# Quillen equivalence of singular model categories

## TL;DR

This paper establishes a Quillen equivalence between two singular model categories over a Gorenstein ring, linking complexes of projectives and injectives through homotopy category equivalences.

## Contribution

It proves a Quillen equivalence between singular contraderived and coderived model categories over Gorenstein rings, connecting projective and injective complexes.

## Key findings

- Quillen equivalence between singular model categories
- Equivalence of homotopy categories of complexes
- Bridging projective and injective complexes

## Abstract

Let $R$ be a left-Gorenstein ring. We show that there is a Quillen equivalence between singular contraderived model category and singular coderived model category. Consequently, an equivalence between the homotopy category of exact complexes of projective modules and the homotopy category of exact complexes of injective modules is given.