Representation theory of graded Artin algebras

Abstract

Having initiated the study of graded Artin algebras A in [ 91, here we initiate the study of their representation theory. Our point of view is to study graded /i-modules, which we believe to be somewhat more tractable than ungraded ones, in order to obtain information about all /i-modules. This point of view leads to the introduction, in Section 1, of the full subcategory mod,@!) of mod/i consisting of the gradable objects of mod/i; that is, the finitely generated /i-modules which support a gradation. In these terms, one of the major results of the paper asserts that mod,@) has finite representation type if and only if mod/i has finite representation type. Thus, in Section 3 we introduce a number G = G(4) designed to measure the size of mod,(/i). If G = co, we show that /i has infinite representation type. If G < co, we show that there is a graded Artin algebra Q of a certain specified form such that /i has infinite representation type precisely when 0 has infinite representation type. The Artin algebra fl has, in particular, desirable diagrammatic properties; and these we will exploit elsewhere. We speculate that when G is finite, every finitely generated /i-module is gradable. Now, in case n has finite representation type, it is obvious that G is finite. We show, indeed, that for every gradation of a given Artin algebra of finite representation type, every module is gradable. This result, and the others cited, are chiefly consequences of a result proved in Section 4: If a component of the Auslander-Reiten graph of a graded Artin algebra contains a gradable module, then the component