Cohomologie rigide et théorie de Dwork: Le cas des sommes exponentielles. (French) Zbl 0577.14013

Cohomologie p-adique, Astérisque 119/120, 17-49 (1984).
[For the entire collection see Zbl 0542.00006.]
In this paper the author first recalls his earlier definition of the rigid cohomology of a scheme of finite type over a finite field, viewed as the residue field of a complete non-archimedean field [Groupe Étude Anal. Ultramétrique, 9e Année: 1981/82, No.3, Exposé J 2 (1983; Zbl 0515.14015)]. The central notion here is that of a superconvergent crystal which involves rigid analysis as well as formal geometry.
The author calculates these cohomology groups in two cases, namely for the Artin-Schreier-extension of the affine line over \({\mathbb{F}}_ q\) defined by the equation \(y=x^ q-x\), and for the n-th power map on \({\mathbb{G}}_{m,{\mathbb{F}}_ q}\). In both cases the higher cohomology groups vanish, while the zero-term splits into a direct sum of crystals, indexed by the additive characters of \({\mathbb{F}}_ q\) in the first case, and by multiplicative characters in the second case.
Finally the author shows that for certain exponential sums his rigid cohomology groups coincide with the cohomology defined by Dwork theory as in Robba’s paper in the same volume [P. Robba, Astérisque 119/120, 191-266 (1984; Zbl 0548.12015)].
Reviewer: F.Herrlich


14F30 \(p\)-adic cohomology, crystalline cohomology
14F40 de Rham cohomology and algebraic geometry
14G20 Local ground fields in algebraic geometry