Big de RhamWitt cohomology

General Introduction
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Online References
Look for something of Borger? Mentioned in a comment here: http://mathoverflow.net/questions/27014/isitpossibletoclassifyallweilcohomologies
arXiv:1211.6006 Big de RhamWitt cohomology: basic results fra arXiv Front: math.NT av Andre Chatzistamatiou Let $X$ be a smooth projective $R$scheme, and let $R$ be an étale $\Z$algebra. As constructed by Hesselholt, we have the absolute big de RhamWitt complex $\W\Omega^_X$ of $X$ at our disposal. There is also a relative version $\W\Omega^_{X/\Z}$ that is characterized by the vanishing of the positive degree part in the case $X=\Spec(\Z)$. In this paper we study the hypercohomology of the relative (big) de RhamWitt complex of $X$. We show that it is a projective module over the ring of (big) Witt vectors of $R$, provided that the de Rham cohomology is torsionfree. In addition, we establish a Poincaré duality theorem. Our results rely on an explicit description of the relative de RhamWitt complex of a smooth $\lambda$ring, which may be of independent interest.
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Paper References
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Definition
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Properties
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Standard theorems
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Open Problems
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Connections to Number Theory
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Computations and Examples
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History and Applications
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Some Research Articles
arXiv:1006.3125 The big de RhamWitt complex from arXiv Front: math.KT by Lars Hesselholt This paper gives a new and direct construction of the multiprime big de RhamWitt 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 2torsion which was not quite correct in the original version.) The new construction is based on the theory of modules and derivations over a lambdaring which is developed first. The main result in this first part of the paper is that the universal derivation of a lambdaring is given by the universal derivation of the underlying ring together with an additional structure depending on the lambdaring 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 RhamWitt complex possible. It is further shown that the big de RhamWitt complex behaves well with respect to étale maps, and finally, the big de RhamWitt 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.
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