  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 $p$.
