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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
Published electronically: February 20, 2003
MathSciNet review: 1974318
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Abstract: Let $S = K[x_1, \ldots, x_n]$ be the polynomial ring in $n \geq 2$variables over a field $K$ and $\mathfrak{m}$ its graded maximal ideal. Let $f_1,\ldots, f_m \in S$ be homogeneous polynomials of degree $d-1\geq 2$ generating an $\mathfrak{m}$-primary ideal, and let $g_1,\ldots,g_r\in S$ be arbitrary homogeneous polynomials of degree $d$. In the present paper it will be proved that the Castelnuovo-Mumford regularity of the standard graded $K$-algebra $A=K[\{f_ix_j\}_{\substack{i=1,\ldots,m\\ j=1,\ldots,n}}, g_1,\ldots, g_r]$is at most $(d-2)(n-1)$. By virtue of this result, it follows that the regularity of a simplicial semigroup ring $K[C]$ with isolated singularity is at most $e(K[C]) - \operatorname{codim}(K[C])$, where $e(K[C])$ is the multiplicity of $K[C]$and $\operatorname{codim}(K[C])$ is the codimension of $K[C]$.

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Additional Information

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

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

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

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