Lyubeznik's resolution and rooted complexes
type: Item
instance of: scholarly article
title: Lyubeznik's resolution and rooted complexes
zbMATH Open document ID: 1040.13010
DOI: 10.1023/A:1020838732281
author: Isabella Novik
published in: Journal of Algebraic Combinatorics
publication date: 2003-03-11T00:00:00Z
review text: Let \(R\) be a polynomial ring, \(I\) be an ideal of \(R\) generated by monomials \(m_1,\dots,m_n\), \(L=L(I)\) be the lattice of the least common multiples for any of \(m_1,\dots,m_n\), \(\Delta\) be a simplicial complex whose faces are labeled with these least common multiples. Based on results of \textit{D. Bayer} and \textit{B. Sturmfels} [J. Reine Angew. Math. 502, 123--140 (1998; Zbl 0909.13011)] concerning \(\Delta\), the author indicates that the routed complex \(RC(L,\pi)\) (with respect to a routed map \(\pi)\), introduced by \textit{A. Björner} and \textit{G. Ziegler} [J. Comb. Theory, Ser. B 51, No. 1, 96--126 (1991; Zbl 0753.05019)], provides a free cellular resolution of \(I\). This resolution is minimal if \(L\) is a geometric lattice. In particular, this result can be applied to matroid ideals [see \textit{I. Novik}, \textit{A. Postnikov} and \textit{B. Sturmfels}, Duke Math. J. 111, No. 2, 287--317 (2002; Zbl 1022.13002)] for modular matroids and thus a generalization of Lyubeznik's resolution [see \textit{G. Lyubeznik}, J. Pure Appl. Algebra 51, No. 1/2, 193--195 (1988; Zbl 0652.13012)] is obtained.
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 05B35
Mathematics Subject Classification ID: 06B99
Mathematics Subject Classification ID: 06C10
Mathematics Subject Classification ID: 13F55
Mathematics Subject Classification ID: 57Q99
zbMATH DE Number: 1879612
zbMATH Keywords: resolutions of monomial ideals
zbMATH Keywords: geometric lattices
zbMATH Keywords: Stanley-Reisner rings
zbMATH Keywords: matroid ideals
MaRDI profile type: Publication
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full work available at URL: a:1020838732281
OpenAlex ID: W1688639084
description: scientific article; zbMATH DE number 1879612