Finiteness and purity of contravariantly finite resolving subcategories of the module categories

Abstract

We provide a characterization of contravariantly finite resolving subcategories of the module category of finite representation type in terms of their functor rings. Furthermore, we characterize contravariantly finite resolving subcategories of the module category Λ-mod of finite type that contain the Jacobson radical of Λ, by their functor categories. We investigate the pure semisimplicity conjecture for a locally finitely presented category

, given that $X$ constitutes a covariantly finite subcategory of Λ-mod and that each simple object within Mod$\left(X^{\mathrm{op}}\right)$ is finitely presented; additionally, we offer a characterization of covariantly finite subcategories of finite representation type through the lens of decomposition properties with respect to their closure under filtered colimits. Consequently, we delve into the finiteness and purity of n-cluster tilting subcategories, along with the Gorenstein projective modules of the module categories.