Regularity over homomorphisms and a Frobenius characterization of Koszul algebras

TL;DR

This paper characterizes Koszul algebras over fields of positive characteristic using Frobenius endomorphisms and Castelnuovo-Mumford regularity, linking homological properties to Frobenius actions.

Contribution

It establishes a new criterion for Koszulness based on the finiteness of regularity of modules under Frobenius, extending the understanding of Frobenius's role in homological algebra.

Findings

R is Koszul iff there exists a module M with finite reg under Frobenius

Introduces a generalized notion of Castelnuovo-Mumford regularity over homomorphisms

Connects Frobenius actions to homological properties of graded algebras

Abstract

Let $R$ be a standard graded algebra over an $F$-finite field of characteristic $p > 0$. Let $\phi:R\to R$ be the Frobenius endomorphism. For each finitely generated graded $R$-module $M$, let ${}^{\phi}\!M$ be the abelian group $M$ with the $R$-module structure induced by the Frobenius endomorphism. The $R$-module ${}^{\phi}\!M$ has a natural grading given by $\text{deg} x=j$ if $x\in M_{jp+i}$ for some $0\le i \le p-1$. In this paper, we prove that $R$ is Koszul if and only if there exists a non-zero finitely generated graded $R$-module $M$ such that $\text{reg}_R\,{}^{\phi}\!M <\infty$. This result supplies another instance for the ability of the Frobenius in detecting homological properties, as exemplified by Kunz's famous regularity criterion. The main technical tool is the notion of Castelnuovo-Mumford regularity over certain homomorphisms between $\mathbb{N}$-graded algebras. The latter notion is a common generalization of the relative and absolute Castelnuovo-Mumford regularity of modules.