On the uniform bound of Frobenius test exponents

TL;DR

This paper establishes a uniform bound for Frobenius test exponents in certain local rings of prime characteristic, simplifying previous proofs and extending results to cases with nilpotent Frobenius actions on lower local cohomologies.

Contribution

It provides a simpler proof for the existence of uniform bounds in generalized Cohen-Macaulay rings and extends results to rings with nilpotent Frobenius actions on lower local cohomologies.

Findings

Existence of a uniform Frobenius test exponent bound in generalized Cohen-Macaulay rings.

Extension of bounds to rings with nilpotent Frobenius actions on lower local cohomologies.

Simplification of previous proofs by Huneke, Katzman, Sharp, and Yao.

Abstract

In this paper we prove the existence of a uniform bound for Frobenius test exponents for parameter ideals of a local ring $(R, \frak m)$ of prime characteristic in the following cases: (1) $R$ is generalized Cohen-Macaulay. Our proof is much more simpler than the original proof of Huneke, Katzman, Sharp and Yao, (2) The Frobenius actions on all lower local cohomologies $H^i_{\frak m}(R)$, $i < \dim R$, are nilpotent.