Notes on the Frobenius test exponents

TL;DR

This paper establishes a lower bound for the Frobenius test exponent in local rings of prime characteristic, relating it to the Hartshorne-Speiser-Lyubeznik number, with a new elementary proof of an isomorphism in local cohomology.

Contribution

It proves that the Frobenius test exponent is always at least the Hartshorne-Speiser-Lyubeznik number, providing an elementary proof of a key isomorphism in local cohomology.

Findings

Frobenius test exponent ≥ Hartshorne-Speiser-Lyubeznik number

Elementary proof of Nagel and Schenzel's local cohomology isomorphism

Establishes a fundamental inequality in prime characteristic local rings

Abstract

In this paper we show that the Frobenius test exponent for parameter ideals of a local ring of prime characteristic is always bigger than or equal to its Hartshorne-Speiser-Lyubeznik number. Our argument is based on an isomorphism of Nagel and Schenzel on local cohomology that we will provide an elementary proof.