Castelnuovo-Mumford regularity of simplicial semigroup rings with isolated singularity

Authors:
Jürgen Herzog and Takayuki Hibi

Journal:
Proc. Amer. Math. Soc. **131** (2003), 2641-2647

MSC (2000):
Primary 13D45; Secondary 52A38

DOI:
https://doi.org/10.1090/S0002-9939-03-06952-1

Published electronically:
February 20, 2003

MathSciNet review:
1974318

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Abstract: Let be the polynomial ring in variables over a field and its graded maximal ideal. Let be homogeneous polynomials of degree generating an -primary ideal, and let be arbitrary homogeneous polynomials of degree . In the present paper it will be proved that the Castelnuovo-Mumford regularity of the standard graded -algebra is at most . By virtue of this result, it follows that the regularity of a simplicial semigroup ring with isolated singularity is at most , where is the multiplicity of and is the codimension of .

**1.**W. Bruns and J. Herzog, ``Cohen-Macaulay rings,'' Revised Edition, Cambridge University Press, 1996.**2.**W. Bruns and U. Vetter, ``Determinantal rings,'' Lecture Note in Mathematics 1237, Springer-Verlag, 1988. MR**89i:13001****3.**D. Eisenbud and S. Goto, Linear free resolutions and minimal multiplicity,*J. Algebra***88**(1984), 89 - 133. MR**85f:13023****4.**L.T. Hoa and J. Stückrad, Castelnuovo-Mumford regularity of simplicial toric rings, preprint, 2001.**5.**L. Gruson, R. Lazarsfeld and C. Peskine, On a theorem of Castelnuovo, and the equations defining space curves,*Inv. Math.***72**(1983), 491 - 506. MR**85g:14033****6.**R. Lazarsfeld, A sharp Castelnuovo bound for smooth surfaces,*Duke Math. J.***55**(1987), 423 - 429. MR**89d:14007**

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Additional Information

**Jürgen Herzog**

Affiliation:
FB6 Mathematik und Informatik, Universität – GHS – Essen, Postfach 103764, 45117 Essen, Germany

Email:
mat300@uni-essen.de

**Takayuki Hibi**

Affiliation:
Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 560–0043, Japan

Email:
hibi@math.sci.osaka-u.ac.jp

DOI:
https://doi.org/10.1090/S0002-9939-03-06952-1

Keywords:
Castelnuovo--Mumford regularity,
Eisenbud--Goto conjecture,
simplicial semigroup ring,
isolated singularity

Received by editor(s):
April 1, 2002

Published electronically:
February 20, 2003

Communicated by:
Bernd Ulrich

Article copyright:
© Copyright 2003
American Mathematical Society