Cohen-Macaulayness of almost complete intersection tangent cones
type: Item
instance of: scholarly article
title: Cohen-Macaulayness of almost complete intersection tangent cones
zbMATH Open document ID: 1154.13008
author: Takafumi Shibuta
published in: Journal of Algebra
publication date: 2008-07-11T00:00:00Z
review text: The author considers affine monomial curves of codimension three, that is, curves which coordinate ring is \(K[[T^{n_1},..., T^{n_1}]]\), where \(n_1,...,n_4\) are nonzero natural integers. A monomial curve can be seen as a quotient \(K[X,Y,Z,W]/I\simeq K[[T^{n_1},..., T^{n_1}]]\), where \(I\) is generated by some binomials. The tangent cone is defined by the ideal \(I^*\subset K[X,Y,Z,W]\) generated by the initial forms of all elements in \(I,\) for the usual degree on \(K[X,Y,Z,W]\). The main theorem in this paper is the following: If \(K[[T^{n_1},..., T^{n_1}]]\) is a complete intersection and if the ideal \(I^*\) defining the tangent cone at the origin is generated by 4 elements then the tangent cone is arithmetically Cohen-Macaulay. The proof consist to give an algorithm to compute a minimal basis of \(I^*\), by extending some results on the theory of standard basis of modules, similar to the theory Gröbner basis.
reviewed by: Marcel Morales
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13A30
Mathematics Subject Classification ID: 13C14
zbMATH DE Number: 5300805
zbMATH Keywords: Cohen-Macaulay
zbMATH Keywords: tangent cone
zbMATH Keywords: monomial curve in codimension three
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2007.11.023
OpenAlex ID: W2081445959
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DOI: 10.1016/J.JALGEBRA.2007.11.023
description: scientific article; zbMATH DE number 5300805