{"title":"Regularity of Spectral Stacks and Discreteness of Weight-Hearts","authors":"Adeel A. Khan, V. Sosnilo","doi":"10.1093/QMATH/HAAB017","DOIUrl":null,"url":null,"abstract":"We study regularity in the context of ring spectra and spectral stacks. Parallel to that, we construct a weight structure on the category of compact quasi-coherent sheaves on spectral quotient stacks of the form $X=[\\operatorname{Spec} R/G]$ defined over a field, where $R$ is a noetherian ${\\mathcal{E}_{\\infty}}$-$k$-algebra and $G$ is a linearly reductive group acting on $R$. In this context we show that regularity of $X$ is equivalent to regularity of $R$. We also show that if $R$ is bounded, such a stack is discrete. This result can be interpreted in terms of weight structures and suggests a general phenomenon: for a symmetric monoidal stable $\\infty$-category with a compatible bounded weight structure, the existence of an adjacent t-structure satisfying a strong boundedness condition should imply discreteness of the weight-heart. \nWe also prove a gluing result for weight structures and adjacent t-structures, in the setting of a semi-orthogonal decomposition of stable $\\infty$-categories.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":"167 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2019-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"8","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: K-Theory and Homology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1093/QMATH/HAAB017","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 8
Abstract
We study regularity in the context of ring spectra and spectral stacks. Parallel to that, we construct a weight structure on the category of compact quasi-coherent sheaves on spectral quotient stacks of the form $X=[\operatorname{Spec} R/G]$ defined over a field, where $R$ is a noetherian ${\mathcal{E}_{\infty}}$-$k$-algebra and $G$ is a linearly reductive group acting on $R$. In this context we show that regularity of $X$ is equivalent to regularity of $R$. We also show that if $R$ is bounded, such a stack is discrete. This result can be interpreted in terms of weight structures and suggests a general phenomenon: for a symmetric monoidal stable $\infty$-category with a compatible bounded weight structure, the existence of an adjacent t-structure satisfying a strong boundedness condition should imply discreteness of the weight-heart.
We also prove a gluing result for weight structures and adjacent t-structures, in the setting of a semi-orthogonal decomposition of stable $\infty$-categories.