{"title":"Generalized tilting theory in functor categories","authors":"Xi Tang","doi":"10.1017/S0017089523000162","DOIUrl":null,"url":null,"abstract":"Abstract This paper is devoted to the study of generalized tilting theory of functor categories in different levels. First, we extend Miyashita’s proof (Math Z 193:113–146,1986) of the generalized Brenner–Butler theorem to arbitrary functor categories \n$\\mathop{\\textrm{Mod}}\\nolimits\\!(\\mathcal{C})$\n with \n$\\mathcal{C}$\n an annuli variety. Second, a hereditary and complete cotorsion pair generated by a generalized tilting subcategory \n$\\mathcal{T}$\n of \n$\\mathop{\\textrm{Mod}}\\nolimits \\!(\\mathcal{C})$\n is constructed. Some applications of these two results include the equivalence of Grothendieck groups \n$K_0(\\mathcal{C})$\n and \n$K_0(\\mathcal{T})$\n , the existences of a new abelian model structure on the category of complexes \n$\\mathop{\\textrm{C}}\\nolimits \\!(\\!\\mathop{\\textrm{Mod}}\\nolimits\\!(\\mathcal{C}))$\n , and a t-structure on the derived category \n$\\mathop{\\textrm{D}}\\nolimits \\!(\\!\\mathop{\\textrm{Mod}}\\nolimits \\!(\\mathcal{C}))$\n .","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"65 1","pages":"595 - 611"},"PeriodicalIF":0.5000,"publicationDate":"2023-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Glasgow Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/S0017089523000162","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
Abstract This paper is devoted to the study of generalized tilting theory of functor categories in different levels. First, we extend Miyashita’s proof (Math Z 193:113–146,1986) of the generalized Brenner–Butler theorem to arbitrary functor categories
$\mathop{\textrm{Mod}}\nolimits\!(\mathcal{C})$
with
$\mathcal{C}$
an annuli variety. Second, a hereditary and complete cotorsion pair generated by a generalized tilting subcategory
$\mathcal{T}$
of
$\mathop{\textrm{Mod}}\nolimits \!(\mathcal{C})$
is constructed. Some applications of these two results include the equivalence of Grothendieck groups
$K_0(\mathcal{C})$
and
$K_0(\mathcal{T})$
, the existences of a new abelian model structure on the category of complexes
$\mathop{\textrm{C}}\nolimits \!(\!\mathop{\textrm{Mod}}\nolimits\!(\mathcal{C}))$
, and a t-structure on the derived category
$\mathop{\textrm{D}}\nolimits \!(\!\mathop{\textrm{Mod}}\nolimits \!(\mathcal{C}))$
.
本文致力于在不同层次上研究函子范畴的广义倾斜理论。首先,我们将广义Brenner–Butler定理的Miyashita证明(Math Z 193:113–1461986)推广到任意函子范畴$\mathop{\textrm{Mod}}\nolimits\!(\mathcal{C})$与$\mathcal{C}$是环状变体。其次,由$\mathop{\textrm{Mod}}\nolimits\的广义倾斜子类别$\mathcal{T}$生成的一个遗传完全余子对!构造了(\mathcal{C})$。这两个结果的一些应用包括Grothendieck群$K_0(\mathcal{C})$和$K_0!(\!\mathop{\textrm{Mod}}\nolimits\!(\!\mathop{\textrm{Mod}}\nolimits\!(\mathcal{C}))$。
期刊介绍:
Glasgow Mathematical Journal publishes original research papers in any branch of pure and applied mathematics. An international journal, its policy is to feature a wide variety of research areas, which in recent issues have included ring theory, group theory, functional analysis, combinatorics, differential equations, differential geometry, number theory, algebraic topology, and the application of such methods in applied mathematics.
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