p-adic étale Tate twists and arithmetic duality

Kanetomo Sato

Annales scientifiques de l'École Normale Supérieure (2007)

  • Volume: 40, Issue: 4, page 519-588
  • ISSN: 0012-9593

How to cite

top

Sato, Kanetomo. "p-adic étale Tate twists and arithmetic duality." Annales scientifiques de l'École Normale Supérieure 40.4 (2007): 519-588. <http://eudml.org/doc/82720>.

@article{Sato2007,
author = {Sato, Kanetomo},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {étale Tate twist; logarithmic Hodge-Witt sheaf; purity; duality},
language = {eng},
number = {4},
pages = {519-588},
publisher = {Elsevier},
title = {p-adic étale Tate twists and arithmetic duality},
url = {http://eudml.org/doc/82720},
volume = {40},
year = {2007},
}

TY - JOUR
AU - Sato, Kanetomo
TI - p-adic étale Tate twists and arithmetic duality
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2007
PB - Elsevier
VL - 40
IS - 4
SP - 519
EP - 588
LA - eng
KW - étale Tate twist; logarithmic Hodge-Witt sheaf; purity; duality
UR - http://eudml.org/doc/82720
ER -

References

top
  1. [1] Altman A., Kleiman S., Introduction to Grothendieck Duality Theory, Lecture Notes in Math., vol. 146, Springer, Berlin, 1970. Zbl0215.37201MR274461
  2. [2] Aritin M., Verdier J.-L., Seminar on étale cohomology of number fields, Woods Hole, 1964 characteristics. 
  3. [3] Bass H., Tate J., The Milnor ring of a global field, in: Bass H. (Ed.), Algebraic K-Theory II, Lecture Notes in Math., vol. 342, Springer, Berlin, 1972, pp. 349-446. Zbl0299.12013MR442061
  4. [4] Beilinson A.A., Height pairings between algebraic cycles, in: Manin Yu.I. (Ed.), K-Theory, Arithmetic and Geometry, Lecture Notes in Math., vol. 1289, Springer, Berlin, 1987, pp. 1-27. Zbl0651.14002MR902590
  5. [5] Beilinson A.A., Bernstein J., Deligne P., Faisceaux pervers, Astérisque, vol. 100, Soc. Math. France, 1982. Zbl0536.14011MR751966
  6. [6] Bloch S., Algebraic K-theory and crystalline cohomology, Inst. Hautes Études Sci. Publ. Math.47 (1977) 187-268. Zbl0388.14010MR488288
  7. [7] Bloch S., Algebraic cycles and higher K-theory, Adv. Math.61 (1986) 267-304. Zbl0608.14004MR852815
  8. [8] Bloch S., Esnault H., The coniveau filtration and non-divisibility for algebraic cycles, Math. Ann.304 (1996) 303-314. Zbl0868.14004MR1371769
  9. [9] Bloch S., Kato K., p-adic étale cohomology, Inst. Hautes Études Sci. Publ. Math.63 (1986) 107-152. Zbl0613.14017MR849653
  10. [10] Bloch S., Ogus A., Gersten's conjecture and the homology of schemes, Ann. Sci. École Norm. Sup. (4)7 (1974) 181-202. Zbl0307.14008MR412191
  11. [11] Cassels J.W.S., Arithmetic on curves of genus 1 (IV). Proof of the Hauptvermutung, J. reine angew. Math.211 (1962) 95-112. Zbl0106.03706MR163915
  12. [12] Deninger C., Duality in the étale cohomology of one-dimensional schemes and generalizations, Math. Ann.277 (1987) 529-541. Zbl0607.14011MR891590
  13. [13] Fesenko I.B., Vostokov S.V., Local Fields and Their Extensions, with a Foreword by Shafarevich, I.R., Transl. Math. Monogr., vol. 121, second ed., Amer. Math. Soc., Providence, 2002. Zbl0781.11042MR1915966
  14. [14] Fontaine J.-M., Messing W., p-adic periods and p-adic étale cohomology, in: Ribet K.A. (Ed.), Current Trends in Arithmetical Algebraic Geometry, Contemp. Math., vol. 67, Amer. Math. Soc., Providence, 1987, pp. 179-207. Zbl0632.14016MR902593
  15. [15] Fujiwara K., A proof of the absolute purity conjecture (after Gabber), in: Usui S., Green M., Illusie L., Kato K., Looijenga E., Mukai S., Saito S. (Eds.), Algebraic Geometry 2000, Azumino, Adv. Stud. Pure Math., vol. 36, Math. Soc. Japan, Tokyo, 2002, pp. 153-183. Zbl1059.14026MR1971516
  16. [16] Gabber O., Some theorems on Azumaya algebras, in: Kervaire M., Ojanguren M. (Eds.), Groupe de Brauer Séminaire, Les Plans-sur-Bex, 1980, Lecture Notes in Math., vol. 844, Springer, Berlin, 1981, pp. 129-209. Zbl0472.14013MR611868
  17. [17] Geisser T., Motivic cohomology over Dedekind rings, Math. Z.248 (2004) 773-794. Zbl1062.14025MR2103541
  18. [18] Gros M., Classes de Chern et classes des cycles en cohomologie logarithmique, Mém. Soc. Math. Fr. (N.S.)21 (1985). Zbl0615.14011
  19. [19] Gros M., Suwa N., La conjecture de Gersten pour les faisceaux de Hodge–Witt logarithmique, Duke Math. J.57 (1988) 615-628. Zbl0715.14011
  20. [20] Grothendieck A., Le groupe de Brauer III, in: Dix exposés sur la cohomologie des schémas, North-Holland, Amsterdam, 1968, pp. 88-188. Zbl0198.25901MR244271
  21. [21] Hartshorne R., Residues and Duality, Lecture Notes in Math., vol. 20, Springer, Berlin, 1966. Zbl0212.26101MR222093
  22. [22] Hartshorne R., Local Cohomology, (a seminar given by Grothendieck, A., Harvard University, Fall, 1961), Lecture Notes in Math., vol. 41, Springer, Berlin, 1967. Zbl0185.49202MR224620
  23. [23] Hartshorne R., Algebraic Geometry, Grad. Texts in Math., vol. 52, Springer, New York, 1977. Zbl0531.14001MR463157
  24. [24] Hasse H., Die Gruppe der p n -primären Zahlen für einen Primteiler p vonp, J. reine angew. Math.176 (1936) 174-183. Zbl0016.05204JFM62.1115.01
  25. [25] Hyodo O., A note on p-adic étale cohomology in the semi-stable reduction case, Invent. Math.91 (1988) 543-557. Zbl0619.14013MR928497
  26. [26] Hyodo O., On the de Rham–Witt complex attached to a semi-stable family, Compositio Math.78 (1991) 241-260. Zbl0742.14015
  27. [27] Illusie L., Complexe de de Rham–Witt et cohomologie cristalline, Ann. Sci. École Norm. Sup. (4)12 (1979) 501-661. 
  28. [28] Illusie L., Réduction semi-stable ordinaire, cohomologie étale p-adique et cohomologie de de Rham d'après Bloch–Kato et Hyodo; Appendice à l'exposé IV, in: Périodes p-adiques, Séminaire de Bures, 1988, Astérisque, vol. 223, Soc. Math. France, Marseille, 1994, pp. 209-220. Zbl1043.11532
  29. [29] Jannsen U., Saito S., Sato K., Étale duality for constructible sheaves on arithmetic schemes, http://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/Jannsen/. Zbl1299.14026
  30. [30] Kato F., Log smooth deformation theory, Tôhoku Math. J.48 (1996) 317-354. Zbl0876.14007MR1404507
  31. [31] Kato F., A generalization of local class field theory using K-groups, II, J. Fac. Sci. Univ. of Tokyo, Sec. IA27 (1980) 602-683. Zbl0463.12006
  32. [32] Kato F., On p-adic vanishing cycles (application of ideas of Fontaine–Messing), in: Algebraic Geometry, Sendai, 1985, Adv. Stud. in Pure Math., vol. 10, Kinokuniya, Tokyo, 1987, pp. 207-251. Zbl0645.14009
  33. [33] Kato K., Logarithmic structures of Fontaine–Illusie, in: Igusa J. (Ed.), Algebraic Analysis, Geometry and Number Theory, The Johns Hopkins Univ. Press, Baltimore, 1988, pp. 191-224. Zbl0776.14004
  34. [34] Kato K., The explicit reciprocity law and the cohomology of Fontaine–Messing, Bull. Soc. Math. France119 (1991) 397-441. Zbl0752.14015
  35. [35] Kato K., Semi-stable reduction and p-adic étale cohomology, in: Périodes p-adiques, Séminaire de Bures, 1988, Astérisque, vol. 223, Soc. Math. France, Marseille, 1994, pp. 269-293. Zbl0847.14009MR1293975
  36. [36] Kato K., A Hasse principle for two-dimensional global fields (with an appendix by Colliot-Thélène, J.-L.), J. reine angew. Math.366 (1986) 142-183. Zbl0576.12012MR833016
  37. [37] Katz N.M., Nilpotent connections and the monodromy theorem: applications of a result of Turrittin, Inst. Hautes Études Sci. Publ. Math.39 (1970) 175-232. Zbl0221.14007MR291177
  38. [38] Kurihara M., A note on p-adic étale cohomology, Proc. Japan Acad. Ser. A63 (1987) 275-278. Zbl0647.14006MR931263
  39. [39] Langer A., Saito S., Torsion zero cycles on the self product of a modular elliptic curve, Duke Math. J.85 (1996) 315-357. Zbl0880.14001MR1417619
  40. [40] Levine M., Techniques of localization in the theory of algebraic cycles, J. Algebraic Geom.10 (2001) 299-363. Zbl1077.14509MR1811558
  41. [41] Levine M., K-theory and motivic cohomology of schemes, Preprint, 1999. MR1744925
  42. [42] Lichtenbaum S., Duality theorems for curves over p-adic fields, Invent. Math.7 (1969) 120-136. Zbl0186.26402MR242831
  43. [43] Lichtenbaum S., Values of zeta functions at non-negative integers, in: Jager H. (Ed.), Number Theory, Noordwijkerhout, 1983, Lecture Notes in Math., vol. 1068, Springer, Berlin, 1984, pp. 127-138. Zbl0591.14014MR756089
  44. [44] Lichtenbaum S., New results on weight-two motivic cohomology, in: Cartier P., Illusie L., Katz N.M., Laumon G., Manin Y., Ribet K.A. (Eds.), The Grothendieck Festschrift III, Progr. Math., vol. 88, Birkhäuser, Boston, 1990, pp. 35-55. Zbl0809.14004MR1106910
  45. [45] Mazur B., Notes on étale cohomology of number fields, Ann. Sci. École Norm. Sup. (4)6 (1973) 521-552. Zbl0282.14004MR344254
  46. [46] Milne J.S., Duality in flat cohomology of a surface, Ann. Sci. École Norm. Sup. (4)9 (1976) 171-202. Zbl0334.14010MR460331
  47. [47] Milne J.S., Values of zeta functions of varieties over finite fields, Amer. J. Math.108 (1986) 297-360. Zbl0611.14020MR833360
  48. [48] Milne J.S., Arithmetic Duality Theorems, Perspectives in Math., vol. 1, Academic Press, Boston, 1986. Zbl0613.14019MR881804
  49. [49] Moser T., A duality theorem for étale p-torsion sheaves on complete varieties over finite fields, Compositio Math.117 (1999) 123-152. Zbl0954.14012MR1695861
  50. [50] Niziol W., Duality in the cohomology of crystalline local systems, Compositio Math.109 (1997) 67-97. Zbl0917.14010MR1473606
  51. [51] Poitou G., Cohomologie galoisienne des modules finis, Dunod, Paris, 1967. Zbl0161.04203MR219591
  52. [52] Raskind W., Abelian class field theory of arithmetic schemes, in: Jacob B., Rosenberg A. (Eds.), Algebraic K-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras (Part 1), Santa Barbara, 1992, Proc. Sympos. Pure Math., vol. 58, Amer. Math. Soc., Providence, 1995, pp. 85-187. Zbl0832.19004MR1327282
  53. [53] Saito S., Arithmetic on two-dimensional local rings, Invent. Math.85 (1986) 379-414. Zbl0609.13003MR846934
  54. [54] Saito S., Arithmetic theory of arithmetic surfaces, Ann. of Math.129 (1989) 547-589. Zbl0688.14019MR997313
  55. [55] Sato K., Logarithmic Hodge–Witt sheaves on normal crossing varieties, Math. Z., in press. Zbl1138.14013
  56. [56] Schneider P., p-adic point of motives, in: Jannsen U. (Ed.), Motives (Part 2), Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, 1994, pp. 225-249. Zbl0814.14023MR1265555
  57. [57] Serre J.-P., Cohomologie galoisienne, Lecture Notes in Math., vol. 5, 5, Springer, Berlin, 1992. Zbl0812.12002
  58. [58] Spiess M., Artin–Verdier duality for arithmetic surfaces, Math. Ann.305 (1996) 705-792. Zbl0887.14008
  59. [59] Tate J., Duality theorems in the Galois cohomology of number fields, in: Proc. Internat. Congress Math., Stockholm, 1962, pp. 234–241. Zbl0126.07002
  60. [60] Tate J., Algebraic cycles and poles of zeta functions, in: Schilling O.F.G. (Ed.), Arithmetical Algebraic Geometry, Harper and Row, New York, 1965, pp. 93-100. Zbl0213.22804MR225778
  61. [61] Tate J., On the conjecture of Birch and Swinnerton–Dyer and a geometric analog, in: Séminaire Bourbaki 1965/66, Exposé 306, Benjamin, New York, 1966, and Collection Hors Série, Société mathématique de France, vol. 9, 1995. Zbl0199.55604
  62. [62] Thomason R.W., Absolute cohomological purity, Bull. Soc. Math. France112 (1984) 397-406. Zbl0584.14007MR794741
  63. [63] Tsuji T., p-adic étale cohomology and crystalline cohomology in the semi-stable reduction case, Invent. Math.137 (1999) 233-411. Zbl0945.14008MR1705837
  64. [64] Tsuji T., On p-adic nearby cycles of log smooth families, Bull. Soc. Math. France128 (2000) 529-575. Zbl0972.14012MR1815397
  65. [65] Urabe T., The bilinear form of the Brauer group of a surface, Invent. Math.125 (1996) 557-585. Zbl0871.13001MR1400317
  66. [66] Grothendieck A., Artin M., Verdier J.-L., Deligne P., Saint-Donat B., Théorie des topos et cohomologie étale des schémas, in: Lecture Notes in Math., vols. 269, 270, 305, Springer, Berlin, 1972, pp. 1972-1973. MR354654
  67. [67] Deligne P., Boutot J.-F., Grothendieck A., Illusie L., Verdier J.-L., Cohomologie étale, Lecture Notes in Math., vol. 569, Springer, Berlin, 1977. Zbl0345.00010MR463174

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.