Inventiones mathematicae

, Volume 137, Issue 3, pp 541–574

Specialization of integral dependence for modules

  • Terence Gaffney
  • Steven L. Kleiman

DOI: 10.1007/s002220050335

Cite this article as:
Gaffney, T. & Kleiman, S. Invent. math. (1999) 137: 541. doi:10.1007/s002220050335


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 Af. 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.

Mathematics Subject Classification (1991): 32S15, 14B05, 13H15

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Terence Gaffney
    • 1
  • Steven L. Kleiman
    • 2
  1. 1.Department of Mathematics, Northeastern University, Boston, MA 02115, USA (e-mail:
  2. 2.Department of Mathematics, 2–278 MIT, Cambridge, MA 02139, USA (e-mail: