Upper bounds for dimensions of singularity categories and their annihilators

TL;DR

This paper investigates bounds on the dimension of the singularity category of a commutative noetherian ring, extending existing theorems to broader classes of rings and exploring properties of Verdier quotients.

Contribution

It provides new upper bounds for the singularity category dimension and extends Liu's theorem beyond local and isolated singularity cases.

Findings

Derived bounds for singularity category dimensions

Extended Liu's theorem to non-local, non-isolated singularities

Analyzed localizations and annihilators of Verdier quotients

Abstract

Let $R$ be a commutative noetherian ring. Denote by $\operatorname{mod} R$ the category of finitely generated $R$-modules and by $\operatorname{D^b}(R)$ the bounded derived category of $\operatorname{mod} R$. In this paper, we first investigate localizations and annihilators of Verdier quotients of $\operatorname{D^b}(R)$. After that, we explore upper bounds for the dimension of the singularity category of $R$ and its (strong) generators. We extend a theorem of Liu to the case where $R$ is neither an isolated singularity nor even a local ring. Some of our results are more generally stated in terms of Spanier--Whitehead category of a resolving subcategory.