no. 4
The explicit reciprocity law and the cohomology of Fontaine-Messing
Bulletin de la Société Mathématique de France, Tome 119 (1991) no. 4, pp. 397-441. 
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     author = {Kato, Kazuya},
     title = {The explicit reciprocity law and the cohomology of {Fontaine-Messing}},
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     pages = {397--441},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {119},
     number = {4},
     year = {1991},
     doi = {10.24033/bsmf.2173},
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     zbl = {0752.14015},
     language = {en},
     url = {http://archive.numdam.org/articles/10.24033/bsmf.2173/}
}
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Kato, Kazuya. The explicit reciprocity law and the cohomology of Fontaine-Messing. Bulletin de la Société Mathématique de France, Tome 119 (1991) no. 4, pp. 397-441. doi : 10.24033/bsmf.2173. http://archive.numdam.org/articles/10.24033/bsmf.2173/

[AH] Artin (E.) and Hasse (H.). - Die beiden Ergänzungssätze zum Reziprozitätsgesetze der ln-ten Potenzreste im Körper der ln-ten Einheiten Wurseln, Abh. Math. Sem. Univ. Hambourg, t. 6, 1928, p. 146-162. | JFM

[BO] Berthelot (P.) and Ogus (A.). - Notes on crystalline cohomology, Princeton University Press, 1978. | MR | Zbl

[BK] Bloch (S.) and Kato (K.). - p-adic etale cohomology, Inst. Hautes Études Sci. Publ. Math., t. 63, 1986, p. 107-152. | Numdam | MR | Zbl

[B1] Brückner (H.). - Hilbertsymbole zum Exponenten pn und Pfaffsche Formen. - Preprint, Hamburg, 1979.

[B2] Brückner (H.). - Explizites reziprozitätsgesetz und anwendungen. - Vorlesungen aus dem Fachbereich Mathematik der Universität Essen, 1979. | Zbl

[C] Coleman (R.F.). - The dilogarithm and the norm residue symbol, Bull. Soc. Math. France, t. 109, 1981, p. 373-402. | Numdam | MR | Zbl

[CSS] Colliot Thélène (J.-L.), Sansuc (J.-J.) and Soulé (C.). - Torsion dans le goupe de Chow de codimension deux, Duke Math. J., t. 50, 1983, p. 763-801. | Zbl

[FM] Fontaine (J.-M.) and Messing (W.). - Contemporary Math., t. 67, 1987, p. 179-207. | MR | Zbl

[Gr] Grothendieck (A.). - Éléments de Géométrie algébrique, IV, première partie, Inst. Hautes Études Sci. Publ. Math., t. 20, 1964. | Numdam | Zbl

[Gu] Guin (D.). - Stabilité de l'homologie du groupe linéaire et K-théorie algébrique, C.R. Acad. Sci. Paris Sér. I, Math., t. 304, 9, 1987, p. 219-222. | MR | Zbl

[H] Hartshorne (R.). - Residues and Duality, in Lecture Notes in Math., 20, Springer, 1966. | MR | Zbl

[I1] Illusie (L.). - Complexe de de Rham-Witt et cohomologie cristalline, Ann. Sci. École Norm. Sup., t. 12, 1979, p. 501-661. | Numdam | MR | Zbl

[IR] Illusie (L.) and Raynaud (M.). - Les suites spectrales associées au complexe de de Rham-Witt, Inst. Hautes Études Sci. Publ. Math., t. 57, 1983, p. 73-212. | Numdam | MR | Zbl

[Iw] Iwasawa (K.). - On explicit formulas for the norm residue symbol, J. Math. Soc. Japan, t. 20, 1968, p. 151-165. | MR | Zbl

[K1] Kato (K.). - A generalization of local class field theory by using K-groups, II, J. Fac. Sci. Univ. Tokyo Sect. IA Math., t. 27, 1980, p. 602-683. | MR | Zbl

[K2] Kato (K.). - A generalization of local class field theory by using K-groups, III, J. Fac. Sci. Univ. Tokyo Sect. IA Math., t. 29, 1982, p. 31-43. | MR | Zbl

[K3] Kato (K.). - Galois cohomology of complete discrete valuation fields, in Lecture Notes in Math., 967, Springer, 1982. | MR | Zbl

[K4] Kato (K.). - Duality theories in p-primary etale cohomology, I, Algebraic and topological theories, Kinokuniya, 1985, p. 127-148. | MR | Zbl

[K5] Kato (K.). - On p-adic vanishing cycles, Preprint, Springer, 1985.

[R] Raynaud (M.). - Anneaux locaux henséliens, Lecture Notes in Math., 169, Springer, 1970. | MR | Zbl

[Sa] Safarevic (I.R.). - A general reciprocity law, Mat. Sb., t. 26, 68, 1950, p. 113-146. | MR

[Sh] Shiratani (K.). - Notes on the Kummer-Hilbert reciprocity law, J. Math. Soc. Japan, t. 12, 1960, p. 412-421. | MR | Zbl

[So] Soulé (C.). - Opérations en K-théorie Algébrique, Canadian J. Math., t. 37, 1985, p. 488-550. | MR | Zbl

[S1] Suslin (A.A.). - Mennicke symbols and their applications in the K-theory of fields, in Lecture Notes in Math., 966, Springer, 1982. | MR | Zbl

[S2] Suslin (A.A.). - Homology of GLn, characteristic classes and Milnor K-theory, in Lecture Notes in Math., 1046, Springer, 1984, p. 357-375. | MR | Zbl

[V1] Vostokov (S.V.). - Explicit form of the law of reciprocity, Izv. Akad. Nauk SSSR Ser. Mat., t. 13, 1979, p. 557-588. | Zbl

[V2] Vostokov (S.V.). - Hilbert symbols in a discrete valuated field, J. Sov. Math., t. 19, 1, 1982, p. 1006-1019. | Zbl

[V3] Vostokov (S.V.). - Explicit construction of class field theory for a multidimensional local field, Izv. Akad. Nauk SSSR Ser. Mat., t. 26, 1986, p. 263-287. | Zbl

[VK] Vostokov (S.V.) and Kirillov (A.N.). - Normed pairing in a two dimensional local field, J. Sov. Math., t. 30, 1, 1985, p. 1847-1853. | Zbl

[W] Wiles (A.). - Higher explicit reciprocity laws, Ann. of Math., t. 107, 1978, p. 235-254. | MR | Zbl

1) The contents of § 1 are treated in the paper : Berthelot (P.) and Messing (W.). - Théorie de Dieudonné cristalline III, The Grothendieck Festschrift, vol. III, Birkhäuser 1990, p. 173-247. | MR | Zbl

2) The modified complex S′n(r)A,B in § 3 is related to the crystalline cohomology theory with logarithmic poles developped in the following papers : Faltings (G.). - F-isocrystals on open varieties : Results and Conjectures, The Grothendieck Festschrift, vol. II, Birkhäuser 1990, p. 219-248. | MR | Zbl

Kato (K.). - Logarithmic structures of Fontaine-Illusie, Proc. of first JAMI conference, John Hopkins Univ. Press, 1990, p. 80-99.

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