论文标题
相对Gorenstein Flat模块和Foxby类及其模型结构
Relative Gorenstein flat modules and Foxby classes and their model structures
论文作者
论文摘要
类别上的模型结构是对该类别引入同质理论的形式方法，如果模型结构是Abelian和Heredinary，则已知其同型类别是三角剖分的。因此，建立和建模三角剖分类别的一个好方法是建立遗传性的阿贝尔模型结构。给定一个环$ r $和（不一定要半单性化的）左$ r $ -module $ c $，我们介绍并研究了相对Gorenstein Cotorsion和Cotorsion模块的新概念：$ \ rm G_C $ -Cotorsion和（强烈）$ \ Mathcal {C} _c} _c $ -cotorsion。 As an application, we prove that there is a unique hereditary abelian model structure on the category of left $R$-modules, in which the cofibrations are the monomorphisms with $\rm G_C$-flat cokernel and the fibrations are the epimorphisms with $\mathcal{C}_C$-cotorsion kernel belonging to the Bass class $ \ MATHCAL {B} _C（R）$。在第二部分中，当$ c $是一个半虚拟化$（r，s）$ - bimodule时，我们调查了左（分别为，右）$ r $ r $模块上的Abelian模型结构的存在，其中联合启动是表达式的（分别是knemorphisms，单态）与Kernel（cokernel）（cokernel）（cokernel）（cokernel）（cokernel）（cokernel）。 $ \ MATHCAL {B} _C（R）$（resp。，$ \ Mathcal {a} _C（r）$）。我们还研究了Auslander-Buchweitz近似理论的观点的$ \ rm G_C $ -FLAT模块和低音类。我们证明它们是弱AB-Contexts的一部分。由于可以双重化弱Ab-context的概念，因此我们还提供了双重结果，涉及$ \ rm g_c $ cotorsion模块和奥斯兰德类别的类别。
A model structure on a category is a formal way of introducing a homotopy theory on that category, and if the model structure is abelian and hereditary, its homotopy category is known to be triangulated. So a good way to both build and model a triangulated category is to build a hereditary abelian model structure. Given a ring $R$ and a (non necessarily semidualizing) left $R$-module $C$, we introduce and study new concepts of relative Gorenstein cotorsion and cotorsion modules: $\rm G_C$-cotorsion and (strongly) $\mathcal{C}_C$-cotorsion. As an application, we prove that there is a unique hereditary abelian model structure on the category of left $R$-modules, in which the cofibrations are the monomorphisms with $\rm G_C$-flat cokernel and the fibrations are the epimorphisms with $\mathcal{C}_C$-cotorsion kernel belonging to the Bass class $\mathcal{B}_C(R)$. In the second part, when $C$ is a semidualizing $(R,S)$-bimodule, we investigate the existence of abelian model structures on the category of left (resp., right) $R$-modules where the cofibrations are the epimorphisms (resp., monomorphisms) with kernel (resp., cokernel) belonging to the Bass (resp., Auslander) class $\mathcal{B}_C(R)$ (resp., $\mathcal{A}_C(R)$). We also study the class of $\rm G_C$-flat modules and the Bass class from the Auslander-Buchweitz approximation theory point of view. We show that they are part of weak AB-contexts. As the concept of weak AB-context can be dualized, we also give dual results that involve the class of $\rm G_C$-cotorsion modules and the Auslander class.
