Parameter Test Ideals of Cohen Macaulay Rings
نویسنده
چکیده
The main aim of this paper is to provide a description of parameter test ideals of local Cohen-Macaulay rings of prime characteristic p. The nature of this description will be such that it will allow us to give an algorithm for producing these ideals. The results in this paper will follow from an analysis of Frobenous maps on injective hulls of the residue fields of the rings under consideration. This analysis is inspired by Gennady Lyubeznik’s work on F -modules and indeed, a crucial tool used here, namely, the functors ∆ defined in section 3 below, are nothing but “the first step” in the construction of Lyubeznik’s H functors in section 4 of [L]. The study of S-modules with Frobenius maps can be elucidated by treating them as left modules over a certain skew polynomial ring S[T ; f ]. A crucial ingredient in this paper is Rodney Sharp’s recent study of these modules in general, and of the S[T ; f ]-module structure of the top local cohomology module in particular. In [S] the parameter test ideal of S was described in terms of certain S[T ; f ]-submodules of the top local cohomology of S, and it is this description on which our explicit description and algorithm is based on. Along the way we gain new insights into the S[T ; f ]-module structure of injective hulls of residue fields which translate into new results. One such result is an algorithm for computing the index of nilpotency (in the sense of section 4 of [L]) of top local cohomology modules, which, together with the results in [KS] translate into an algorithm for computing the Frobenius closure of parameter ideals in Cohen-Macaulay local rings and in view of [HKSY] provide an important ingredient for the corresponding computation in generalized Cohen-Macaulay rings as well. Another spinoff is a very simple proof of a crucial ingredient in [ABL] which together with Corollary 3.6 there gives an alternative proof of the fact that for a power series ring R of prime characteristic, for all nonzero f ∈ R, 1/f generates Rf as a DR-module.
منابع مشابه
Results on Generalization of Burch’s Inequality and the Depth of Rees Algebra and Associated Graded Rings of an Ideal with Respect to a Cohen-Macaulay Module
Let be a local Cohen-Macaulay ring with infinite residue field, an Cohen - Macaulay module and an ideal of Consider and , respectively, the Rees Algebra and associated graded ring of , and denote by the analytic spread of Burch’s inequality says that and equality holds if is Cohen-Macaulay. Thus, in that case one can compute the depth of associated graded ring of as In this paper we ...
متن کاملCOHEN-MACAULAY HOMOLOGICAL DIMENSIONS WITH RESPECT TO AMALGAMATED DUPLICATION
In this paper we use "ring changed'' Gorenstein homologicaldimensions to define Cohen-Macaulay injective, projective and flatdimensions. For doing this we use the amalgamated duplication of thebase ring with semi-dualizing ideals. Among other results, we prove that finiteness of these new dimensions characterizes Cohen-Macaulay rings with dualizing ideals.
متن کاملFrobenius Test Exponents for Parameter Ideals in Generalized Cohen–macaulay Local Rings
This paper studies Frobenius powers of parameter ideals in a commutative Noetherian local ring R of prime characteristic p. For a given ideal a of R, there is a power Q of p, depending on a, such that the Q-th Frobenius power of the Frobenius closure of a is equal to the Q-th Frobenius power of a. The paper addresses the question as to whether there exists a uniform Q0 which ‘works’ in this con...
متن کاملThe Equality I = Qi in Buchsbaum Rings
Let A be a Noetherian local ring with the maximal ideal m and d = dim A. Let Q be a parameter ideal in A. Let I = Q : m. The problem of when the equality I = QI holds true is explored. When A is a Cohen-Macaulay ring, this problem was completely solved by A. Corso, C. Huneke, C. Polini, and W. Vasconcelos [CHV, CP, CPV], while nothing is known when A is not a Cohen-Macaulay ring. The present pu...
متن کاملTHE CONCEPT OF (I; J)-COHEN MACAULAY MODULES
We introduce a generalization of the notion of depth of an ideal on a module by applying the concept of local cohomology modules with respect to a pair of ideals. We also introduce the concept of $(I,J)$-Cohen--Macaulay modules as a generalization of concept of Cohen--Macaulay modules. These kind of modules are different from Cohen--Macaulay modules, as an example shows. Also an art...
متن کامل