- Acta Math.
- Volume 214, Number 1 (2015), 135-207.
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 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 directly 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.
Generous assistance from DNRF Niels Bohr Professorship, JSPS Grant-in-Aid 23340016, and CMI Senior Scholarship is gratefully acknowledged.
Acta Math., Volume 214, Number 1 (2015), 135-207.
Received: 8 June 2013
Revised: 3 December 2014
First available in Project Euclid: 30 January 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 19D35: Negative $K$-theory, NK and Nil
Secondary: 14F20: Étale and other Grothendieck topologies and (co)homologies 19D55: $K$-theory and homology; cyclic homology and cohomology [See also 18G60]
2015 © Institut Mittag-Leffler
Hesselholt, Lars. The big de Rham–Witt complex. Acta Math. 214 (2015), no. 1, 135--207. doi:10.1007/s11511-015-0124-y. https://projecteuclid.org/euclid.acta/1485802413