Computing the support of monomial iterated mapping cones
type: Item
instance of: scholarly article
title: Computing the support of monomial iterated mapping cones
zbMATH Open document ID: 1277.13011
author: Eduardo Saenz-de-Cabezon
published in: Journal of Symbolic Computation
publication date: 2010-09-10T00:00:00Z
review text: Let \(I=\langle f_1,\dots,f_r\rangle\subseteq \mathbb{K}[x_1,\dots,x_n]\) and let \(I_i=\langle f_1,\dots,f_i\rangle\) be the subideal of \(I\) generated by the first \(i\) generators of \(I\). The following short exact sequence \[ 0\rightarrow \overline{I}_i \overset{\varphi}{\rightarrow} I_{i-1}\oplus \langle f_i\rangle \overset{i}{\rightarrow} I_i\rightarrow 0 \] is defined as Mayer-Vietoris sequence, where \(\overline{I}_i=I_{i-1}\cap \langle f_i\rangle\). In this paper the author constructs the corresponding Mayer-Vietoris tree, in order to study the monomial iterated mapping cones. By constructing this tree the author succeeds in computing the monomial support of a free resolution of a monomial ideal, avoiding the hard computation of the resolution itself. Moreover by this approach algorithms are provided to analyze this support to obtain homological and numerical invariants of a monomial ideal, such as Betti numbers, Hilbert series, irreducible decompositions, regularity and Euler characteristic of simplicial complexes. It is also worth mentioning that the author in the last section of the article, concludes in the effectiveness of the Mayer-Vietoris tree, by comparing it with other commonly used computer algebras systems, such as CoCoA, Singular, and Macaulay2.
reviewed by: Christos Tatakis
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 13P20
Mathematics Subject Classification ID: 13F55
zbMATH DE Number: 5782199
zbMATH Keywords: Mayer-Vietoris trees
zbMATH Keywords: iterated mapping cones
zbMATH Keywords: monomial ideals
zbMATH Keywords: free resolutions
Wikidata QID: Q60246187
describes a project that uses: SINGULAR
describes a project that uses: Macaulay2
describes a project that uses: CoCoA
describes a project that uses: CoCoALib
MaRDI profile type: Publication
full work available at URL: j.jsc.2010.06.001
OpenAlex ID: W2062250216
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DOI: 10.1016/J.JSC.2010.06.001
description: scientific article; zbMATH DE number 5782199