Graded annihilators of modules over the Frobenius skew polynomial ring, and tight closure

Abstract

Abstract. This paper is concerned with the tight closure of an ideal a in a commutative Noetherian local ring R of prime characteristic p. Several authors, including R. Fedder, K-i. Watanabe, K. E. Smith, N. Hara and F. Enescu, have used the natural Frobenius action on the top local cohomology module of such an R to good effect in the study of tight closure, and this paper uses that device. The main part of the paper develops a theory of what are here called 'special annihilator submodules' of a left module over the Frobenius skew polynomial ring associated to R; this theory is then applied in the later sections of the paper to the top local cohomology module of R and used to show that, if R is Cohen-Macaulay, then it must have a weak parameter test element, even if it is not excellent. IntroductionThroughout the paper, R will denote a commutative Noetherian ring of prime characteristic p. We shall always denote by f : R −→ R the Frobenius homomorphism, for which f (r) = r p for all r ∈ R. Let a be an ideal of R. The n-th Frobenius power a [p n ] of a is the ideal of R generated by all p n -th powers of elements of a. We use R• to denote the complement in R of the union of the minimal prime ideals of R. An element r ∈ R belongs to the tight closure a * of a if and only if there exists c ∈ R• such that crWe say that a is tightly closed precisely when a * = a. The theory of tight closure was invented by M. Hochster and C. Huneke [8], and many applications have been found for the theory; see [10] and [11], for example.In the case when R is local, several authors have used, as an aid to the study of tight closure, the natural Frobenius action on the top local cohomology module of R; see, for example, R.

The Frobenius structure of local cohomology

“…For example, if R is complete, reduced, and Gorenstein, the largest proper F-stable submodule of H d m (R) corresponds to the tight closure of 0 (in the finitistic sense, see [Hochster and Huneke 1990, §8]), and its annihilator is the test ideal of R. Also see Discussion 2.10 here. We would like to note here that other results related to F-stable submodules of local cohomology may be found in [Enescu 2001;Katzman 2006;Sharp 2007].…”

“…The following proposition can be seen as a consequence of the more general Theorem 3.6 and its corollary in [Sharp 2007]. However, its proof is not very difficult and we include it here for the convenience of the reader.…”

$D$-module and $F$-module length of local cohomology modules

“…We first collect definitions and facts from [Sha07] and [Kat08]. Given an Artinian A{f }-module W , a special ideal of W is an ideal of A that is also the annihilator of some A{f }-submodule V ⊆ W , a special prime is a special ideal that is also a prime ideal (note that the special ideals depend on the A{f }-module structure on W , i.e., the Frobenius action f on W ).…”

A sufficient condition for F-purity

“…We also provide examples to show that not all complete F -pure Cohen-Macaulay rings satisfy this condition. In fact, if R is Cohen-Macaulay and F -injective, we show that this property is closely related to whether the natural injective Frobenius action on H d m (R) can be "lifted" to an injective Frobenius action on E R , the injective hull of the residue field of R. And instead of using pseudocanonical covers, our treatment uses the anti-nilpotent condition for modules with Frobenius action introduced in [EH08] and [Sha07].…”