# From subcategories to the entire module categories

## TL;DR

This paper establishes a connection between the representation theory of certain subcategories of modules over Artin algebras and the entire module category, simplifying the computation of almost split sequences and Auslander-Reiten quivers.

## Contribution

It introduces functors linking subcategories to module categories over Artin algebras, enabling easier computation of almost split sequences and Auslander-Reiten quivers.

## Key findings

- Almost split sequences in subcategories can be computed via functors to module categories.
- Most of the Auslander-Reiten quiver of subcategories can be derived from that of an algebra.
- Special cases include estimates for Gorenstein Artin algebras of finite representation type.

## 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 Ausalander-Reiten quiver of an appropriate algebra and next adding the remaining vertices and arrows in an obvious way. As a special case, whenever $\Lambda$ is a Gorenstein Artin algebra of finite representation type, then the subcategories of Gorenstein projective modules over the $2 \times 2$ upper triangular matrix algebra over $\Lambda$ and the stable Auslander algebra of $\Lambda$ can be estimated by the category of modules over the stable Cohen-Macaulay Auslander algebra of $\Lambda$.