# Behavior of the Auslander condition with respect to regradings

## TL;DR

This paper investigates the behavior of the Auslander condition in noetherian graded rings, establishing equivalences with graded versions and exploring related homological properties such as injective and global dimensions.

## Contribution

It demonstrates that the Auslander condition in noetherian rings graded by finite rank abelian groups is equivalent to the graded Auslander condition, and explores related homological properties.

## Key findings

- Equivalence of Auslander condition and graded Auslander condition in graded rings.
- Homological relations between a ring, its quotient, and localization.
- Insights into injective dimension, global dimension, and Cohen-Macaulay property.

## Abstract

We show that a noetherian ring graded by an abelian group of finite rank satisfies the Auslander condition if and only if it satisfies the graded Auslander condition. In addition, we also study the injective dimension, the global dimension and the Cohen-Macaulay property from the same perspective of that for the Auslander condtion. A key step of our approach is to establish homological relations between a graded ring $R$, its quotient ring modulo the ideal $\hbar R$ and its localization ring with respect to the Ore set $\{\, \hbar^i\, \}_{i\geq0}$, where $\hbar$ is a homogeneous regular normal non-invertible element of $R$.