$F$-finiteness of homomorphisms and its descent

Abstract

Let $p$ be a prime number. We define the notion of $F$-finiteness of homomorphisms of $\mathbb{F}_{p}$-algebras, and discuss some basic properties. In particular, we prove a sort of descent theorem on $F$-finiteness of homomorphisms of $\mathbb{F}_{p}$-algebras. As a corollary, we prove the following. Let $g\colon B \to C$ be a homomorphism of Noetherian $\mathbb{F}_{p}$-algebras. If $g$ is faithfully flat reduced and $C$ is $F$-finite, then $B$ is $F$-finite. This is a generalization of Seydi's result on excellent local rings of characteristic $p$.