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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
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