arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2403.05232math.RT
收藏

由有限Gorenstein维数模产生的模型结构链

Chains of model structures arising from modules of finite Gorenstein dimension

Nan Gao, Xue-Song Lu, Pu Zhang

阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

AI总结：

本文证明PGF模类和Gorenstein投射模类分别构成完全遗传余挠对，诱导出Hovey三元组链，并推广到精确范畴，给出PGF与GP相等的充要条件。

摘要：

对于任意整数$n\ge 0$和任意环$R$，$(\mathcal {PGF}_n, \\ \mathcal P_n^\perp \cap \mathcal {PGF}^{\perp})$被证明是$R$-Mod中的一个完全遗传的余挠对，其中$\mathcal {PGF}$是由J. Šaroch和J. Štovíček引入的PGF模类，而$\mathcal {PGF}_n$是PGF维数$\le n$的$R$-模类。对于任意Artin代数$R$，$(\mathcal {GP}_n, \\ \mathcal P_n^\perp \cap \mathcal {GP}^{\perp})$被证明是$R$-Mod中的一个完全且遗传的余挠对，其中$\mathcal {GP}_n$是Gorenstein投射维数$\le n$的模类。这些余挠对诱导出两个遗传的Hovey三元组链$(\mathcal {PGF}_n, \\ \mathcal P_n^\perp, \\ \mathcal {PGF}^{\perp})$和$(\mathcal {GP}_n, \\ \mathcal P_n^\perp, \\ \mathcal {GP}^{\perp})$，并且同一链中相应的同伦范畴是相同的。观察到$R$-Mod中的某些完全余挠对可以诱导$R$-Mod的某些特殊扩张封闭子范畴中的完全余挠对。然后，在精确范畴$\mathcal {PGF}_n$、$\mathcal {GP}_n$、$\mathcal {GF}_n$、$\mathcal {PGF}^{<\infty}$、$\mathcal {GP}^{<\infty}$和$\mathcal {GF}^{<\infty}$中也得到了相应的结果。作为副产品，对于环$R$，$\mathcal{PGF} = \mathcal {GP}$当且仅当$\mathcal{PGF}^\perp\cap\mathcal{GP}_n=\mathcal P_n$对某个$n$成立。

英文摘要：

For any integer $n\ge 0$ and any ring $R$, \ $(\mathcal {PGF}_n, \ \mathcal P_n^\perp \cap \mathcal {PGF}^{\perp})$ proves to be a complete hereditary cotorsion pair in $R$-Mod, where $\mathcal {PGF}$ is the class of PGF modules, introduced by J. Šaroch and J. Štovíček, and \ $\mathcal {PGF}_n$ is the class of $R$-modules of PGF dimension $\le n$. For any Artin algebra $R$, \ $(\mathcal {GP}_n, \ \mathcal P_n^\perp \cap \mathcal {GP}^{\perp})$ proves to be a complete and hereditary cotorsion pair in $R$-Mod, where $\mathcal {GP}_n$ is the class of modules of Gorenstein projective dimension $\le n$. These cotorsion pairs induce two chains of hereditary Hovey triples \ $(\mathcal {PGF}_n, \ \mathcal P_n^\perp, \ \mathcal {PGF}^{\perp})$ and \ $(\mathcal {GP}_n, \ \mathcal P_n^\perp, \ \mathcal {GP}^{\perp})$, and the corresponding homotopy categories in the same chain are the same. It is observed that some complete cotorsion pairs in $R$-Mod can induce complete cotorsion pairs in some special extension closed subcategories of $R$-Mod. Then corresponding results in exact categories $\mathcal {PGF}_n$, \ $\mathcal {GP}_n$, \ $\mathcal {GF}_n$, \ $\mathcal {PGF}^{<\infty}$, \ $\mathcal {GP}^{<\infty}$ and $\mathcal {GF}^{<\infty}$, are also obtained. As a byproduct, $\mathcal{PGF} = \mathcal {GP}$ for a ring $R$ if and only if $\mathcal{PGF}^\perp\cap\mathcal{GP}_n=\mathcal P_n$ for some $n$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑