On asymptotic socle degrees of local cohomology modules

TL;DR

This paper investigates the asymptotic behavior of socle degrees in local cohomology modules of powers of ideals in graded algebras, linking it to gauge-boundedness in prime characteristic.

Contribution

It introduces a new perspective on the asymptotic socle degrees of local cohomology modules and connects this to gauge-boundedness in prime characteristic.

Findings

Existence of a constant c controlling socle degrees

Connection established between socle behavior and gauge-boundedness

Insights into the asymptotic properties of local cohomology modules

Abstract

Let $R$ be a standard graded algebra over a field $k$ and $I$ be a homogeneous ideal of $R$. We study the question whether there is a constant $c$ such that $\Soc(H^{j}_{\fm}(R/I^t))_{<-ct}=0$ for all $t\geq 1$ and a variation of this question. We also draw a connection between this question and the notion of gauge-boundedness in prime characteristic.