## Kyoto Journal of Mathematics

- Kyoto J. Math.
- Volume 57, Number 3 (2017), 585-612.

### Quadratic numerical semigroups and the Koszul property

Jürgen Herzog and Dumitru I. Stamate

#### Abstract

Let $H$ be a numerical semigroup. We give effective bounds for the multiplicity $e\left(H\right)$ when the associated graded ring ${gr}_{\mathfrak{m}}K\left[H\right]$ is defined by quadrics. We classify Koszul complete intersection semigroups in terms of gluings. Furthermore, for several classes of numerical semigroups considered in the literature (arithmetic, compound, special almost-complete intersections, $3$-semigroups, symmetric or pseudosymmetric $4$-semigroups) we classify those which are Koszul.

#### Article information

**Source**

Kyoto J. Math., Volume 57, Number 3 (2017), 585-612.

**Dates**

Received: 30 October 2015

Accepted: 25 April 2016

First available in Project Euclid: 22 April 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.kjm/1492826434

**Digital Object Identifier**

doi:10.1215/21562261-2017-0007

**Mathematical Reviews number (MathSciNet)**

MR3685056

**Zentralblatt MATH identifier**

06774048

**Subjects**

Primary: 13A30: Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics

Secondary: 16S37: Quadratic and Koszul algebras 16S36: Ordinary and skew polynomial rings and semigroup rings [See also 20M25] 13C40: Linkage, complete intersections and determinantal ideals [See also 14M06, 14M10, 14M12] 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05] 13P10: Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)

**Keywords**

Koszul ring quadratic ring numerical semigroup tangent cone complete intersection gluing standard basis arithmetic sequence symmetric and pseudosymmetric semigroups

#### Citation

Herzog, Jürgen; Stamate, Dumitru I. Quadratic numerical semigroups and the Koszul property. Kyoto J. Math. 57 (2017), no. 3, 585--612. doi:10.1215/21562261-2017-0007. https://projecteuclid.org/euclid.kjm/1492826434