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

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 _ν} ]] 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.

Key Words:

• Associated graded ring
• Gorenstein
• Integral closure
• Numerical semigroup
• Quasi-socle ideal

2000 Mathematics Subject Classification:

• Primary 13A30
• Secondary 13C40

ACKNOWLEDGMENTS

The author would like to thank William Heinzer for his guidance and help during the writing of this article. He would also like to thank YiHuang Shen for many useful conversations and examples. The computer algebra system Singular was used for all computer calculations.

Notes

Communicated by I. Swanson.