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

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.