Galois covering of pure-semisimple categories

Abstract

Let C be a locally bounded k-category, where k is a field.We prove that C is pure-semisimple, that is, every object of Mod-C is pure-projective if and only if every family of morphisms between indecomposable finitely generated C-modules is Noetherian. Our formalism establishes the pure-semisimplicity of Galois coverings, that is, if C is a G-category with a free G-action on ind-C, then C is pure-semisimple if and only if C/G is so. © 2023 by Kyoto University.