The Frobenius structure of local cohomology

Abstract

Given a local ring of positive prime characteristic there is a natural Frobenius action on its local cohomology modules with support at its maximal ideal. In this paper we study the local rings for which the local cohomology modules have only finitely many submodules invariant under the Frobenius action. In particular we prove that F-pure Gorenstein local rings as well as the face ring of a finite simplicial complex localized or completed at its homogeneous maximal ideal have this property. We also introduce the notion of an antinilpotent Frobenius action on an Artinian module over a local ring and use it to study those rings for which the lattice of submodules of the local cohomology that are invariant under Frobenius satisfies the Ascending Chain Condition.

On Some Local Cohomology Spectral Sequences

“…The interest for studying these F-stable submodules arises naturally in the study of singularities of algebraic varieties in prime characteristic, due to its connection with test ideals. In Theorem 7.20, we show that Brun-BrunsRömer decomposition of Stanley toric face rings of prime characteristic is compatible with this natural Frobenius action, and therefore they only have a finite number of F-stable submodules; in particular, we recover and extend [EH08,Theorem 5.1], because Stanley-Reisner rings are a particular case of Stanley toric face rings [Ngu12, Example 2.2 (i)].…”

A sufficient condition for F-purity

“…A local ring (R, m) is called F -pure if the Frobenius endomorphism F : R → R is pure. The Frobenius endomorphism on R induces a natural Frobenius action on each local cohomology module H i m (R) (see Discussion 2.2 and 2.4 in [EH08] for a detailed explanation of this). We say a local ring is F -injective if F acts injectively on all of the local cohomology modules of R with support in m. We note that F -pure implies F -injective [HR76].…”

“…We will also use some notations introduced in [EH08] (see also [Ma12]). We say an Rmodule M is an R{F }-module if there is a Frobenius action F : M → M such that for all u ∈ M, F (ru) = r p u.…”

“…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].…”

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

“…One important example that we will use repeatedly is that H R,A (H 3 This is not the original definition of F -rationality, but is shown to be equivalent ( [Smi97]). [HR76], [Fed83], [EH08].…”