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Journal of the London Mathematical Society 2005 72(1):205-224; doi:10.1112/S0024610705006575
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© The London Mathematical Society

Tjurina and Milnor Numbers of Matrix Singularities

V. Goryunov and D. Mond

Department of Mathematical Sciences, University of Liverpool Liverpool L69 3BX, United Kingdom goryunov{at}liv.ac.uk
Mathematics Institute, University of Warwick Coventry CV4 7AL, United Kingdom mond{at}maths.warwick.ac.uk

Received 22 April 2004.

To gain understanding of the deformations of determinants and Pfaffians resulting from deformations of matrices, the deformation theory of composites f {circ} 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

{tau} = µ(f {circ} F) – ß0 + ß1,

where {tau} is the length of T1(F) and ßi 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 {tau} and the rank of the vanishing homology of the zero locus of f {circ} F are obtained.


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