### The big de Rham-Witt complex

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 Madsen and myself 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 λ-ring which is
developed first. The main result in this first part of the paper,
which also gives a comprehensive review of the theory of Witt
vectors, is that the universal derivation of a λ-ring is given
by the universal derivation of the underlying ring together with an
additional structure depending on the λ-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
Kähler 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 étale maps, and
finally, the big de Rham-Witt complex of the ring of integers is
explicitly evaluated.

Lars Hesselholt
<larsh@math.nagoya-u.ac.jp>