G-dimension over local homomorphisms. Applications to the Frobenius endomorphism

Abstract

We develop a theory of G-dimension over local homomorphisms which encompasses the classical theory of G-dimension for finitely generated modules over local rings. As an application, we prove that a local ring $R$ of characteristic $p$ is Gorenstein if and only if it possesses a nonzero finitely generated module of finite projective dimension that has finite G-dimension when considered as an $R$-module via some power of the Frobenius endomorphism of $R$. We also prove results that track the behavior of Gorenstein properties of local homomorphisms under composition and decomposition.