## de Rham-Witt complex. (Complexe de de Rham-Witt.)(French)Zbl 0446.14008

Astérisque 63, 83-112 (1979).
Summary: It is known that one can compute the crystalline cohomology of a smooth algebraic variety $$X$$ over a field $$k$$ of characteristic $$p>0$$ as the de Rham cohomology of a smooth lifting of $$X$$ (if one exists). The author constructs a complex of “characteristic 0 differential forms” $$W\Omega_X^*$$ over $$X$$ – the de Rham-Witt complex, and proves that the crystalline cohomology is isomorphic to the Zariski hypercohomology of $$W\Omega_X^*$$. This is a generalization of a result of Bloch. It is a slope spectral sequence
$E_1^{ij} = H^j(W\Omega_x^i) \Rightarrow H^*(W\Omega_X^*) \xrightarrow{\sim} H_{\text{cris}}(X/W).$
In the paper there are some results also about degeneration of the spectral sequence and applications.
For the entire collection see [Zbl 0404.00008].
Reviewer: A. N. Rudakov

### MSC:

 14F40 de Rham cohomology and algebraic geometry 14F30 $$p$$-adic cohomology, crystalline cohomology