Goryunov, Victor V. and Mond, D. (David). (2005) Tjurina and Milnor numbers of matrix singularities. Journal of the London Mathematical Society, Vol.72 (No.1). pp. 205-224. ISSN 0024-6107

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Abstract

To gain understanding of the deformations of determinants and Pfaffians resulting from deformations of matrices, the deformation theory of composites f ◦ F with isolated singularities is studied, where f : Y −→C is a function with (possibly non-isolated) singularity and F : X −→Y is a map into the domain of f, and F only is deformed. The corresponding T1(F) is identified as (something like) the cohomology of a derived functor, and a canonical long exact sequence is constructed from which it follows that τ = μ(f ◦ F) − β0 + β1, where τ is the length of T1(F) and βi is the length of ToriOY(OY/Jf, OX). This explains numerical coincidences observed in lists of simple matrix singularities due to Bruce, Tari, Goryunov, Zakalyukin and Haslinger. When f has Cohen–Macaulay singular locus (for example when f is the determinant function), relations between τ and the rank of the vanishing homology of the zero locus of f ◦ F are obtained.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Pfaffian systems, Deformations of singularities, Geometry, Algebraic, Matrices, Singularities (Mathematics)
Journal or Publication Title:	Journal of the London Mathematical Society
Publisher:	Cambridge University Press
ISSN:	0024-6107
Date:	August 2005
Volume:	Vol.72
Number:	No.1
Page Range:	pp. 205-224
Identification Number:	10.1112/S0024610705006575
Status:	Peer Reviewed
Access rights to Published version:	Open Access
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URI:	http://wrap.warwick.ac.uk/id/eprint/739

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