论文标题
相对奇异性类别和奇异等价
Relative Singularity categories and singular equivalences
论文作者
论文摘要
让$ r $为正确的诺德人戒指。我们介绍了相对奇异性类别的概念$δ_{\ Mathcal {x}}（r）$ r $的$相对于违反有限的子类别$ \ MATHCAL {x} $ $ \ rm {mod \ rm {mod} \ rm {mod} \ mbox { - mbox { - } $ $ $ $ $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \米{ $δ_ {\ Mathcal {x}}（r）$是三角形等同于同型类别的子类别$ \ MATHBB {k} _ {\ rm {ac}}}（\ rm {ac}}}（\ Mathcal {x}）$，超过$ \ Mathcal {x} $。作为一个应用程序，给出了经典奇点类别的新描述$ \ mathbb {d} _ {\ rm {sg}}}（r）$。使用相对奇异性类别来提高两个给定的右环的模块类别的两个合适的子类别之间的稳定等效性，以在环之间获得奇异的对等。在不同类型的环中，包括路径环，三角形矩阵环，微不足道的延伸环和张量环，我们为它们的奇异性类别提供了一些后果。
Let $R$ be a right notherian ring. We introduce the concept of relative singularity category $Δ_{\mathcal{X}}(R)$ of $R$ with respect to a contravariantly finite subcategory $\mathcal{X}$ of $\rm{mod}\mbox{-}R.$ Along with some finiteness conditions on $\mathcal{X}$, we prove that $Δ_{\mathcal{X}}(R)$ is triangle equivalent to a subcategory of the homotopy category $\mathbb{K}_{\rm{ac}}(\mathcal{X})$ of exact complexes over $\mathcal{X}$. As an application, a new description of the classical singularity category $\mathbb{D}_{\rm{sg}}(R)$ is given. The relative singularity categories are applied to lift a stable equivalence between two suitable subcategories of the module categories of two given right notherian ring to get a singular equivalence between the rings. In different types of rings, including path rings, triangular matrix rings, trivial extension rings and tensor rings, we provide some consequences for their singularity categories.
