# Change of grading, injective dimension and dualizing complexes

## TL;DR

This paper investigates how changing the grading group affects the injective dimension of modules over graded algebras and applies these findings to the stability of dualizing complexes.

## Contribution

It provides bounds for the injective dimension of modules under grading change and applies these results to dualizing complexes stability.

## Key findings

- Bounds for injective dimension under grading change
- Application to stability of dualizing complexes
- Extension of Van den Bergh's ideas

## Abstract

Let $G,H$ be groups, $\phi: G \rightarrow H$ a group morphism, and $A$ a $G$-graded algebra. The morphism $\phi$ 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.