Generalized $F$-depth and graded nilpotent singularities

Rees algebras and generalized depth‐like conditions in prime characteristic

“…In various specializations, this invariant recovers Lyubeznik's 𝐹-depth [15] and the generalized 𝐹-depth introduced in [18]. Also, as zero modules are obviously nilpotent, it generalizes a similar invariant introduced in [9] in the context of generalized Cohen-Macaulay rings.…”

“…We define the former below. The following lemma helps us understand how nilpotence is tracked along short exact sequences; see [18,Lemma 2.11] for a proof. We conclude this subsection by describing the most important example of an 𝑅[𝐹]-module for our purposes, that is, how a Frobenius action on a module 𝑀 induces a Frobenius action on the local cohomology modules of 𝑀.…”

“…Introduced in [1], the classes of 𝐹-nilpotent singularities and its variants have gathered significant recent attention [12, 17-19, 21, 22, 25]. Critical tools for studying these singularities are Lyubeznik's 𝐹-depth, which tracks the smallest cohomological index of a non-nilpotent local cohomology module of a local ring, and variations of this invariant as introduced in [18]. We show in this paper that 𝐹-depth, and indeed many other depth-like invariants, can be interpreted as different instances of a common generalized invariant.…”

“…Likewise, generalized weakly 𝐹-nilpotent singularities were identified in [17] as analogs of generalized Cohen-Macaulay singularities. All of these singularities can be equivalently described as rings for which natural depth-like invariants are as large as possible [18]. Additionally, building on the framework introduced in [18], these notions are considered naturally in graded settings as well.…”

“…All of these singularities can be equivalently described as rings for which natural depth-like invariants are as large as possible [18]. Additionally, building on the framework introduced in [18], these notions are considered naturally in graded settings as well.…”