Generalized $F$-depth and graded nilpotent singularities

TL;DR

This paper introduces the generalized $F$-depth invariant to study weakly $F$-nilpotent singularities, providing explicit constructions, behavior under algebraic operations, and bounds on Frobenius test exponents.

Contribution

It defines the generalized $F$-depth, analyzes its properties under various algebraic operations, and produces explicit examples with bounds on Frobenius test exponents.

Findings

Generalized $F$-depth effectively tracks weakly $F$-nilpotent singularities.

Explicit examples of $F$-nilpotent singularities are constructed.

Bounds on Frobenius test exponents are established.

Abstract

We address explicit constructions of new variants of $F$-nilpotent singularities. In particular, we explore how (generalized) weakly $F$-nilpotent singularities behave under gluing, Segre products, Veronese subrings, and the formation of diagonal hypersurface algebras. From these results, explicit examples are produced and we provide bounds on their Frobenius test exponents. To accomplish these tasks, we introduce the {\it generalized $F$-depth} in analogy to Lyubeznik's $F$-depth. These depth-like invariants track (generalized) weakly $F$-nilpotent singularities in a similar fashion as (generalized) depth tracks (generalized) Cohen-Macaulay singularities.