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Kummer-Artin-Schreier-Witt theories. (Théories de Kummer-Artin-Schreier-Witt.) (French. Abridged English version) Zbl 0845.14023
In this note it is proved the existence of some exact sequences of group schemes which unify the Kummer and Artin-Schreier-Witt theories for cyclic étale coverings of degree $$p^n$$ over a base of mixed characteristics 0 and $$p$$. Let $$\zeta_r$$ be a primitive $$p^r$$-root of unity such that $$\zeta^p_{r + 1} = \zeta_r$$, $$r \geq 1$$, and let $$A = \mathbb{Z}_{ (p)} [\zeta_n]$$ and $$K = \mathbb{Q} (\zeta_n)$$. By Kummer theory the isogeny $$p^n : G_m \to G_m$$ is universal for the cyclic extensions of degree $$p^n$$ of $$K$$. By the Artin-Schreier-Witt theory, the isogeny $$F - 1 : W_n \to W_n$$ is universal for the cyclic extensions of degree $$p^n$$, $$W_n$$ being the group scheme of Witt vectors of length $$n$$. In this note it is announced the following theorem: There exists an exact sequence of group $$A$$-schemes $0 \to \mathbb{Z}/p^n \to {\mathcal W}_n \to {\mathcal V}_n \to 0$ such that the generic fibre is isomorphic to the sequence $0 \to \mu_{p^n,k} \to (G_{m,k})^n \to(G_{m,k})^n \to 0$ and the closed fibre is isomorphic to the sequence of Artin-Schreier-Witt: $0 \to \mathbb{Z}/p^n \to W_{n, \mathbb{F}_p} @>F - 1>> W_{n, \mathbb{F}_p} \to 0.$ Some details are given, but a complete proof is announced for a forthcoming paper [cf. J. Théor. Nombres Bordx. 7, No. 1, 177-189 (1995)].

##### MSC:
 14L15 Group schemes 13K05 Witt vectors and related rings (MSC2000) 11S45 Algebras and orders, and their zeta functions