Change of grading, injective dimension and dualizing complexes

Abstract

Let $G,H$ be groups, $ϕ: G \rightarrow H$ a group morphism, and $A$ a $G$-graded algebra. The morphism $ϕ$ induces an $H$-grading on $A$, and on any $G$-graded $A$-module, which thus becomes an $H$-graded $A$-module. Given an injective $G$-graded $A$-module, we give bounds for its injective dimension when seen as $H$-graded $A$-module. Following ideas by Van den Bergh, we give an application of our results to the stability of dualizing complexes through change of grading.

Introduction rewritten, corrections suggested by referee added. Accepted for publications at Comm. in Alg