Characteristic cycles of local cohomology modules of monomial ideals
type: Item
instance of: scholarly article
title: Characteristic cycles of local cohomology modules of monomial ideals
zbMATH Open document ID: 0974.13020
DOI: 10.1016/S0022-4049(98)00171-6
published in: Journal of Pure and Applied Algebra
publication date: 2001-11-11T00:00:00Z
review text: This paper studies the local cohomology modules \(\text{H}_I^i(R)\), where \(I\) is a square free monomial ideal in \(R=k[x_1,\dots,x_n]\) (or \(R=kx_1,\dots,x_n\)), \(k\) a field of characteristic 0. Such an ideal \(I\) has a minimal primary decomposition of the form \(I=I_1\cap\cdots\cap I_m\), where the \(I_i\) are ``face ideals'', i.e. ideals of the form \((x_{i_1},\dots,x_{i_h})\). The \(\text{H}^i_I(R)\) have a natural structure of finitely generated \({\mathcal D}=D(R,k)\)-module. Furthermore these local cohomology modules are holonomic \(\mathcal D\)-modules (i.e. of dimension \(n\)), hence have a characteristic cycle \(CC(\text{H}^i_I(R))\). The characteristic cycle \(CC(\text{H}^i_I(R))\) is first described if \(I\) is a face ideal. Then a closed form for \(CC(\text{H}^i_I(R))\) is given for \(I\) a square free monomial ideal in terms of local cohomology modules \(CC(\text{H}^j_{I_d}(R))\) where \(I_d\) ranges over the face ideals occurring in the primary decomposition of \(I\), and sums of such face ideals (which will again be face ideals). This is used to determine when \(\text{H}^i_I(R)=0\) and to find the cohomological dimension \(\text{cd}(R,I)\) in terms of the primary decomposition of \(I\). The invariants \(\lambda_{p,i}(R/I)\) introduced by \textit{G. Lyubeznik} [in Invent. Math. 113, 41-55 (1993; Zbl 0795.13004)] are found and interpreted as the multiplicities of the characteristic cycle of \(\text{H}^p_m(\text{H}^{n-i}_I(R))\) (where \(m\) is the graded maximal ideal of \(R\)). Finally this material is related to the \(f\)-vectors of the associated simplicial complex of a Stanley-Reisner ring.
Mathematics Subject Classification ID: 13F55
Mathematics Subject Classification ID: 13D45
Mathematics Subject Classification ID: 13N10
Mathematics Subject Classification ID: 14B15
Mathematics Subject Classification ID: 13D03
zbMATH DE Number: 1488530
zbMATH Keywords: local cohomology
zbMATH Keywords: monomial ideal
zbMATH Keywords: \(\mathcal D\)-module
zbMATH Keywords: face ideals
zbMATH Keywords: characteristic cycle
zbMATH Keywords: cohomological dimension
zbMATH Keywords: Stanley-Reisner ring
author: Josep Àlvarez Montaner
reviewed by: Leslie G. Roberts
MaRDI profile type: Publication
full work available at URL: s0022-4049(98)00171-6
OpenAlex ID: W2074642019
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description: scientific article; zbMATH DE number 1488530