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.