Singular equivalences of commutative noetherian rings and reconstruction of singular loci

TL;DR

This paper investigates conditions under which commutative noetherian rings are singularly equivalent by developing support theory for their singularity categories, aiding in understanding their singular loci.

Contribution

It introduces a necessary condition for singular equivalence of commutative noetherian rings through support theory without tensor structure.

Findings

Established a support theory for triangulated categories without tensor structure.

Provided a necessary condition for singular equivalence of rings.

Enhanced understanding of the relationship between singularity categories and singular loci.

Abstract

Two left noetherian rings $R$ and $S$ are said to be {\it singularly equivalent} if their singularity categories are equivalent as triangulated categories. The aim of this paper is to give a necessary condition for two commutative noetherian rings to be singularly equivalent. To do this, we develop the support theory for triangulated categories without tensor structure.