Acta Mathematica

The big de Rham–Witt complex

Lars Hesselholt

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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 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.

Note

Generous assistance from DNRF Niels Bohr Professorship, JSPS Grant-in-Aid 23340016, and CMI Senior Scholarship is gratefully acknowledged.

Article information

Source
Acta Math., Volume 214, Number 1 (2015), 135-207.

Dates
Received: 8 June 2013
Revised: 3 December 2014
First available in Project Euclid: 30 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485802413

Digital Object Identifier
doi:10.1007/s11511-015-0124-y

Mathematical Reviews number (MathSciNet)
MR3316757

Zentralblatt MATH identifier
1316.13028

Subjects
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]

Keywords
de Rham–Witt complex lambda-rings derivations foliations

Rights
2015 © Institut Mittag-Leffler

Citation

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


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