F-finiteness of homomorphisms and its descent

Let p be a prime number. We define the notion of F-finiteness of homomorphisms of F_p-algebras, and discuss some basic properties. In particular, we prove a sort of descent theorem on F-finiteness of homomorphisms of F_p-algebras. As a corollary, we prove the following. Let g: B C be a homomorphism of Noetherian 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.