Graded artin algebras, rational series, and bounds for homological dimensions

Abstract

is a minimal injective resolution of A, then all the I,, are projective. Then n is selfinjecrive. It is of interest to note that the Nakayama conjecture follows in a trivial way from the finitistic dimension conjecture. This one says the following: Let n be an artin algebra. Then sup{proj dim M( proj dimM<oc$ is finite where M ranges over the finitely generated Amodules. The finitistic projective dimension has its origins in the early 1960s and was formulated from results of Rosenberg and Zelinsky. In this paper we study the behavior or certain rational power series associated to every graded module over a graded finite dimensional algebra, series which are slightly different from the usual Poincare-Betti series. It is interesting to note that rationality is enough to enable us to prove that over a graded finite dimensional algebra, the linitistic projective dimension is finite over the class of finitely generated graded modules whose nonprojective syzygies have no projective summands. This implies the Nakayama conjecture for graded algebras. (See also Wilson [3].) Also using rationality we find bounds for global dimension in terms of graded length and the number of nonisomorphic simple modules. Finally let A be an algebra of finite representation type having n nonisomorphic indecomposable modules. We prove that every composition of more than 2n 2 nonisomorphisms of Amodules is zero. This is done by using the result of Auslander-Reiten which states that every nonzero homomorphism which is not an isomorphism is a sum of compositions of irreducible morphisms.