Holmstrom Big de Rham-Witt cohomology

Big de Rham-Witt cohomology

arXiv:1006.3125 The big de Rham-Witt complex from arXiv Front: math.KT by Lars Hesselholt 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 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. The latter complex may be interpreted as the complex of differentials along the leaves of a foliation of Spec Z.

Big de Rham-Witt cohomology

Look for something of Borger? Mentioned in a comment here: http://mathoverflow.net/questions/27014/is-it-possible-to-classify-all-weil-cohomologies

Webpage of Borger

arXiv:1211.6006 Big de Rham-Witt 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 Rham-Witt 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 Rham-Witt 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 torsion-free. In addition, we establish a Poincaré duality theorem. Our results rely on an explicit description of the relative de Rham-Witt complex of a smooth $\lambda$-ring, which may be of independent interest.

nLab page on Big de Rham-Witt cohomology