Goto Numbers of a Numerical Semigroup Ring and the Gorensteiness of Associated Graded Rings

TL;DR

This paper characterizes Goto numbers of monomial parameter ideals in numerical semigroup rings using semigroup properties, and explores conditions for associated graded rings to be Gorenstein, linking these invariants.

Contribution

It provides a semigroup-based characterization of Goto numbers and establishes criteria for Gorensteinness of associated graded rings in numerical semigroup rings.

Findings

Characterization of Goto numbers via semigroup properties

Explicit formulas for Goto numbers in certain classes of rings

Conditions for associated graded rings to be Gorenstein

Abstract

The Goto number of a parameter ideal Q in a Noetherian local ring (R,m) is the largest integer q such that Q : m^q is integral over Q. The Goto numbers of the monomial parameter ideals of R = k[[x^{a_1}, x^{a_2},..., x_{a_{\nu}}]] are characterized using the semigroup of R. This helps in computing them for classes of numerical semigroup rings, as well as on a case-by-case basis. The minimal Goto number of R and its connection to other invariants is explored. Necessary and sufficient conditions for the associated graded rings of R and R/x^{a_1}R to be Gorenstein are also given, again using the semigroup of R.