{"title":"Projective bundle formula for Heller's relative $K_{0}$","authors":"V. Sadhu","doi":"10.2996/kmj/1605063630","DOIUrl":null,"url":null,"abstract":"In this article, we study the Heller relative $K_{0}$ group of the map $\\mathbb{P}_{X}^{r} \\to \\mathbb{P}_{S}^{r},$ where $X$ and $S$ are quasi-projective schemes over a commutative ring. More precisely, we prove that the projective bundle formula holds for Heller's relative $K_{0},$ provided $X$ is flat over $S.$ As a corollary, we get a description of the relative group $K_{0}(\\mathbb{P}_{X}^{r} \\to \\mathbb{P}_{S}^{r})$ in terms of generators and relations, provided $X$ is affine and flat over $S.$","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"29 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2020-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: Algebraic Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2996/kmj/1605063630","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
In this article, we study the Heller relative $K_{0}$ group of the map $\mathbb{P}_{X}^{r} \to \mathbb{P}_{S}^{r},$ where $X$ and $S$ are quasi-projective schemes over a commutative ring. More precisely, we prove that the projective bundle formula holds for Heller's relative $K_{0},$ provided $X$ is flat over $S.$ As a corollary, we get a description of the relative group $K_{0}(\mathbb{P}_{X}^{r} \to \mathbb{P}_{S}^{r})$ in terms of generators and relations, provided $X$ is affine and flat over $S.$