Cohomological Hasse principle and motivic cohomology for arithmetic schemes
Publications Mathématiques de l'IHÉS, Tome 115 (2012), pp. 123-183.

In 1985 Kazuya Kato formulated a fascinating framework of conjectures which generalizes the Hasse principle for the Brauer group of a global field to the so-called cohomological Hasse principle for an arithmetic scheme X. In this paper we prove the prime-to-characteristic part of the cohomological Hasse principle. We also explain its implications on finiteness of motivic cohomology and special values of zeta functions.

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     title = {Cohomological {Hasse} principle and motivic cohomology for arithmetic schemes},
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Kerz, Moritz; Saito, Shuji. Cohomological Hasse principle and motivic cohomology for arithmetic schemes. Publications Mathématiques de l'IHÉS, Tome 115 (2012), pp. 123-183. doi : 10.1007/s10240-011-0038-y. http://archive.numdam.org/articles/10.1007/s10240-011-0038-y/

[SGA4] Artin, M.; Grothendieck, A.; Verdier, J. L. Theorie des Topos et Cohomologie étale des Schemas, vol. 3, Lecture Notes in Math., Springer, Berlin, 1973 | Zbl 0245.00002

[B] Bloch, S. Algebraic cycles and higher algebraic K-theory, Adv. Math., Volume 61 (1986), pp. 267-304 | Article | MR 852815 | Zbl 0608.14004

[BK] Bloch, S.; Kato, K. p-adic etale cohomology, Publ. Math. IHES, Volume 63 (1986), pp. 107-152 | MR 849653 | Zbl 0613.14017

[BO] Bloch, S.; Ogus, A. Gersten’s conjecture and the homology of schemes, Ann. Sci. Éc. Norm. Super. 4 ser., Volume 7 (1974), pp. 181-202 | MR 412191 | Zbl 0307.14008

[CT] Colliot-Thélène, J.-L. On the reciprocity sequence in the higher class field theory of function fields, Algebraic K-Theory and Algebraic Topology (1993), pp. 35-55 | MR 1367291 | Zbl 0885.19002

[CTSS] Colliot-Thélène, J.-L.; Sansuc, J.-J.; Soulé, C. Torsion dans le groupe de Chow de codimension deux, Duke Math. J., Volume 50 (1983), pp. 763-801 | Article | MR 714830 | Zbl 0574.14004

[CTHK] Colliot-Thélène, J.-L.; Hoobler, R.; Kahn, B. The Bloch-Ogus-Gabber theorem, Algebraic K-theory (Fields Inst. Commun.) (1997), pp. 31-94 | MR 1466971 | Zbl 0911.14004

[CJS] V. Cossart, U. Jannsen, and S. Saito, Resolution of singularities for embedded surfaces, in preparation (see www.mathematik.uni-regensburg.de/Jannsen).

[Deg] Déglise, F. Transferts sur les groupes de Chow á coefficients, Math. Z., Volume 252 (2006), pp. 315-343 | Article | MR 2207800 | Zbl 1095.14009

[D] Deligne, P. La conjecture de Weil II, Publ. Math. IHES, Volume 52 (1981), pp. 313-428 | Numdam | MR 601520 | Zbl 0456.14014

[FG] Fujiwara, K. A proof of the absolute purity conjecture (after Gabber), Algebraic Geometry, Azumino 2001 (Adv. Stud. in Pure Math.) (2002), pp. 153-184 | MR 1971516 | Zbl 1059.14026

[F] Fulton, W. Intersection Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer, Berlin, 1998 | Article | Zbl 0885.14002

[Ga] Gabber, O. On space filling curves and Albanese varieties, Geom. Funct. Anal., Volume 11 (2001), pp. 1192-1200 | Article | MR 1878318 | Zbl 1072.14513

[Ge1] Geisser, T. Motivic cohomology over Dedekind rings, Math. Z., Volume 248 (2004), pp. 773-794 | Article | MR 2103541 | Zbl 1062.14025

[Ge2] Geisser, T. Weil-étale cohomology over finite fields, Math. Ann., Volume 330 (2004), pp. 665-692 | Article | MR 2102307 | Zbl 1069.14021

[Ge3] Geisser, T. Arithmetic cohomology over finite fields and special values of ζ-functions, Duke Math. J., Volume 133 (2006), pp. 27-57 | Article | MR 2219269 | Zbl 1104.14011

[Ge4] Geisser, T. Arithmetic homology and an integral version of Kato’s conjecture, J. Reine Angew. Math., Volume 644 (2010), pp. 1-22 | Article | MR 2671773 | Zbl 1211.14014

[GL] Geisser, T.; Levine, M. The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky, J. Reine Angew., Volume 530 (2001), pp. 55-103 | Article | MR 1807268 | Zbl 1023.14003

[GS] Gillet, H.; Soulé, C. Descent, motives and K-theory, J. Reine Angew. Math., Volume 478 (1996), pp. 127-176 | MR 1409056 | Zbl 0863.19002

[Gr] Gros, M. Sur la partie p-primaire du groupe de Chow de codimension deux, Commun. Algebra, Volume 13 (1985), pp. 2407-2420 | Article | MR 807481 | Zbl 0591.14003

[GrSw] Gros, M.; Suwa, N. Application d’Abel-Jacobi p-adique et cycles algébriques, Duke Math. J., Volume 57 (1988), pp. 579-613 | Article | MR 962521 | Zbl 0697.14005

[EGAII] A. Grothendieck and J. Dieudonné, Éléments de Géométrie Algébrique, II Étude globale élémentaire de quelques classes de morphismes, Publ. Math. IHES, 8 (1961). | Numdam | MR 217084 | Zbl 0118.36206

[HW] Haesemeyer, C.; Weibel, C. Norm Varieties and the Chain Lemma (after Markus Rost), Abel Symposium (2009), pp. 95-130 | MR 2597737 | Zbl 1244.19003

[H] Hironaka, H. Resolution of singularities of an algebraic variety over a field of characteristic zero: I-II, Ann. Math., Volume 79 (1964), pp. 109-326 | Article | MR 199184 | Zbl 0122.38603

[Il1] Illusie, L. Complexe de De Rham-Witt et cohomologie cristalline, Ann. Sci. Éc. Norm. Super., Volume 12 (1979), pp. 501-661 | MR 565469 | Zbl 0436.14007

[Il2] L. Illusie, On Gabber’s refined uniformization, preprint available at http://www.math.u-psud.fr/~illusie/.

[Il3] Illusie, L. Perversité et variation, Manuscr. Math., Volume 112 (2003), pp. 271-295 | Article | MR 2067039 | Zbl 1036.14008

[J1] Jannsen, U. Mixed Motives and Algebraic K-theory, Lecture Notes in Mathematics, Springer, Berlin, 1990 | MR 1043451 | Zbl 0691.14001

[J2] U. Jannsen, Hasse principles for higher dimensional fields, arXiv:0910.2803.

[J3] U. Jannsen, Rigidity Results on K-cohomology and Other Functors, in preparation (see www.mathematik.uni-regensburg.de/Jannsen).

[JS1] Jannsen, U.; Saito, S. Kato homology of arithmetic schemes and higher class field theory, Documenta Math. Extra Volume: Kazuya Kato’s Fiftieth Birthday (2003), pp. 479-538 | MR 2046606 | Zbl 1092.14504

[JS2] U. Jannsen and S. Saito, Kato homology and motivic cohomology over finite fields, arXiv:0910.2815.

[JS3] U. Jannsen and S. Saito, Bertini theorems and Lefschetz pencils over discrete valuation rings, with applications to higher class field theory, J. Algebr. Geom., to appear. | MR 2957692 | Zbl 1267.14010

[JSS] U. Jannsen, S. Saito, and K. Sato, Etale duality for constructible sheaves on arithmetic schemes, arXiv:0910.3759. | Zbl 1299.14026

[K] Kato, K. A Hasse principle for two dimensional global fields, J. Reine Angew. Math., Volume 366 (1986), pp. 142-183 | MR 833016 | Zbl 0576.12012

[KS] Kato, K.; Saito, S. Unramified class field theory of arithmetic surfaces, Ann. Math., Volume 118 (1985), pp. 241-275 | Article | MR 717824 | Zbl 0562.14011

[KeS] M. Kerz and S. Saito, Cohomological Hasse principle and McKay principle for weight homology, preprint available at http://arxiv.org/abs/1111.7177.

[Le] M. Levine, K-theory and motivic cohomology of schemes, preprint.

[Li1] Lichtenbaum, S. Values of zeta functions at non-negative integers, Number Theory, Noordwijkerhout (Lecture Notes in Math.) (1983), pp. 127-138 | Article | MR 756089 | Zbl 0591.14014

[Li2] Lichtenbaum, S. Weil-étale topology on schemes over finite fields, Compos. Math., Volume 141 (2005), pp. 689-702 (127–138) | Article | MR 2135283 | Zbl 1073.14024

[M1] Milne, J. S. Duality in the flat cohomology of a surface, Ann. Sci. Éc. Norm. Super. 4 ser., Volume 9 (1976), pp. 171-201 | Numdam | MR 460331 | Zbl 0334.14010

[M2] Milne, J. S. Values of zeta-functions of varieties over finite fields, Am. J. Math., Volume 108 (1986), pp. 297-360 | Article | MR 833360 | Zbl 0611.14020

[M3] Milne, J. S. Motivic cohomology and values of zeta-functions, Compos. Math., Volume 68 (1988), pp. 59-102 | MR 962505 | Zbl 0681.14007

[Mi] Milnor, J. Algebraic K-theory and quadratic forms, Invent. Math., Volume 9 (1970), pp. 318-344 | Article | MR 260844 | Zbl 0199.55501

[NS] Nesterenko, Yu.; Suslin, A. Homology of the general linear group over a local ring, and Milnor’s K-theory, Math. USSR, Izv., Volume 34 (1990), pp. 121-145 | Article | MR 992981 | Zbl 0684.18001

[Pa] Paranjape, K. H. Some spectral sequences for filtered complexes, J. Algebra, Volume 186 (1996), pp. 793-806 | Article | MR 1424593 | Zbl 0876.18007

[P] Poonen, B. Bertini theorems over finite fields, Ann. Math., Volume 160 (2004), pp. 1099-1127 | Article | MR 2144974 | Zbl 1084.14026

[R] Rost, M. Chow groups with coefficients, Doc. Math., Volume 1 (1996), pp. 319-393 | MR 1418952 | Zbl 0864.14002

[Sa1] Saito, S. Unramified class field theory of arithmetic schemes, Ann. Math., Volume 121 (1985), pp. 251-281 | Article | MR 786349 | Zbl 0593.14001

[Sa2] Saito, S. Cohomological Hasse principle for a threefold over a finite field, Algebraic K-Theory and Algebraic Topology (NATO ASI Series) (1994), pp. 229-241 | MR 1367301 | Zbl 0899.14004

[Sa3] Saito, S. Recent progress on the Kato conjecture, Quadratic Forms, Linear Algebraic Groups, and Cohomology (Developments in Math.) (2010), pp. 109-124 | Article | MR 2648722 | Zbl 1225.14013

[SS] Saito, S.; Sato, K. A finiteness theorem for zero-cycles over p-adic fields, Ann. Math., Volume 172 (2010), pp. 1593-1639 | Article | MR 2726095 | Zbl 1210.14012

[Sat] K. Sato, Characteristic classes for p-adic étale Tate twists and the image of p-adic regulators, preprint available at http://arxiv.org/abs/1004.1357. | MR 3064983 | Zbl 1276.19004

[SV] Suslin, A.; Voevodsky, V. Bloch-Kato conjecture and motivic cohomology with finite coefficients, Cycles, Transfer, and Motivic Homology Theories (Annals of Math. Studies) (1999) | Zbl 1021.14006

[SJ] Suslin, A.; Joukhovitski, S. Norm varieties, J. Pure Appl. Algebra, Volume 206 (2006), pp. 245-276 | Article | MR 2220090 | Zbl 1091.19002

[Sw] Suwa, N. A note on Gersten’s conjecture for logarithmic Hodge-Witt sheaves, K-Theory, Volume 9 (1995), pp. 245-271 | Article | MR 1344141 | Zbl 0838.14014

[To] Totaro, B. Milnor K-theory is the simplest part of algebraic K-theory, K-Theory, Volume 6 (1992), pp. 177-189 | Article | MR 1187705 | Zbl 0776.19003

[V1] Voevodsky, V. On motivic cohomology with ℤ/ℓ-coefficients, Ann. Math., Volume 174 (2011), pp. 401-438 | Article | MR 2811603 | Zbl 1236.14026

[V2] Voevodsky, V. Motivic Eilenberg-MacLane spaces, Publ. Math. IHES, Volume 112 (2010), pp. 1-99 | Article | Numdam | MR 2737977 | Zbl 1227.14025

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