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arXiv · 2008.01428

Canonical trace ideal and residue for numerical semigroup rings

Jürgen Herzog·Takayuki Hibi·Dumitru I. Stamate
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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.

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Jürgen Herzog, Takayuki Hibi, Dumitru I. Stamate. 2020-08-04. Canonical trace ideal and residue for numerical semigroup rings. https://doi.org/10.1007/s00233-021-10205-x

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