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

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

2003-03-20

arxiv.org/abs/math/0303265v3

Mathematics

Commutative Algebra

31 pages, uses Xy-pic

Scientific paper

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.

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