Frobenius actions on local cohomology modules and deformation

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.

14 pages, final version

References in corpus (2)

• The weak ordinarity conjecture and $F$-singularities
• $F$-injectivity and Frobenius closure of ideals in Noetherian rings of characteristic $p>0$

Cited by in corpus (1)

• Surjectivity of some local cohomology map and the second vanishing theorem