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Tjurina and Milnor numbers of matrix singularities
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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. 205224. ISSN 00246107

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Official URL: http://dx.doi.org/10.1112/S0024610705006575
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 nonisolated) 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:  00246107 
Official Date:  August 2005 
Volume:  Vol.72 
Number:  No.1 
Page Range:  pp. 205224 
Identification Number:  10.1112/S0024610705006575 
Status:  Peer Reviewed 
Access rights to Published version:  Open Access 
References:  1. J. W. Bruce, ‘Families of symmetric matrices’, Preprint, University of Liverpool, 1999. 
URI:  http://wrap.warwick.ac.uk/id/eprint/739 
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