Parameter-test-ideals of Cohen–Macaulay rings

Abstract

We describe an algorithm for computing parameter-test-ideals in certain local CohenMacaulay rings. The algorithm is based on the study of a Frobenius map on the injective hull of the residue field of the ring and on the application of Sharp's notion of 'special ideals'. Our techniques also provide an algorithm for computing indices of nilpotency of Frobenius actions on top local cohomology modules of the ring and on the injective hull of its residue field. The study of nilpotent elements on injective hulls of residue fields also yields a great simplification of the proof of the celebrated result in the article

Frobenius maps on injective hulls and their applications to tight closure

“…The first such result is Theorem 3.4, which gives a formula for weak parameter test ideals of complete local rings. This is a generalization of Theorem 8.2 in [6], which gave a similar description of the parameter test ideals of complete local rings under the assumption that a certain Frobenius map on the injective hull of the residue field is injective.…”

“…Section 2 studies basic properties of submodules of E S and their annihilators that are used throughout this paper. Section 3 generalizes Theorem 8.2 in [6] and gives an explicit description of the weak parameter test ideals of S in the case where S is Cohen-Macaulay with canonical module ω ⊆ S but where the Frobenius map on E S induced from the natural Frobenius map on H dim S mS (S) is not necessarily injective. Section 4 introduces a certain operation on an E S -ideal and applies it to the description of quasi-maximal filtrations of E S .…”

“…This paper uses mutually inverse functors Δ e : C e → D e and Ψ e : D e → C e (originally introduced in [6]), which are defined as follows. For M ∈ C e we have an R-linear map α M :…”

$D$-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 ).…”

Test ideals of non-principal ideals: Computations, jumping numbers, alterations and division theorems

“…The trace map can be described in this case via the Cartier isomorphism (see Section 2 for details). The ( ) [1/p e ] -operation, alternately denoted I e ( ) in [Kat08], is highly computable and has been implemented in Macaulay2 by M. Katzman (along with generalizations to not necessarily smooth ambient spaces using [BSTZ10]). However, the recipe τ (R, a t ) = (a a ) [1/p e ] fails when a is not principal, and the lack of a similarly effective description has been an obstruction for computing examples of τ (R, a t ) with non-principal a.…”