Nilpotence of Frobenius action and the Hodge filtration on local cohomology

Abstract

An F-nilpotent local ring is a local ring (R,m) of prime characteristic defined by the nilpotence of the Frobenius action on its local cohomology modules Hmi(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.