一个左右凝聚环满足$\boldsymbol{\text{PGF}}(R)\boldsymbol{\neq}\boldsymbol{\text{GP}}(R)$
Two-sided coherent algebras over any field with $\mathcal{PGF}(R)\subsetneq\mathcal{GP}(R)$
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中文总结 AI 辅助
该研究构造了一个左右凝聚环$T$及一个非Gorenstein平坦的强Gorenstein投射左$T$-模$G$，证明了$\text{PGF}(T)\boldsymbol{\neq}\text{GP}(T)$，澄清了相关Gorenstein模类的包含关系。
中文摘要 AI 辅助
对于环$R$，设$\boldsymbol{\text{GP}}(R)$、$\boldsymbol{\text{GF}}(R)$和$\boldsymbol{\text{PGF}}(R)$分别表示Gorenstein投射、Gorenstein平坦和投射余分解Gorenstein平坦左$R$-模类。我们构造了一个左右凝聚环$T$及一个强Gorenstein投射左$T$-模$G$，它不是Gorenstein平坦的，因此$\boldsymbol{\text{PGF}}(T)\boldsymbol{\neq}\boldsymbol{\text{GP}}(T)$。
英文摘要
For a ring $R$, let $\mathcal{GP}(R)$, $\mathcal{GF}(R)$, and $\mathcal{PGF}(R)$ denote the classes of Gorenstein projective, Gorenstein flat, and projectively coresolved Gorenstein flat left $R$-modules, respectively. We answer negatively the question whether $\mathcal{GP}(R)=\mathcal{PGF}(R)$ for every ring. More precisely, over every field $k$ we construct a left and right coherent central $k$-algebra $T$ and a strongly Gorenstein projective left $T$-module which is not Gorenstein flat; hence $\mathcal{PGF}(T)\subsetneq\mathcal{GP}(T)$.
