Finiteness and purity of subcategories of the module categories

TL;DR

This paper investigates the finiteness and purity properties of subcategories within module categories using functor rings and categories, providing new characterizations and extending existing results in representation theory.

Contribution

It introduces novel characterizations of contravariantly finite resolving subcategories and explores the pure semisimplicity conjecture, unifying various known results in module category theory.

Findings

Characterization of contravariantly finite resolving subcategories of finite type

Conditions for covariantly finite subcategories containing the Jacobson radical

Results on finiteness and purity of n-cluster tilting and Gorenstein projective modules

Abstract

In this paper, by using functor rings and functor categories, we study finiteness and purity of subcategories of the module categories. We give a characterisation of contravariantly finite resolving subcategories of the module category of finite representation type in terms of their functor rings. We also characterize contravariantly finite resolving subcategories of the module category $\Lambda${\rm -mod} that contain the Jacobson radical of $\Lambda$ of finite type, by their functor categories. We study the pure semisimplicity conjecture for a locally finitely presented category ${\underrightarrow{\lim}}\mathscr{X}$ when $\mathscr{X}$ is a covariantly finite subcategory of $\Lambda$-mod and every simple object in Mod$(\mathscr{X}^{\rm op})$ is finitely presented and give a characterization of covariantly finite subcategories of finite representation type in terms of decomposition properties of their closure under filtered colimits. As a consequence we study finiteness and purity of $n$-cluster tilting subcategories and the subcategory of the Gorenstein projective modules of the module categories. These results extend and unify some known results.