Canonical Degrees of Cohen-Macaulay Rings and Modules: a Survey

TL;DR

This survey explores invariants of Cohen-Macaulay local rings with canonical modules, focusing on canonical degrees and related metrics that measure how far these rings are from being Gorenstein, including new extensions of these degrees.

Contribution

It introduces and unifies various canonical degrees and invariants, and proposes methods to extend these concepts beyond rings with canonical ideals.

Findings

Canonical degrees are multiplicity-based functions of the Rees algebra.

Unified presentation of three degrees arising from common roots.

Proposals for extending degrees to rings without canonical ideals.

Abstract

The aim of this survey is to discuss invariants of Cohen-Macaulay local rings that admit a canonical module. 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. We give a uniform presentation of three degrees arising from common roots. Finally we experiment with ways to extend one of these degrees to rings where C is not necessarily an ideal.