Anglais
Let S be the spectrum of a discrete valuation ring R with generic point η=SpecK and closed point s=SpecK. Let X be a smooth S-scheme with generic fiber Xη, and closed fiber Xs. We construct a specialization map of Zariski-K-cohomology groups f:H1(Xη,K2)→H1(Xs,K2), which depends on the choice of a uniforinizing element π∈R. Then we show that f is compatible with the natural reduction map from H1(X,K2) to H1(Xs,K2). This observation is exploited to prove the following THEOREM. - The notations are as above, in particular let K be a local field of characteristic zero, which is unramified over Qp ; let Xn, be a smooth projective variety with ordinary good reduction, such that dimXη≤p−2 and Xs is ordinary.Then, if the condition Pic(Xˉs)(p)≡0 is satisfied, the map f:H1(Xˉη,K2)→H1(Xˉs,K2), induced by f by passing to the geometric fibers, is surjective on the p-primary torsion groups. The proof combines results of Bloch and Kato on p-adic étale cohomology in the case of ordinary reduction with assertions of Suslin, Lichtenbaum, Colliot-Thélène and Raskind on Zariski-K-cohomology, especially in characteristic p.
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