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

Illusie, Luc

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

Keywords:

complex of characteristic zero differential forms; de Rham-Witt complex; Zariski hypercohomology; degeneration of spectral sequence