Almost split sequences for abelian group graded rings

Abstract

While existence theorems for almost split sequences have been developed in a wide variety of settings, it is only recently that any attention has been given to the question of the existence of almost split sequences for categories of graded modules. Not only is this an interesting question in its own right, but the close connection between finitely generated graded modules and coherent sheaves on projective varieties also yields information concerning almost split sequences for categories of coherent sheaves. In [4, 51 we dealt with the question for Z-graded rings. In this paper we deal with rings graded by arbitrary abelian groups. We note that Geigle and Lenzing have studied some special two-dimensional situations in connection with their work on the relationship between vector bundles on weighted projective curves and modules over some particular finite dimensional algebras. Our results and method of proof are for the most part graded versions of those given in [3] for Cohen-Macaulay modules over isolated singularities. However, in some instances we need to provide new arguments. In particular, there is an equivalence of functor categories which plays an important role. In Section 1 we develop the necessary preliminaries on graded categories and functors to be able to state and prove this theorem. In Section 2 we apply this result to obtain our existence theorem for almost split sequences for modules graded by an arbitrary abelian group.