Correspondence between trace ideals and birational extensions with application to the analysis of the Gorenstein property of rings

Abstract

Over an arbitrary commutative ring, correspondences among three sets, the set of trace ideals, the set of stable ideals, and the set of birational extensions of the base ring, are studied. The correspondences are well-behaved, if the base ring is a Gorenstein ring of dimension one. It is shown that with one extremal exception, the surjectivity of one of the correspondences characterizes the Gorenstein property of the base ring, provided it is a Cohen-Macaulay local ring of dimension one. Over a commutative Noetherian ring, a characterization of modules in which every submodule is a trace module is given. The notion of anti-stable rings is introduced, exploring their basic properties.2010 Mathematics Subject Classification. 13H10, 13H15, 13A30.

On reflexive and $I$-Ulrich modules over curve singularities

“…We also make frequent use of birational extensions of R and trace ideals. This is heavily inspired by some recent interesting work from Kobayashi [Kob17], Goto-Isobe-Kumashiro [GIK20], Faber [Fab19] and Herzog-Hibi-Stamate [HHS19].…”

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].…”

Stable trace ideals and applications

“…Stable ideals can be thought of as ideals with simplest blow-ups, and Lipman exploited their nice properties in his seminal ( [12])on Arf rings. Trace ideals have long been useful technical tools in commutative algebra, but recently they have attracted new attention as interesting objects in their own right, see [5,11,6,7].…”

“…Let R = C[[t 4 , t 5 , t 6 ]] with I a = (t 4 − at 5 , t 6 ) for a ∈ C and m = (t 4 , t 5 , t6 ). Then T (m) = {m} ∪ {I a } a∈C , see[6, Example 3.4]. The minimal elements are I a , each of them is stable.…”