Gorenstein dimensions of unbounded complexes and faithfully flat change of base (With an appendix by Driss Bennis)

TL;DR

This paper investigates how Gorenstein homological dimensions of unbounded complexes over a commutative ring change under faithfully flat base change, establishing equivalences and stability results.

Contribution

It proves that Gorenstein flat and injective properties are preserved under faithfully flat base change for modules, extending to unbounded complexes and their homological dimensions.

Findings

Gorenstein flatness is preserved under faithfully flat base change.

Gorenstein injectivity is characterized by cotorsion and base change properties.

Stability of Gorenstein homological dimensions under faithfully flat base change.

Abstract

For a commutative ring R and a faithfully flat R-algebra S we prove, under mild extra assumptions, that an R-module M is Gorenstein flat if and only if the left S-module S\otimes M is Gorenstein flat, and that an R-module N is Gorenstein injective if and only if it is cotorsion and the left S-module Hom(S,N) is Gorenstein injective. We apply these results to the study of Gorenstein homological dimensions of unbounded complexes. In particular, we prove two theorems on stability of these dimensions under faithfully flat (co-)base change.