Canonical trace ideal and residue for numerical semigroup rings

Abstract

For a numerical semigroup ring $K[H]$ we study the trace of its canonical ideal. The colength of this ideal is called the residue of $H$. This invariant measures how far is $H$ from being symmetric, i.e. $K[H]$ from being a Gorenstein ring. We remark that the canonical trace ideal contains the conductor ideal, and we study bounds for the residue. For $3$-generated numerical semigroups we give explicit formulas for the canonical trace ideal and the residue of $H$. Thus, in this setting we can classify those whose residue is at most one (the nearly-Gorenstein ones), and we show the eventual periodic behaviour of the residue in a shifted family.

16 pages; this paper contains most of the results related to numerical semigroups from the initial version of our paper arXiv:1612.02723 [math.AC]

References in corpus (2)

• Periodic Occurance of Complete Intersection Monomial Curves
• Canonical Degrees of Cohen-Macaulay Rings and Modules: a Survey

Cited by in corpus (3)

• Trace of canonical modules, annihilator of Ext, and classes of rings close to being Gorenstein
• Nearly Gorenstein and almost symmetric properties in shifted numerical semigroups
• Stable set rings which are Gorenstein on the punctured spectrum