The bi-canonical degree of a Cohen–Macaulay ring

Abstract

Dedicated to Professor Craig Huneke on the occasion of his birthday for his groundbreaking contributions to Algebra, particularly to Commutative Algebra.Abstract. This paper is a sequel to [8] where we introduced an invariant, called canonical degree, of Cohen-Macaulay local rings that admit a canonical ideal. Here to each such ring R with a canonical ideal, we attach a different invariant, called bi-canonical degree, which in dimension 1 appears also in [12] as the residue of R. The minimal values of these functions characterize specific classes of Cohen-Macaulay rings. We give a uniform presentation of such degrees and discuss some computational opportunities offered by the bi-canonical degree.

Generalization of bi-canonical degrees

“…This is helpful in computing bi-canonical degrees in explicit examples. In Proposition 3.8 we extend a result of [7] relating the bi-canonical degree to the trace of the canonical module. We also propose some open questions, inspired by results of [7].…”

“…Let (R, m) be a Cohen-Macaulay local ring of dimension d that has a canonical module ω. Our central viewpoint is to look at the properties of ω as a way to refine our understanding of R. We recall that R is Gorenstein when ω is isomorphic to R. In [6] and [7], if the canonical module ω is an ideal, we treated metrics aimed at measuring the deviation from R being Gorenstein. More precisely, two integers arise by considering the following lenghts (when R has dimension one):…”

“…The canonical degree cdeg(R) is the focus of [6] and the bi-canonical degree bideg(R) is the focus of [7]. They are extended to higher dimensions (see Definition 2.1 and Definition 2.9).…”

“…Attached to each such ring R, when ω is an ideal, there are integers-the type of R, the reduction number of ω-that provide valuable metrics to express the deviation of R from being a Gorenstein ring. In [6] and [7] we enlarged this list with the canonical degree and the bi-canonical degree. In this work we extend the bi-canonical degree to rings where ω is not necessarily an ideal.…”

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

“…(iii) bideg(R) was introduced in [19] and called the bi-canonical degree of R: bideg(R) = 1 if and only if R is a so-called Goto ring.…”