Rees algebras and generalized depth-like conditions in prime characteristic

TL;DR

This paper investigates nilpotent Frobenius actions on Rees algebras in prime characteristic, introducing a new depth-like invariant that generalizes existing concepts and proves a related Cohen-Macaulay singularity theorem.

Contribution

It introduces a novel depth-like invariant unifying several existing invariants and extends a theorem of Huneke to Cohen-Macaulay singularities in prime characteristic.

Findings

Introduces a new depth-like invariant related to Frobenius actions.

Proves a nilpotent analog of Huneke's theorem for Cohen-Macaulay singularities.

Identifies properties of the invariant and conditions for deformation of weak F-nilpotence.

Abstract

In this article we address a question concerning nilpotent Frobenius actions on Rees algebras and associated graded rings. We prove a nilpotent analog of a theorem of Huneke for Cohen-Macaulay singularities. This is achieved by introducing a depth-like invariant which captures as special cases Lyubeznik's F-depth and the generalized F-depth from Maddox-Miller and is related to the generalized depth with respect to an ideal. We also describe several properties of this new invariant and identify a class of regular elements for which weak F-nilpotence deforms.