Graded artin algebras

Abstract

A graded Artin algebra is a graded ring which, neglecting the grading, is an Artin algebra. Although this definition has the merit of brevity, it does not expose some of the basic properties of graded Artin algebras; for instance, that there is a bound on absolute values of degrees of homogeneous elements. Indeed, a different definition is given in Section 1, and the discrepancy is rectified by Theory 1.4. We admit that our chief interest in graded Artin algebras lies in applications to the representation theory of Artin algebras-see [ 5 1. We feel, nonetheless, that graded Artin algebras are interesting objects of study in their own right, and some of our techniques (especially in Sections 5 and 6) are equally applicable to arbitrary graded rings. Thus, this paper, unlike its sequel [ 51, is designed to be read by a general ring theoretical audience. It is, in addition, self-contained from the standpoint of knowledge concerning graded rings required. The major concern of the paper is the existence of gradings on given modules over a graded Artin algebra-we use the term gradable when a gradation exists. Our transcendent result along these lines is that direct summands of finitely generated gradable modules are gradable. It follows immediately (see Section 3) that simple modules and projective modules over a graded Artin algebra are gradable. Also, inasmuch as we do not restrict ourselves to positively graded rings, endomorphism rings of graded finitely generated modules over graded Artin algebras are graded Artin algebras. In a similar vein, duality and horn