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created: 2022-04-05T23:42
updated: 2024-04-19T16:18
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date: 2022-03-23 15:21
modification date: Wednesday 23rd March 2022 15:21:12
title: "de Rham-Witt"
aliases: ["de Rham-Witt complex"]

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Last modified date: <%+ tp.file.last_modified_date() %>


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- Tags 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [formal scheme](formal%20scheme)
	- [Spf](Spf.md)
	- [Witt vector](Archive/AWS2019/Witt%20vectors.md)
	- [mixed characteristic](Unsorted/mixed%20characteristic.md)

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# de Rham-Witt

![](attachments/Pasted%20image%2020220514181458.png)

# de Rham-Witt cohomology

Motivation: take $X\in \smooth\Alg\Var\slice k$, and define $\cocomplex{H}_\dR(X\slice k) \da \cocomplex{\HH}(\cocomplex{\Omega}_{X/k})$ using [hypercohomology](Unsorted/hypercohomology.md). Over $\CC$ this will be isomorphic to singular cohomology and take values in $\kmod$, but if $\characteristic k = p$ then the cohomology is entirely $p\dash$torsion.
So Grothendieck/Berthelot define [crystalline cohomology](Unsorted/crystalline%20cohomology.md) which takes values in $\mods{W(k)}$, the $p\dash$typical [Witt vectors](Archive/AWS2019/Witt%20vectors.md). 
Originally this was defined in terms of the structure sheaf of the crystalline topos of $X$, but a more modern definition realizes it as $\cocomplex{H}_\crys(X\slice k) = \cocomplex{\HH}(\cocomplex{\Omega}_{X, \mathrm{drW}})$, the hypercohomology of the de Rham-Witt complex.
This is a lift of algebraic de Rham in the following sense: $\cofib(p \cocomplex{\Omega}_{X, \mathrm{drW}} \to \cocomplex{\Omega}_{X, \mathrm{drW}}) \homotopic \cocomplex{\Omega}_{X/k}$ is a quasi-isomorphism of cochain complexes of sheaves.