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 Tor
O
Y
i (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.
