  It has been shown by Nistor that given any extension of associative algebras
over C, the connecting morphism in periodic cyclic homology is compatible,
under the Chern-Connes character, with the index morphism in lower algebraic
K-theory. The proof relies on the abstract properties of cyclic theory,
essentially excision, which does not provide explicit formulas a priori.
Avoiding the use of excision, we explain in this article how to get explicit
formulas in a wide range of situations. The method is connected to the
renormalization procedure introduced in our previous work on the bivariant
Chern character for quasihomomorphisms, leading to "local" index formulas in
the sense of non-commutative geometry. We illustrate these principles with the
example of the classical family index theorem: we find that the characteristic
numbers of the index bundle associated to a family of elliptic
pseudodifferential operators are expressed in terms of the (fiberwise) Wodzicki
residue.
