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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. 205-224. doi:10.1112/S0024610705006575 ISSN 0024-6107.
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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 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 | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > 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 | ||||
Official Date: | August 2005 | ||||
Dates: |
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Volume: | Vol.72 | ||||
Number: | No.1 | ||||
Page Range: | pp. 205-224 | ||||
DOI: | 10.1112/S0024610705006575 | ||||
Status: | Peer Reviewed | ||||
Access rights to Published version: | Open Access (Creative Commons) |
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