The reduction number of canonical ideals

Abstract

In this paper, we introduce a new invariant of Cohen-Macaulay local rings in terms of canonical ideals. The invariant measures how close to be Gorenstein, and preserved by localizations, dividing non-zerodivisors, and flat local homomorphisms. Furthermore it builds bridges between almost Gorenstein and nearly Gorenstein in dimension one. We also explore the invariant in numerical semigroup rings and rings arising from idealizations.

When are trace ideals finite?

“…On the other hand, Herzog and Rahimbeigi proved that for one-dimensional analytically irreducible Gorenstein local K-algebras, where K is an infinite field, the finiteness of trace ideals is equivalent to the finiteness of indecomposable maximal Cohen-Macaulay modules ( [19,Corollary 2.16]). Other progresses on trace ideals are seen in, for examples, [7,8,9,10,11,15,17,18,20,22,23,27]. Among them, in this paper, we study the following finiteness problem on trace ideals, which is also posed by several papers [9,Question 7.16(1)], [10,Question 3.7], and [19].…”