A localisation theorem for singularity categories of proper dg algebras

TL;DR

This paper proves a localization theorem for singularity categories of proper dg algebras, establishing a short exact sequence that generalizes known results for finite-dimensional algebras, thus advancing the understanding of their categorical structure.

Contribution

It introduces a new localization theorem for singularity categories of proper dg algebras, extending existing results to a broader class of algebras over noetherian rings.

Findings

Establishes a short exact sequence of singularity categories for proper dg algebras in a recollement.

Generalizes known results from finite-dimensional algebras to proper dg algebras.

Provides a framework for understanding the categorical structure of dg algebras over noetherian rings.

Abstract

Given a recollement of three proper dg algebras over a noetherian commutative ring, e.g. three algebras which are finitely generated over the base ring, which extends one step downwards, it is shown that there is a short exact sequence of their singularity categories. This allows us to recover and generalise some known results on singularity categories of finite-dimensional algebras.