In the second part, we prove that in the homotopy category of a regular
scheme S, the set of endomorphisms of the infinite Grassmannian (which
gives a model for algebraic K-theory by a theorem by Morel and Voevodsky)
is naturally isomorphic to the set of natural transformations from the
Grothendieck group functor (considered as presheaf of sets on the category
of smooth schemes over S) to itself.
This enables us to build a special λ-ring structure on
higher K-groups and to check that this construction is the same as the
ones that were constructed before. Additive operations on algebraic
K-theory are studied carefully and stable versions of the theorems are
provided either with integer or rational coefficients. The technique also
allows us to define Chern classes on higher K-groups with values in
motivic cohomology (and other cohomological theories) and to assert the
existence of superphantom

maps in the homotopy theory of schemes.

Joël Riou <joel.riou@normalesup.org>