On cohomological invariants of local rings in positive characteristic

TL;DR

This paper investigates the Frobenius depth of local rings in positive characteristic, comparing it with formal grade and depth to deepen understanding of Frobenius invariants.

Contribution

It provides a comparative analysis of Frobenius depth, formal grade, and depth, offering new insights into their relationships in positive characteristic rings.

Findings

Frobenius depth is compared with formal grade and depth.

New relationships between Frobenius invariants are established.

Enhanced understanding of Frobenius depth's role in local ring theory.

Abstract

The Frobenius depth denoted by $\Fdepth$ defined by Hartshorne-Speiser in 1977 and later by Lyubeznik in 2006, in a different way, for rings of positive characteristic. The aim of the present paper is to compare the $\Fdepth$ with formal grade, and $\depth$ to shed more light on the notion of Frobenius depth from a different point of view.