$F$-finiteness of homomorphisms and its descent

TL;DR

This paper introduces the concept of F-finiteness for homomorphisms of F_p-algebras, explores its fundamental properties, and establishes a descent theorem that generalizes previous results on F-finiteness in algebraic structures.

Contribution

The paper defines F-finiteness for algebra homomorphisms, proves a descent theorem, and extends known results to a broader class of Noetherian F_p-algebras.

Findings

F-finiteness of homomorphisms is well-defined and has basic properties.

A descent theorem for F-finiteness is established.

If a faithfully flat reduced homomorphism has F-finite codomain, then the domain is F-finite.

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 $g:B\to C$ 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 $p$.