Invariants of Cohen-Macaulay rings associated to their canonical ideals

TL;DR

This paper introduces new invariants for Cohen-Macaulay local rings with canonical ideals, providing metrics to measure their deviation from Gorenstein rings, including type, reduction number, roots, and canonical degrees.

Contribution

The paper develops new invariants for Cohen-Macaulay rings with canonical ideals, expanding the tools to analyze their structure and Gorenstein deviation.

Findings

Defined the type and reduction number as invariants.

Introduced roots and canonical degrees based on Rees algebra multiplicities.

Provided a framework to quantify deviation from Gorenstein property.

Abstract

The purpose of this paper is to introduce new invariants of Cohen-Macaulay local rings. Our focus is the class of Cohen-Macaulay local rings that admit a canonical ideal. Attached to each such ring R with a canonical ideal C, there are integers--the type of R, the reduction number of C--that provide valuable metrics to express the deviation of R from being a Gorenstein ring. We enlarge this list with other integers--the roots of R and several canonical degrees. The latter are multiplicity based functions of the Rees algebra of C.