Annihilators and decompositions of singularity categories

TL;DR

This paper explores the structure of singularity categories over commutative Noetherian rings, establishing relations between subcategories defined by ring elements and providing a decomposition that bounds the category's dimension.

Contribution

It introduces a novel relation between subcategories of the singularity category associated with ring elements and derives a decomposition method.

Findings

Established relations between (x), (y), and (xy)

Decomposed the singularity category into simpler components

Provided an upper bound on the dimension of the singularity category

Abstract

Given any commutative Noetherian ring $R$ and an element $x$ in $R$, we consider the full subcategory $\C(x)$ of its singularity category consisting of objects for which the morphism that is given by the multiplication by $x$ is zero. Our main observation is that we can establish a relation between $\C(x), \C(y)$ and $\C(xy)$ for any two ring elements $x$ and $y$. Utilizing this observation, we obtain a decomposition of the singularity category and consequently an upper bound on the dimension of the singularity category.