SOME PROPERTIES OF GRADED LOCAL COHOMOLOGY MODULES

CHRISTEL ROTTHAUS AND LIANA M. ŞEGA

Abstract. We consider a finitely generated graded module M over a standard graded commutative Noetherian ring R = ⨁_{d≥0} R_{d} and we study the local cohomology modules H_{R_{+}}^{i}(M) with respect to the irrelevant ideal R_{+} of R. We prove that the top nonvanishing local cohomology is tame, and the set of its minimal associated primes is finite. When M is Cohen-Macaulay and R_{0} is local, we establish new formulas for the index of the top, respectively bottom, nonvanishing local cohomology. As a consequence, we obtain that the (S_{k})-loci of a Cohen-Macaulay R-module M, regarded as an R_{0}-module, are open in Spec(R_{0}). Also, when dim(R_{0}) ≤ 2 and M is a Cohen-Macaulay R-module, we prove that H_{R_{+}}^{i}(M) is tame, and its set of minimal associated primes is finite for all i.

introduction

Let R = ⨁_{d≥0} R_{d} be a positively graded commutative Noetherian ring which is standard in the sense that R = R_{0}[R_{1}], and set R_{+} = ⨁_{d>0} R_{d}, the irrelevant ideal of R. Let M = ⨁_{d∈ℤ} M_{d} be a finitely generated graded R-module. In this paper we study the graded local cohomology modules H_{R_{+}}^{i}(M).

It is known that each of the graded components H_{R_{+}}^{i}(M)_{j} is finitely generated over R_{0} and H_{R_{+}}^{i}(M)_{j} = 0 for all j ≫ 0. Brodmann and Hellus [3] have recently raised the question whether the modules H_{R_{+}}^{i}(M) are tame (or asymptotically gap free), meaning that either H_{R_{+}}^{i}(M)_{j} = 0 for all j ≪ 0 or H_{R_{+}}^{i}(M)_{j} ≠ 0 for all j ≪ 0. A positive answer is known in several cases, cf. [1], [3], [9], [12], [13].

In order to understand the finiteness properties of the modules H_{R_{+}}^{i}(M), Huneke [7] asked whether the set of their associated primes is finite; this was answered negatively by Singh [17]. However, as noted by Katzman in [8], it is not known whether the set of minimal associated primes is finite, or in other words, whether the support of H_{R_{+}}^{i}(M) is Zariski-closed.

When i is the index of the bottom nonvanishing local cohomology, it is known that the set Ass_{R} (H_{R_{+}}^{i}(M)) is finite, and it follows that H_{R_{+}}^{i}(M) is tame, cf [3]. When i is the index of the top nonvanishing local cohomology and M = R, it is proved in [9] that H_{R_{+}}^{i}(R) has only finitely many minimal associated primes. In this paper we prove:

Theorem 1. If H_{R_{+}}^{n}(M) ≠ 0 and H_{R_{+}}^{i}(M) = 0 for all i > n, then:

(1) H_{R_{+}}^{n}(M) is tame

(2) H_{R_{+}}^{n}(M) has finitely many minimal associated primes.

Date: June 3, 2004.

To study local cohomology for all indices i, some particular cases are treated.

Theorem 2. Assume that M is a Cohen-Macaulay R-module, and either dim(R_{0}) ≤ 2 or dim(R_{0}) ≤ 3 and R_{0} is semilocal. The modules H_{R_{+}}^{i}(M) have then finitely many minimal associated primes for all i.

When R_{0} is semilocal of dimension at most 2, Brodmann, Fumasoli and Lim [1] proved that H_{R_{+}}^{i}(M) is tame for all i. Assuming that M is Cohen-Macaulay, we eliminate the condition that R_{0} is semilocal. This recovers a result of Lim [13].

Theorem 3. If M is a Cohen-Macaulay R-module and dim(R_{0}) ≤ 2, then H_{R_{+}}^{i}(M) is tame for all i.

When M is a Cohen-Macaulay R-module and R_{0} is local we prove:

sup{i | H_{R_{+}}^{i}(M) = 0 for all j < i} = dim_{R} M − dim_{R_{0}} M
inf{i | H_{R_{+}}^{i}(M) = 0 for all j > i} = dim_{R} M − depth_{R_{0}} M

Note that the notion of depth of M over R_{0} is meaningful, cf. [16] for details. In particular, when R_{0} is local and M is Cohen-Macaulay over R, the following statements are equivalent:

(a) depth_{R_{0}}(M) = dim_{R_{0}}(M)

(b) M_{i} is a Cohen-Macaulay R_{0}-module with dim_{R_{0}}(M_{i}) = dim_{R_{0}}(M) for all i.

(c) There exists j ≥ 0 so that H_{R_{+}}^{i}(M) = 0 for all i ≠ j.

Another consequence of the formulas is somewhat surprising. Recall that a finite module N over a commutative Noetherian ring A satisfies the Serre condition (S_{k}) if depth_{A_{𝔭}}(N_{𝔭}) ≥ min{k, dim N_{𝔭}} for all 𝔭 ∈ Spec(A). When the ring A is excellent, Grothendieck [6] proved that the set

U_{S_{k}}(N) = {𝔭 ∈ Spec(A) | the A_{𝔭}-module N_{𝔭} satisfies (S_{k})}

is open in Spec(A). Noting that the above notions make sense for A = R_{0} and N = M, the authors extended this result in [16] and proved that the set

U_{S_{k}}^{0}(M) = {𝔭 ∈ Spec(R_{0}) | the (R_{0})_{𝔭}-module M_{𝔭} satisfies (S_{k})}

is open in Spec(R_{0}) whenever R is excellent. In general, the condition that the ring is excellent is necessary for the (S_{k})-loci to be open. However, when M is Cohen-Macaulay, it can be removed:

Theorem 4. If M is Cohen-Macaulay over R, then for any k ∈ ℕ the set U_{S_{k}}^{0}(M) is open in Spec(R_{0}).

The paper is organized as follows: In the first section we collect definitions and several known results on local cohomology that are used throughout the paper.

In the second section we prove Theorem 1(1) as Theorem 2.8. A stronger result is obtained when R_{0} is local, with maximal ideal 𝔪_{0}. In this case, we prove that if n denotes the largest integer i with H_{R_{+}}^{i}(M) ≠ 0, then the R-module H_{R_{+}}^{n}(M)/𝔪_{0}H_{R_{+}}^{n}(M) is Artinian; in particular, this shows that the minimal number of generators of H_{R_{+}}^{n}(M)_{j} has polynomial growth for j ≪ 0.

In the third section we prove that certain subsets of Spec(R_{0}) are open. In particular, Theorem 1(2) is proved as Theorem 3.5. (See also 1.1.)

In Section 4 we obtain the formulas above for the top and bottom nonvanishing local cohomology, and prove Theorem 4 as Corollary 4.9.

In Section 5 we prove Theorem 2 as Theorems 5.3 and 5.4, and Theorem 3 as Theorem 5.6.

1. preliminaries

Throughout the whole paper, we let R denote a positively graded commutative Noetherian ring R = ⨁_{d≥0} R_{d}, which is standard in the sense that R = R_{0}[R_{1}], and set R_{+} = ⨁_{i>0} R_{i}, the irrelevant ideal of R. Let M = ⨁_{d∈ℤ} M_{d} be a finitely generated graded R-module.

For any M as above and any integer a we let M[a] denote the a-shift of M, defined as the graded R-module with M[a]_{i} = M_{i−a}.

1.1. As noted in [3, 5.5] there is a bijection between the sets Ass_{R} (H_{R_{+}}^{i}(M)) and Ass_{R_{0}} (H_{R_{+}}^{i}(M)) given by 𝔭 + R_{+} ↦ 𝔭. Since over a Noetherian ring the set of minimal primes of the support of a module coincides with the set of minimal associated primes of the module, the following statements are equivalent:

(a) The R-module H_{R_{+}}^{i}(M) has finitely many minimal associated primes.

(b) The R_{0}-module H_{R_{+}}^{i}(M) has finitely many minimal associated primes.

(c) The set Supp_{R} (H_{R_{+}}^{i}(M)) is closed in Spec(R).

(d) The set Supp_{R_{0}} (H_{R_{+}}^{i}(M)) is closed in Spec(R_{0}).

1.2. We set

g_{R}(M) = grade(R_{+}, M)

Recall that grade(R_{+}, M) = ∞ if and only if R_{+}M = M. In our case, this is equivalent to M = 0.

The following relation is known, cf. [4, 6.2.7]:

(1.2.1)  g_{R}(M) = sup{i | H_{R_{+}}^{j}(M) = 0 for all j < i}

When R_{0} is local with maximal ideal 𝔪_{0} we set

n_{R}(M) = dim_{R}(M/𝔪_{0}M)

We make the convention that dim_{R}(0) = −∞, and note that n_{R}(M) = −∞ if and only if M = 0. By [3, 3.4], we have

(1.2.2)  n_{R}(M) = inf{i | H_{R_{+}}^{j}(M) = 0 for all j > i}

1.3. We recall the following properties (see for example [4]) :

1.3.1. (Homogeneous Prime Avoidance Lemma) If P_{1}, …, P_{s} are prime ideals in Spec(R) and R_{+} is not contained in P_{i} for all i, then there exists a homogeneous element in R_{+} ∖ (P_{1} ∪ ⋯ ∪ P_{s}).

In particular, if 0 < g_{R}(M) < ∞, then there exists a homogeneous element x ∈ R_{+} which is M-regular.

1.3.2. H_{R_{+}}^{i}(M) ≅ H_{R_{+}}^{i}(M/Γ_{R_{+}}(M)) for all i > 0.

1.3.3. (Flat Base Change) If R → R′ is a flat homomorphism of Noetherian rings, then H_{R_{+}}^{i}(M) ⊗_{R} R′ ≅ H_{R_{+}R′}^{i}(M′) for all i, where M′ = M ⊗_{R} R′.

In particular, if H_{R_{+}R_{𝔮}}^{i}(M_{𝔮}) = 0 for some 𝔮 ∈ Spec(R_{0}), then H_{R_{+}R_{𝔭}}^{i}(M_{𝔭}) = 0 for all 𝔭 ∈ Spec(R_{0}) with 𝔭 ⊆ 𝔮.

1.3.4. (Independence Theorem) If R → R′ is a homomorphism of Noetherian rings and N is a finite R′-module (with the induced structure of R-module) then H_{R_{+}R′}^{i}(N) ≅ H_{R_{+}}^{i}(N) for all i.

Moreover, when R_{0} is local with maximal ideal 𝔪_{0} we have, cf. [3]:

1.3.5. n_{R}(M) ≤ 0 if and only if M = Γ_{R_{+}}(M).

1.3.6. n_{R}(M) = n_{R}(M/Γ_{R_{+}}(M)), provided that n_{R}(M) > 0.

1.3.7. If n_{R}(M) > 0, then there exists a homogeneous element x ∈ R_{+} such that n_{R}(M/xM) = n_{R}(M) − 1. This follows from 1.3.1, by choosing x to avoid all the minimal primes of (𝔪_{0}M :_{R} M). Moreover, if g_{R}(M) > 0, then the element x may be chosen to be also M-regular.

2. the top local cohomology is tame

The assumptions on R and M are as in the first section.

2.1. Theorem. Assume that R_{0} is local with maximal ideal 𝔪_{0}.

If n = n_{R}(M), then the R-module H_{R_{+}}^{n}(M)/𝔪_{0}H_{R_{+}}^{n}(M) is Artinian.

Proof. We will prove the statement by induction on n_{R}(M).

Assume that n_{R}(M) = 0. By 1.3.5, we have then M = Γ_{R_{+}}(M). It follows that Supp_{R}(M/𝔪_{0}M) = {𝔪_{0} + R_{+}}. Since M is finitely generated as an R-module, we conclude that that M/𝔪_{0}M is Artinian.

Now assume that we proved the statement for any finitely generated graded Rmodule N with n_{R}(N) = k − 1 ≥ 0. We want to prove it for n_{R}(M) = k. In view of 1.3.2 and 1.3.6 we may replace M with M/Γ_{R_{+}}(M), so that we may assume Γ_{R_{+}}(M) = 0. Let x ∈ R_{+} be a homogeneous M-regular element with deg(x) = a such that n_{R}(M/xM) = n_{R}(M) − 1 = k − 1 (see 1.3.7).

The short exact sequence

0 → M ⟶^{x} M[−a] → (M/xM)[−a] → 0

yields a long exact sequence

⋯ → H_{R_{+}}^{k−1}(M/xM) → H_{R_{+}}^{k}(M) ⟶^{x} H_{R_{+}}^{k}(M)[−a] → H_{R_{+}}^{k}(M/xM)[−a] → ⋯

Since n_{R}(M/xM) = k − 1, we have H_{R_{+}}^{k}(M/xM) = 0 by (1.2.2). Let L denote the kernel of the multiplication by x on H_{R_{+}}^{k}(M). The induction hypothesis yields that the R-module H_{R_{+}}^{k−1}(M/xM)/𝔪_{0}H_{R_{+}}^{k−1}(M/xM) is Artinian. As a homomorphic image of this module, L/𝔪_{0}L is also Artinian.

We have then an exact sequence

L/𝔪_{0}L → H_{R_{+}}^{k}(M)/𝔪_{0}H_{R_{+}}^{k}(M) ⟶^{x} H_{R_{+}}^{k}(M)/𝔪_{0}H_{R_{+}}^{k}(M)[−a] → 0

which shows that the kernel of multiplication by x on H_{R_{+}}^{k}(M)/𝔪_{0}H_{R_{+}}^{k}(M) is an Artinian R-module. Since H_{R_{+}}^{k}(M)/𝔪_{0}H_{R_{+}}^{k}(M) is an (x)-torsion R-module, we conclude that it is Artinian using for example a result of Melkersson [15, 1.3]. □

2.2. Recall from [4, 15.1.5] that for all i and n the R_{0}-module H_{R_{+}}^{n}(M)_{i} is finitely generated. When (R_{0}, 𝔪_{0}) is local, it makes thus sense to introduce the numbers

ℓ_{R}^{n}(M)_{i} := length_{R_{0}} (H_{R_{+}}^{n}(M)/𝔪_{0}H_{R_{+}}^{n}(M))_{i}

Recall also that H_{R_{+}}^{n}(M)_{i} = 0 for i ≫ 0.

2.3. Since R is finitely generated (in degree 1), it is isomorphic to a quotient of a polynomial ring S = R_{0}[x_{1}, … x_{s}], with variables in degree 1. By 1.3.4, we have H_{R_{+}}^{i}(M) ≅ H_{S_{+}}^{i}(M) for all i.

When (R_{0}, 𝔪_{0}) is local, it follows from 1.2.2 that n_{R}(M) = n_{S}(M). Note that for every i and n we also have ℓ_{R}^{n}(M)_{i} = ℓ_{S}^{n}(M)_{i}.

2.4. Corollary. Set n = n_{R}(M) and assume that R is generated over R_{0} by s elements. There exists then a polynomial q(t) ∈ ℚ[t] of degree at most s such that

ℓ_{R}^{n}(M)_{i} = q(i) for all i ≪ 0 .

Proof. By 2.3 we may assume R = R_{0}[x_{1}, … x_{s}]. The existence of the polynomial q(t) is given for example by [10, 2], using Theorem 2.1. □

We recall a terminology introduced in [3]: A graded R-module T = ⊕_{d∈ℤ}T_{d} is said to be tame (or asymptotically gap free) if the set

{d ∈ ℤ | T_{d} ≠ 0, T_{d+1} = 0}

is finite. Clearly, all Artinian and Noetherian R-modules are tame. Theorem 2.1 shows thus that the R-module H_{R}^{n}(M)/𝔪_{0}H_{R}^{n}(M) is tame when n = n_{R}(M). Using Nakayama’s Lemma, we note:

2.5. Corollary. If n = n_{R}(M), then H_{R_{+}}^{n}(M) is tame. □

We note that the top local cohomology module is almost never Noetherian:

2.6. Remark. Assume that R_{0} is local and set n = n_{R}(M). If n > 0, then H_{R_{+}}^{n}(M)_{j} ≠ 0 for all j ≪ 0.

Indeed, by 1.3.2, we may assume Γ_{R_{+}}(M) = 0 and thus g_{R}(M) > 0. By 1.3.7 there exists then a homogeneous M-regular element x ∈ R_{+} with deg(x) = a such that n_{R}(M/xM) = n_{R}(M) − 1. In particular, this implies that H_{R_{+}}^{n}(M/xM) = 0. The long exact sequence in homology

⋯ → H_{R_{+}}^{n}(M) ⟶^{x} H_{R_{+}}^{n}(M)[−a] → H_{R_{+}}^{n}(M/xM)[−a] → ⋯

shows that multiplication by x on H_{R_{+}}^{n}(M) is surjective, hence there exist infinitely many indices j with H_{R_{+}}^{n}(M)_{j} ≠ 0. In view of Corollary 2.5, we have H_{R_{+}}^{n}(M)_{j} ≠ 0 for all j ≪ 0. (Alternatively, this result can be proved by reducing to the case when the residue field of R_{0} is infinite, in which case we may assume a = 1.)

For the rest of the section we remove the assumption that R_{0} is local.

2.7. Remark. Let n be an integer such that H_{R_{+}}^{i}(M) = 0 for all i > n and H_{R_{+}}^{n}(M) ≠ 0. Using (1.2.2) and 1.3.3 we see that n_{R_{𝔭}}(M_{𝔭}) ≤ n for all 𝔭 ∈ Spec(R_{0}). In particular, this implies that H_{R_{+}R_{𝔭}}^{n}(M_{𝔭}) ≠ 0 if and only if n = n_{R_{𝔭}}(M_{𝔭}). If n ≠ n_{R_{𝔭}}(M_{𝔭}) for all 𝔭, then H_{R_{+}R_{𝔭}}^{n}(M_{𝔭}) = 0 for all 𝔭, hence H_{R_{+}}^{n}(M) = 0, a contradiction. In conclusion, there exists some 𝔭 ∈ Spec(R_{0}) such that n = n_{R_{𝔭}}(M_{𝔭}).

2.8. Theorem. If n is an integer such that H_{R_{+}}^{i}(M) = 0 for all i > n and H_{R_{+}}^{n}(M) ≠ 0, then H_{R_{+}}^{n}(M) is tame.

Proof. If n = 0, then H_{R_{+}}^{n}(M) is a finite R_{0}-module, and thus H_{R_{+}}^{n}(M)_{j} = 0 for all j ≪ 0. So we may assume n > 0.

By Remark 2.7, there exists 𝔭 ∈ Spec(R_{0}) such that n = n_{R_{𝔭}}(M_{𝔭}). By Remark 2.6, it follows that H_{R_{+}R_{𝔭}}^{n}(M_{𝔭})_{j} = 0 for all j ≪ 0, and thus H_{R_{+}}^{n}(M) = 0 for all j ≪ 0. □

3. the support of the top local cohomology is closed

In this section we prove several results regarding open loci, which culminate with the one announced in the title. First, we record a basic lemma:

3.1. Lemma. Let N = ⨁_{d∈ℤ} N_{d} be a finitely generated R-module and 𝔭 ∈ Spec(R_{0}) be a prime ideal.

(1) If N_{𝔭} = 0 then there exists an open set U of Spec(R_{0}) such that 𝔭 ∈ U and N_{𝔮} = 0 for all 𝔮 ∈ U.

(2) Let x ∈ R_{+} be a homogeneous element such that x/1 ∈ R_{+}R_{𝔭} is N_{𝔭}-regular. There exists then an open set U of Spec(R_{0}) such that 𝔭 ∈ U and N_{𝔮} = 0 or x/1 ∈ R_{+}R_{𝔮} is N_{q}-regular for al q ∈ U.

Proof. (1) Let n_{1}, … n_{s} be a set of generators of N over R and let a ∈ R_{0} ∖ 𝔭 such that n_{1}/1, …, n_{s}/1 = 0 in N_{a}. Consider then

U = U_{a} := {𝔭 ∈ Spec(R_{0}) | a ∉ 𝔭} .

(2) Let K be the kernel of multiplication by x on N. Since x/1 is N_{𝔭}-regular, we have K_{𝔭} = 0. Use then (1) to find an open set U such that K_{𝔮} = 0 for all 𝔮 ∈ U. □

3.2. Lemma. For any 𝔮 ∈ Spec(R_{0}) we set n_{𝔮} = n_{R_{𝔮}}(M_{𝔮}).

(1) If 𝔭 and 𝔮 are prime ideals in Spec(R_{0}) such that 𝔭 ⊆ 𝔮, then n_{𝔭} ≤ n_{𝔮}.

(2) For any 𝔭 ∈ Spec(R_{0}) there exists an open set U ⊆ Spec(R_{0}) such that 𝔭 ∈ U and n_{𝔮} ≤ n_{𝔭} for all 𝔮 ∈ U; in particular, n_{𝔮} = n_{𝔭} for all 𝔮 ∈ U ∩ V (𝔭).

Proof. (1) By localizing at 𝔮, we may assume that R_{0} is local, with maximal ideal 𝔮, and hence n_{𝔮} = n_{R}(M). If n_{𝔭} = −∞, then the inequality is clear. Assume now that n_{𝔭} ≥ 0. From (1.2.2) we have H_{R_{+}R_{𝔭}}^{n_{𝔭}}(M_{𝔭}) ≠ 0 and from 1.3.3 it follows that H_{R_{+}}^{n_{𝔭}}(M) ≠ 0. Using again (1.2.2), we conclude n_{𝔭} ≤ n_{R}(M) = n_{𝔮}.

(2) We proceed by induction on n_{𝔭}. If n_{𝔭} = −∞, then M_{𝔭} = 0 and we choose then U as in Lemma 3.1(1) so that 𝔭 ∈ U and for all 𝔮 ∈ U we have M_{𝔮} = 0, and thus n_{𝔮} = −∞.

If n_{𝔭} = 0 then (M/Γ_{R_{+}}(M))_{𝔭} = 0 by 1.3.5. Using Lema 3.1(1) we choose then U so that 𝔭 ∈ U and (M/Γ_{R_{+}}(M))_{𝔮} = 0 for all q ∈ U. For such 𝔮 it follows that n_{𝔮} ≤ 0 using again 1.3.5.

Assume now that n_{𝔭} = n > 0 and that the statement is proved for all graded finitely generated R modules N with n_{R_{𝔭}}(N) = n−1. Set \overline{M} = M/Γ_{R_{+}}(M). Since n_{𝔭} > 0, we have n_{𝔭} = n_{R_{𝔭}}(\overline{M}_{𝔭}) by 1.3.6. Let x ∈ R_{+} be a homogeneous \overline{M}_{𝔭}-regular element such that n_{R_{𝔭}}(\overline{M}_{𝔭}/x\overline{M}_{𝔭}) = n_{𝔭} − 1, cf. 1.3.7. By the induction hypothesis, there exists an open set U such that 𝔭 ∈ U and n_{R_{𝔭}}(\overline{M}_{𝔭}/x\overline{M}_{𝔭}) ≥ n_{R_{𝔮}}(\overline{M}_{𝔮}/x\overline{M}_{𝔮}) for all 𝔮 ∈ U. The inequality in the statement is clearly satisfied for all 𝔮 ∈ U with n_{𝔮} ≤ 0. For all 𝔮 ∈ U with n_{𝔮} > 0 we have:

n_{𝔭} = n_{R_{𝔭}}(\overline{M}_{𝔭}/x\overline{M}_{𝔭}) + 1 ≥ n_{R_{𝔮}}(\overline{M}_{𝔮}/x\overline{M}_{𝔮}) + 1 ≥ n_{𝔮}(\overline{M}_{𝔮}) = n_{𝔮}(M_{𝔮})

For the second inequality, note that if S is a graded ring with unique graded maximal ideal 𝔫, then for any nonzero finitely generated graded S-module N, and any homogeneous element z ∈ 𝔫 we have dim_{R}(N/zN) ≥ dim_{R}(N)−1; this is the graded version of [5, A.4]. To prove this, it suffices to reduce the problem to the local case, using the fact dim_{R}(N) = dim_{R_{𝔫}}(N_{𝔫}) and dim_{R}(N/zN) = dim_{R_{𝔫}}(N_{𝔫}/zN_{𝔫}). □

We obtain a similar lemma for the grade:

3.3. Lemma. For any 𝔭 ∈ Spec(R_{0}) we set g_{𝔭} = g_{R_{𝔭}}(M_{𝔭}). .

(1) If 𝔭, 𝔮 are prime ideals in Spec(R_{0}) such that 𝔭 ⊆ 𝔮, then g_{𝔭} ≥ g_{𝔮}.

(2) For any 𝔭 ∈ Spec(R_{0}) there exists an open set U ⊆ Spec(R_{0}) such that 𝔭 ∈ U and g_{𝔮} ≥ g_{𝔭} for all 𝔮 ∈ U. In particular, g_{𝔮} = g_{𝔭} for all 𝔮 ∈ U ∩ V (𝔭).

Proof. (1) By localizing at 𝔮 we may assume that R_{0} is local with maximal ideal 𝔮 and thus g_{𝔮} = g_{R}(M). If g_{𝔭} = ∞, then the inequality is clear. Assume now that g_{𝔭} ≠ ∞. By (1.2.1) we have H_{R_{+}R_{𝔭}}^{g_{𝔭}}(M_{𝔭}) ≠ 0. From 1.3.3 it follows that H_{R_{+}}^{g_{𝔭}}(M) ≠ 0. Using again (1.2.1) we conclude g_{𝔭} ≥ g_{R}(M) = g_{𝔮}.

(2) We prove the statement by induction on g_{𝔭}. If g_{𝔭} = ∞, then M_{𝔭} = 0 and choose U so that 𝔭 ∈ U and M_{𝔮} = 0 for all 𝔮 ∈ U, as in Lemma 3.1(1).

If g_{𝔭} = 0 then the assertion is clear.

If 0 < g_{𝔭} < ∞ then choose a homogeneous element x/1 ∈ R_{+}R_{𝔭} which is regular on M_{𝔭}, cf. 1.3.1. By Lemma 3.1(2) we can choose an open set U_{1} such that 𝔭 ∈ U_{1} and x/1 ∈ R_{+}R_{𝔮} is regular on M_{𝔮} or M_{𝔮} = 0 for all 𝔮 ∈ U_{1}. Since g_{R_{𝔭}}(M_{𝔭}/xM_{𝔭}) = g_{𝔭} − 1, we can use the induction hypothesis to obtain an open set U_{2} so that 𝔭 ∈ U_{2} and

g_{R_{𝔭}}(M_{𝔭}/xM_{𝔭}) ≤ g_{R_{𝔮}}(M_{𝔮}/xM_{𝔮}) for all 𝔮 ∈ U_{2} .

Setting U = U_{1} ∩ U_{2} we have thus for all 𝔮 ∈ U with M_{𝔮} ≠ 0:

g_{𝔭} = g_{R_{𝔭}}(M_{𝔭}/xM_{𝔭}) + 1 ≤ g_{R_{𝔮}}(M_{𝔮}/xM_{𝔮}) + 1 = g_{𝔮}

When M_{𝔮} = 0 we have g_{𝔮} = ∞, hence the inequality is also satisfied. □

The next proposition can be deduced immediately from the above lemmas.

3.4. Proposition. For any integer k the following sets are open in Spec(R_{0}):

D_{1}^{k}(M) : = {𝔮 ∈ Spec(R_{0}) | g_{R_{𝔮}}(M_{𝔮}) ≥ k}
D_{2}^{k}(M) : = {𝔮 ∈ Spec(R_{0}) | n_{R_{𝔮}}(M_{𝔮}) ≤ k}
D_{3}^{k}(M) : = {𝔮 ∈ Spec(R_{0}) | n_{R_{𝔮}}(M_{𝔮}) − g_{R_{𝔮}}(M_{𝔮}) ≤ k}

Our main theorem in this section generalizes a result of Katzman and Sharp in [9, 1.8]; they prove the case M = R of Theorem 3.5 below.

3.5. Theorem. Let n be an integer such that H_{R_{+}}^{i}(M) = 0 for all i > n and H_{R_{+}}^{n}(M) ≠ 0. The set Supp_{R_{0}}(H_{R_{+}}^{n}(M)) is then closed in Spec(R_{0}).

In view of 1.1, this gives Theorem 1(2) in the introduction.

Proof. Using 1.2, we conclude

Supp_{R_{0}} (H_{R_{+}}^{n}(M)) = {𝔮 ∈ Spec(R_{0}) | n_{R_{𝔮}}(M_{𝔮}) = n}

We also have n_{R_{𝔮}}(M_{𝔮}) ≤ n for all 𝔮 ∈ Spec(R_{0}), hence, in the notation of Proposition 3.4, the complement in Spec(R_{0}) of the set above is precisely the set D_{2}^{n−1}(M), which is open. □

3.6. Remark. Arguments similar to those in the proof above show that the support of the bottom nonvanishing local cohomology is closed, too. However the result would be weaker than what is known, since Ass_{R_{0}}(H_{R_{+}}^{g}(M)) is finite for g = grade(R_{+}, M), cf. [2].

4. Cohen-Macaulay modules

In this section we consider the case when M is a Cohen-Macaulay R-module. We recall below several known facts on Cohen-Macaulay graded modules, for which we refer to [5].

4.1. Assume that (R_{0}, 𝔪_{0}) is local and set 𝔪 = 𝔪_{0}+R_{+}, the unique graded maximal ideal of R. The R-module M is then Cohen-Macaulay if and only if the R_{𝔪}-module M_{𝔪} is Cohen-Macaulay, and in this case we have:

dim_{R}(M) = dim_{R_{𝔪}}(M_{𝔪}) = depth_{R_{𝔪}}(M_{𝔪}) = grade(𝔪, M)

4.2. Proposition. Assume (R_{0}, 𝔪_{0}) is local, and M is a nonzero Cohen-Macaulay R-module. The following then holds:

grade(R_{+}, M) = dim_{R} M − dim_{R_{0}} M

Proof. Since R is isomorphic to a quotient of a polynomial ring over R_{0}, we may actually assume by 1.3.4 that R is a polynomial ring R = R_{0}[x_{1}, … x_{s}]. (Note that dimension is also invariant under the change of the ring.) Furthermore, by 1.3.3 we may assume that R_{0} is complete, and hence it is a quotient S_{0}/I, where S_{0} is a regular local ring. It follows that R is a quotient of the polynomial ring S_{0}[x_{1}, … x_{s}]. Replacing R with this ring, we may thus assume that both R and R_{0} are regular (in particular, Cohen-Macaulay)

In his thesis, Lim [11, 1.2.9] proved that the following holds:

(∗) grade(R_{+}, M) = ht(R_{+}) + ht(I ∩ R_{0}) − ht I

where I = \sqrt{ann_{R} M}. Note that I ∩ R_{0} = \sqrt{ann_{R_{0}} M}, using for example [16, 1.1.2(1)]. Since R_{0} and R are Cohen-Macaulay, we have

ht(I ∩ R_{0}) = dim R_{0} − dim_{R_{0}} M
ht(I) = dim R − dim_{R} M

Since ht(R_{+}) = dim(R) − dim(R_{0}), the formula (∗) gives the equality in the statement. □

When R_{0} is local, the notion of depth of M over R_{0} can be introduced in the usual way (even if M is not necessarily finitely generated over R_{0}), namely as being equal to the length of a maximal regular sequence. (See [16] for more details.)

4.3. Proposition. If (R_{0}, 𝔪_{0}) is local and M is a Cohen-Macaulay R-module, then

dim_{R}(M/𝔪_{0}M) = dim_{R}(M) − depth_{R_{0}}(M)

Proof. Let 𝔪 = 𝔪_{0} + R_{+} be the unique graded maximal ideal of R. Consider a maximal M-regular sequence q_{1}, … q_{s} in 𝔪_{0}, and choose t_{1}, … t_{r} ∈ 𝔪 such that q_{1}, … q_{s}, t_{1}, …, t_{r} is a maximal M-regular sequence in 𝔪. We have thus:

depth_{R_{0}}(M) = s and grade(𝔪, M) = r + s

Using 4.1 we evaluate the right-hand part of the equality in the statement:

dim_{R}(M) − depth_{R_{0}}(M) = grade(𝔪, M) − depth_{R_{0}}(M) = r

Set \overline{M} = M/(q_{1}, … q_{s})M. Note that \overline{M} is a graded Cohen-Macaulay R-module and in view of 4.1 we have

dim_{R}(\overline{M}) = grade(𝔪,\overline{M}) = r

To prove the statement, it suffices thus to show that the R-modules M/𝔪_{0}M and \overline{M} have the same dimension. Since M/𝔪_{0}M is a homomorphic image of \overline{M}, we have dim_{R}(M/𝔪_{0}M) ≤ dim_{R}(\overline{M}).

To prove the reverse inequality, consider Q ∈ Ass_{R}(\overline{M}) such that Q ∩ R_{0} = 𝔪_{0}. We can choose such a prime because depth_{R_{0}}(\overline{M}) = 0, hence 𝔪_{0} ∈ Ass_{R_{0}}(\overline{M}), and we can apply for example [16, 2.1.2].

The R-module \overline{M} is graded Cohen-Macaulay, hence the R_{𝔪}-module \overline{M}_{𝔪} is Cohen-Macaulay. Since Q_{𝔪} is an associated prime of this last module, we have

(4.3.1)  dim_{R}(\overline{M}) = dim_{R_{𝔪}}(\overline{M}_{𝔪}) = dim_{R_{𝔪}}(R_{𝔪}/Q_{𝔪}) = dim_{R}(R/Q)

Let \overline{N} be a Q-primary submodule \overline{N} ⊆ \overline{M}, and let N denote the preimage of \overline{N} in M. We have Ass_{R}(M/N) = Ass_{R}(\overline{M}/\overline{N}) = \{Q\}, hence:

(4.3.2)  dim_{R}(M/N) = dim_{R}(R/Q)

It also follows that rad(ann_{R}(M/N)) = Q, and since 𝔪_{0} ⊆ Q, we conclude that there exists an integer r such that 𝔪_{0}^{r} ⊆ ann_{R}(M/N), and hence 𝔪_{0}^{r}M ⊆ N. We have thus:

(4.3.3)  dim_{R}(M/𝔪_{0}^{r}M) ≥ dim_{R}(M/N)

On the other hand, by [16, 5.1] we have:

(4.3.4)  dim_{R}(M/𝔪_{0}^{r}M) = dim_{R}(M/𝔪_{0}M)

Putting together the four equations displayed above, we obtain

dim_{R}(M/𝔪_{0}M) ≥ dim_{R}(\overline{M})

and this finishes the proof. □

4.4. When R_{0} is local, we set

codepth_{R_{0}}(M) = dim_{R_{0}}(M) − depth_{R_{0}}(M)

Recall from [16, 1.2.2] that we have the following formulas:

depth_{R_{0}}(M) = inf{depth_{R_{0}}(M_{i}) | i ∈ ℤ with M_{i} ≠ 0}
dim_{R_{0}}(M) = sup{dim_{R_{0}}(M_{i}) | i ∈ ℤ}

Note that if M ≠ 0 then 0 ≤ codepth_{R_{0}}(M) ≤ dim(R_{0}) and if M = 0 then codepth_{R_{0}}(M) = −∞.

Using the notation introduced earlier, Propositions 4.2 and 4.3 prove that the following formula holds whenever R_{0} is local, and M is a nonzero Cohen-Macaulay R-module:

(4.4.1)  n_{R}(M) − g_{R}(M) = codepth_{R_{0}}(M)

Note that the formula also holds when M = 0.

4.5. Remark. The R_{0}-module M is Cohen-Macaulay (meaning that codepth_{R_{0}} M ≤ 0) if and only if for each i the R_{0}-module M_{i} is Cohen-Macaulay and dim_{R_{0}} M_{i} = dim_{R_{0}} M.

Indeed, if M is Cohen-Macaulay, then, using 4.4, we have for all i:

depth_{R_{0}} M_{i} ≥ depth_{R_{0}} M = dim_{R_{0}} M ≥ dim_{R_{0}} M_{i}

It follows that equalities hold above, and in particular M_{i} is Cohen-Macaulay.

Conversely, choose i be such that depth_{R_{0}} M = depth_{R_{0}} M_{i}, cf. 4.4. Since depth_{R_{0}} M_{i} = dim_{R_{0}} M_{i} = dim_{R_{0}} M, it follows that M is Cohen-Macaulay.

In view of 1.2, we can use the formula (4.4.1) and Remark 4.5 to give a necessary and sufficient condition for a Cohen-Macaulay R-module to have only one nonvanishing local cohomology.

4.6. Corollary. Assume that M is a Cohen-Macaulay R-module and R_{0} is local. The following statements are then equivalent:

(1) M is Cohen-Macaulay as an R_{0}-module.

(2) The R_{0}-module M_{i} is Cohen-Macaulay, with dim_{R_{0}} M_{i} = dim_{R_{0}} M for all i.

(3) There exists an integer j ≥ 0 such that H_{R_{+}}^{i}(M) = 0 for all i ≠ j. □

In the remaining of the section we eliminate the condition that R_{0} is local. Note that the formula (4.4.1) gives:

4.7. Corollary. Assume that M is a Cohen-Macaulay R-module. For any 𝔭 ∈ Spec(R_{0}) the following equality holds:

n_{R_{𝔭}}(M_{𝔭}) − g_{R_{𝔭}}(M_{𝔭}) = codepth_{(R_{0})_{𝔭}}(M_{𝔭})  □

Combining Corollary 4.7 with Proposition 3.4, we obtain:

4.8. Corollary. If M is Cohen-Macaulay over R, then for any k ∈ ℕ the set

U_{C_{n}}^{0}(M) = {𝔭 ∈ Spec(R_{0}) | codepth_{(R_{0})_{𝔭}}(M_{𝔭}) ≤ k}

is open in Spec(R_{0}). □

Furthermore, by [16, 3.3] we have:

4.9. Corollary. If M is Cohen-Macaulay over R, then for any k ∈ ℕ the set

U_{S_{n}}^{0}(M) = {𝔭 ∈ Spec(R_{0}) | the (R_{0})_{𝔭}-module M_{𝔭} satisfies (S_{k})}

is open in Spec(R_{0}). □

5. base rings of small dimension

In this section we prove in several cases that the support of the local cohomology is closed. To prove that a set is open, we use the topological Nagata criterion, cf. [14, 24.2], as recalled below:

5.1. A set D in Spec(R_{0}) is open if and only if the following two conditions are satisfied:

(1) If 𝔮 ∈ D and 𝔭 ⊆ 𝔮, then 𝔭 ∈ D.

(2) For any prime 𝔭 ∈ D there exists an open non-empty subset of V (𝔭) contained in D, that is, there exists an open set U in Spec(R_{0}) such that ∅ ≠ U ∩ V (𝔭) ⊆ D.

The assumptions on R and M are as in Section 1. To simplify the notation, for every 𝔭 ∈ Spec(R_{0}) we set:

g_{𝔭} = g_{R_{𝔭}}(M_{𝔭}) and n_{𝔭} = n_{R_{𝔭}}(M_{𝔭})

5.2. Proposition. Assume that M is a Cohen-Macaulay R-module and the inequality codepth_{(R_{0})_{𝔭}}(M_{𝔭}) ≤ 1 holds for all 𝔭 ∈ Spec(R_{0}).

The set Supp_{R_{0}}(H_{R_{+}}^{i}(M)) is then closed in Spec(R_{0}) for all i.

Proof. Fix some i. We will prove the conditions (1) and (2) of 5.1 for the set D = Spec(R_{0}) ∖ Supp_{R_{0}}(H_{R_{+}}^{i}(M)). By Flat Base Change, we have

D = {𝔭 ∈ Spec(R_{0}) | H_{R_{+}R_{𝔭}}^{i}(M_{𝔭}) = 0}

and we see that (1) is satisfied. To check (2), let 𝔭 ∈ D. If M_{𝔭}=0, then we choose U as in Lemma 3.1(1).

If M_{𝔭} ≠ 0, then Corollary 4.7 gives g_{𝔭} ≤ n_{𝔭} ≤ g_{𝔭} + 1. Since H_{R_{+}R_{𝔭}}^{i}(M_{𝔭}) = 0, we have i ∉ {g_{𝔭}, n_{𝔭}} by 1.2. Using Theorems 3.2 and 3.3 we can choose an open set U such that 𝔭 ∈ U and n_{𝔮} = n_{𝔭} and g_{𝔮} = g_{𝔭} for all 𝔮 ∈ U ∩ V (𝔭). Thus, for all 𝔮 ∈ U ∩ V (𝔭) we have g_{𝔮} ≤ n_{𝔮} ≤ g_{𝔮} + 1 and i ∉ {g_{𝔮}, n_{𝔮}}. Using again 1.2 it follows that H_{R_{+}R_{𝔮}}^{i}(M_{𝔮}) = 0, that is, 𝔮 ∈ D. □

5.3. Theorem. Assume that M is a Cohen-Macaulay R-module.

If dim R_{0} ≤ 2, then Supp_{R_{0}}(H_{R_{+}}^{i}(M)) is closed for all i.

Proof. Fix some i. As above, we only need to check condition (2) of 5.1 for the set D = Spec(R_{0}) ∖ Supp_{R_{0}}(H_{R_{+}}^{i}(M)). Let 𝔭 ∈ D. We distinguish the following two cases:

(a) ht 𝔭 = 2. In this case we take U = Spec(R_{0}), noting that U ∩ V (𝔭) = {𝔭}.

(b) ht 𝔭 ≤ 1. In this case we have codepth_{(R_{0})_{𝔭}}(M_{𝔭}) ≤ dim(R_{0})_{𝔭} ≤ 1, hence g_{𝔭} ≤ n_{𝔭} ≤ g_{𝔭} + 1 when M_{𝔭} ≠ 0, by Corollary 4.7. We proceed as in the proof of Proposition 5.2. □

5.4. Theorem. Assume that R_{0} is semilocal, and M is a Cohen-Macaulay Rmodule. If dim R_{0} ≤ 3, then Supp_{R_{0}}(H_{R_{+}}^{i}(M)) is closed for all i.

Proof. Fix some i. As above, we only need to check condition (2) of 5.1 for the set D = Spec(R_{0}) ∖ Supp_{R_{0}}(H_{R_{+}}^{i}(M)). Let 𝔭 ∈ D. We distinguish the following three cases:

(a) If ht 𝔭 = 3, then we take U = Spec(R_{0}).

(b) If ht 𝔭 = 2, then we take U = Spec(R_{0}) ∖ {𝔪_{1}, …, 𝔪_{s}}, where 𝔪_{1}, …, 𝔪_{s} are the maximal ideals of R_{0}.

(c) If ht 𝔭 ≤ 1, then by Corollary 4.7 we have g_{𝔭} ≤ n_{𝔭} ≤ g_{𝔭} + 1 when M_{𝔭} ≠ 0 and we proceed as in the proof of Proposition 5.2. □

We recall a result of Brodmann, Fumasoli and Lim [1]:

5.5. If R_{0} is semilocal of dimension at most 2, then H_{R_{+}}^{i}(M) is tame for all i.

Lim [13] has proved tameness of the local cohomology for any ring of dimension at most 2, under the additional assumption that M is Cohen-Macaulay. Using our methods, we recover below Lim’s result.

5.6. Theorem. Assume that M is a Cohen-Macaulay R-module.

If dim R_{0} ≤ 2, then H_{R_{+}}^{i}(M) is tame for all i.

Proof. Fix some i. If there exists some 𝔭 ∈ Spec(R_{0}) such that H_{R_{+}R_{𝔭}}^{i}(M_{𝔭})_{d} ≠ 0 for infinitely many d < 0, then 5.5 yields that H_{R_{+}R_{𝔭}}^{i}(M_{𝔭})_{j} ≠ 0 for all j ≪ 0, hence H_{R_{+}}^{i}(M)_{j} ≠ 0 for all j ≪ 0 , and thus H_{R_{+}}^{i}(M) is tame.

It remains thus to study the case when H_{R_{+}R_{𝔭}}^{i}(M_{𝔭}) is a finite (R_{0})_{𝔭}-module for all 𝔭 ∈ Spec(R_{0}). Note that if i = 0, then H_{R_{+}}^{0}(M) is itself a finite R_{0}-module.

Also, if i = 1, then it follows that H_{R_{+}}^{1}(M) is a finite R_{0}-module as well, using a result of Faltings, cf. [4, 9.6.1]. Thus, we may also assume i > 1.

Claim: For every integer n the following set is open:

D_{n} = {𝔭 ∈ Spec(R_{0}) | H_{R_{+}R_{𝔭}}^{i}(M_{𝔭})_{j} = 0 for all j ≤ −n}

Assuming the claim, we will prove the theorem. Set Z_{n} = Spec(R_{0}) ∖ D_{n}.

If Z_{n} = ∅ for some n, then H_{R_{+}R_{𝔭}}^{i}(M_{𝔭})_{j} = 0 for all j ≤ −n and all 𝔭 ∈ Spec(R_{0}), hence H_{R_{+}}^{i}(M)_{j} = 0 for all j ≤ −n, and thus H_{R_{+}}^{i}(M) is tame.

Assume now that for each n we have Z_{n} ≠ ∅. The following chain of closed subsets:

⋯ ⊆ Z_{n} ⊆ Z_{n−1} ⊆ ⋯ ⊆ Z_{0}

shows that there exists an n_{0} such that Z_{n} = Z_{n_{0}} for all n ≥ n_{0}. If 𝔭 ∈ Z_{n_{0}}, then 𝔭 ∈ Z_{n} for all n ≥ n_{0}, and it follows that H_{R_{+}R_{𝔭}}^{i}(M_{𝔭}) is nonzero in infinitely many degrees; this case was treated in the beginning.

Proof of the claim: To prove that the set D_{n} is open we will use again the criterion 5.1. We only need to prove (2), that is: for each 𝔭 ∈ D_{n} we need to find an open set U of Spec(R_{0}) such that ∅ ≠ U ∩ V (𝔭) ⊆ D_{n}. We have the following two cases:

(a) If ht 𝔭 = 2, then take U = Spec(R_{0}), and note that U ∩ V (𝔭) = {𝔭}.

(b) Assume now ht 𝔭 ≤ 1. If M_{𝔭} = 0, then choose U as in Lemma 3.1(1). We may assume thus M_{𝔭} ≠ 0. Since codepth_{(R_{0})_{𝔭}}(M_{𝔭}) ≤ 1, Corollary 4.7 implies

(5.6.1)  g_{𝔭} ≤ n_{𝔭} ≤ g_{𝔭} + 1

If g_{𝔭} = 0, then n_{𝔭} ≤ 1. Let U be an open set containing 𝔭 such that n_{𝔮} ≤ n_{𝔭}, and thus n_{𝔮} ≤ 1, for all 𝔮 ∈ U. Since i > 1, we have then H_{R_{+}R_{𝔮}}^{i}(M_{𝔮}) = 0 for all 𝔮 ∈ U, hence U ⊆ D_{n}.

Assume now that g_{𝔭} > 0. Note that i ≠ n_{𝔭}. Indeed, if i = n_{𝔭}, then H_{R_{+}R_{𝔭}}^{i}(M_{𝔭}), and thus H_{R_{+}}^{i}(M), is nonzero in infinitely many degrees by Remark 2.6. This contradicts 𝔭 ∈ D_{n}.

If i ≠ g_{𝔭} then it suffices to choose an open set U such that 𝔭 ∈ U and g_{𝔭} = g_{𝔮} and n_{𝔭} = n_{𝔮} for all 𝔮 ∈ U ∩ V (𝔭); this can be done using Lemmas 3.2 and 3.3. For all such 𝔮 we have i ≠ n_{𝔮} and i ≠ g_{𝔮}. In view of (5.6.1) and 1.2 we conclude H_{R_{+}R_{𝔮}}^{i}(M_{𝔮}) = 0, and thus 𝔮 ∈ D_{n}.

If i = g_{𝔭}, consider an open set U_{1} = {𝔮 ∈ Spec(R_{0}) | b ∉ 𝔮} with b ∈ R_{0} such that 𝔭 ∈ U_{1} and g_{𝔮} ≥ g_{𝔭} for all 𝔮 ∈ U_{1} (by Lemma 3.3). The primes of the ring (R_{0})_{b} correspond bijectively with the elements of U_{1}. By assumption H_{R_{+}R_{𝔮}}^{i}(M_{𝔮}) is finite over R_{0} for all 𝔮 ∈ U_{1}. Since g_{𝔮} ≥ i for all such 𝔮, it actually follows that H_{R_{+}R_{𝔮}}^{j}(M_{𝔮}) is finitely generated for all j < i+1. By a theorem a Faltings [4, 9.6.1], it follows that H_{R_{+}R_{b}}^{j}(M_{b}) is finitely generated for all j < i + 1, and hence there exists an integer d such that H_{R_{+}R_{b}}^{i}(M_{b})_{l} = 0 for all l ≤ d. In particular, it follows that H_{R_{+}R_{𝔮}}^{i}(M_{𝔮})_{l} = 0 for all 𝔮 ∈ U_{1} and all l ≤ d. If d ≥ −n then we take U = U_{1}. If d < −n then we take U = U_{1} ∩ U_{2} where

U_{2} = ⋂_{d<j≤−n} Spec(R_{0}) ∖ Supp_{R_{0}}(H_{R_{+}}^{i}(M)_{j})  □

Note. After the submission of this paper, we learned that the results of Section 2 have been obtained independently by Sazeedeh. Also, M. Brodmann informed us that he has a different proof of Theorem 3.5.

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Department of Mathematics, Michigan State University, East Lansing, MI 48824

E-mail address: rotthaus@math.msu.edu

Department of Mathematics, Michigan State University, East Lansing, MI 48824

E-mail address: lsega@math.msu.edu