# Totally acyclic complexes and locally Gorenstein rings

## TL;DR

This paper extends a characterization of Gorenstein rings, showing that a commutative noetherian ring is locally Gorenstein if and only if every acyclic complex of injective modules is totally acyclic, removing the need for a dualizing complex.

## Contribution

It generalizes the known characterization of Gorenstein rings to all commutative noetherian rings without requiring a dualizing complex.

## Key findings

- Characterization of Gorenstein rings extended to arbitrary noetherian rings
- Equivalence between locally Gorenstein property and totally acyclic complexes
- Removal of dualizing complex assumption in the characterization

## Abstract

A commutative noetherian ring with a dualizing complex is Gorenstein if and only if every acyclic complex of injective modules is totally acyclic. We extend this characterization, which is due to Iyengar and Krause, to arbitrary commutative noetherian rings, i.e. we remove the assumption about a dualizing complex. In this context Gorenstein, of course, means locally Gorenstein at every prime.