Resumen de Tjurina and Milnor numbers of matrix singularities
V. Goryunov, David Mond 
To gain understanding of the deformations of determinants and Pfaffians resulting from deformations of matrices, the deformation theory of composites with isolated singularities is studied, where is a function with (possibly non-isolated) singularity and is a map into the domain of , and only is deformed. The corresponding is identified as (something like) the cohomology of a derived functor, and a canonical long exact sequence is constructed from which it follows that , where is the length of and is the length of . This explains numerical coincidences observed in lists of simple matrix singularities due to Bruce, Tari, Goryunov, Zakalyukin and Haslinger. When has Cohen-Macaulay singular locus (for example when is the determinant function), relations between and the rank of the vanishing homology of the zero locus of are obtained.
