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.
Citation
Srikanth Iyengar. Sean Sather-Wagstaff. "G-dimension over local homomorphisms. Applications to the Frobenius endomorphism." Illinois J. Math. 48 (1) 241 - 272, Spring 2004. https://doi.org/10.1215/ijm/1258136183
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