Direct sums of exact covers of complexes

Abstract

A ring $R$ is left noetherian if and only if the direct sum of injective envelopes of any family of left $R$-modules is the injective envelope of the direct sum of the given family of modules (or equivalently, if and only if the direct sum of any family of injective left $R$-modules is also injective). This result of Bass ([2]) led to a series of similar closure questions concerning classes of modules and classes of envelopes and covers (Chase in [4] considers the question of the closure of the class of flat modules with respect to products). Motivated by Bass' result we consider the question of direct sums of exact covers of complexes. From the close connection between minimal injective resolutions of modules and exact covers of complexes it seemed reasonable to conjecture that we get this closure over left noetherian rings. In this paper we show that this is not the case and that under various additional hypotheses on the ring that in fact the ring must have finite left global dimension for this to happen. Our results raise what we consider an interesting question about characterizing the local rings of finite global dimension in terms of a certain property of minimal projective resolutions of finitely generated modules over the local ring. We also consider the closely related question of when the direct sum of DG-injective complexes is DG-injective.