{"title":"An elementary description of K1(R) without elementary matrices","authors":"T. Huettemann, Zuhong Zhang","doi":"10.12958/adm1568","DOIUrl":null,"url":null,"abstract":"Let $R$ be a ring with unit. Passing to the colimit with respect to the standard inclusions $GL(n,R) \\to GL(n+1,R)$ (which add a unit vector as new last row and column) yields, by definition, the stable linear group $GL(R)$; the same result is obtained, up to isomorphism, when using the \"opposite\" inclusions (which add a unit vector as new first row and column). In this note it is shown that passing to the colimit along both these families of inclusions simultaneously recovers the algebraic $K$-group $K_1(R) = GL(R)/E(R)$ of~$R$, giving an elementary description that does not involve elementary matrices explicitly.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":"30 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2019-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: K-Theory and Homology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.12958/adm1568","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Let $R$ be a ring with unit. Passing to the colimit with respect to the standard inclusions $GL(n,R) \to GL(n+1,R)$ (which add a unit vector as new last row and column) yields, by definition, the stable linear group $GL(R)$; the same result is obtained, up to isomorphism, when using the "opposite" inclusions (which add a unit vector as new first row and column). In this note it is shown that passing to the colimit along both these families of inclusions simultaneously recovers the algebraic $K$-group $K_1(R) = GL(R)/E(R)$ of~$R$, giving an elementary description that does not involve elementary matrices explicitly.