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Let $R$ be a standard graded algebra over an $F$-finite field ofcharacteristic $p > 0$. Let $\phi:R\to R$ be the Frobenius endomorphism. Foreach finitely generated graded $R$-module $M$, let ${}^{\phi}\!M$ be theabelian group $M$ with the $R$-module structure induced by the Frobeniusendomorphism. 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 finitelygenerated graded $R$-module $M$ such that $\text{reg}_R\,{}^{\phi}\!M <\infty$.This result supplies another instance for the ability of the Frobenius indetecting homological properties, as exemplified by Kunz's famous regularitycriterion. The main technical tool is the notion of Castelnuovo-Mumfordregularity over certain homomorphisms between $\mathbb{N}$-graded algebras. Thelatter notion is a common generalization of the relative and absoluteCastelnuovo-Mumford regularity of modules.

Regularity over homomorphisms and a Frobenius characterization of Koszul algebras