Reverse search for monomial ideals
type: Item
instance of: scholarly article
title: Reverse search for monomial ideals
zbMATH Open document ID: 1221.13037
author: Amelia Taylor
author: Dave Bayer
published in: Journal of Symbolic Computation
publication date: 2009-09-14T00:00:00Z
review text: Let \(I\) be a monomial ideal minimally generated by \(x^{\mathbf a_1},\dots, x^{\mathbf a_r}\) in a polynomial ring \(S=k[x_1,\dots, x_n]\) over a field \(k\). For an element \(\mathbf b\) in \(\mathbb N^n\), set \[ K_\mathbf b=\{F\subset\{1,2,\dots,n\}\mid x^{\mathbf b-F}\in I\}, \] where we identify a subset \(F\) of \(\{1,2,\dots,n\}\) with the 0-1 vector whose support is \(F\). Then \(K_\mathbf b\) is a simplicial complex (or the null complex, i.e., \(\emptyset\not\in K_\mathbf b\), if \(x^\mathbf b\not\in I\)). It is known that the multigraded Betti numbers can be expressed as \[ \beta_{i,\mathbf b}(I)=\beta_{i+1,\mathbf b}(S/I)=\dim_k \tilde H_{i-1}(K_\mathbf b;k), \] where \(\tilde H_{i}(K_\mathbf b;k)\) is the reduced homology [see \textit{D. Bayer} and \textit{B. Sturmfels}, ``Cellular resolutions of monomial modules'', J. Reine Angew. Math. 502, 123--140 (1998; Zbl 0909.13011) or \textit{E. Miller} and \textit{B. Sturmfels}, Combinatorial commutative algebra. New York, NY: Springer (2005; Zbl 1066.13001 and Zbl 1090.13001)]. Since the reduced homologies of a simplicial complex whose geometric realization is contractible are all zero, in order to compute Betti numbers of \(I\), it is enough to consider \(\mathbf b\in\mathbb N^n\) such that \(K_\mathbf b\) is a non-cone, i.e., there is no vertex \(i\) such that \(K_\mathbf b=\text{star}_{K_\mathbf b}(\{i\})\). The authors call \(\mathbf b\in\mathbb N^n\) with \(K_\mathbf b\) is not a cone a corner of \(I\). Let us denote by \(\mathbf a\vee\mathbf a'\) the componentwise maximum of \(\mathbf a\) and \(\mathbf a'\) for \(\mathbf a, \mathbf a'\in\mathbb N^n\). One way to find all corners of \(I\) is first consider \(\mathbf a_1\vee\mathbf a_2\vee\cdots\vee\mathbf a_r\), the exponent vector of the least common multiple of the generators of \(I\), and use the full \(\text{lcm}\) lattice of the generators of \(I\). In this paper, the authors propose the following algorithm to find all the corners of \(I\). First c…
reviewed by: Mitsuhiro Miyazaki
Mathematics Subject Classification ID: 13F55
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 13P99
zbMATH DE Number: 5603617
zbMATH Keywords: monomial ideal
zbMATH Keywords: reverse search
zbMATH Keywords: simplicial complex
describes a project that uses: SINGULAR
describes a project that uses: Homology
describes a project that uses: CoCoA
describes a project that uses: Macaulay2
MaRDI profile type: Publication
full work available at URL: j.jsc.2009.05.002
OpenAlex ID: W2153919403
cites work: Computing irredundant irreducible decompositions of large scale monomial ideals
cites work: Reverse search for enumeration
cites work: Cellular resolutions of monomial modules
cites work: Computation of Hilbert functions
cites work: Computation of Hilbert-Poincaré series
cites work: Q4279508
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cites work: The lcm-lattice in monomial resolutions
cites work: Combinatorial Commutative Algebra
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DOI: 10.1016/J.JSC.2009.05.002
description: scientific article; zbMATH DE number 5603617