{"title":"Countably Generated Matrix Algebras","authors":"Arvid Siqveland","doi":"arxiv-2408.01034","DOIUrl":null,"url":null,"abstract":"We define the completion of an associative algebra $A$ in a set\n$M=\\{M_1,\\dots,M_r\\}$ of $r$ right $A$-modules in such a way that if $\\mathfrak\na\\subseteq A$ is an ideal in a commutative ring $A$ the completion $A$ in the\n(right) module $A/\\mathfrak a$ is $\\hat A^M\\simeq \\hat A^{\\mathfrak a}.$ This\nworks by defining $\\hat A^M$ as a formal algebra determined up to a computation\nin a category called GMMP-algebras. From deformation theory we get that the\ncomputation results in a formal algebra which is the prorepresenting hull of\nthe noncommutative deformation functor, and this hull is unique up to\nisomorphism.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"41 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Rings and Algebras","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.01034","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
We define the completion of an associative algebra $A$ in a set
$M=\{M_1,\dots,M_r\}$ of $r$ right $A$-modules in such a way that if $\mathfrak
a\subseteq A$ is an ideal in a commutative ring $A$ the completion $A$ in the
(right) module $A/\mathfrak a$ is $\hat A^M\simeq \hat A^{\mathfrak a}.$ This
works by defining $\hat A^M$ as a formal algebra determined up to a computation
in a category called GMMP-algebras. From deformation theory we get that the
computation results in a formal algebra which is the prorepresenting hull of
the noncommutative deformation functor, and this hull is unique up to
isomorphism.