The explicit reciprocity law and the cohomology of Fontaine-Messing

Kazuya Kato

Bulletin de la Société Mathématique de France (1991)

  • Volume: 119, Issue: 4, page 397-441
  • ISSN: 0037-9484

How to cite

top

Kato, Kazuya. "The explicit reciprocity law and the cohomology of Fontaine-Messing." Bulletin de la Société Mathématique de France 119.4 (1991): 397-441. <http://eudml.org/doc/87631>.

@article{Kato1991,
author = {Kato, Kazuya},
journal = {Bulletin de la Société Mathématique de France},
keywords = {regular local rings; symbol maps; Milnor -groups; Fontaine-Messing cohomology; reciprocity law},
language = {eng},
number = {4},
pages = {397-441},
publisher = {Société mathématique de France},
title = {The explicit reciprocity law and the cohomology of Fontaine-Messing},
url = {http://eudml.org/doc/87631},
volume = {119},
year = {1991},
}

TY - JOUR
AU - Kato, Kazuya
TI - The explicit reciprocity law and the cohomology of Fontaine-Messing
JO - Bulletin de la Société Mathématique de France
PY - 1991
PB - Société mathématique de France
VL - 119
IS - 4
SP - 397
EP - 441
LA - eng
KW - regular local rings; symbol maps; Milnor -groups; Fontaine-Messing cohomology; reciprocity law
UR - http://eudml.org/doc/87631
ER -

References

top
  1. [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. Zbl54.0191.05JFM54.0191.05
  2. [BO] BERTHELOT (P.) and OGUS (A.). — Notes on crystalline cohomology, Princeton University Press, 1978. Zbl0383.14010MR58 #10908
  3. [BK] BLOCH (S.) and KATO (K.). — p-adic etale cohomology, Inst. Hautes Études Sci. Publ. Math., t. 63, 1986, p. 107-152. Zbl0613.14017MR87k:14018
  4. [B1] BRÜCKNER (H.). — Hilbertsymbole zum Exponenten pn und Pfaffsche Formen. — Preprint, Hamburg, 1979. 
  5. [B2] BRÜCKNER (H.). — Explizites reziprozitätsgesetz und anwendungen. — Vorlesungen aus dem Fachbereich Mathematik der Universität Essen, 1979. Zbl0415.12004
  6. [C] COLEMAN (R.F.). — The dilogarithm and the norm residue symbol, Bull. Soc. Math. France, t. 109, 1981, p. 373-402. Zbl0493.12019MR84j:12017
  7. [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. Zbl0574.14004
  8. [FM] FONTAINE (J.-M.) and MESSING (W.). — Contemporary Math., t. 67, 1987, p. 179-207. Zbl0632.14016MR89g:14009
  9. [Gr] GROTHENDIECK (A.). — Éléments de Géométrie algébrique, IV, première partie, Inst. Hautes Études Sci. Publ. Math., t. 20, 1964. Zbl0136.15901
  10. [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. Zbl0609.18005MR88f:18015
  11. [H] HARTSHORNE (R.). — Residues and Duality, in Lecture Notes in Math., 20, Springer, 1966. Zbl0212.26101MR36 #5145
  12. [I1] ILLUSIE (L.). — Complexe de de Rham-Witt et cohomologie cristalline, Ann. Sci. École Norm. Sup., t. 12, 1979, p. 501-661. Zbl0436.14007MR82d:14013
  13. [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. Zbl0538.14012MR85e:14026
  14. [Iw] IWASAWA (K.). — On explicit formulas for the norm residue symbol, J. Math. Soc. Japan, t. 20, 1968, p. 151-165. Zbl0256.12013MR37 #5183
  15. [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. Zbl0463.12006MR83g:12020a
  16. [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. Zbl0503.12004MR83g:12020b
  17. [K3] KATO (K.). — Galois cohomology of complete discrete valuation fields, in Lecture Notes in Math., 967, Springer, 1982. Zbl0506.12022MR84k:12006
  18. [K4] KATO (K.). — Duality theories in p-primary etale cohomology, I, Algebraic and topological theories, Kinokuniya, 1985, p. 127-148. Zbl0800.14009MR1102256
  19. [K5] KATO (K.). — On p-adic vanishing cycles, Preprint, Springer, 1985. 
  20. [R] RAYNAUD (M.). — Anneaux locaux henséliens, Lecture Notes in Math., 169, Springer, 1970. Zbl0203.05102MR43 #3252
  21. [Sa] SAFAREVIC (I.R.). — A general reciprocity law, Mat. Sb., t. 26, 68, 1950, p. 113-146. Zbl0036.15901MR11,230f
  22. [Sh] SHIRATANI (K.). — Notes on the Kummer-Hilbert reciprocity law, J. Math. Soc. Japan, t. 12, 1960, p. 412-421. Zbl0111.04502MR24 #A2575
  23. [So] SOULÉ (C.). — Opérations en K-théorie Algébrique, Canadian J. Math., t. 37, 1985, p. 488-550. Zbl0575.14015MR87b:18013
  24. [S1] SUSLIN (A.A.). — Mennicke symbols and their applications in the K-theory of fields, in Lecture Notes in Math., 966, Springer, 1982. Zbl0502.18004MR84f:18023
  25. [S2] SUSLIN (A.A.). — Homology of GLn, characteristic classes and Milnor K-theory, in Lecture Notes in Math., 1046, Springer, 1984, p. 357-375. Zbl0528.18007MR86f:11090a
  26. [V1] VOSTOKOV (S.V.). — Explicit form of the law of reciprocity, Izv. Akad. Nauk SSSR Ser. Mat., t. 13, 1979, p. 557-588. Zbl0467.12018
  27. [V2] VOSTOKOV (S.V.). — Hilbert symbols in a discrete valuated field, J. Sov. Math., t. 19, 1, 1982, p. 1006-1019. Zbl0486.12007
  28. [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. Zbl0608.12017
  29. [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. Zbl0566.12006
  30. [W] WILES (A.). — Higher explicit reciprocity laws, Ann. of Math., t. 107, 1978, p. 235-254. Zbl0378.12006MR58 #605
  31. 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. Zbl0753.14041MR92h:14012
  32. 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. Zbl0736.14004MR92f:14015
  33. KATO (K.). — Logarithmic structures of Fontaine-Illusie, Proc. of first JAMI conference, John Hopkins Univ. Press, 1990, p. 80-99. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.