{"title":"论\\(\\mathbb {Z}\\) -范畴的k理论","authors":"Eugenia Ellis, Rafael Parra","doi":"10.1007/s40062-023-00333-2","DOIUrl":null,"url":null,"abstract":"<div><p>We establish connections between the concepts of Noetherian, regular coherent, and regular <i>n</i>-coherent categories for <span>\\(\\mathbb {Z}\\)</span>-linear categories with finitely many objects and the corresponding notions for unital rings. These connections enable us to obtain a negative <i>K</i>-theory vanishing result, a fundamental theorem, and a homotopy invariance result for the <i>K</i>-theory of <span>\\(\\mathbb {Z}\\)</span>-linear categories.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"18 4","pages":"455 - 476"},"PeriodicalIF":0.7000,"publicationDate":"2023-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the K-theory of \\\\(\\\\mathbb {Z}\\\\)-categories\",\"authors\":\"Eugenia Ellis, Rafael Parra\",\"doi\":\"10.1007/s40062-023-00333-2\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We establish connections between the concepts of Noetherian, regular coherent, and regular <i>n</i>-coherent categories for <span>\\\\(\\\\mathbb {Z}\\\\)</span>-linear categories with finitely many objects and the corresponding notions for unital rings. These connections enable us to obtain a negative <i>K</i>-theory vanishing result, a fundamental theorem, and a homotopy invariance result for the <i>K</i>-theory of <span>\\\\(\\\\mathbb {Z}\\\\)</span>-linear categories.</p></div>\",\"PeriodicalId\":49034,\"journal\":{\"name\":\"Journal of Homotopy and Related Structures\",\"volume\":\"18 4\",\"pages\":\"455 - 476\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2023-11-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"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/s40062-023-00333-2\",\"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/s40062-023-00333-2","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
We establish connections between the concepts of Noetherian, regular coherent, and regular n-coherent categories for \(\mathbb {Z}\)-linear categories with finitely many objects and the corresponding notions for unital rings. These connections enable us to obtain a negative K-theory vanishing result, a fundamental theorem, and a homotopy invariance result for the K-theory of \(\mathbb {Z}\)-linear 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.
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