{"title":"Homologically Smooth Connected Cochain DGAs","authors":"X.-F. Mao","doi":"10.1007/s10468-024-10287-5","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span>\\(\\mathscr {A}\\)</span> be a connected cochain DG algebra such that <span>\\(H(\\mathscr {A})\\)</span> is a Noetherian graded algebra. We give some criteria for <span>\\(\\mathscr {A}\\)</span> to be homologically smooth in terms of the singularity category, the cone length of the canonical module <i>k</i> and the global dimension of <span>\\(\\mathscr {A}\\)</span>. For any cohomologically finite DG <span>\\(\\mathscr {A}\\)</span>-module <i>M</i>, we show that it is compact when <span>\\(\\mathscr {A}\\)</span> is homologically smooth. If <span>\\(\\mathscr {A}\\)</span> is in addition Gorenstein, we get </p><div><div><span>$$\\begin{aligned} \\textrm{CMreg}M = \\textrm{depth}_{\\mathscr {A}}\\mathscr {A} + \\mathrm {Ext.reg}\\, M<\\infty , \\end{aligned}$$</span></div></div><p>where <span>\\(\\textrm{CMreg}M\\)</span> is the Castelnuovo-Mumford regularity of <i>M</i>, <span>\\(\\textrm{depth}_{\\mathscr {A}}\\mathscr {A}\\)</span> is the depth of <span>\\(\\mathscr {A}\\)</span> and <span>\\( \\mathrm {Ext.reg}\\, M\\)</span> is the Ext-regularity of <i>M</i>.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 5","pages":"1931 - 1955"},"PeriodicalIF":0.5000,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebras and Representation Theory","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10468-024-10287-5","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let \(\mathscr {A}\) be a connected cochain DG algebra such that \(H(\mathscr {A})\) is a Noetherian graded algebra. We give some criteria for \(\mathscr {A}\) to be homologically smooth in terms of the singularity category, the cone length of the canonical module k and the global dimension of \(\mathscr {A}\). For any cohomologically finite DG \(\mathscr {A}\)-module M, we show that it is compact when \(\mathscr {A}\) is homologically smooth. If \(\mathscr {A}\) is in addition Gorenstein, we get
where \(\textrm{CMreg}M\) is the Castelnuovo-Mumford regularity of M, \(\textrm{depth}_{\mathscr {A}}\mathscr {A}\) is the depth of \(\mathscr {A}\) and \( \mathrm {Ext.reg}\, M\) is the Ext-regularity of M.
期刊介绍:
Algebras and Representation Theory features carefully refereed papers relating, in its broadest sense, to the structure and representation theory of algebras, including Lie algebras and superalgebras, rings of differential operators, group rings and algebras, C*-algebras and Hopf algebras, with particular emphasis on quantum groups.
The journal contains high level, significant and original research papers, as well as expository survey papers written by specialists who present the state-of-the-art of well-defined subjects or subdomains. Occasionally, special issues on specific subjects are published as well, the latter allowing specialists and non-specialists to quickly get acquainted with new developments and topics within the field of rings, algebras and their applications.