{"title":"Grothendieck’s Vanishing and Non-vanishing Theorems in an Abstract Module Category","authors":"Divya Ahuja, Surjeet Kour","doi":"10.1007/s10485-024-09767-y","DOIUrl":null,"url":null,"abstract":"<div><p>In this article, we prove Grothendieck’s Vanishing and Non-vanishing Theorems of local cohomology objects in the non-commutative algebraic geometry framework of Artin and Zhang. Let <i>k</i> be a field of characteristic zero and <span>\\({\\mathscr {S}}_{k}\\)</span> be a strongly locally noetherian <i>k</i>-linear Grothendieck category. For a commutative noetherian <i>k</i>-algebra <i>R</i>, let <span>\\({\\mathscr {S}}_R\\)</span> denote the category of <i>R</i>-objects in <span>\\({\\mathscr {S}}_k\\)</span> obtained through a non-commutative base change by <i>R</i> of the abelian category <span>\\({\\mathscr {S}}_{k}\\)</span>. First, we establish Grothendieck’s Vanishing Theorem for any object <span>\\({\\mathscr {M}}\\)</span> in <span>\\({\\mathscr {S}}_{R}\\)</span>. Further, if <i>R</i> is local and <span>\\({\\mathscr {S}}_{k}\\)</span> is Hom-finite, we prove Non-vanishing Theorem for any finitely generated flat object <span>\\({\\mathscr {M}}\\)</span> in <span>\\({\\mathscr {S}}_R\\)</span>.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10485-024-09767-y.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Categorical Structures","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10485-024-09767-y","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
In this article, we prove Grothendieck’s Vanishing and Non-vanishing Theorems of local cohomology objects in the non-commutative algebraic geometry framework of Artin and Zhang. Let k be a field of characteristic zero and \({\mathscr {S}}_{k}\) be a strongly locally noetherian k-linear Grothendieck category. For a commutative noetherian k-algebra R, let \({\mathscr {S}}_R\) denote the category of R-objects in \({\mathscr {S}}_k\) obtained through a non-commutative base change by R of the abelian category \({\mathscr {S}}_{k}\). First, we establish Grothendieck’s Vanishing Theorem for any object \({\mathscr {M}}\) in \({\mathscr {S}}_{R}\). Further, if R is local and \({\mathscr {S}}_{k}\) is Hom-finite, we prove Non-vanishing Theorem for any finitely generated flat object \({\mathscr {M}}\) in \({\mathscr {S}}_R\).
期刊介绍:
Applied Categorical Structures focuses on applications of results, techniques and ideas from category theory to mathematics, physics and computer science. These include the study of topological and algebraic categories, representation theory, algebraic geometry, homological and homotopical algebra, derived and triangulated categories, categorification of (geometric) invariants, categorical investigations in mathematical physics, higher category theory and applications, categorical investigations in functional analysis, in continuous order theory and in theoretical computer science. In addition, the journal also follows the development of emerging fields in which the application of categorical methods proves to be relevant.
Applied Categorical Structures publishes both carefully refereed research papers and survey papers. It promotes communication and increases the dissemination of new results and ideas among mathematicians and computer scientists who use categorical methods in their research.