Pure-semisimplicity of the category of graded modules over graded artin algebras

Abstract

Let Λ be a Z-graded artin algebra. It is proved that the category of graded Λ-modules is pure-semisimple if and only if there are only finitely many nonisomorphic indecomposable finitely generated graded Λ-modules. As a consequence of this result together with a known result of Gordon and Green (which states that Λ is of finite representation type if and only if there are only finitely many non-isomorphic indecomposable finitely generated graded Λ-modules), we see that the category of all Λ-modules is pure-semisimple if and only if the category of all graded Λ-modules is so. © World Scientific Publishing Company.