Behaviour of injective dimension with respect to regradings

TL;DR

This paper investigates how the injective dimension of graded modules over a noetherian algebra changes under different gradings, using functors and Hopf algebra frameworks.

Contribution

It introduces bounds for the injective dimension under regradings and extends the theory to H-comodule algebras with new functor constructions.

Findings

Provides bounds for injective dimensions under regradings

Develops functors applicable to H-comodule algebras

Extends grading change techniques to a broader algebraic context

Abstract

Given a left noetherian k-algebra A graded by a group G, an injective object I in the category of G-graded A-modules and a morphism from G to another group G', we provide bounds for the injective dimension of I as a G'-graded A-module. For this, we use three change of grading functors. Most of the constructions concerning these functors work in the context of H-comodule algebras, where H is a Hopf algebra, so we develop them in this general context.