Frobenius actions on local cohomology modules and deformation

Abstract

Abstract: Let $(R, m) $ be a Noetherian local ring of characteristic $ p> 0$. We introduce and study $ F $-full and $ F $-anti-nilpotent singularities, both are defined in terms of the Frobenius actions on the local cohomology modules of $ R $ supported at the maximal ideal. We prove that if $ R/(x) $ is $ F $-full or $ F $-anti-nilpotent for a nonzerodivisor $ x\ in R $, then so is $ R $. We use these results to obtain new cases on the deformation of $ F $- injectivity.