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2015 The big de Rham–Witt complex
Lars Hesselholt
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Acta Math. 214(1): 135-207 (2015). DOI: 10.1007/s11511-015-0124-y


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.

Funding Statement

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


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Lars Hesselholt. "The big de Rham–Witt complex." Acta Math. 214 (1) 135 - 207, 2015.


Received: 8 June 2013; Revised: 3 December 2014; Published: 2015
First available in Project Euclid: 30 January 2017

zbMATH: 1316.13028
MathSciNet: MR3316757
Digital Object Identifier: 10.1007/s11511-015-0124-y

Primary: 19D35
Secondary: 14F20 , 19D55

Keywords: de Rham–Witt complex , derivations , foliations , lambda-rings

Rights: 2015 © Institut Mittag-Leffler

Vol.214 • No. 1 • 2015
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