{"title":"Homological dimensions for endomorphism algebras of Gorenstein projective modules","authors":"Aiping Zhang, Xueping Lei","doi":"10.21136/cmj.2024.0199-23","DOIUrl":null,"url":null,"abstract":"<p>Let <i>A</i> be a CM-finite Artin algebra with a Gorenstein-Auslander generator <i>E, M</i> be a Gorenstein projective <i>A</i>-module and <i>B</i> = End<sub><i>A</i></sub><i>M</i>. We give an upper bound for the finitistic dimension of <i>B</i> in terms of homological data of <i>M</i>. Furthermore, if <i>A</i> is <i>n</i>-Gorenstein for 2 ⩽ <i>n</i> < ∞, then we show the global dimension of <i>B</i> is less than or equal to <i>n</i> plus the <i>B</i>-projective dimension of Hom<sub><i>A</i></sub>(<i>M, E</i>). As an application, the global dimension of End<sub><i>A</i></sub><i>E</i> is less than or equal to <i>n</i>.</p>","PeriodicalId":50596,"journal":{"name":"Czechoslovak Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Czechoslovak Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.21136/cmj.2024.0199-23","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let A be a CM-finite Artin algebra with a Gorenstein-Auslander generator E, M be a Gorenstein projective A-module and B = EndAM. We give an upper bound for the finitistic dimension of B in terms of homological data of M. Furthermore, if A is n-Gorenstein for 2 ⩽ n < ∞, then we show the global dimension of B is less than or equal to n plus the B-projective dimension of HomA(M, E). As an application, the global dimension of EndAE is less than or equal to n.
设 A 是具有戈伦斯坦-奥斯兰德生成器 E 的 CM 有限阿尔丁代数，M 是戈伦斯坦投影 A 模块，B = EndAM。此外，如果 A 在 2 ⩽ n < ∞ 时是 n-Gorenstein 的，那么我们将证明 B 的全局维度小于或等于 n 加上 HomA(M, E) 的 B 投影维度。作为应用，EndAE 的全局维度小于或等于 n。
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