D40, 13F15, 13H10.

Research partially supported by NSF (L. L. A. and J. D. S.) and NSERC (R.-O. B.). .

1. Hilbert series

When M and N are graded R–modules Hom_{R}(M, N)_{n} stands for the K–vector subspace of Hom_{R}(M, N) consisting of homomorphisms α of degree n, that is, such that α(M_{j}) ⊆ N_{n+j} for j ∈ ℤ. For a finite M the inclusion ⨁_{n∈ℤ} Hom_{R}(M, N)_{n} ⊆ Hom_{R}(M, N) is an equality, and thus Hom_{R}(M, N) is graded; as M has a resolution by finite free graded R– modules and homomorphisms of degree 0, this induces for each i a grading on Ext_{R}^{i}(M, N).

Theorem 1. If M and N are finite graded R–modules such that Ext_{R}^{i}(M, N) = 0 for i ≫ 0, then there is an equality of rational functions

∑_{i}(−1)^{i}H_{Ext_{R}^{i}(M,N)}(t) = \frac{H_{M}(t^{−1}) · H_{N}(t)}{H_{R}(t^{−1})} .

Remark. This is a far reaching generalization of the known expression H_{Ext_{R}^{d−m}(M,ω)}(t) = (−1)^{m}H_{M}(t^{−1}) which holds for an m-dimensional graded Cohen-Macaulay module M over a d-dimensional Cohen-Macaulay ring R with canonical module ω, cf. [4; (4.3.7)].

Indeed, Ext_{R}^{i}(M, ω) = 0 unless i = d − m by [4; (3.3.10)], so we only need to show that H_{ω}(t) = (−1)^{d}H_{R}(t^{−1}). Consider the polynomial ring Q = K[X_{1}, …, X_{e}] graded by deg X_{j} = d_{j} and the surjective homomorphism of K–algebras Q ⟶ R with X_{j} ↦ x_{j} for 1 ≤ j ≤ e. By [4; (3.6.12), (3.6.10)], for W = Q(− ∑_{i=1}^{e} d_{i}) we have ω ≅ Ext_{Q}^{e−d}(R, W), hence we get H_{ω}(t) by applying the theorem to the Q–modules R and W.

When f(t) is a function of a complex variable t, we denote by [f(t)]_{z} the expansion of f(t) as a Laurent series ∑_{j∈ℤ} a_{j}(t − z)^{j} when z ∈ ℂ, and as a Laurent series ∑_{j∈ℤ} a_{j}t^{−j} when z = ∞. The order of [f(t)]_{z} is introduced by ord [f(t)]_{z} = inf{j ∈ ℤ | a_{j} ≠ 0}.

For a finite graded R–module M we often write [M]_{z} in place of [H_{M}(t)]_{z}. Clearly, [M]_{0} ∈ ℤ((t)) and [M]_{∞} ∈ ℤ((t^{−1})), where 𝔸((t)) = 𝔸[[t]][t^{−1}] and 𝔸((t^{−1})) = 𝔸[[t^{−1}]][t] denote rings of formal Laurent series of finite order with coefficients in 𝔸, with topologies defined by the powers of the respective indeterminate. Note that if {g_{i}}_{i⩾0} is a sequence of formal Laurent series in either ring, then ∑_{i⩾0} g_{i} converges when lim_{i→∞} ord g_{i} = ∞.

Much of the paper deals with properties of the rational function

ϕ_{R}(M, N)(t) = \frac{H_{M}(t^{−1}) · H_{N}(t)}{H_{R}(t^{−1})} .

Without homological assumptions on M or N, we establish the following limited

Proposition 2. If M and N are finite graded R–modules, and N has finite length, then H_{Ext_{R}^{i}(M,N)}(t) ∈ ℤ[t, t^{−1}], the order of [Ext_{R}^{i}(M, N)]_{∞} goes to infinity with i, and ∑_{i}(−1)^{i} [Ext_{R}^{i}(M, N)]_{∞} ∈ ℤ((t^{−1})) is the Laurent expansion around ∞ of ϕ_{R}(M, N)(t).

For comparison and for later application, we recall the situation in homology. The grading of M ⊗_{K} N by (M ⊗_{K} N)_{n} = ⨁_{j∈ℤ}(M_{j} ⊗_{K} N_{n−j}) induces a grading of M ⊗_{R} N. As above, this defines a grading of Tor_{i}^{R}(M, N) for each i, and by [1; Lemma 7] we have Lemma 3. If M and N are finite graded R–modules, then the order of [Tor_{i}^{R}(M, N)]_{0} ∈ ℤ((t)) goes to ∞ with i, and ∑_{i}(−1)^{i} [Tor_{i}^{R}(M, N)]_{0} ∈ ℤ((t)) is the Laurent expansion around 0 of the rational function

χ^{R}(M, N)(t) = \frac{H_{M}(t) · H_{N}(t)}{H_{R}(t)} .  □

Before starting on the proofs we record some general properties of Laurent expansions.

Remark. If 0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0 is an exact sequence of graded R–modules, then [M′]_{z} + [M′′]_{z} = [M]_{z}. Indeed, additivity is obvious for z = 0, hence yields an equality of rational functions H_{M′}(t) + H_{M′′}(t) = H_{M}(t), which implies additivity for arbitrary z.

Similarly, one sees that [M(a)]_{z} = [t^{−a}]_{z} [M]_{z} for any a ∈ ℤ, where M(a) is the a’th translate of M, that is, the graded R–module with M(a)_{n} = M_{a+n} for n ∈ ℤ.

Proof of Theorem 1. Let Q ⟶ R and W be as in the remark following the statement of the theorem, let I^{•} be a minimal (hence finite, cf. [4, (3.6.6)]) graded injective resolution of the Q–module W, and let F^{•} be a minimal graded free resolution of the R–module M. As in [2, (4.4.I)], there is a canonical isomorphism of complexes of graded R–modules

θ: F^{•} ⊗_{R} Hom_{Q}(N, I^{•}) ⟶ Hom_{Q}(Hom_{R}(F^{•}, N), I^{•})

given by θ(f ⊗_{R}α)(β) = (−1)^{i(j−h)}αβ(f) for f ∈ F^{i}, α ∈ Hom_{Q}(N, I^{j}), β ∈ Hom_{Q}(F^{h}, N). Filtering the complex on the left by the cohomological degree of F^{•}, and the one on the right by that of I^{•}, we obtain two spectral sequences which converge to a common limit:

2E_{p}^{q} = Tor_{p}^{R}(M, Ext_{Q}^{q}(N, W)) ⟹ E_{p−q} ⟸ Ext_{Q}^{q}(Ext_{R}^{p}(M, N), W) = ^{2}E^{p,q} .

In the first spectral sequence we have _{r}d_{p}^{q}: _{r}E_{p}^{q} ⟶ _{r}E_{p−r}^{q−r+1} for r ≥ 2, and _{2}E_{p}^{q} = 0 unless p ≥ 0 and 0 ≤ q ≤ e. It follows that E_{i} = 0 for i < −q, and that _{e+2}E_{p}^{q} = _{∞}E_{p}^{q}.

By Lemma 3 we know that for any fixed q the order of [Tor_{p}^{R}(M, Ext_{Q}^{q}(N, W))]_{0} ∈ ℤ((t))

goes to infinity together with p, hence ∑_{p}(−1)^{p} [_{2}E_{p}^{q}]_{0} is in ℤ((t)). As _{r}E_{p}^{q} is a subquotient of _{2}E_{p}^{q}, we further have ∑_{p}(−1)^{p} [_{r}E_{p}^{q}]_{0} ∈ ℤ((t)) for each r ≥ 2, and we set

χ(_{r}E) = ∑_{q=0}^{e}(−1)^{q} ∑_{p}(−1)^{p} [_{r}E_{p}^{q}]_{0} .

Next we show that χ(_{r}E) = χ(_{r+1}E) for r ≥ 2. To this end, we first note that up to any given degree n each series is determined by a finite number of the summands used to define it, hence when looking at the coefficient of t^{n} we may restrict both summations to the same finite set of indices p and q. In χ(_{r}E) this coefficient is the Euler characteristic of the degree n part of a finite complex of graded R–modules. As the differential _{r}d_{p}^{q} preserves the grading of these modules, the coefficient of t^{n} in χ(_{r+1}E) is the Euler characteristic of the degree n part of the homology of this complex. The classical permanence property of Euler characteristics implies that these coefficients are equal.

By the finite convergence of the spectral sequence, we thus see that the series χ(E) = ∑_{i}(−1)^{i} [E_{i}]_{0} is equal to χ(_{2}E). Together with Lemma 3 this implies

χ(E) = ∑_{q=0}^{e}(−1)^{q} ∑_{p}(−1)^{p} [Tor_{p}^{R}(M, Ext_{Q}^{q}(N, W))]_{0}

= ∑_{q=0}^{e}(−1)^{q} [χ_{R}(M, Ext_{Q}^{q}(N, W))(t)]_{0}

= [\frac{H_{M}(t)}{H_{R}(t)} ∑_{q=0}^{e}(−1)^{q}H_{Ext_{Q}^{q}(N,W))}(t)]_{0} .

In order to finish the computation we use the equality of rational functions

(2)  ∑_{q=0}^{e}(−1)^{q}H_{Ext_{Q}^{q}(A,W)}(t) = (−1)^{e}H_{A}(t^{−1}) .

which holds for each finite Q–module A. Indeed, if A = Q(b), then the only non-vanishing Ext is Ext_{Q}^{0}(Q(b), W)) ≅ Q(−b − ∑_{j=1}^{e} d_{j}), and the equality is checked by a direct computation which uses the expression H_{Q}(t) = 1/ ∏_{j=1}^{e}(1 − t^{d_{j}}). The general case follows, as each A has a finite resolution by finite direct sums of translates of Q, and both sides of the formula are additive functions on the Grothendieck group G(Q) of the category of finite graded Q–modules and homomorphisms of degree 0. Thus, we have established that

χ(E) = (−1)^{e} [\frac{H_{M}(t) · H_{N}(t^{−1})}{H_{R}(t)}]_{0} = (−1)^{e} [ϕ_{R}(M, N)(t^{−1})]_{0} .

Now we turn to the second spectral sequence. It has^{2}E^{p,q} = 0 unless p ≥ 0 and 0 ≤ q ≤ e, and differentials ^{r}d^{p,q}: ^{r}E^{p,q} ⟶ ^{r}E^{p−r+1,q−r}, hence ^{e+2}E^{p,q} = ^{∞}E^{p,q}. Furthermore, our assumption implies that ^{r}E^{p,q} = 0 for p ≫ 0, hence for each r ≥ 2 the sum

χ(^{r}E) = ∑_{p}(−1)^{p} ∑_{q=0}^{e}(−1)^{q} [^{r}E^{p,q}]_{0}

is actually finite. This time we may apply directly the classical argument on Euler characteristics. In view of formula (2) it yields

χ(E) = ∑_{p}(−1)^{p} ∑_{q=0}^{e}(−1)^{q} [Ext_{Q}^{q}(Ext_{R}^{p}(M, N), W)]_{0}

= ∑_{p}(−1)^{p} [∑_{q=0}^{e}(−1)^{q}H_{Ext_{Q}^{q}(Ext_{R}^{p}(M,N),W)}(t)]_{0}

= (−1)^{e} [∑_{p}(−1)^{p}H_{Ext_{R}^{p}(M,N)}(t^{−1})]_{0} .

The overall result of the preceding computations now reads

[∑_{p}(−1)^{p}H_{Ext_{R}^{p}(M,N)}(t^{−1})]_{0} = [ϕ_{R}(M, N)(t^{−1})]_{0}

and this clearly implies the desired equality of rational functions. □

Proof of Proposition 2. Consider N^{∨} = Hom_{K}(N, K) with the induced structure of graded R–module. By Lemma 3, the expansion around 0 of the rational function χ^{R}(M, N)(t) is equal to ∑_{i}(−1)^{i} [Tor_{i}^{R}(M, N^{∨})]_{0} ∈ ℤ((t)). As rank_{K} N is finite, we have H_{N^{∨}}(t) = H_{N}(t^{−1}) ∈ ℤ[t, t^{−1}], and thus an equality of formal Laurent series

∑_{i⩾0}(−1)^{i} [H_{Tor_{i}^{R}(M,N^{∨})}(t)]_{0} = [\frac{H_{M}(t) · H_{N}(t^{−1})}{H_{R}(t)}]_{0} .

In the commutative diagram of homomorphisms of rings

ℂ(t) ⟶^{t↦t^{−1}} ℂ(t)
[ ]_{0}↓  ↓^{[ ]}_{∞}
ℂ((t)) ⟶_{t↦t^{−1}} ℂ((t^{−1}))

the lower row is an isomorphism of topological fields, hence the preceding equality yields

∑_{i⩾0}(−1)^{i} [H_{Tor_{i}^{R}(M,N^{∨})}(t^{−1})]_{∞} = [\frac{H_{M}(t^{−1}) · H_{N}(t)}{H_{R}(t^{−1})}]_{∞} .

The canonical isomorphisms (M ⊗_{R} (N^{∨}))^{∨} ≅ Hom_{R}(M, N^{∨∨}) ≅ Hom_{R}(M, N) extend to isomorphisms of graded R–modules (Tor_{i}^{R}(M, N^{∨}))^{∨} ≅ Ext_{R}^{i}(M, N) for i ≥ 0. Thus, in ℤ[t, t^{−1}] we have H_{Tor_{i}^{R}(M,N^{∨})}(t^{−1}) = H_{Ext_{R}^{i}(M,N)}(t), and this finishes the proof. □

2. Bass numbers

Let N be a finite graded R–module. For an integer i and a prime ideal 𝔭 of R, the rank of Ext_{R_{𝔭}}^{i}(k(𝔭), N_{𝔭}) over the field k(𝔭) = R_{𝔭}/𝔭R_{𝔭} is known as the i’th Bass number of N and is denoted μ_{R}^{i}(𝔭, N). We establish nontrivial lower bounds for the Bass numbers μ_{R}^{i}(N) = μ_{R}^{i}(R_{+}, N) of “many” graded modules of finite injective dimension.

A graded ring R is said to be standard if it is generated by elements of degree 1. In this case H_{N}(t) has a unique expression of the form

(3)  H_{N}(t) = \frac{e_{N}(t)}{(1 − t)^{n}} ,

where n = dim N, and e_{N}(t) is a Laurent polynomial in ℤ[t, t^{−1}] such that e_{N}(1) is a positive integer, known as the multiplicity of N.

Theorem 4. Let R be a standard graded ring of Krull dimension d.

If N ≠ 0 is a finite graded R–module of finite injective dimension and Krull dimension n, then e_{R}(t^{−1}) divides e_{N}(t) in ℤ[t, t^{−1}]. If p is a prime number and q is defined by

q = max \{p^{r} ∈ ℤ | ∑_{s=0}^{p^{r}} t^{s} divides \frac{e_{N}(t)}{e_{R}(t^{−1})} in ℤ[t, t^{−1}]\} ,

then the Bass numbers of N satisfy the inequality

∑_{i} μ_{R}^{i}(N) ≥ p^{\frac{d−n+q−1}{q(p−1)}} .

Proof. Since each Ext_{R}^{i}(K, N) is a finite graded K–vector space, and is trivial for i > inj dim_{R} N, the rational function ∑_{i}(−1)^{i}H_{Ext_{R}^{i}(K,N)}(t) is a Laurent polynomial, which we denote by ϕ_{R}^{N}(t). Theorem 1 and formula (3) yield an equality in ℤ[t, t^{−1}]:

(1 − t)^{d−n}e_{N}(t) = (−t)^{d}e_{R}(t^{−1})ϕ_{R}^{N}(t) .

As d ≥ n and e_{R}(1) ≠ 0, unique factorization in ℤ[t, t^{−1}] shows that e_{R}(t^{−1}) divides e_{N}(t). This proves the first assertion. The second one follows by the argument of [1; §4]. □

Under the hypotheses of the next result the theorem applies with p = 2 and q = 1:

Corollary 5. If −1 is not a root of the Laurent polynomial e_{N}(t)/e_{R}(t^{−1}), then

∑_{i} μ_{R}^{i}(N) ≥ 2^{d−n} .

This inequality holds, in particular, when the multiplicity of N is an odd number. □

Remark. If N is a finite module over a local ring (S, 𝔫), and depth_{S} N = g < d = dim S, then a result of Bruns [4; (9.6.1.a)] implies that ∑_{i} μ_{S}^{i}(N) ≥ (d − g)(d − g − 1) + 3.

The quadratic lower bound above is in general weaker than the exponential bound given by the corollary, so we raise the question whether an inequality ∑_{i} μ_{S}^{i}(𝔫, N) ≥ 2^{d−n} holds for each finite S–module N of Krull dimension n and of finite injective dimension. Such a bound would be best possible: if x is a system of parameters of S, E is the injective envelope of S/𝔫, and N is the finite length module Hom_{S}(S/(x), E), then μ_{S}^{i}(𝔫, N) = \binom{d}{i} since S is Cohen–Macaulay by the Bass Conjecture proved by Peskine, Szpiro, Hochster, and P. Roberts, cf. [4; (9.6.2), (9.6.4.b)].

3. Laurent coefficients

We turn to properties of the coefficients of Laurent expansions around 1 of rational functions representing the Hilbert series of finite R–modules. For purposes of calibration, it is convenient to write such an expansion in the form

[M]_{1} = ∑_{j⩾0} \frac{f_{R}^{j}(M)}{(1 − t)^{d−j}}

with d equal to the Krull dimension of R, and to set f_{R}^{j}(M) = 0 for j < 0. We call f_{R}^{j}(M) the j’th Laurent coefficient of M and note that it is a rational number.

Remark. When R is standard and d−dim M = h, comparison with (3) yields rank_{K}(M_{n}) = ∑_{j=h}^{d−1} f_{R}^{j}(M)\binom{n+d−j−1}{d−j−1} for n ≫ 0, hence the Laurent coefficients of the principal part of [M]_{1} are directly related to the coefficients of the Hilbert polynomial of M in the usual “binomial” format, as in [4; (4.1.5)]: f_{R}^{j}(M) = (−1)^{h−j}e_{h−j}(M) for h ≤ j ≤ d − 1.

Under the assumptions of Theorem 1, the j’th coefficient in the Laurent expansion around 1 of the rational function ϕ_{R}(M, N)(t) is an alternating sum of the corresponding Laurent coefficients of Ext modules. The number

ε_{R}^{j}(M, N) = ∑_{i=0}^{j}(−1)^{i}f_{R}^{j}(Ext_{R}^{i}(M, N))

represents a portion of such a sum. The next result shows that under a homological finiteness hypothesis it contains the entire information.

Proposition 6. If the projective dimension of a finite graded R–module M or the injective dimension of a finite graded R–module N is finite, then

[ϕ_{R}(M, N)(t)]_{1} = ∑_{j⩾0} \frac{ε_{R}^{j}(M, N)}{(1 − t)^{d−j}} .

The proposition applies to all finite modules over a graded polynomial ring R, and raises the question of extending its validity—at least on part—to other graded rings. When

ϕ_{R}(M, N)(t) = ∑_{j=0}^{n} \frac{ε_{R}^{j}(M, N)}{(1 − t)^{d−j}} + O (\frac{1}{(1 − t)^{d−n−1}} )

for some integer n ≥ 0, we say that the sequences ϕ_{R}^{∗}(M, N) (of coefficients of the Laurent expansion of ϕ_{R}(M, N)(t) around 1) and ε_{R}^{∗}(M, N) = {ε_{R}^{j}(M, N)}_{j∈ℤ} agree up to level n.

The next result shows that an agreement of ε^{∗} and ϕ^{∗} imposes strong restrictions on the local structure of R. We adopt the convention that the codimension of a prime ideal 𝔭 of R is the non-negative integer codim 𝔭 = dim R − dim(R/𝔭) and note that height 𝔭 ≤ codim 𝔭, with equality when R is an integral domain. The ring R is regular in codimension c if R_{𝔭} is regular for each prime 𝔭 such that codim 𝔭 ≤ c; by Matijevic [8; (2.1)] it suffices to impose the condition only on the primes in Proj R, the set of homogeneous prime ideals of R.

Proposition 7. Let 𝔭 be a homogeneous prime ideal of R with codim 𝔭 = h. The sequences ε_{R}^{∗}(R/𝔭, R/𝔭) and ϕ_{R}^{∗}(R/𝔭, R/𝔭) agree up to level h precisely when the local ring R_{𝔭} is regular, and if furthermore h = 0 then 𝔭 is the unique codimension 0 prime of R.

The condition that ε_{R}^{∗}(M, N) and ϕ_{R}^{∗}(M, N) agree up to level n can be rewritten as

H_{R}(t^{−1}) · ∑_{j=0}^{n} \frac{ε_{R}^{j}(M, N)}{(1 − t)^{d−j}} = H_{M}(t^{−1}) · H_{N}(t) + O (\frac{1}{(1 − t)^{2d−n−1}} ) .

Taking on both sides Laurent expansions around 1, we can further reformulate it in terms of a system of n + 1 numerical equalities involving the numbers f_{R}^{j}(M), f_{R}^{j}(N), f_{R}^{j}(R), and ε_{R}^{j}(M, N) for 0 ≤ j ≤ n. Equations (4.0), (4.1), (4.2) display such a system for n = 2.

Theorem 8. Let M and N be finite graded modules over a graded ring R.

If R has a unique prime 𝔯 of codimension 0 and R_{𝔯} is a field, then

(4.0)  f_{R}^{0}(R)ε_{R}^{0}(M, N) = f_{R}^{0}(M)f_{R}^{0}(N) .

If R is an integral domain which is regular in codimension 1, then

(4.1)  f_{R}^{0}(R)ε_{R}^{1}(M, N) − f_{R}^{1}(R)ε_{R}^{0}(M, N) = f_{R}^{0}(M)f_{R}^{1}(N) − f_{R}^{1}(M)f_{R}^{0}(N) .

If R is a unique factorization domain which is regular in codimension 2, then

(4.2)  \begin{matrix}f_{R}^{0}(R)ε_{R}^{2}(M, N) − f_{R}^{1}(R)ε_{R}^{1}(M, N) + (f_{R}^{2}(R) − f_{R}^{1}(R)) ε_{R}^{0}(M, N) = \\ f_{R}^{0}(M)f_{R}^{2}(N) − f_{R}^{1}(M)f_{R}^{1}(N) + (f_{R}^{2}(M) − f_{R}^{1}(M)) f_{R}^{0}(N) . \end{matrix}

Remark. When R is a domain for any finite graded R–module M one has

(5)  f_{R}^{0}(M) = rank_{R}(M)f_{R}^{0}(R) ,

cf. Lemma 10(c) below. If furthermore R is regular in codimension 1, then formula (1), due to Benson and Crawley-Boevey [3; (2.4)], is seen to follow from (4.0) and (4.1).

Recall that a graded complete intersection is a quotient of a graded polynomial ring by a regular sequence of homogeneous elements. By a classical result of Grothendieck [7; (XI.3.14)], cf. also [5], a complete intersection which is regular in codimension 3 is factorial, hence we get the following easily applicable

Corollary 9. If R is a graded complete intersection which is regular in codimension 3, then the equalities of the theorem hold for all finite graded R–modules. □

In the next section we show that neither the UFD hypothesis of the theorem nor the codimension 3 hypothesis of the corollary can be significantly weakened. The proofs of the results above depend on a few lemmas; the first one (for ψ = f_{R}^{1}) is in [3; (2.2), (2.3)].

Lemma 10. Let M ≠ 0 be a finite R–modules with h = dim R − dim M.

(a) f_{R}^{j}(−) is an additive function on G(R) for each j ∈ ℤ.

(b) f_{R}^{j}(M) = 0 for j < h and f_{R}^{h}(M) > 0.

(c) f_{R}^{h}(M) = ∑\begin{matrix}𝔭∈Proj R \\ codim 𝔭=h \end{matrix} length_{R_{𝔭}}(M_{𝔭})f_{R}^{h}(R/𝔭) .

Proof. (a) results from the additivity of [ ]_{1} and the uniqueness of Laurent expansions. (b) is a fact of dimension theory.

(c) is a formal consequence of the preceding two because M admits a finite filtration with subquotients of the form (R/𝔮)(a) for appropriate 𝔮 ∈ Proj R and a ∈ ℤ. □

Lemma 11. Let M and N be finite R–modules, such that for some c ∈ ℕ and for all 𝔭 ∈ Proj R with codim 𝔭 ≤ c either proj dim_{R_{𝔭}}(M_{𝔭}) or inj dim_{R_{𝔭}}(N_{𝔭}) is finite. For each integer j ≤ c and any submodule L of Ext_{R}^{i}(M, N), we have f_{R}^{j}(L) = 0 when i > j.

Proof. By Lemma 10(b) it suffices to show that j < dim R − dim L = g, hence we may assume that g ≤ c. If 𝔭 ∈ Supp L is a homogeneous prime with codim 𝔭 = g, then

Ext_{R_{𝔭}}^{i}(M_{𝔭}, N_{𝔭}) ≅ Ext_{R}^{i}(M, N)_{𝔭} ⊇ L_{𝔭} ≠ 0

implies that i ≤ min{proj dim_{R_{𝔭}}(M_{𝔭}), inj dim_{R_{𝔭}}(N_{𝔭})}. One of these dimensions is finite by assumption, and thus is bounded above by depth R_{𝔭}, cf. [4; (1.3.3) and (3.1.17)]. Thus, we have j < i ≤ depth R_{𝔭} ≤ codim 𝔭 = g, as desired. □

Proof of Proposition 6 : Equate the Laurent expansions around 1 of the rational functions in Theorem 1 and use Lemma 11 to regroup the terms in the sum of Ext’s. □

The argument for part (a) of the next lemma is abstracted from the proof of [3; (2.4)].

Lemma 12. The numbers ε_{R}^{j}(M, N) have the following properties.

(a) ε_{R}^{j}(−, −) is a biadditive function on G(R) if R is regular in codimension c and j ≤ c.

(b) ε_{R}^{j}(M, N) = 0 when j < dim R − dim(R/(ann M + ann N)) .

Proof. (a) Fix an integer j between 0 and c, take a short exact sequence of finite R–modules 0 ⟶ N′ ⟶ N ⟶ N′′ ⟶ 0, and consider the exact sequence

0 ⟶ Hom_{R}(M, N′) ⟶ Hom_{R}(M, N) ⟶ Hom_{R}(M, N′′) ⟶
Ext_{R}^{1}(M, N′) ⟶  …  ⟶ Ext_{R}^{j−1}(M, N′′) ⟶
Ext_{R}^{j}(M, N′) ⟶ Ext_{R}^{j}(M, N) ⟶ Ext_{R}^{j}(M, N′′) ⟶ N^{j+1} ⟶ 0 .

By assumption, the ring R_{𝔭} has global dimension ≤ j whenever codim 𝔭 ≤ c, hence Lemma 11 shows that f_{R}^{j}(N^{j+1}) = 0. The additivity of ε_{R}^{j}(M, −) now follows by applying f_{R}^{j} to the long exact sequence. The additivity of ε_{R}^{j}(−, N) yields to a similar approach.

(b) For each i the R–module Ext_{R}^{i}(M, N) is annihilated by (ann M + ann N), hence we have dim R − dim Ext_{R}^{i}(M, N) ≥ dim R − dim R/(ann M + ann N) > j, and thus ε_{R}^{j}(M, N) vanishes in the indicated range by Lemma 10(b). □

Proof of Proposition 7. Lemma 10(b) shows that f_{R}^{h}(R/𝔭) and f_{R}^{0}(R) are positive, and that

[ϕ_{R}(R/𝔭, R/𝔭)(t)]_{1} = (−1)^{h} · \frac{f_{R}^{h}(R/𝔭)}{f_{R}^{0}(R)} · \frac{f_{R}^{h}(R/𝔭)}{(1 − t)^{d−2h}} + O (\frac{1}{(1 − t)^{d−2h−1}} ) .

With η_{h}(𝔭) = ∑_{i=0}^{h}(−1)^{i} rank_{k(𝔭)}(Ext_{R_{𝔭}}^{i}(k(𝔭), k(𝔭))), Lemmas 12(b) and 10(c) give

∑_{j=0}^{h} \frac{ε_{R}^{j}(R/𝔭, R/𝔭)}{(1 − t)^{d−j}} = η_{h}(𝔭) · \frac{f_{R}^{h}(R/𝔭)}{(1 − t)^{d−h}} .

Thus, ε_{R}^{∗}(R/𝔭, R/𝔭) and ϕ_{R}^{∗}(R/𝔭, R/𝔭) agree up to level 0 if and only if f_{R}^{0}(R) = f_{R}^{0}(R/𝔭). In view of Lemma 10(c), this is equivalent to the statement that 𝔭 is the only prime of R with codim 𝔭 = 0 and that R_{𝔭} is a field.

If h > 0, then ε_{R}^{∗}(R/𝔭, R/𝔭) and ϕ_{R}^{∗}(R/𝔭, R/𝔭) agree up to level h if and only if η_{h}(𝔭) = 0. When the local ring R_{𝔭} is regular the rank of Ext_{R_{𝔭}}^{i}(k(𝔭), k(𝔭)) over k(𝔭) is equal to \binom{h}{i}, hence η_{h}(𝔭) = 0. When R_{𝔭} is not regular its embedding dimension is at least 2 because dim R_{𝔭} = h ≥ 1, and then η_{h}(𝔭) ≠ 0 by Okiyama [9; Proposition 8]. □

Lemma 13. (a) If M and N are finite graded R–modules, then for a, b ∈ ℤ the sequences ε_{R}^{∗}(M(a), N(b)) and ϕ_{R}^{∗}(M(a), N(b)) agree up to level n if and only if ε_{R}^{∗}(M, N) and ϕ_{R}^{∗}(M, N) agree up to the same level.

(b) If 𝔭 is a homogeneous prime ideal such that codim 𝔭 = h and the local ring R_{𝔭} is Gorenstein, then ε_{R}^{∗}(R/𝔭, R) and ϕ_{R}^{∗}(R/𝔭, R) agree up to level h.

Proof. (a) The canonical isomorphisms Ext_{R}^{i}(M(a), N(b)) ≅ Ext_{R}^{i}(M, N)(b − a) show that

∑_{j=0}^{n} \frac{ε_{R}^{j}(M(a), N(b))}{(1 − t)^{d−j}} = t^{a−b} · ∑_{j=0}^{n} \frac{ε_{R}^{j}(M, N)}{(1 − t)^{d−j}}

holds for each n ≥ 0, while it is clear that ϕ_{R}(M(a), N(b))(t) = t^{a−b} · ϕ_{R}(M, N)(t).

(b) By a direct computation as in the proof of Proposition 7, we get

[ϕ_{R}(R/𝔭, R)(t)]_{1} = (−1)^{h} · \frac{f_{R}^{h}(R/𝔭)}{(1 − t)^{d−h}} + O (\frac{1}{(1 − t)^{d−h−1}} ) .

On the other hand, the Gorenstein hypothesis yields μ_{R}^{i}(𝔭, R) = δ_{ih}, hence

∑_{j=0}^{h} \frac{ε_{R}^{j}(R/𝔭, R)}{(1 − t)^{d−j}} = (−1)^{h} · \frac{f_{R}^{h}(R/𝔭)}{(1 − t)^{d−h}} .  □

Proof of Theorem 8. Due to Lemmas 10(a) and 12(a), both sides of (4.n) are biadditive functions on G(R) for n = 0, 1, 2. Thus, it suffices to consider M and N which are translates of quotients of R by homogeneous prime ideals. By Lemma 13(a), we can even assume that M = R/𝔭 and N = R/𝔮 with 𝔭, 𝔮 ∈ Proj R.

If codim 𝔭 > n, or codim 𝔮 > n, or codim 𝔭 = n = codim 𝔮 and 𝔭 ≠ 𝔮, then the left hand side of (4.n) vanishes by Lemma 12(b), and its right hand side vanishes by Lemma 10(b). If codim 𝔭 = n and 𝔭 = 𝔮 (respectively, if 𝔭 = 0 or if 𝔮 = 0), then (4.n) holds by Proposition 7 (respectively, by Proposition 6 or by Lemma 13(b)).

It remains to prove (4.2) when codim 𝔰 = 1, where 𝔰 = 𝔭 or 𝔰 = 𝔮. As R is factorial, 𝔰 = (x) for a homogeneous x ∈ R, hence by additivity and Lemma 13(a) we get the result from the exact sequence 0 ⟶ R(− deg x) ⟶ R ⟶ R/𝔰 ⟶ 0 and already settled cases. □

Remark 14. Let M be a finite graded module over a graded integral domain R. If R is Gorenstein in codimension 1, then

(6.1)  f_{R}^{1}(Hom_{R}(M, R)) − f_{R}^{1}(Ext_{R}^{1}(M, R)) = 2f_{R}^{1}(R) rank_{R}(M) − f_{R}^{1}(M) .

If R is factorial, then in addition

(6.2) f_{R}^{2}(Hom_{R}(M, R)) − f_{R}^{2}(Ext_{R}^{1}(M, R)) + f_{R}^{2}(Ext_{R}^{2}(M, R)) =

(1 + 2\frac{f_{R}^{1}(R)}{f_{R}^{0}(R)})(f_{R}^{1}(R) rank_{R}(M) − f_{R}^{1}(M)) + f_{R}^{2}(M) .

Indeed, the formulas above are obtained by rewriting (4.1) and (4.2) for N = R with the help of (5). A factorial domain is Cohen-Macaulay in codimension 2 (due to normality), and thus is Gorenstein in this codimension by a result of Murthy, cf. [4; (3.3.19)]. The argument for Lemma 12(a) shows that ε_{R}^{j}(−, R) is an additive function on G(R) for j = 0, 1, 2. At this point we may retrace the preceding proof, ignoring any discussion of primes 𝔮 ≠ 0.

4. Normal domains

Our examples are built from the ring R = K[X, Y, U, V]/(XV −Y U) = K[x, y, u, v] with the standard grading, and the R–module M = R/(u, v) ≅ K[X, Y].

Example 15. Equality (4.2) may fail over a 3-dimensional graded hypersurface ring R which is regular in codimension 2 (and hence is a normal domain).

Over K[X, Y, U, V] the quadratic form XV −Y U has a homogeneous matrix factorization

( \begin{matrix}V & −Y \\ −U & X \end{matrix} ) ( \begin{matrix}X & Y \\ U & V \end{matrix} ) = ( \begin{matrix}XV − Y U & 0 \\ 0 & XV − Y U \end{matrix} ) = ( \begin{matrix}X & Y \\ U & V \end{matrix} ) ( \begin{matrix}V & −Y \\ −U & X \end{matrix} ) ,

hence Eisenbud [6; (4.1)] provides the following minimal graded free resolution of M:

… ⟶ R^{2}(−4) \underrightarrow{(\begin{matrix}v & −y \\ −u & x \end{matrix} )}R^{2}(−3) \underrightarrow{( \begin{matrix}x & y \\ u & v \end{matrix} )}R^{2}(−2) \underrightarrow{(\begin{matrix}v & −y \\ −u & x \end{matrix} )}R^{2}(−1) \underrightarrow{(u, v)}R ⟶ 0 .

Applying to it the functor Hom_{R}(−, M) we get a complex of graded R–modules

0 ⟶ M \underrightarrow{0} M^{2}(1) \underrightarrow{( \begin{matrix}0 & 0 \\ −y & x \end{matrix} )} M^{2}(2) \underrightarrow{( \begin{matrix}x & 0 \\ y & 0 \end{matrix} )} M^{2}(3) \underrightarrow{( \begin{matrix}0 & 0 \\ −y & x \end{matrix} )} M^{2}(4) ⟶ ….

Direct computations based on the M–regularity of the sequence x, y yield

M ≅ Hom_{R}(M, M) ≅ Ext_{R}^{1}(M, M) ;
Ext_{R}^{2i}(M, M) ≅ k(2i) for i ≥ 1 ;
Ext_{R}^{2i+1}(M, M) = 0 for i ≥ 1 .

Accordingly, we get Laurent expansions

H_{R}(t) = \frac{1 − t^{2}}{(1 − t)^{4}} = \frac{2}{(1 − t)^{3}} − \frac{1}{(1 − t)^{2}} ;

H_{M}(t) = H_{Hom_{R}(M,M)}(t) = H_{Ext_{R}^{1}(M,M)}(t) = \frac{1}{(1 − t)^{2}} ;

H_{Ext_{R}^{2}(M,M)}(t) = \frac{1}{t^{2}} = ∑_{j⩾0}(j + 1)(1 − t)^{j} .

It follows that for M = N the expression on the left hand side of (4.2) is equal to 0, while the one on its right hand side is equal to −1.

Example 16. Equality (6.2) may fail over a 4-dimensional graded Cohen-Macaulay ring R′ of type 2 which is regular in codimension 3 (and hence is a normal domain).

Let R′ = R[Z, W]/(xW − uZ, yW − vZ) = K[x, y, z, u, v, w] with the standard grading, and set M′ = R′/(u, v, w) ≅ K[X, Y, Z]. Since w is R′–regular and annihilates M′, for S = R′/(w) ≅ R[Z]/(uZ, vZ) and for each i we have Ext_{R′}^{i}(M′, R′) ≅ Ext_{S}^{i−1}(M′, S)(1). As M′ = M[Z], and the minimal free resolution of M[Z] over R[Z] is “the same as” that of M over R, it is easy to get a beginning of a minimal S–free resolution of M′ in the form

S^{4}(−2) \underrightarrow{( \begin{matrix}v & −y & z & 0 \\ −u & x & 0 & z \end{matrix} )} S^{2}(−1) \underrightarrow{(u, v)} S ⟶ 0 .

Thus, we are interested in the homology of the complex

0 ⟶ S ⟶^{α=( \begin{matrix}u \\ v \end{matrix} )} S^{2}(1) ⟶^{β=( \begin{matrix}v & −u \\ −y & x \\ z & 0 \\ 0 & z \end{matrix} )} S^{4}(2) .

In degree 0 we have Ker α = (0 :_{S} (u, v)) = zS ≅ M′(−1), hence Ext_{R′}^{1}(M′, R′) ≅ M′.

In degree 1 the complex is exact, hence Ext_{R′}^{2}(M′, R′) = 0. Indeed, writing b ∈ S^{2}(1) in the form (r_{1} + s_{1}z, r_{2} + s_{2}z) with uniquely determined r_{j} ∈ R, we see that β(b) = 0 implies

vr_{1} − ur_{2} = 0  −yr_{1} + xr_{2} = 0  z^{2}s_{1} = 0  z^{2}s_{2} = 0 .

¿From the last two equations we get

s_{j} ∈ (0 :_{S} z^{2}) = ((0 :_{S} z) :_{S} z) = ((u, v) :_{S} z) = (u, v)S ,

hence b = (r_{1}, r_{2}). In view of the preceding example, the first two equations yield an a ∈ R such that r_{1} = au and r_{2} = av, hence b = a(u, v) ∈ Im α.

We can now collect the relevant data on Laurent expansions:

H_{R′}(t) = \frac{1 − 3t^{2} + 2t^{3}}{(1 − t)^{6}} = \frac{3}{(1 − t)^{4}} − \frac{2}{(1 − t)^{3}} ;

H_{M′}(t) = H_{Ext_{R′}^{1} (M′,R′)}(t) = \frac{1}{(1 − t)^{3}} ;

H_{Hom_{R′} (M′,R′)}(t) = H_{Ext_{R′}^{2} (M′,R′)}(t) = 0 .

They show that the left hand side of (6.2) is equal to 0, and its right hand side to \frac{1}{3}.

References

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Department of Mathematics, Purdue University, West Lafayette, Indiana 47907, U. S. A. E-mail: avramov@math.purdue.edu

Department of Mathematics, University of Toronto, Toronto, Ontario M5S 1A1, Canada E-mail: ragnar@lake.scar.utoronto.ca

Department of Mathematics, Northwestern University, Evanston, Illinois 60208, U. S. A. E-mail: judy@math.nwu.edu