Local cohomology modules with infinite dimensional socles

Let T be a commutative Noetherian local ring of dimension at least two and R=T[x_1,...,x_n] a polynomial ring in n variables over T. Consider R as a graded ring with deg T = 0 and deg x_i = 1 for all i. Let I=R_+ and f a homogeneous polynomial whose coefficients form a system of parameters for T. We show that the socle of H^n_I(R/fR) is infinite dimensional, generalizing an example due to Hartshorne.


Introduction
The third of Huneke's four problems in local cohomology [Hu] is to determine when H i I (M) is Artinian for a given ideal I of a commutative Noetherian local ring R and finitely generated R-module M. An R-module N is Artinian if and only Supp R N ⊆ {m} and Hom R (R/m, N) is finitely generated, where m is the maximal ideal of R. Thus, Huneke's problem may be separated into two subproblems: • When is Supp R H i I (M) ⊆ {m}? • When is Hom R (R/m, H i I (M)) finitely generated? This article is concerned with the second question. For an R-module N, one may identify Hom R (R/m, N) with the submodule {x ∈ N | mx = 0}, which is an R/m-vector space called the socle of N (denoted soc R N). It is known that if R is an unramified regular local ring then the local cohomology modules H i I (R) have finite dimensional socles for all i ≥ 0 and all ideals I of R ( [HS], [L1], [L2]). The first example of a local cohomology module with an infinite dimensional socle was given in 1970 by Hartshorne [Ha]: Let k be a field, R = k [[u, v]][x, y], P = (u, v, x, y)R, I = (x, y)R, and f = ux + vy. Then soc R P H 2 IR P (R P /f R P ) is infinite dimensional. Of course, since I and f are homogeneous, this is equivalent to saying that Hom R (R/P, is infinite dimensional. Hartshorne proved this by exhibiting an infinite set of linearly independent elements in the * socle of H 2 I (R). In the last 30 years there have been few results in the literature which explain or generalize Harthshorne's example. For affine semigroup rings, a remarkable result proved by Helm and Miller [HM] gives necessary and sufficient conditions (on the semigroup) for the ring to possess a local cohomology module (of a finitely generated module) having infinite dimensional socle. Beyond that work, however, little has been done.
In this paper we prove the following: Theorem 1.1. Let (T, m) be a Noetherian local of dimension at least two. Let R = T [x 1 , . . . , x n ] be a polynomial ring in n variables over T , I = (x 1 , . . . , x n ), and f ∈ R a homogeneous polynomial whose coefficients form a system of parameters for T . Then the * socle of H n I (R/f R) is infinite dimensional. Hartshorne's example is obtained by letting T = k [[u, v]], n = 2, and f = ux+vy (homogeneous of degree 1). Note, however, that we do not require the coefficient ring to be regular, or even Cohen-Macaulay. As a further illustration, consider the following: Part of the proof of Theorem 1.1 was inspired by the recent work of Katzman [Ka] where information on the graded pieces of H n I (R/f R) is obtained by examining matrices of a particular form. We apply this technique in the proof of Lemma 2.8.
Throughout all rings are assumed to be commutative with identity. The reader should consult [Mat] or [BH] for any unexplained terms or notation and [BS] for the basic properties of local cohomology.

The Main Result
Let R = ⊕R ℓ be a Noetherian ring graded by the nonnegative integers. Assume R 0 is local and let P be the homogeneous maximal ideal of R. Given a finitely generated graded R-module M we define the * socle of M by Clearly, * soc R M ∼ = soc R P M P . An interesting special case of Huneke's third problem is the following: Question 2.1. Let n := µ R (R + /P R + ), the minimal number of generators of R + . When is * soc H n R + (R) finitely generated?
For i ∈ N it is well known that H i R+ (R) is a graded R-module, each graded piece H i R + (R) ℓ is a finitely generated R 0 -module, and H i R + (R) ℓ = 0 for all sufficiently large integers ℓ ( [BS,15.1.5]). If we know a priori that H n R+ (R) ℓ has finite length for all ℓ (e.g., if Supp R H n R + (R) ⊆ {P }), then Question 2.1 is equivalent to: Question 2.2. When is Hom R (R/R + , H n R + (R)) finitely generated?
We give a partial answer to these questions for hypersurfaces. For the remainder of this section we adopt the following notation: Let (T, m) be a local ring of dimension d and R = T [x 1 , . . . , x n ] a polynomial ring in n variables over T . We endow R with an N-grading by setting deg T = 0 and deg x i = 1 for all i. Let I = R + = (x 1 , . . . , x n )R and P = m + I the homogeneous maximal ideal of R. Let f ∈ R be a homogeneous element of degree p and C f the ideal of T generated by the nonzero coefficients of f .
Our main result is the following: Theorem 2.3. Assume d ≥ 2 and the (nonzero) coefficients of f form a system of parameters for T . Then * soc R H n I (R/f R) is not finitely generated. The proof of this theorem will be given in a series of lemmas below. Before proceeding with the proof we make a couple of remarks: Remark 2.4. (a) If d ≤ 1 in Theorem 2.3 then * soc H n I (R/f R) is finitely generated. This follows from [DM, Corollary 2] since dim R/I = dim T ≤ 1. (b) The hypothesis that the nonzero coefficients of f form a system of parameters for T is stronger than our proof requires. One only needs that C f be mprimary and that there exists a dimension 2 ideal containing all but two of the coefficients of f . (See the proof of Lemma 2.8.) The following lemma identifies the support of H n I (R/f R) for a homogeneous element f ∈ R. This lemma also follows from a much more general result recently proved by Katzman and Sharp [KS,Theorem 1.5].
Lemma 2.5. Let f ∈ R be a homogeneous element. Then It is enough to prove that H n I (R/f R) = 0 if and only if C f = T . As H n I (R/f R) k is a finitely generated T -module for all k, we have by Nakayama that . . . , x n ] is a polynomial ring in n variables over a field and N = (x 1 , . . . , x n )S. As dim S = n, we see that H n N (S/f S) = 0 if and only if the image of f modulo m is nonzero. Hence, H n I (R/f R) = 0 if and only if at least one coefficient of f is a unit, i.e., C f = T .
We are mainly interested in the case the coefficients of f generate an m-primary ideal: Corollary 2.6. Let f ∈ R be homogeneous and suppose C f is m-primary. Then Supp R H n I (R/f R) = {P }. Our next lemma is the key technical result in the proof of Theorem 2.3.
Lemma 2.7. Suppose u, v ∈ T such that ht(u, v)T = 2. For each integer n ≥ 1 let M n be the cokernel of φ n : T n+1 → T n where φ n is represented by the matrix .
Let J = ∩ n≥1 ann T M n . Then dim T /J = dim T .
Proof: LetT denote the m-adic completion of T . Then ht(u, v)T = 2, ann T M n = annT (M n ⊗ TT ) ∩ T , and dim T /(I ∩ T ) ≥ dimT /I for all ideals I ofT . Thus, we may assume T is complete. Now let p be a prime ideal of T such that dim T /p = dim T . Since T is catenary, ht(u, v)T /p = 2. Assume the lemma is true for complete domains. Then ∩ n≥1 ann T /p (M n ⊗ T T /p) = p/p. Hence which implies that dim T /J ≥ dim T /p = dim T . Thus, it suffices to prove the lemma for complete domains.
As T is complete, the integral closure S of T is a finite R-module. Since ht(u, v)S = 2 ( [Mat,Theorem 15.6]) and S is normal, {u, v} is a regular sequence on S. It is easily seen that I n (A n ), the ideal of n × n minors of A n , is (u, v) n T . By the main result of [BE] we obtain ann S (M n ⊗ T S) = (u, v) n S. Hence ann T M n ⊆ (u, v) n S ∩ T . As S is a finite T -module there exists an integer k such that ann T M n ⊆ (u, v) n−k T for all n ≥ k. Therefore, ∩ n≥1 ann T M n = (0), which completes the proof.
Lemma 2.8. Assume d ≥ 2 and let f ∈ R be a homogeneous element of degree p such that the coefficients of f form a system of parameters for T . Then dim T / ann T H n I (R/f R) ≥ 2. Proof: Let c 1 , . . . , c d be the nonzero coefficients of f .
, we may assume that dim T = 2 and f has exactly two nonzero terms.
For any w ∈ R there is a surjective map H n I (R/wf R) → H n I (R/f R). Hence, ann T H n I (R/wf R) ⊆ ann T H n I (R/f R). Thus, we may assume that the terms of f have no (nonunit) common factor. Without loss of generality, we may write R = T [x 1 , . . . , x k , y 1 , . . . y r ] and f = ux d 1 1 · · · x d k k + vy e 1 1 · · · y er r = ux d + vy e , where {u, v} is a system of parameters for T . As f is homogeneous, p = i d i = i e i . is nonzero for infinitely many k. Hence * soc R (H n I (R/f R)) = Hom R (R/P, H n I (R/f R)) is not finitely generated.