{"title":"On graded \\({\\mathbb {E}}_{\\infty }\\)-rings and projective schemes in spectral algebraic geometry","authors":"Mariko Ohara, Takeshi Torii","doi":"10.1007/s40062-021-00298-0","DOIUrl":null,"url":null,"abstract":"<div><p>We introduce graded <span>\\({\\mathbb {E}}_{\\infty }\\)</span>-rings and graded modules over them, and study their properties. We construct projective schemes associated to connective <span>\\({\\mathbb {N}}\\)</span>-graded <span>\\({\\mathbb {E}}_{\\infty }\\)</span>-rings in spectral algebraic geometry. Under some finiteness conditions, we show that the <span>\\(\\infty \\)</span>-category of almost perfect quasi-coherent sheaves over a spectral projective scheme <span>\\(\\text { {Proj}}\\,(A)\\)</span> associated to a connective <span>\\({\\mathbb {N}}\\)</span>-graded <span>\\({\\mathbb {E}}_{\\infty }\\)</span>-ring <i>A</i> can be described in terms of <span>\\({{\\mathbb {Z}}}\\)</span>-graded <i>A</i>-modules.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"17 1","pages":"105 - 144"},"PeriodicalIF":0.7000,"publicationDate":"2022-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-021-00298-0.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Homotopy and Related Structures","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s40062-021-00298-0","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We introduce graded \({\mathbb {E}}_{\infty }\)-rings and graded modules over them, and study their properties. We construct projective schemes associated to connective \({\mathbb {N}}\)-graded \({\mathbb {E}}_{\infty }\)-rings in spectral algebraic geometry. Under some finiteness conditions, we show that the \(\infty \)-category of almost perfect quasi-coherent sheaves over a spectral projective scheme \(\text { {Proj}}\,(A)\) associated to a connective \({\mathbb {N}}\)-graded \({\mathbb {E}}_{\infty }\)-ring A can be described in terms of \({{\mathbb {Z}}}\)-graded A-modules.
期刊介绍:
Journal of Homotopy and Related Structures (JHRS) is a fully refereed international journal dealing with homotopy and related structures of mathematical and physical sciences.
Journal of Homotopy and Related Structures is intended to publish papers on
Homotopy in the broad sense and its related areas like Homological and homotopical algebra, K-theory, topology of manifolds, geometric and categorical structures, homology theories, topological groups and algebras, stable homotopy theory, group actions, algebraic varieties, category theory, cobordism theory, controlled topology, noncommutative geometry, motivic cohomology, differential topology, algebraic geometry.