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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 of characteristic 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 -module via some power of the Frobenius endomorphism of . We also prove results that track the behavior of Gorenstein properties of local homomorphisms under composition and decomposition.