  We develop a theory of G-dimension for modules over local homomorphisms which
encompasses the classical theory of G-dimension for finite 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 finite 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
(de)composition.
