Canonical trace ideal and residue for numerical semigroup rings

TL;DR

This paper investigates the trace of the canonical ideal in numerical semigroup rings, introducing the residue as a measure of deviation from Gorenstein property, with explicit formulas and classifications for 3-generated cases.

Contribution

It provides explicit formulas for the canonical trace ideal and residue in 3-generated numerical semigroups, and characterizes nearly-Gorenstein semigroups.

Findings

Explicit formulas for 3-generated cases

Classification of nearly-Gorenstein semigroups

Residue exhibits eventual periodicity

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.