论文标题
在$τ$上 - 使用子类别
On $τ$-tilting subcategories
论文作者
论文摘要
本文的主要主题是研究$τ$ - 在Abelian类别中使用$ \ Mathscr {a} $的子类别，并具有足够的投影对象。我们介绍了$τ$ - 体现扭转的三倍，并在$ \ Mathscr {a} $中的$τ$ - cotorsion扭转三倍的收集与$ \τ$的集合之间进行了$ \ mathscr {a} $的$ \τ$的集合，从扭转三元组和$ \ Mathscr {a} $的倾斜子类别的集合。使用持久模块说明了一般定义和结果。如果$ \ mathscr {a} = {\ rm {mod \ mbox {}} r} $，其中$ r $是一个单一的关联戒指，我们表征所有支持$τ$ - tilting，resp。所有支持$τ^ - $ - 倾斜，$ {\ rm {mod \ mbox {}} r} $的子类别在finendo Quasitilting，resp。准材料，模块。结果，将显示每个锡模块分别诱导$τ$ tilting，分别支持$τ^{ - } $ - 倾斜，$ {\ rm {mod \ mbox \ mbox {}}} r} r} r} $。我们还研究了$ {\ rm rep}（q，\ mathscr {a}）$的理论，其中$ q $是有限的和无环的颤动。特别是，我们提供了一种算法来构建支持$τ$ - 以$ {\ rm rep}（q，q，q，\ mathscr {a}）$中的子类别，从某些支持$τ$中置于$ \ mathscr {a} $的子类别，并构建$（n+1）$ - $ 1） \ Mathscr {a}）$从$ n $ - tilting $ \ mathscr {a} $中的子类别。
The main theme of this paper is to study $τ$-tilting subcategories in an abelian category $\mathscr{A}$ with enough projective objects. We introduce the notion of $τ$-cotorsion torsion triples and show a bijection between the collection of $τ$-cotorsion torsion triples in $\mathscr{A}$ and the collection of $τ$-tilting subcategories of $\mathscr{A}$, generalizing the bijection by Bauer, Botnan, Oppermann and Steen between the collection of cotorsion torsion triples and the collection of tilting subcategories of $\mathscr{A}$. General definitions and results are exemplified using persistent modules. If $\mathscr{A}={\rm{Mod\mbox{}}R}$, where $R$ is an unitary associative ring, we characterize all support $τ$-tilting, resp. all support $τ^-$-tilting, subcategories of ${\rm{Mod\mbox{}}R}$ in term of finendo quasitilting, resp. quasicotilting, modules. As a result, it will be shown that every silting module, respectively every cosilting module, induces a support $τ$-tilting, respectively support $τ^{-}$-tilting, subcategory of ${\rm{Mod\mbox{}}R}$. We also study the theory in ${\rm Rep}(Q, \mathscr{A})$, where $Q$ is a finite and acyclic quiver. In particular, we give an algorithm to construct support $τ$-tilting subcategories in ${\rm Rep}(Q, \mathscr{A})$ from certain support $τ$-tilting subcategories of $\mathscr{A}$ and present a systematic way to construct $(n+1)$-tilting subcategories in ${\rm Rep}(Q, \mathscr{A})$ from $n$-tilting subcategories in $\mathscr{A}$.
