Nilpotence of Frobenius action and the Hodge filtration on local cohomology

TL;DR

This paper explores the relationship between Frobenius nilpotence, Hodge filtration, and local cohomology in algebraic geometry, providing new characterizations of certain singularities in prime characteristic and their reductions.

Contribution

It offers a Hodge-theoretic interpretation of three-dimensional F-nilpotent singularities and characterizes them via divisor class groups and Brauer groups in the graded case.

Findings

Hodge-theoretic interpretation of F-nilpotent singularities

Characterization of singularities using divisor class groups

Connection between Frobenius action and Hodge filtration

Abstract

An $F$-nilpotent local ring is a local ring $(R, \mathfrak{m})$ of prime characteristic defined by the nilpotence of the Frobenius action on its local cohomology modules $H^i_{\mathfrak{m}}(R)$. A singularity in characteristic zero is said to be of $F$-nilpotent type if its modulo $p$ reduction is $F$-nilpotent for almost all $p$. In this paper, we give a Hodge-theoretic interpretation of three-dimensional normal isolated singularities of $F$-nilpotent type. In the graded case, this yields a characterization of these singularities in terms of divisor class groups and Brauer groups.