{"title":"相对奇异范畴和奇异等价","authors":"Rasool Hafezi","doi":"10.1007/s40062-021-00289-1","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>R</i> be a right noetherian ring. We introduce the concept of relative singularity category <span>\\(\\Delta _{\\mathcal {X} }(R)\\)</span> of <i>R</i> with respect to a contravariantly finite subcategory <span>\\(\\mathcal {X} \\)</span> of <span>\\({\\text {{mod{-}}}}R.\\)</span> Along with some finiteness conditions on <span>\\(\\mathcal {X} \\)</span>, we prove that <span>\\(\\Delta _{\\mathcal {X} }(R)\\)</span> is triangle equivalent to a subcategory of the homotopy category <span>\\(\\mathbb {K} _\\mathrm{{ac}}(\\mathcal {X} )\\)</span> of exact complexes over <span>\\(\\mathcal {X} \\)</span>. As an application, a new description of the classical singularity category <span>\\(\\mathbb {D} _\\mathrm{{sg}}(R)\\)</span> is given. The relative singularity categories are applied to lift a stable equivalence between two suitable subcategories of the module categories of two given right noetherian rings to get a singular equivalence between the rings. In different types of rings, including path rings, triangular matrix rings, trivial extension rings and tensor rings, we provide some consequences for their singularity categories.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"16 3","pages":"487 - 516"},"PeriodicalIF":0.7000,"publicationDate":"2021-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-021-00289-1","citationCount":"0","resultStr":"{\"title\":\"Relative singularity categories and singular equivalences\",\"authors\":\"Rasool Hafezi\",\"doi\":\"10.1007/s40062-021-00289-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <i>R</i> be a right noetherian ring. We introduce the concept of relative singularity category <span>\\\\(\\\\Delta _{\\\\mathcal {X} }(R)\\\\)</span> of <i>R</i> with respect to a contravariantly finite subcategory <span>\\\\(\\\\mathcal {X} \\\\)</span> of <span>\\\\({\\\\text {{mod{-}}}}R.\\\\)</span> Along with some finiteness conditions on <span>\\\\(\\\\mathcal {X} \\\\)</span>, we prove that <span>\\\\(\\\\Delta _{\\\\mathcal {X} }(R)\\\\)</span> is triangle equivalent to a subcategory of the homotopy category <span>\\\\(\\\\mathbb {K} _\\\\mathrm{{ac}}(\\\\mathcal {X} )\\\\)</span> of exact complexes over <span>\\\\(\\\\mathcal {X} \\\\)</span>. As an application, a new description of the classical singularity category <span>\\\\(\\\\mathbb {D} _\\\\mathrm{{sg}}(R)\\\\)</span> is given. The relative singularity categories are applied to lift a stable equivalence between two suitable subcategories of the module categories of two given right noetherian rings to get a singular equivalence between the rings. In different types of rings, including path rings, triangular matrix rings, trivial extension rings and tensor rings, we provide some consequences for their singularity categories.</p></div>\",\"PeriodicalId\":49034,\"journal\":{\"name\":\"Journal of Homotopy and Related Structures\",\"volume\":\"16 3\",\"pages\":\"487 - 516\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2021-08-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1007/s40062-021-00289-1\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Homotopy and Related Structures\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s40062-021-00289-1\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Homotopy and Related Structures","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s40062-021-00289-1","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Relative singularity categories and singular equivalences
Let R be a right noetherian ring. We introduce the concept of relative singularity category \(\Delta _{\mathcal {X} }(R)\) of R with respect to a contravariantly finite subcategory \(\mathcal {X} \) of \({\text {{mod{-}}}}R.\) Along with some finiteness conditions on \(\mathcal {X} \), we prove that \(\Delta _{\mathcal {X} }(R)\) is triangle equivalent to a subcategory of the homotopy category \(\mathbb {K} _\mathrm{{ac}}(\mathcal {X} )\) of exact complexes over \(\mathcal {X} \). As an application, a new description of the classical singularity category \(\mathbb {D} _\mathrm{{sg}}(R)\) is given. The relative singularity categories are applied to lift a stable equivalence between two suitable subcategories of the module categories of two given right noetherian rings to get a singular equivalence between the rings. In different types of rings, including path rings, triangular matrix rings, trivial extension rings and tensor rings, we provide some consequences for their singularity categories.
期刊介绍:
Journal of Homotopy and Related Structures (JHRS) is a fully refereed international journal dealing with homotopy and related structures of mathematical and physical sciences.
Journal of Homotopy and Related Structures is intended to publish papers on
Homotopy in the broad sense and its related areas like Homological and homotopical algebra, K-theory, topology of manifolds, geometric and categorical structures, homology theories, topological groups and algebras, stable homotopy theory, group actions, algebraic varieties, category theory, cobordism theory, controlled topology, noncommutative geometry, motivic cohomology, differential topology, algebraic geometry.