Specialization of integral dependence for modules (Q1298086)

The authors study analytic families of germs of isolated complete-intersection singularieties (ICIS), or ICIS germs. Their main goal is to develop some new algebraic tools and a geometric point of view that enables them to describe some standard equisingularity conditions (Whitney's condition A and Thom's Condition \(A_f)\) in terms of suitable numerical invariants of isolated singularities. The basic numerical invariants used in this context are certain homological Buchsbaum-Rim multiplicities for modules, the general theory of which is recalled and considerably advanced in Sections 1 and 2 of this paper, culminating in a generalization (from ideals to modules) of B. Teissier's so-called ``principle of specialization of integral dependence'' [cf. \textit{B. Teissier}, Astérisque 7-8, 285-362 (1974; Zbl 0295.14003)]. Originally, B. Teissier had established this principle as an equivalent condition for Whitney equisingularity, and formulated it in terms of the Jacobian ideal of an analytic family of hypersurface germs with isolated singularities. In the sequel, the authors apply their generalization of this principle of specialization of integral dependence to the study of ICIS germs, i.e, to higher codimension. For this purpose, they introduce the more general concept of the Jacobian module of an ICIS germ, the integral closure of which is then used to describe equisingularity conditions based on the behavior of the limit tangent hyperplanes to the general member of the family. After a thorough treatment of strict dependence (à la M. Lejeune-Jalabert and B. Teissier) in the more general context, which is carried out in Section 3 of the paper, the authors study Whitney's Equisingularity Condition A and establish a generalization of it to families of ICIS germs in Section 4. The concluding Section 5 concerns Thom's Condition \(A_f\) for function germs \(f\) and culminates in a generalization of the Lê-Saito theorem to families of ICIS germs. This generalization is based upon a more recent construction by \textit{A. J. Parameswaran} [Compos. Math. 80, No. 3, 323-336 (1991; Zbl 0751.14005)] and yields, among other important results, a refinement of an earlier theorem of \textit{J. Briançon}, \textit{P. Maisonobe} and \textit{M. Merle} [Invent. Math. 117, 531-550 (1994; Zbl 0920.32010)]. Altogether, the methods and results developed in the paper under review must be seen as a major step toward the general study of equisingularity.