Uniformly dominant local rings and Orlov spectra of singularity categories

TL;DR

This paper introduces the concept of uniformly dominant local rings, providing conditions for their existence, analyzing their properties, and exploring implications for the Orlov spectrum of singularity categories.

Contribution

It defines uniformly dominant local rings, establishes their properties, and connects them to bounds on the Orlov spectrum in singularity categories.

Findings

Burch rings are uniformly dominant.

Local rings with quasi-decomposable maximal ideal are uniformly dominant.

Upper bounds on the Orlov spectrum are obtained for certain singularities.

Abstract

We define a uniformly dominant local ring as a commutative noetherian local ring with an integer r such that the residue field is built from any nonzero object in the singularity category by direct summands, shifts and at most r mapping cones. We find sufficient conditions for uniform dominance, by which we show Burch rings and local rings with quasi-decomposable maximal ideal are uniformly dominant. For a uniformly dominant excellent equicharacteristic isolated singularity, we get an upper bound of the Orlov spectrum of the singularity category. We prove uniform dominance is preserved under basic operations, and give techniques to construct uniformly dominant local rings. An application of our methods to local rings with decomposable maximal ideal is provided as well.