Hasse principles for higher-dimensional fields. (English) Zbl 1346.14057

The paper is devoted to the proof of several seminal conjectures on local-global principles for higher-dimensional fields. These conjectures, proposed by K. Kato [J. Reine Angew. Math. 366, 142–183 (1986; Zbl 0576.12012)], are formulated in terms of Galois cohomology of a function field \(F\) in \(d\) variables over a global field \(K\) under the assumption that \(K/F\) is primary, i.e., \(K\) is separably closed in \(F\).
Conjecture 1 states that the restriction map \[ \alpha_n: H^{d+2}(F,\mathbb Z/n\mathbb Z(d+1)) \to \bigoplus_v H^{d+2}(F_v,\mathbb Z/n\mathbb Z(d+1)) \] is injective (here the sum is taken over all places of \(K\), \(F_v=K_v(V\times_K K_v)\), where \(V\) is a geometrically integral model of the field \(F\)). It was proved by Kato (loc. cit.) for \(d=1\).
The first principal result of the paper under review states that this conjecture holds provided \(n\) is invertible in \(K\). Here are the main steps of the proof. First, the assertion of the conjecture is reduced to another one, where finite coefficients \(\mathbb Z/n\mathbb Z(j)\) are replaced with infinite ones \(\mathbb Q_\ell/\mathbb Z_\ell (j)\). A similar step was made by Kato using the Merkurjev-Suslin theorem whereas Jannsen uses the Bloch-Kato conjecture (now the Voevodsky-Rost theorem). To prove the injectivity theorem for the restriction map with infinite coefficients, one has to use weights, i.e., Deligne’s proof of the Weil conjectures, and resolution of singularities to control the weights. In the case of positive characteristic, some problems are to overcome in absence of Hironaka’s theorem. For \(\ell\) invertible in \(K\), the author uses a weaker form of resolution, namely the de Jong alteration in a refined version established by Gabber.
Similarly to the cases \(d=1\) (Colliot-Thélène’s appendix to Kato’s paper quoted above), and \(d=2\) (J.-L. Colliot-Thélène and U. Jannsen [C. R. Acad. Sci. Paris Sér. I Math. 312, 759–762 (1991; Zbl 0743.11020)]), natural applications to quadratic forms arise, more precisely to sum of squares. Namely, the author proves that if \(F\) is a finitely generated field of characteristic zero, then the Pythagoras number of \(F\) is finite, and if \(F\) is of transcendence degree \(d\) over \(\mathbb Q\), then any sum of squares in \(F\) is a sum of \(2^{d+1}\) squares, provided \(d\geq 2\). The proof uses a weaker form of Conjecture 1, whose proof only needs the Milnor conjecture (now the Voevodsky theorem). (More elementary methods give the finiteness and the weaker bound \(2^{d+2}\), see A. Pfister [Jahresber. DMV 102, 15–41 (2000; Zbl 1140.19300)].)
The next Kato’s conjecture refers to the cokernel of the above restriction map. Kato proved it for \(d=1\). It is proved unconditionally in the case where \(K\) is a number field, and conditionally on resolution of singularities for \(n\) invertible in \(K\). (Unconditional proofs were given by M. Kerz and S. Saito [Publ. Math. IHES 115, 123–183 (2012; Zbl 1263.14026)] and by the author [arXiv:0910.2803].)
Yet another Kato’s conjecture, for schemes over finite fields, is proved conditionally on resolution of singularities.


14G05 Rational points
11R34 Galois cohomology
14F20 Étale and other Grothendieck topologies and (co)homologies
Full Text: DOI arXiv


[1] L. Barbieri Viale, ”\(\mathcal H\)-cohomologies versus algebraic cycles,” Math. Nachr., vol. 184, pp. 5-57, 1997. · Zbl 0889.14003 · doi:10.1002/mana.19971840102
[2] S. Bloch and K. Kato, ”\(p\)-adic étale cohomology,” Inst. Hautes Études Sci. Publ. Math., vol. 63, pp. 107-152, 1986. · Zbl 0613.14017 · doi:10.1007/BF02831624
[3] S. Bloch and A. Ogus, ”Gersten’s conjecture and the homology of schemes,” Ann. Sci. École Norm. Sup., vol. 7, pp. 181-201 (1975), 1974. · Zbl 0307.14008
[4] J. Colliot-Thélène and U. Jannsen, ”Sommes de carrés dans les corps de fonctions,” C. R. Acad. Sci. Paris Sér. I Math., vol. 312, iss. 11, pp. 759-762, 1991. · Zbl 0743.11020
[5] J. -L. Colliot-Thélène, ”On the reciprocity sequence in the higher class field theory of function fields,” in Algebraic \(K\)-Theory and Algebraic Topology, Dordrecht: Kluwer Acad. Publ., 1993, vol. 407, pp. 35-55. · Zbl 0885.19002 · doi:10.1007/978-94-017-0695-7
[6] J. -L. Colliot-Thélène, ”Birational invariants, purity and the Gersten conjecture,” in \(K\)-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras, Providence, RI: Amer. Math. Soc., 1995, vol. 58, pp. 1-64. · Zbl 0834.14009
[7] J. Colliot-Thélène, J. Sansuc, and C. Soulé, ”Torsion dans le groupe de Chow de codimension deux,” Duke Math. J., vol. 50, iss. 3, pp. 763-801, 1983. · Zbl 0574.14004 · doi:10.1215/S0012-7094-83-05038-X
[8] J. Colliot-Thélène, R. T. Hoobler, and B. Kahn, ”The Bloch-Ogus-Gabber theorem,” in Algebraic \(K\)-Theory, Providence, RI: Amer. Math. Soc., 1997, vol. 16, pp. 31-94. · Zbl 0911.14004
[9] P. Deligne, ”La conjecture de Weil. I,” Inst. Hautes Études Sci. Publ. Math., vol. 43, pp. 273-307, 1974. · Zbl 0287.14001 · doi:10.1007/BF02684373
[10] P. Deligne, ”La conjecture de Weil. II,” Inst. Hautes Études Sci. Publ. Math., vol. 52, pp. 137-252, 1980. · Zbl 0456.14014 · doi:10.1007/BF02684780
[11] H. Gillet, ”Homological descent for the \(K\)-theory of coherent sheaves,” in Algebraic \(K\)-Theory, Number Theory, Geometry and Analysis, New York: Springer-Verlag, 1984, vol. 1046, pp. 80-103. · Zbl 0557.14009
[12] H. Gillet, ”\(K\)-theory and intersection theory revisited,” \(K\)-Theory, vol. 1, iss. 4, pp. 405-415, 1987. · Zbl 0651.14001 · doi:10.1007/BF00539625
[13] H. Gillet and C. Soulé, ”Descent, motives and \(K\)-theory,” J. reine angew. Math., vol. 478, pp. 127-176, 1996. · Zbl 0863.19002 · doi:10.1515/crll.1996.478.127
[14] M. J. Greenberg, ”Rational points in Henselian discrete valuation rings,” Inst. Hautes Études Sci. Publ. Math., vol. 31, pp. 59-64, 1966. · Zbl 0146.42201 · doi:10.1007/BF02684802
[15] M. Gros, Classes de Chern et Classes de Cycles en Cohomologie de Hodge-Witt Logarithmique, Paris: Soc. Math. France, 1985, vol. 21. · Zbl 0615.14011
[16] M. Gros, ”Sur la partie \(p\)-primaire du groupe de Chow de codimension deux,” Comm. Algebra, vol. 13, iss. 11, pp. 2407-2420, 1985. · Zbl 0591.14003 · doi:10.1080/00927878508823280
[17] M. Gros and N. Suwa, ”Application d’Abel-Jacobi \(p\)-adique et cycles algébriques,” Duke Math. J., vol. 57, iss. 2, pp. 579-613, 1988. · Zbl 0697.14005 · doi:10.1215/S0012-7094-88-05726-2
[18] C. Haesemeyer and C. Weibel, ”Norm varieties and the Chain Lemma (after Markus Rost),” in Algebraic Topology, New York: Springer-Verlag, 2009, vol. 4, pp. 95-130. · Zbl 1244.19003 · doi:10.1007/978-3-642-01200-6_6
[19] H. Hironaka, ”Resolution of singularities of an algebraic variety over a field of characteristic zero. I,” Ann. of Math., vol. 79, pp. 109-203, 1964. · Zbl 0122.38603 · doi:10.2307/1970486
[20] H. Hironaka, ”Resolution of singularities of an algebraic variety over a field of characteristic zero. II,” Ann. of Math., vol. 79, pp. 205-326, 1964. · Zbl 0122.38603 · doi:10.2307/1970486
[21] L. Illusie, ”Complexe de deRham-Witt et cohomologie cristalline,” Ann. Sci. École Norm. Sup., vol. 12, iss. 4, pp. 501-661, 1979. · Zbl 0436.14007
[22] Travaux de Gabber sur l’Uniformisation Locale et la Cohomologie Étale des Schémas Quasi-Excellents, Illusie, L., Laszlo, Y., and Orgogozo, F., Eds., Paris: Société Mathématique de France, 2014, vol. 363-364. · Zbl 1297.14003
[23] U. Jannsen, ”On the Galois cohomology of \(l\)-adic representations attached to varieties over local or global fields,” in Séminaire de Théorie des Nombres, Paris 1986-87, Boston: Birkhäuser, 1988, vol. 75, pp. 165-182. · Zbl 0687.14017
[24] U. Jannsen, ”On the \(l\)-adic cohomology of varieties over number fields and its Galois cohomology,” in Galois Groups over \({\mathbf Q}\), New York: Springer-Verlag, 1989, vol. 16, pp. 315-360. · Zbl 0703.14010 · doi:10.1007/978-1-4613-9649-9_5
[25] U. Jannsen, Mixed Motives and Algebraic \(K\)-Theory, New York: Springer-Verlag, 1990. · Zbl 0691.14001 · doi:10.1007/BFb0085080
[26] U. Jannsen, ”Principe de Hasse cohomologique,” in Séminaire de Théorie des Nombres, Paris, 1989-90, Boston: Birkhäuser, 1992, vol. 102, pp. 121-140. · Zbl 0745.11053 · doi:10.1007/978-1-4757-4269-5_10
[27] U. Jannsen, Rigidity theorems for \(\mathcal K\)- and \(\mathcal H\)-homology and other functors, 2015.
[28] U. Jannsen, ”Weights in arithmetic geometry,” Jpn. J. Math., vol. 5, iss. 1, pp. 73-102, 2010. · Zbl 1204.14011 · doi:10.1007/s11537-010-0947-4
[29] U. Jannsen, Hasse principles for higher-dimensional fields, 2009. · Zbl 1346.14057 · doi:10.4007/annals.2016.183.1.1
[30] U. Jannsen and S. Saito, ”Kato homology of arithmetic schemes and higher class field theory over local fields,” Doc. Math., iss. Extra Vol., pp. 479-538, 2003. · Zbl 1092.14504
[31] U. Jannsen and S. Saito, ”Algebraic \(K\)-theory and motivic cohomology,” Oberwolfach Rep., vol. 6, iss. 2, pp. 1731-1773, 2009. · Zbl 1177.14016 · doi:10.4171/OWR/2009/31
[32] U. Jannsen, S. Saito, and K. Sato, ”Étale duality for constructible sheaves on arithmetic schemes,” J. reine angew. Math., vol. 688, pp. 1-65, 2014. · Zbl 1299.14026 · doi:10.1515/crelle-2012-0043
[33] B. Kahn, J. P. Murre, and C. Pedrini, ”On the transcendental part of the motive of a surface,” in Algebraic Cycles and Motives. Vol. 2, Cambridge: Cambridge Univ. Press, 2007, vol. 344, pp. 143-202. · Zbl 1130.14008
[34] K. Kato, ”A Hasse principle for two-dimensional global fields,” J. reine angew. Math., vol. 366, pp. 142-183, 1986. · Zbl 0576.12012 · doi:10.1515/crll.1986.366.142
[35] N. M. Katz and W. Messing, ”Some consequences of the Riemann hypothesis for varieties over finite fields,” Invent. Math., vol. 23, pp. 73-77, 1974. · Zbl 0275.14011 · doi:10.1007/BF01405203
[36] M. Kerz and S. Saito, ”Cohomological Hasse principle and motivic cohomology for arithmetic schemes,” Publ. Math. Inst. Hautes Études Sci., vol. 115, pp. 123-183, 2012. · Zbl 1263.14026 · doi:10.1007/s10240-011-0038-y
[37] J. Milnor, ”Algebraic \(K\)-theory and quadratic forms,” Invent. Math., vol. 9, pp. 318-344, 1970. · Zbl 0199.55501 · doi:10.1007/BF01425486
[38] J. S. Milne, Étale Cohomology, Princeton, N.J.: Princeton Univ. Press, 1980, vol. 33. · Zbl 0433.14012
[39] J. S. Milne, ”Values of zeta functions of varieties over finite fields,” Amer. J. Math., vol. 108, iss. 2, pp. 297-360, 1986. · Zbl 0611.14020 · doi:10.2307/2374676
[40] A. S. Merkurjev and A. A. Suslin, ”\(K\)-cohomology of Severi-Brauer varieties and the norm residue homomorphism,” Math. USSR Izvestiya, vol. 21, pp. 307-340, 1983. · Zbl 0525.18008 · doi:10.1070/IM1983v021n02ABEH001793
[41] A. Pfister, ”On the Milnor conjectures: history, influence, applications,” Jahresber. Deutsch. Math.-Verein., vol. 102, iss. 1, pp. 15-41, 2000. · Zbl 1140.19300
[42] M. Rost, ”Norm varieties and algebraic cobordism,” in Proceedings of the International Congress of Mathematicians, Vol. II, Beijing, 2002, pp. 77-85. · Zbl 1042.19002
[43] S. Saito, ”Cohomological Hasse principle for a threefold over a finite field,” in Algebraic \(K\)-Theory and Algebraic Topology, Dordrecht: Kluwer Acad. Publ., 1993, vol. 407, pp. 229-241. · Zbl 0899.14004
[44] A. Suslin and S. Joukhovitski, ”Norm varieties,” J. Pure Appl. Algebra, vol. 206, iss. 1-2, pp. 245-276, 2006. · Zbl 1091.19002 · doi:10.1016/j.jpaa.2005.12.012
[45] N. Suwa, ”A note on Gersten’s conjecture for logarithmic Hodge-Witt sheaves,” \(K\)-Theory, vol. 9, iss. 3, pp. 245-271, 1995. · Zbl 0838.14014 · doi:10.1007/BF00961667
[46] V. Voevodsky, ”Motivic cohomology with \({\mathbf Z}/2\)-coefficients,” Publ. Math. Inst. Hautes Études Sci., vol. 98, pp. 59-104, 2003. · Zbl 1057.14028 · doi:10.1007/s10240-003-0010-6
[47] V. Voevodsky, ”On motivic cohomology with \(\mathbb Z/l\)-coefficients,” Ann. of Math., vol. 174, iss. 1, pp. 401-438, 2011. · Zbl 1236.14026 · doi:10.4007/annals.2011.174.1.11
[48] V. Voevodsky, ”Motivic Eilenberg-Maclane spaces,” Publ. Math. Inst. Hautes Études Sci., vol. 112, pp. 1-99, 2010. · Zbl 1227.14025 · doi:10.1007/s10240-010-0024-9
[49] A. Grothendieck, Éléments de Géométrie Algébrique. IV. Étude Locale des Schémas et des Morphismes de Schémas. II, , 1965, vol. 24. · Zbl 0135.39701 · doi:10.1007/BF02684322
[50] Théorie des topos et cohomologie étale des schémas. Tome 2, Artin, M., Grothendieck, A., and Verdier, J. L., Eds., New York: Springer-Verlag, 1972, vol. 270. · Zbl 0237.00012 · doi:10.1007/BFb0061319
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.