Annihilating ideals for an algebraic local cohomology class
type: Item
instance of: scholarly article
title: Annihilating ideals for an algebraic local cohomology class
zbMATH Open document ID: 1171.32020
author: Shinichi Tajima
author: Yayoi Nakamura
published in: Journal of Symbolic Computation
publication date: 2009-04-14T00:00:00Z
review text: The authors consider isolated hypersurface singularities in the context of algebraic analysis. Given a holomorphic function \(f: ({\mathbb C}^n,0)\to({\mathbb C},0)\), let \(W_f\) be the space of algebraic local cohomology classes annihilated by the Jacobi ideal \(J_f\); it is the dual vector space to the Jacobi algebra \({\mathcal O}/J_f\). Let \({\mathcal A}nn^{(k)}_{{\mathcal D}_0}(\omega)\) be the right \(\mathcal D_0\)-module, generated by the the set of linear differential operators of order at most \(k\) annihilating a generator of \(W_f\). Then \( {\mathcal D}_0/ {\mathcal A}nn^{(k)}_{{\mathcal D}_0}(\omega)\) is a holonomic system, supported at the origin. The multiplicity \(\mu^{(k)}\) of its characteristic cycle gives an invariant of the singularity, with \(\mu^{(0)}\) the Milnor number. The purpose of this paper is to give algorithms for computing the relevant objects. The first one concerns the computation of a basis of \(W_f\), in terms of relative Čech cohomology. This leads to a test for the membership problem for \(J_f\) using residues. The second algorithm computes the annihilators \({\mathcal A}nn^{(k)}_{{\mathcal D}_0}(\omega)\). The algorithms are illustrated with the singularity \(Z_{12}\).
reviewed by: Jan Stevens
Mathematics Subject Classification ID: 32S40
zbMATH DE Number: 5543813
zbMATH Keywords: hypersurface isolated singularities
zbMATH Keywords: Grothendieck local duality
zbMATH Keywords: algebraic local cohomology
zbMATH Keywords: holonomic \(\mathcal D\)-modules
describes a project that uses: Kan
MaRDI profile type: Publication
full work available at URL: j.jsc.2007.07.017
OpenAlex ID: W1975105594
cites work: Q3973332
cites work: Isolated points, duality and residues
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cites work: Algorithms for \(D\)-modules -- restriction, tensor product, localization, and local cohomology groups
cites work: Algebraic local cohomology classes attached to quasi-homogeneous hypersurface isolated singularities
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cites work: Algorithmic computation of local cohomology modules and the local cohomological dimension of algebraic varieties
DOI: 10.1016/J.JSC.2007.07.017
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description: scientific article; zbMATH DE number 5543813