{"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":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"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\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2023-11-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s40062-023-00333-2\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s40062-023-00333-2","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","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.
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