The big de Rham-Witt complex

TL;DR

This paper introduces a new, direct construction of the multi-prime big de Rham-Witt complex for any commutative ring, correcting previous issues and utilizing lambda-ring modules and derivations.

Contribution

It provides a novel, explicit construction of the big de Rham-Witt complex based on lambda-ring theory, improving upon the original by Madsen and author.

Findings

Corrects 2-torsion issues in the original construction

Establishes the behavior of the complex under étale maps

Explicitly evaluates the complex for the ring of integers

Abstract

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.