{"title":"Topological endomorphism rings of tilting complexes","authors":"Michal Hrbek","doi":"10.1112/jlms.12939","DOIUrl":null,"url":null,"abstract":"<p>In a compactly generated triangulated category, we introduce a class of tilting objects satisfying a certain purity condition. We call these the decent tilting objects and show that the tilting heart induced by any such object is equivalent to a category of contramodules over the endomorphism ring of the tilting object endowed with a natural linear topology. This extends the recent result for <i>n</i>-tilting modules by Positselski and Št'ovíček. In the setting of the derived category of modules over a ring, we show that the decent tilting complexes are precisely the silting complexes such that their character dual is cotilting. The hearts of cotilting complexes of cofinite type turn out to be equivalent to the category of discrete modules with respect to the same topological ring. Finally, we provide a kind of Morita theory in this setting: Decent tilting complexes correspond to pairs consisting of a tilting and a cotilting-derived equivalence as described above tied together by a tensor compatibility condition.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"109 6","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12939","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12939","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In a compactly generated triangulated category, we introduce a class of tilting objects satisfying a certain purity condition. We call these the decent tilting objects and show that the tilting heart induced by any such object is equivalent to a category of contramodules over the endomorphism ring of the tilting object endowed with a natural linear topology. This extends the recent result for n-tilting modules by Positselski and Št'ovíček. In the setting of the derived category of modules over a ring, we show that the decent tilting complexes are precisely the silting complexes such that their character dual is cotilting. The hearts of cotilting complexes of cofinite type turn out to be equivalent to the category of discrete modules with respect to the same topological ring. Finally, we provide a kind of Morita theory in this setting: Decent tilting complexes correspond to pairs consisting of a tilting and a cotilting-derived equivalence as described above tied together by a tensor compatibility condition.
在一个紧凑生成的三角范畴中,我们引入了一类满足特定纯度条件的倾斜对象。我们称这些对象为体面倾斜对象,并证明任何这类对象所诱导的倾斜心都等价于倾斜对象的内形环上的一个禀赋了自然线性拓扑的等价模范畴。这扩展了波西泽尔斯基和什托维契克最近关于 n 倾斜模块的结果。在环上模块的派生类中,我们证明了体面的倾斜复数正是淤积复数,它们的特征对偶是同向的。结果表明,对于同一拓扑环,共穷类的共穷复数之心等价于离散模块范畴。最后,我们提供了这种情况下的一种莫里塔理论:体面的倾斜复数对应于由上述倾斜和同调派生等价关系组成的对,它们通过张量相容条件联系在一起。
期刊介绍:
The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.