Nilpotence of Frobenius action and the Hodge filtration on local cohomology

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 $H_{\mathfrak{m}}^i(R)$. 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.

18 pages; v2: several typos fixed

References in corpus (1)

• The weak ordinarity conjecture and $F$-singularities

Cited by in corpus (1)

• A Gorenstein criterion for strongly $F$-regular and log terminal singularities