Gorenstein projective dimensions of complexes under base change with respect to a semidualizing module
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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 that the Auslander-Buchsbaum formula holds for $G_{C}$-projective dimension.
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 that the Auslander-Buchsbaum formula holds for $G_{C}$-projective dimension.
참고문헌 (18)
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