A sufficient condition for finiteness of Frobenius test exponents

TL;DR

This paper provides a sufficient condition related to local cohomology modules under which the Frobenius test exponent of a local ring in prime characteristic is finite, advancing understanding of Frobenius closure properties.

Contribution

It establishes a new criterion involving local cohomology for the finiteness of the Frobenius test exponent in local rings.

Findings

Fte(R) is finite under the new condition.

Finiteness of Frobenius closure in local cohomology implies finite Fte.

Provides a link between local cohomology and Frobenius test exponents.

Abstract

The Frobenius test exponent $\operatorname{Fte}(R)$ of a local ring $(R,\mathfrak{m})$ of prime characteristic $p > 0$ is the smallest $e_0 \in \mathbb{N}$ such that for every ideal $\mathfrak{q}$ generated by a (full) system of parameters, the Frobenius closure $\mathfrak{q}^F$ has $(\mathfrak{q}^F)^{[p^{e_0}]} = \mathfrak{q}^{[p^{e_0}]}$. We establish a suffcient condition for $\operatorname{Fte}(R)<\infty$ and use it to show that if $R$ is such that the Frobenius closure of the zero submodule in the lower local cohomology modules has finite colength, i.e. $H^j_{\mathfrak{m}}(R) / 0^F_{H^j_{\mathfrak{m}}(R)}$ is finite length for $0 \le j < \dim(R)$, then $\operatorname{Fte}(R)<\infty$.