p-adic étale Tate twists and arithmetic duality
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 40 (2007) no. 4, pp. 519-588.
@article{ASENS_2007_4_40_4_519_0,
     author = {Sato, Kanetomo},
     title = {$p$-adic \'etale {Tate} twists and arithmetic duality},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {519--588},
     publisher = {Elsevier},
     volume = {Ser. 4, 40},
     number = {4},
     year = {2007},
     doi = {10.1016/j.ansens.2007.04.002},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.ansens.2007.04.002/}
}
TY  - JOUR
AU  - Sato, Kanetomo
TI  - $p$-adic étale Tate twists and arithmetic duality
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2007
DA  - 2007///
SP  - 519
EP  - 588
VL  - Ser. 4, 40
IS  - 4
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.ansens.2007.04.002/
UR  - https://doi.org/10.1016/j.ansens.2007.04.002
DO  - 10.1016/j.ansens.2007.04.002
LA  - en
ID  - ASENS_2007_4_40_4_519_0
ER  - 
%0 Journal Article
%A Sato, Kanetomo
%T $p$-adic étale Tate twists and arithmetic duality
%J Annales scientifiques de l'École Normale Supérieure
%D 2007
%P 519-588
%V Ser. 4, 40
%N 4
%I Elsevier
%U https://doi.org/10.1016/j.ansens.2007.04.002
%R 10.1016/j.ansens.2007.04.002
%G en
%F ASENS_2007_4_40_4_519_0
Sato, Kanetomo. $p$-adic étale Tate twists and arithmetic duality. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 40 (2007) no. 4, pp. 519-588. doi : 10.1016/j.ansens.2007.04.002. http://www.numdam.org/articles/10.1016/j.ansens.2007.04.002/

[1] Altman A., Kleiman S., Introduction to Grothendieck Duality Theory, Lecture Notes in Math., vol. 146, Springer, Berlin, 1970. | MR | Zbl

[2] Aritin M., Verdier J.-L., Seminar on étale cohomology of number fields, Woods Hole, 1964 characteristics.

[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. | MR | Zbl

[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. | MR | Zbl

[5] Beilinson A.A., Bernstein J., Deligne P., Faisceaux pervers, Astérisque, vol. 100, Soc. Math. France, 1982. | MR | Zbl

[6] Bloch S., Algebraic K-theory and crystalline cohomology, Inst. Hautes Études Sci. Publ. Math. 47 (1977) 187-268. | Numdam | MR | Zbl

[7] Bloch S., Algebraic cycles and higher K-theory, Adv. Math. 61 (1986) 267-304. | MR | Zbl

[8] Bloch S., Esnault H., The coniveau filtration and non-divisibility for algebraic cycles, Math. Ann. 304 (1996) 303-314. | MR | Zbl

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

[10] Bloch S., Ogus A., Gersten's conjecture and the homology of schemes, Ann. Sci. École Norm. Sup. (4) 7 (1974) 181-202. | Numdam | MR | Zbl

[11] Cassels J.W.S., Arithmetic on curves of genus 1 (IV). Proof of the Hauptvermutung, J. reine angew. Math. 211 (1962) 95-112. | MR | Zbl

[12] Deninger C., Duality in the étale cohomology of one-dimensional schemes and generalizations, Math. Ann. 277 (1987) 529-541. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[17] Geisser T., Motivic cohomology over Dedekind rings, Math. Z. 248 (2004) 773-794. | MR | Zbl

[18] Gros M., Classes de Chern et classes des cycles en cohomologie logarithmique, Mém. Soc. Math. Fr. (N.S.) 21 (1985). | Numdam | Zbl

[19] Gros M., Suwa N., La conjecture de Gersten pour les faisceaux de Hodge-Witt logarithmique, Duke Math. J. 57 (1988) 615-628. | Zbl

[20] Grothendieck A., Le groupe de Brauer III, in: Dix exposés sur la cohomologie des schémas, North-Holland, Amsterdam, 1968, pp. 88-188. | MR | Zbl

[21] Hartshorne R., Residues and Duality, Lecture Notes in Math., vol. 20, Springer, Berlin, 1966. | MR | Zbl

[22] Hartshorne R., Local Cohomology, (a seminar given by Grothendieck, A., Harvard University, Fall, 1961), Lecture Notes in Math., vol. 41, Springer, Berlin, 1967. | MR | Zbl

[23] Hartshorne R., Algebraic Geometry, Grad. Texts in Math., vol. 52, Springer, New York, 1977. | MR | Zbl

[24] Hasse H., Die Gruppe der p n -primären Zahlen für einen Primteiler p von p, J. reine angew. Math. 176 (1936) 174-183. | JFM

[25] Hyodo O., A note on p-adic étale cohomology in the semi-stable reduction case, Invent. Math. 91 (1988) 543-557. | MR | Zbl

[26] Hyodo O., On the de Rham-Witt complex attached to a semi-stable family, Compositio Math. 78 (1991) 241-260. | Numdam | Zbl

[27] Illusie L., Complexe de de Rham-Witt et cohomologie cristalline, Ann. Sci. École Norm. Sup. (4) 12 (1979) 501-661. | Numdam

[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. | Zbl

[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/.

[30] Kato F., Log smooth deformation theory, Tôhoku Math. J. 48 (1996) 317-354. | MR | Zbl

[31] Kato F., A generalization of local class field theory using K-groups, II, J. Fac. Sci. Univ. of Tokyo, Sec. IA 27 (1980) 602-683. | Zbl

[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. | Zbl

[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. | Zbl

[34] Kato K., The explicit reciprocity law and the cohomology of Fontaine-Messing, Bull. Soc. Math. France 119 (1991) 397-441. | Numdam | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | Numdam | MR | Zbl

[38] Kurihara M., A note on p-adic étale cohomology, Proc. Japan Acad. Ser. A 63 (1987) 275-278. | MR | Zbl

[39] Langer A., Saito S., Torsion zero cycles on the self product of a modular elliptic curve, Duke Math. J. 85 (1996) 315-357. | MR | Zbl

[40] Levine M., Techniques of localization in the theory of algebraic cycles, J. Algebraic Geom. 10 (2001) 299-363. | MR | Zbl

[41] Levine M., K-theory and motivic cohomology of schemes, Preprint, 1999. | MR

[42] Lichtenbaum S., Duality theorems for curves over p-adic fields, Invent. Math. 7 (1969) 120-136. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[45] Mazur B., Notes on étale cohomology of number fields, Ann. Sci. École Norm. Sup. (4) 6 (1973) 521-552. | Numdam | MR | Zbl

[46] Milne J.S., Duality in flat cohomology of a surface, Ann. Sci. École Norm. Sup. (4) 9 (1976) 171-202. | Numdam | MR | Zbl

[47] Milne J.S., Values of zeta functions of varieties over finite fields, Amer. J. Math. 108 (1986) 297-360. | MR | Zbl

[48] Milne J.S., Arithmetic Duality Theorems, Perspectives in Math., vol. 1, Academic Press, Boston, 1986. | MR | Zbl

[49] Moser T., A duality theorem for étale p-torsion sheaves on complete varieties over finite fields, Compositio Math. 117 (1999) 123-152. | MR | Zbl

[50] Niziol W., Duality in the cohomology of crystalline local systems, Compositio Math. 109 (1997) 67-97. | MR | Zbl

[51] Poitou G., Cohomologie galoisienne des modules finis, Dunod, Paris, 1967. | MR | Zbl

[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. | MR | Zbl

[53] Saito S., Arithmetic on two-dimensional local rings, Invent. Math. 85 (1986) 379-414. | MR | Zbl

[54] Saito S., Arithmetic theory of arithmetic surfaces, Ann. of Math. 129 (1989) 547-589. | MR | Zbl

[55] Sato K., Logarithmic Hodge-Witt sheaves on normal crossing varieties, Math. Z., in press.

[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. | MR | Zbl

[57] Serre J.-P., Cohomologie galoisienne, Lecture Notes in Math., vol. 5, 5, Springer, Berlin, 1992. | Zbl

[58] Spiess M., Artin-Verdier duality for arithmetic surfaces, Math. Ann. 305 (1996) 705-792.

[59] Tate J., Duality theorems in the Galois cohomology of number fields, in: Proc. Internat. Congress Math., Stockholm, 1962, pp. 234-241. | Zbl

[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. | MR | Zbl

[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. | Numdam | Zbl

[62] Thomason R.W., Absolute cohomological purity, Bull. Soc. Math. France 112 (1984) 397-406. | Numdam | MR | Zbl

[63] Tsuji T., p-adic étale cohomology and crystalline cohomology in the semi-stable reduction case, Invent. Math. 137 (1999) 233-411. | MR | Zbl

[64] Tsuji T., On p-adic nearby cycles of log smooth families, Bull. Soc. Math. France 128 (2000) 529-575. | Numdam | MR | Zbl

[65] Urabe T., The bilinear form of the Brauer group of a surface, Invent. Math. 125 (1996) 557-585. | MR | Zbl

[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. | MR

[67] Deligne P., Boutot J.-F., Grothendieck A., Illusie L., Verdier J.-L., Cohomologie étale, Lecture Notes in Math., vol. 569, Springer, Berlin, 1977. | MR | Zbl

Cited by Sources: