# Local duality for the singularity category of a finite dimensional Gorenstein algebra

## TL;DR

This paper establishes a duality theorem for the singularity category of finite dimensional Gorenstein algebras, extending existing dualities and revealing new structural properties related to Serre duality and Auslander-Reiten triangles.

## Contribution

It introduces a novel duality for the singularity category that complements Happel's duality on perfect complexes, with implications for local and torsion subcategories.

## Key findings

- Proves a duality theorem for the singularity category.
- Establishes an analogue of Serre duality.
- Shows existence of Auslander-Reiten triangles in specific subcategories.

## Abstract

A duality theorem for the singularity category of a finite dimensional Gorenstein algebra is proved. It complements a duality on the category of perfect complexes, discovered by Happel. One of its consequences is an analogue of Serre duality, and the existence of Auslander-Reiten triangles for the $\mathfrak{p}$-local and $\mathfrak{p}$-torsion subcategories of the derived category, for each homogeneous prime ideal $\mathfrak{p}$ arising from the action of a commutative ring via Hochschild cohomology.