Invariants of Cohen–Macaulay rings associated to their canonical ideals

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.

Generalized Gorenstein Arf rings

“…A generalized Gorenstein ring [11] is one of the generalization of a Gorenstein ring, defined by a certain embedding of the rings into their canonical modules; see 2.2 for the precise definition. The class of generalized Gorenstein rings is a new class of Cohen-Macaulay rings, which naturally covers the class of Gorenstein rings and fills the gap in-between Cohen-Macaulay and Gorenstein properties; see [6,8,11,12,13,15,16,17,18,21,23,24,26,34]. In fact such rings extend the definition of almost Gorenstein rings which were initially defined by Barucci and Fröberg [4] over one-dimensional analytically unramified local rings, and further developed and defined by Goto, Matsuoka, and Phuong [13] over arbitrary Cohen-Macaulay local rings of dimension one.…”

The bi-canonical degree of a Cohen–Macaulay ring

“…(1) The canonical degree cdeg(R) of R was introduced in [8]. In dimension 1, we select an appropriate regular element c of C and define cdeg(R) = deg(C/(c)).…”

“…Let (R, m) be a Cohen-Macaulay local ring of dimension d that has a canonical ideal C. We will use [4] as our reference for the basic properties of these rings and their terminology. Our central viewpoint is to look at the properties of C as a way to refine our understanding of R. Recall that R is Gorenstein when C is isomorphic to R. In [8] we treated metrics aimed at measuring the deviation from R being Gorenstein. Here we explore another pathway but still with the same overall goal.…”

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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.

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Ulrich ideals and 2-AGL rings

“…The series [3,6,11,12,13,14,15,19,20,21] of researches are motivated and supported by the strong desire to stratify Cohen-Macaulay rings, finding new and interesting classes which naturally include that of Gorenstein rings. As is already pointed out by these works, the class of almost Gorenstein local rings (AGL rings for short) could be a very nice candidate for such classes.…”