Another Proof of a Theorem of Van den Bergh about Graded-Injective Modules

Abstract

M. Van den Bergh proved that if A is a left noetherian N-graded ring, and I is a graded-injective left A-module, then the injective dimension of I in the ungraded sense is at most 1. In this note we give another proof of this result. Our proof follows the same strategy as the original proof by Van den Bergh, yet it is somewhat more conceptual, in that it isolates the precise "reason" for the dimension jump.