The minimal free resolution of a class of square-free monomial ideals.
type: Item
instance of: scholarly article
title: The minimal free resolution of a class of square-free monomial ideals.
zbMATH Open document ID: 1058.13008
author: Rahim Zaare-Nahandi
author: Rashid Zaare-Nahandi
published in: Journal of Pure and Applied Algebra
publication date: 2004-07-12T00:00:00Z
review text: In this very well written paper one finds a thorough study of monomial ideals generated by minors of the direct sum of several vectors of indeterminates. The authors call such an ideal transversal and denote it by \(I_t\) if it is generated by degree \(t\) monomials. A transversal ideal is square-free, so it may be associated to a simplicial complex in the usual way. A first result asserts that a transversal ideal is Gorenstein exactly when it is principal. Although the simplicial complex naturally associated to a transversal ideal is pure and shellable if and only if all vectors of indeterminates have length one (cf. proposition 1.1), its Alexander dual is always pure and shellable (cf. proposition 2.1). Therefore, \(I_t\) has a linear resolution and the quotient ring \(S/I_t\) is Golod. Moreover, the Buchsbaum property of a quadratic transversal ideal is equivalent to the purity of the associated simplicial complex. The Hilbert series of a transversal ideal is obtained explicitly in theorem~2.4 from an exact sequence which is usually considered for shellable complexes. The minimal free resolution of the Stanley-Reisner ring associated to a transversal ideal is computed in section 3. The construction relies on the Eagon-Northcott complex associated to maximal minors of matrices of indeterminates and on the minimal free resolution of determinantal rings defined by generic diagonal matrices. The final sections contains applications to the study of generic multiple points in algebraic geometry. It is shown that transversal ideals generalize the ideals of minors of a generic pluri-circulant matrix over a field with convenient properties.
reviewed by: Mihai Cipu
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 13F55
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 13D25
Mathematics Subject Classification ID: 14M05
Mathematics Subject Classification ID: 14M07
Mathematics Subject Classification ID: 13C40
zbMATH DE Number: 2078862
zbMATH Keywords: transversal monomial ideal
zbMATH Keywords: Gorenstein complex
zbMATH Keywords: Buchsbaum complex
zbMATH Keywords: shellable simplicial complex
zbMATH Keywords: Eagon-Northcott resolution
zbMATH Keywords: Hilbert series
zbMATH Keywords: finite free resolution
zbMATH Keywords: pluri-circulant matrix
zbMATH Keywords: generic multiple points
MaRDI profile type: Publication
full work available at URL: j.jpaa.2003.10.026
OpenAlex ID: W1996831140
cites work: Q3708148
cites work: Cellular resolutions of monomial modules
cites work: Q3874222
cites work: The resolution of the ideal of \(2\times 2\) minors of a \(2\times n\) matrix of linear forms
cites work: Ideals defined by matrices and a certain complex associated with them
cites work: Resolutions of Stanley-Reisner rings and Alexander duality
cites work: Q4317713
cites work: Minimal resolutions of some monomial ideals
cites work: Componentwise linear ideals and Golod rings
cites work: Q4120658
cites work: The Alexander duality functors and local duality with monomial support
cites work: Q5462454
cites work: Generic and cogeneric monomial ideals
cites work: Q4694082
cites work: IDEALS OF MINORS DEFINING GENERIC SINGULARITIES AND THEIR GRöBNER BASES
cites work: ON THE IDEALS OF MINORS OF PLURI-CIRCULANT MATRICES
cites work: Gröbner basis and free resolution of the ideal of 2-minors of a 2 ×nmatrix of linear forms*
DOI: 10.1016/J.JPAA.2003.10.026
description: scientific article; zbMATH DE number 2078862