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 $K[C]$.References
- W. Bruns and J. Herzog, “Cohen–Macaulay rings,” Revised Edition, Cambridge University Press, 1996.
- Winfried Bruns and Udo Vetter, Determinantal rings, Lecture Notes in Mathematics, vol. 1327, Springer-Verlag, Berlin, 1988. MR 953963, DOI 10.1007/BFb0080378
- David Eisenbud and Shiro Goto, Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984), no. 1, 89–133. MR 741934, DOI 10.1016/0021-8693(84)90092-9
- L.T. Hoa and J. Stückrad, Castelnuovo–Mumford regularity of simplicial toric rings, preprint, 2001.
- L. Gruson, R. Lazarsfeld, and C. Peskine, On a theorem of Castelnuovo, and the equations defining space curves, Invent. Math. 72 (1983), no. 3, 491–506. MR 704401, DOI 10.1007/BF01398398
- Robert Lazarsfeld, A sharp Castelnuovo bound for smooth surfaces, Duke Math. J. 55 (1987), no. 2, 423–429. MR 894589, DOI 10.1215/S0012-7094-87-05523-2
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