Initial simplicial complexes of prime ideals
type: Item
instance of: scholarly article
title: Initial simplicial complexes of prime ideals
zbMATH Open document ID: 1073.13013
published in: Journal of Algebra
publication date: 2005-04-21T00:00:00Z
review text: Let \(I\) be an ideal in a polynomial ring \(R = k[x_0, \dots, x_n]\) over a field~\(k\) and \(<\) a term order on \(R\). The initial complex of~\(I\) is a simplicial complex \(\Delta = \Delta_<(I)\) such that the Stanley-Reisner ideal \(I_\Delta\) is equal to the radical of the initial ideal \(\text{in}_< I\) of~\(I\). The initial ideal and hence the initial complex of \(I\) is useful to study the residue ring~\(R/I\). \textit{M. Kalkbrener} and \textit{B. Sturmfels} [Adv. Math. 116, No. 2, 365--376 (1995; Zbl 0877.13025)] proved that \(\Delta_<(I)\) is pure and strongly connected if \(I\) is a prime ideal. Is the converse true? What simplicial complex can be the initial complex of a prime ideal? The present article is an approach to these questions. The main theorem of this article is the following: Assume that \(k\) is algebraically closed. For any integer \(n \geq d > 0\), there are a prime ideal \(I \subset R\) and a term order~\(>\) such that \(R/I\) is complete intersection and the initial complex of~\(I\) is the \(d\)-skeleton of the \(n\)-simplex, that is, the subcomplex of the \(n\)-simplex generated by all the \(d\)-faces.
reviewed by: Takesi Kawasaki
Mathematics Subject Classification ID: 13F55
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 14M05
zbMATH DE Number: 2159745
zbMATH Keywords: initial ideal
zbMATH Keywords: simplicial complex
zbMATH Keywords: Stanley-Reisner ideal
zbMATH Keywords: initial complex
zbMATH Keywords: set-theoretic complete intersection
author: John P. Dalbec
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2005.01.013
OpenAlex ID: W2058755885
cites work: Q5690438
cites work: Q3328651
cites work: Q3943065
cites work: Two remarks on independent sets
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cites work: On hyperplane sections of reduced irreducible varieties of low codimension
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cites work: Initial complexes of prime ideals
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cites work: Q4293528
DOI: 10.1016/J.JALGEBRA.2005.01.013
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description: scientific article; zbMATH DE number 2159745