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\(p\)-adic étale cohomology. (English) Zbl 0613.14017
This paper gives the complete proofs of the results summarized by S. Bloch in Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. I: Arithmetic, Prog. Math. 35, 13-26 (1983; Zbl 0584.14009). Let \(K\) be a field, complete with respect to a discrete valuation, of characteristic 0 and residue characteristic \(p>0\). Let \(V\) be a complete smooth variety over \(K\) and \(X\) a proper smooth model of \(V\) over the valuation ring \(\Lambda\) of \(K\). The central object of the paper are the étale cohomology groups \(H^ q(\bar V,{\mathbb{Q}}_ p)\), where \(\bar V\) is the extension of \(V\) to the algebraic closure \(\bar K\) of \(K\). They (more precisely: certain subquotients of them) are compared as Gal\((\bar K/K)\)-modules to suitable de Rham-Witt cohomology groups and crystalline cohomology groups. As a corollary the authors prove that \(H^ q(\bar V,{\mathbb{Q}}_ p)\) has a Hodge-Tate decomposition under the assumption that the reduction of \(\bar X\) be “ordinary”, a notion which for an abelian variety \(A\) coincides with the usual one (the group of \(p\)-torsion points has order \(p^{\dim A}\)).
Reviewer: F.Herrlich

MSC:
14F30 \(p\)-adic cohomology, crystalline cohomology
14F40 de Rham cohomology and algebraic geometry
14G20 Local ground fields in algebraic geometry
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
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References:
[1] Bass, H., andTate, J., The Milnor ring of a global field, inAlgebraic K-theory, II,Springer Lecture Notes in Math.,342 (1972), 349–442.
[2] Berthelot, P.,Cohomologie cristalline des schémas de charactéristique p > o, Springer Lecture Notes in Math.,407 (1974). · Zbl 0298.14012
[3] Bloch, S., Algebraic K-theory and crystalline cohomology,Publ. Math. I.H.E.S.,47 (1974), 187–268. · Zbl 0388.14010
[4] Bloch, S., Torsion algebraic cycles, K2, and Brauer groups of function fields, inGroupe de Brauer, Springer Lecture Notes in Math.,844 (1981), 75–102.
[5] Bloch, S.,p-adic étale cohomology, inArithmetic and Geometry, vol. I, 13–27, Birkhäuser (1983).
[6] Deligne, P. (withIllusie, L.), Cristaux ordinaires et coordonnées canoniques, inSurfaces algébriques, Springer Lecture Notes in Math.,868 (1981), 80–137.
[7] Gabber, O.,An injectivity property for étale cohomology, preprint, 1981. · Zbl 0828.14011
[8] Gabber, O.,Gersten’s conjecture for some complexes of vanishing cycles, preprint, 1981. · Zbl 0827.19002
[9] Gabber, O.,Affine analog of proper base change theorem, preprint, 1981. · Zbl 0492.16002
[10] Illusie, L., Complexe de De Rham-Witt et cohomologie cristalline,Ann. Sci. Ec. Norm. Sup., 4e sér.,12 (1979), 501–661. · Zbl 0436.14007
[11] Kato, K., A generalization of local class field theory by using K-groups, II,J. Fac. Sci. Univ. Tokyo,27 (1980), 603–683. · Zbl 0463.12006
[12] Kato, K., Galois cohomology of complete discrete valuation fields, inAlgebraic K-theory, Springer Lecture Notes in Math.,967 (1982), 215–238.
[13] Katz, N., Serre-Tate local moduli, inSurfaces algébriques, Springer Lecture Notes in Math.,868 (1981), 138–202.
[14] Merkurjev, A. S., andSuslin, A. A., K-cohomology of Severi-Brauer varieties and norm residue homomorphism,Math. U.S.S.R. Izvestiya,21 (1983), 307–340. · Zbl 0525.18008 · doi:10.1070/IM1983v021n02ABEH001793
[15] Milnor, J., Algebraic K-theory and quadratic forms,Inv. Math.,9 (1970), 318–344. · Zbl 0199.55501 · doi:10.1007/BF01425486
[16] Serre, J.-P.,Cohomologie Galoisienne, Springer Lecture Notes in Math.,5 (1965).
[17] Soulé, C., K-théorie des anneaux d’entiers de corps de nombres et cohomologie étale,Inv. Math.,55 (1979), 251–295. · Zbl 0437.12008 · doi:10.1007/BF01406843
[18] Tate, J.,p-divisible groups, inProceedings of a Conference on local fields, Springer-Verlag, 1967, 158–183. · Zbl 0157.27601
[19] Tate, J., Relation between K2 Galois cohomology,Inv. Math.,36 (1976), 257–274. · Zbl 0359.12011 · doi:10.1007/BF01390012
[20] Delione, P., La conjecture de Weil, II,Publ. Math. I.H.E.S.,52 (1980), 313–428.
[21] Illusie, L., andRaynaud, M., Les suites spectrales associées au complexe de De Rham-Witt,Publ. Math. I.H.E.S.,57 (1983), 73–212. · Zbl 0538.14012
[22] Serre, J.-P., Sur les corps locaux à corps résiduel algébriquement clos,Bull. Soc. Math. France,8 (1961), 105–154. · Zbl 0166.31103
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