This paper gives a new and direct construction of the multi-prime big de
Rham-Witt complex which is defined for every commutative and unital ring; the
original construction by the author and Madsen relied on the adjoint functor
theorem and accordingly was very indirect. (The construction given here also
corrects the 2-torsion which was not quite correct in the original version.)
The new construction is based on the theory of modules and derivations over a
lambda-ring which is developed first. The main result in this first part of the
paper is that the universal derivation of a lambda-ring is given by the
universal derivation of the underlying ring together with an additional
structure depending on the lambda-ring structure in question. In the case of
the ring of big Witt vectors, this additional structure gives rise to divided
Frobenius operators on the module of K\"ahler differentials. It is the
existence of these divided Frobenius operators that makes the new construction
of the big de Rham-Witt complex possible. It is further shown that the big de
Rham-Witt complex behaves well with respect to \'etale maps, and finally, the
big de Rham-Witt complex of the ring of integers is explicitly evaluated. The
latter complex may be interpreted as the complex of differentials along the
leaves of a foliation of Spec Z.Comment: 63 page
