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Some observations on motivic cohomology of arithmetic schemes. (English) Zbl 0694.14005
Following S. Lichtenbaum [Invent. Math. 88, 183-215 (1987; Zbl 0615.14004)], one defines for arithmetic schemes X the motivic, i.e. universal, cohomology groups denoted by $$H^{i+2}(K,{\mathbb{Z}}(i))$$, $$i\geq -1$$, where K is the function field of X. For instance, $$H^ 3(K,{\mathbb{Z}}(1))$$ is defined as $$H^ 2(K,{\mathbb{G}}_ m)$$, the Brauer group of K. A wide range of known as well as new facts are obtained in a unified way with a general reciprocity homomorphism, which, if it is an isomorphism, establishes a duality between motivic cohomology group and an idele class group of X.
For $$i=0$$, this duality amounts to higher dimensional class field theory; for $$i=-1$$, it is a reciprocity uniqueness theorem. If X is regular of dimension 2, for $$i=2$$ one has a certain Hasse principle for the existence of O-cycles on X of degree 1 and information about the kernel and cokernel of the reciprocity homomorphism for $$i=1$$ is related to the Brauer-Grothendieck group of X.
Reviewer: J.H.de Boer

##### MSC:
 14F99 (Co)homology theory in algebraic geometry 14A20 Generalizations (algebraic spaces, stacks) 14G25 Global ground fields in algebraic geometry
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##### References:
 [1] [A] Artin, M.: Dimension cohomologique: Premiers résultats, in: Théorie des topos et cohomologie étale des schémas, Tome 3. (Lecture Notes in Math., Vol. 305, pp. 43-63 Berlin, Heidelberg, New York: Springer 1973 [2] [Be] Beilinson, A.: Height pairings between algebraic cycles To appear in Proc. Boulder Conf. on algebraicK-theory [3] [B1] Bloch, S.: Algebraic cycles and higherK-theory. Preprint, IHES (1985) [4] [B-M-S] Beilinson, A., MacPherson, R., Schechtman, V.: Notes on motivic cohomology. Preprint · Zbl 0632.14010 [5] [B-T] Bass, H., Tate, J.: The Milnor ring of a global field. (Lecture Notes in Math., Vol. 342, pp. 349-446) Berlin, Heidelberg, New York: Springer 1972 [6] [C-S-S] Colliot-Thélèane, J.-L., Sansuc, J.-J., Swinnerton-Dyer, P.: Intersections of two quardrics and Châtelet surfaces. J. Reine Angew. Math.373, 37-107 (1987),374, 72-168 (1987) · Zbl 0622.14029 [7] [G] Grothendieck, A.: Le groupe de Brauer I et II, in: Dix exposés sur la cohomologie des schémas. Amsterdam: North-Holland 1968 [8] [I] Illusie, L.: Complexe de De Rham-Witt et cohomologie cristalline. Ann. Sci. Norm. Super. Pisa, Cl. Sci., IV. Ser. 12, 501-661 (1979) · Zbl 0436.14007 [9] [K-1] Kato, K.: A generalization of local class field theory by usingK-groups I. J. Fac. Sci. Univ. Tokyo, Sec. IA,26, 303-376 (1979); II. ibid. Kato, K.: A generalization of local class field theory by usingK-groups I. J. Fac. Sci. Univ. Tokyo, Sec. IA27, 603-683 (1980); III. ibid. Kato, K.: A generalization of local class field theory by usingK-groups I. J. Fac. Sci. Univ. Tokyo, Sec. IA29, 31-43 (1982) · Zbl 0428.12013 [10] [K-2] Kato, K.: Galois cohomology of complete discrete valuation fields. (Lecture Notes in Math. Vol. 967, pp. 215-238. Berlin, Heidelberg, New York: Springer 1982 · Zbl 0506.12022 [11] [K-3] Kato, K.: The existence theorem for higher local class field theory. Preprint · Zbl 1008.11061 [12] [K-4] Kato, K.: MilnorK-theory and the Chow group of zero-cycles. Contemp. Math.55, 241-253 (1986) [13] [K-5] Kato, K.: A Hasse principle for two dimensional global fields. J. Reine, Angew. Math. 366, 142-183 (1986) · Zbl 0576.12012 [14] [K-S] Kato, K., Saito, S.: Global class field theory of arithmetic schemes. Contemp. Math.55, 255-331 (1986) [15] [L-1] Lichtenbaum, S.: Values of zeta functions at non-negative integers. In: Number Theory. (Lecture Notes in Math., Vol. 1068, pp. 127-138). Berlin, Heidelberg, New York: Springer 1984 · Zbl 0591.14014 [16] [L-2] Lichtenbaum, S.: The construction of weight-two arithmetic cohomology. Invent. Math.88, 183-215 (1987) · Zbl 0615.14004 [17] [L-3] Lichtenbaum, S.: Duality theorems for curves overp-adic fields. Invent. Math.7, 120-136 (1969) · Zbl 0186.26402 [18] [M-1] Milne, J.: Etale Cohomology. Princeton Univ. Press. 1980 [19] [M-2] Milne, J.: Motivic cohomology and values of zeta functions. Compos., Math.68, 59-102 (1988) · Zbl 0681.14007 [20] [Ma] Manin, Yu.I.: Le groupe de Brauer-Grothendieck en géométrie diophatienne. Actes du congrès intern. Math. Nice,1, 401-411 (1970) [21] [Me] Merkurjev, A.S.: On the torsion ofK 2 of local fields. Ann. Math.118, 375-381 (1983) · Zbl 0519.12010 [22] [N] Nisnevich, A.: Arithmetic and cohomology invariants of semisimple group schemes and compactifications of locally symmetric spaces. Funct. Anal. appl.14, 61-62 (1980) · Zbl 0526.14030 [23] [Sa-1] Saito, S.: Class field theory for curves over local fields. J. Number Theory21, 44-80 (1985) · Zbl 0599.14008 [24] [Sa-2] Saito, S.: Arithmetic on two dimensional local rings. Invent. Math.85, 379-414 (1986) · Zbl 0609.13003 [25] [Sa-3] Saito, S.: Unramified class field theory of arithmetical schemes. Ann. Math.121, 251-281 (1985) · Zbl 0593.14001 [26] [Sa-4] Saito, S.: Arithmetic theory on an arithmetic surface. Ann. Math.129, 547-589 (1989) · Zbl 0688.14019 [27] [Sal] Salberger, P.: Zero-cycles on rational surfaces over number fields. Invent. Math.91, 505-524 (1988) · Zbl 0688.14008 [28] [San] Sansuc, J.-J.: Principe de Hasse, surfaces cubiques et intersections de deux quadriques, exposé aux Journées arithmétiques de Besançon (1985). Astèrisque 147-148 183-207 (1987) [29] [Se-1] Serre, J.-P.: Cohomologie Galoisienne. (Lecture Notes in Math., Vol. 5). Berlin, Heidelberg, New York: Springer 1965 [30] [Se-2] Serre, J.-P.: Corps Locaux. Paris: Hermann 1962 [31] [Su] Suslin, A.A.: Torsion inK 2 of fields, Lomi Preprint E-2-82, USSR Academy of Sciences, Steklov Mathematical Institute, Leningrad Department [32] [T-1] Tate, J.: On the conjectures of Birch and Swinnerton-Dyer and a geometric analog. In: Dix Exposés sur la cohomologie des schémas, pp. 189-214. Amsterdam North-Holland 1968 [33] [T-2] Tate, J.: Algebraic cycles and poles of zeta functions, Arithmetic Algebraic Geometry. New York: Harper and Row 1965 · Zbl 0213.22804 [34] [T-3] Tate, J.: Symbols in arithmetic Actes Congrès Intern. Math. Nice, 1, 201-211 (1970) [35] [T-4] Tate, J.: On the torsion inK 2 of fields. In: Algebraic Number Theory, Papers contributed for the International Symposium, Kyoto (1976), pp. 243-261
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