{"title":"代数有界扩展的分类属性和同调猜想","authors":"Yongyun Qin, Xiaoxiao Xu, Jinbi Zhang, Guodong Zhou","doi":"arxiv-2407.21480","DOIUrl":null,"url":null,"abstract":"An extension $B\\subset A$ of finite dimensional algebras is bounded if the\n$B$-$B$-bimodule $A/B$ is $B$-tensor nilpotent, its projective dimension is\nfinite and $\\mathrm{Tor}_i^B(A/B, (A/B)^{\\otimes_B j})=0$ for all $i, j\\geq 1$.\nWe show that for a bounded extension $B\\subset A$, the algebras $A$ and $B$ are\nsingularly equivalent of Morita type with level. Additively, under some\nconditions, their stable categories of Gorenstein projective modules and\nGorenstein defect categories are equivalent, respectively. Some homological\nconjectures are also investigated for bounded extensions, including\nAuslander-Reiten conjecture, finististic dimension conjecture, Fg condition,\nHan's conjecture, and Keller's conjecture. Applications to trivial extensions\nand triangular matrix algebras are given. In course of proofs, we give some handy criteria for a functor between module\ncategories induces triangle functors between stable categories of Gorenstein\nprojective modules and Gorenstein defect categories, which generalise some\nknown criteria and hence, might be of independent interest.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"78 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Categorical properties and homological conjectures for bounded extensions of algebras\",\"authors\":\"Yongyun Qin, Xiaoxiao Xu, Jinbi Zhang, Guodong Zhou\",\"doi\":\"arxiv-2407.21480\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"An extension $B\\\\subset A$ of finite dimensional algebras is bounded if the\\n$B$-$B$-bimodule $A/B$ is $B$-tensor nilpotent, its projective dimension is\\nfinite and $\\\\mathrm{Tor}_i^B(A/B, (A/B)^{\\\\otimes_B j})=0$ for all $i, j\\\\geq 1$.\\nWe show that for a bounded extension $B\\\\subset A$, the algebras $A$ and $B$ are\\nsingularly equivalent of Morita type with level. Additively, under some\\nconditions, their stable categories of Gorenstein projective modules and\\nGorenstein defect categories are equivalent, respectively. Some homological\\nconjectures are also investigated for bounded extensions, including\\nAuslander-Reiten conjecture, finististic dimension conjecture, Fg condition,\\nHan's conjecture, and Keller's conjecture. Applications to trivial extensions\\nand triangular matrix algebras are given. In course of proofs, we give some handy criteria for a functor between module\\ncategories induces triangle functors between stable categories of Gorenstein\\nprojective modules and Gorenstein defect categories, which generalise some\\nknown criteria and hence, might be of independent interest.\",\"PeriodicalId\":501136,\"journal\":{\"name\":\"arXiv - MATH - Rings and Algebras\",\"volume\":\"78 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-07-31\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Rings and Algebras\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2407.21480\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Rings and Algebras","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2407.21480","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Categorical properties and homological conjectures for bounded extensions of algebras
An extension $B\subset A$ of finite dimensional algebras is bounded if the
$B$-$B$-bimodule $A/B$ is $B$-tensor nilpotent, its projective dimension is
finite and $\mathrm{Tor}_i^B(A/B, (A/B)^{\otimes_B j})=0$ for all $i, j\geq 1$.
We show that for a bounded extension $B\subset A$, the algebras $A$ and $B$ are
singularly equivalent of Morita type with level. Additively, under some
conditions, their stable categories of Gorenstein projective modules and
Gorenstein defect categories are equivalent, respectively. Some homological
conjectures are also investigated for bounded extensions, including
Auslander-Reiten conjecture, finististic dimension conjecture, Fg condition,
Han's conjecture, and Keller's conjecture. Applications to trivial extensions
and triangular matrix algebras are given. In course of proofs, we give some handy criteria for a functor between module
categories induces triangle functors between stable categories of Gorenstein
projective modules and Gorenstein defect categories, which generalise some
known criteria and hence, might be of independent interest.