## Illinois Journal of Mathematics

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

#### 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.

#### Article information

Source
Illinois J. Math., Volume 48, Number 1 (2004), 241-272.

Dates
First available in Project Euclid: 13 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258136183

Digital Object Identifier
doi:10.1215/ijm/1258136183

Mathematical Reviews number (MathSciNet)
MR2048224

Zentralblatt MATH identifier
1103.13009

#### Citation

Iyengar, Srikanth; Sather-Wagstaff, Sean. G-dimension over local homomorphisms. Applications to the Frobenius endomorphism. Illinois J. Math. 48 (2004), no. 1, 241--272. doi:10.1215/ijm/1258136183. https://projecteuclid.org/euclid.ijm/1258136183