Gorenstein projective dimensions of complexes under base change with respect to a semidualizing module

Bull. Korean Math. Soc. Published online December 29, 2020

Chunxia Zhang
Chongqing Normal University

Abstract : Let $R\rightarrow S$ be a ring homomorphism. The relations of Gorenstein projective dimension with respect to a semidualizing module of homologically bounded complexes between $U\otimes_{R}^{\mathbf{L}}X$ and $X$ are considered, where $X$ is an $R$-complex and $U$ is an $S$-complex. Some sufficient conditions are given under which the equality
$\mathcal{GP}_{\widetilde{C}}\text{-}\mathrm{pd}_{S}(S\otimes^{\mathbf{L}}_{R}X)=\mathcal{GP}_{C}\text{-}\mathrm{pd}_{R}(X)$ holds. As an application it is shown the Auslander-Buchsbaum formula for $G_{C}$-projective dimension.

Keywords : semidualizing module; $G_{C}$-projective module; $G_{C}$-projective dimension; ring homomorphism; depth of complex.