Rees algebras and generalized depth‐like conditions in prime characteristic

Abstract

In this paper, we address a question concerning nilpotent Frobenius actions on Rees algebras and associated graded rings. We prove a nilpotent analog of a theorem of Huneke for Cohen–Macaulay singularities. This is achieved by introducing a depth‐like invariant which captures as special cases Lyubeznik's F‐depth and the generalized F‐depth from Maddox–Miller and is related to the generalized depth with respect to an ideal. We also describe several properties of this new invariant and identify a class of regular elements for which weak F‐nilpotence deforms.