论文标题
三角类别中的理想突变和概括的澳大利亚理论
Ideal mutations in triangulated categories and generalized Auslander-Reiten theory
论文作者
论文摘要
我们介绍了三角形类别中理想突变的概念，该类别概括了Iyama和Yoshino \ cite {iyama2008Moint}的版本，通过用一个理想的形态学的近似值来代替近似值的对象，通过近似。作为应用程序，对于hom-Finite Krull-Schmidt三角形类别$ \ MATHCAL {t} $上的代数关闭字段$ k $。 （1）我们将Jorgensen \ cite [theorem 3.3] {Jorgensen2010 Quotients}定理概括为更通用的设置； （2）我们提供了一种方法来检测$ \ Mathcal {t} $是否具有Auslander-Reiten三角形，通过检查其Jacobson激进$ \ MATHCAL {J} $：（I）$ \ \ \ MATHCAL {J} $是功能上有限的（II）gh $ ____c { cogh} _ {\ Mathcal {J}} $，以及（iii）gh $ _ {\ Mathcal {J}} $ - 源地图与GH $ _ {\ MathCal {J}} $ - $ - sink Maps cosigine （3）我们通过使用理想突变概括了经典的Auslander-Reiten理论。
We introduce the notion of ideal mutations in a triangulated category, which generalizes the version of Iyama and Yoshino \cite{iyama2008mutation} by replacing approximations by objects of a subcategory with approximations by morphisms of an ideal. As applications, for a Hom-finite Krull-Schmidt triangulated category $\mathcal{T}$ over an algebraically closed field $K$. (1) We generalize a theorem of Jorgensen \cite[Theorem 3.3]{jorgensen2010quotients} to a more general setting; (2) We provide a method to detect whether $\mathcal{T}$ has Auslander-Reiten triangles or not by checking the necessary and sufficient conditions on its Jacobson radical $\mathcal{J}$: (i) $\mathcal{J}$ is functorially finite, (ii) Gh$_{\mathcal{J}}= {\rm CoGh}_{\mathcal{J}}$, and (iii) Gh$_{\mathcal{J}}$-source maps coincide with Gh$_{\mathcal{J}}$-sink maps; (3) We generalize the classical Auslander-Reiten theory by using ideal mutations.
