Specialization of integral dependence for modules

Abstract.

We establish the principle of specialization of integral dependence for submodules of finite colength of free modules, as part of the general algebraic-geometric theory of the Buchsbaum–Rim multiplicity. Then we apply the principle to the study of equisingularity of ICIS germs, obtaining results for Whitney’s Condition A and Thom’s Condition A_f. Notably, we describe these equisingularity conditions for analytic families in terms of various numerical invariants, which, for the most part, depend only on the members of a family, not on its total space.