Abstract In this paper we show that how the representation theory of subcategories (of the category of modules over an Artin algebra) can be connected to the representation theory of all modules over some algebra. The subcategories dealing with are some certain subcategories of the morphism categories (including submodule categories studied recently by Ringel and Schmidmeier) and of the Gorenstein projective modules over (relative) stable Auslander algebras. These two kinds of subcategories, as will be seen, are closely related to each other. To make such a connection, we will define a functor from each type of the subcategories to the category of modules over some Artin algebra. It is shown that to compute the almost split sequences in the subcategories it is enough to do the computation with help of the corresponding functors in the category of modules over some Artin algebra which is known and easier to work. Then as an application the most part of Auslander–Reiten quiver of the subcategories is obtained only by the Auslander–Reiten quiver of an appropriate algebra and next adding the remaining vertices and arrows in an obvious way. As a special case, when Λ is a Gorenstein Artin algebra of finite representation type, then the subcategories of Gorenstein projective modules over the 2 × 2 {2\times 2} lower triangular matrix algebra over Λ and the stable Auslander algebra of Λ can be estimated by the category of modules over the stable Cohen–Macaulay Auslander algebra of Λ.

中文翻译：

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从子类别到整个模块类别

摘要 在本文中，我们展示了子类别（Artin 代数上的模块类别）的表示理论如何与某些代数上的所有模块的表示理论联系起来。处理的子范畴是态射范畴（包括 Ringel 和 Schmidmeier 最近研究的子模范畴）和（相对）稳定 Auslander 代数上的 Gorenstein 射影模的某些子范畴。正如将要看到的，这两种子类别彼此密切相关。为了建立这种联系，我们将在某些 Artin 代数上定义从每个类型的子类别到模块类别的函子。结果表明，为了计算子类别中几乎分裂的序列，在模块类别中的相应函子的帮助下，在一些已知且更容易工作的 Artin 代数上进行计算就足够了。然后作为应用，子类别的 Auslander-Reiten quiver 的大部分仅通过适当代数的 Auslander-Reiten quiver 获得，然后以明显的方式添加剩余的顶点和箭头。作为特例，当Λ是有限表示类型的Gorenstein Artin代数时，则Λ上的2×2{2\times 2}下三角矩阵代数和Λ的稳定Auslander代数上的Gorenstein射影模的子范畴可以为通过 Λ 的稳定 Cohen-Macaulay Auslander 代数上的模块类别估计。