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

Abstract

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