SMF

On a specialization map in K2-cohomology

On a specialization map in K2-cohomology

Andreas LANGER
     
                
  • Année : 1994
  • Tome : 226
  • Format : Électronique
  • Langue de l'ouvrage :
    Anglais
  • Pages : 211-233
  • DOI : 10.24033/ast.279

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 πR. 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 dimXηp2 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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