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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Castelnuovo–Mumford regularity of simplicial semigroup rings with isolated singularity
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by Jürgen Herzog and Takayuki Hibi PDF
Proc. Amer. Math. Soc. 131 (2003), 2641-2647 Request permission

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 $A=K[\{f_ix_j\}_{\substack {i=1,\ldots ,mj=1,\ldots ,n}}, g_1,\ldots , g_r]$ 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
  • MR Author ID: 189999
  • 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
  • MR Author ID: 219759
  • Email: hibi@math.sci.osaka-u.ac.jp
  • Received by editor(s): April 1, 2002
  • Published electronically: February 20, 2003
  • Communicated by: Bernd Ulrich
  • © Copyright 2003 American Mathematical Society
  • 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
  • MathSciNet review: 1974318