On the ubiquity of uniformly dominant local rings

TL;DR

This paper investigates the property of uniform dominance in local rings, establishing bounds on the dominant index for various classes of Cohen-Macaulay rings and refining existing results in the field.

Contribution

It provides new bounds on the dominant index for non-complete intersection rings, Burch rings, and quasi-fiber product rings, extending and refining previous results.

Findings

Bound of 6d+5 for codimension 2 non-complete intersection rings.

Bound of d+1 for Burch rings.

Bound of d for quasi-fiber product rings or rings with multiplicity ≤ 5.

Abstract

Let R be a d-dimensional Cohen-Macaulay complete local ring with infinite residue field k. The dominant index $\operatorname{dx}(R)$ is by definition the least number of extensions necessary to build k in the singularity category $\operatorname{D^{sg}}$ out of each nonzero object, up to finite direct sums, direct summands and shifts. The local ring R is called uniformly dominant if $\operatorname{dx}(R)$ is finite. In this paper, we prove that R is uniformly dominant with $\operatorname{dx}(R)\le6d+5$ if R has codimension 2 and is not a complete intersection. Also, we show that R is uniformly dominant with $\operatorname{dx}(R)\le d+1$ if R is Burch, and with $\operatorname{dx}(R)\le d$ if R is either a quasi-fiber product ring, or has multiplicity at most 5 and is not Gorenstein. A result on hypersurfaces by Ballard, Favero and Katzarkov is recovered, and results on Burch rings and quasi-fiber product rings by Takahashi are refined.