2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/147209We 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 such equisingularity conditions as Whitney's Condition A, Thom's Condition A_f, and Henry, Merle and Sabbah's Condition W_f. Notably, we describe these 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.42 pages, PLAIN Tex (with switches for AFour, 2-across). Significant improvements have been made in the statements and proofs of many results in Sections 3 to 6Algebraic Geometrytext