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Duality via cycle complexes. (English) Zbl 1215.19001
In this paper the author proves a very hard theorem on the adjointness for a constructible sheaf and for the complex of étale sheaves constructed in a manner of Bloch’s defining higher Chow groups. By this theorem, the corresponding dualities of perfect pairings are also obtained, respectively, over a finite field, a local field, and a number ring.
The applications of these results can be taken, respectively, as generalizations of several known related theorems.

##### MSC:
 19E15 Algebraic cycles and motivic cohomology ($$K$$-theoretic aspects) 14F42 Motivic cohomology; motivic homotopy theory
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##### References:
 [1] S. Bloch, ”Torsion algebraic cycles and a theorem of Roitman,” Compositio Math., vol. 39, iss. 1, pp. 107-127, 1979. · Zbl 0463.14002 [2] S. Bloch, ”Algebraic cycles and higher $$K$$-theory,” Adv. in Math., vol. 61, iss. 3, pp. 267-304, 1986. · Zbl 0608.14004 [3] S. Bloch, ”The moving lemma for higher Chow groups,” J. Algebraic Geom., vol. 3, iss. 3, pp. 537-568, 1994. · Zbl 0830.14003 [4] S. Bloch and K. Kato, ”$$p$$-adic étale cohomology,” Inst. Hautes Études Sci. Publ. Math., iss. 63, pp. 107-152, 1986. · Zbl 0613.14017 [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 [6] P. Deligne, ”Théorie de Hodge. III,” Inst. Hautes Études Sci. Publ. Math., iss. 44, pp. 5-77, 1974. · Zbl 0237.14003 [7] C. Deninger, ”Duality in the étale cohomology of one-dimensional proper schemes and generalizations,” Math. Ann., vol. 277, iss. 3, pp. 529-541, 1987. · Zbl 0607.14011 [8] C. Deninger and K. Wingberg, ”Artin-Verdier duality for $$n$$-dimensional local fields involving higher algebraic $$K$$-sheaves,” J. Pure Appl. Algebra, vol. 43, iss. 3, pp. 243-255, 1986. · Zbl 0608.12016 [9] T. Geisser, ”Motivic cohomology over Dedekind rings,” Math. Z., vol. 248, iss. 4, pp. 773-794, 2004. · Zbl 1062.14025 [10] T. Geisser, ”Motivic cohomology, $$K$$-theory and topological cyclic homology,” in Handbook of $$K$$-Theory. Vol. 1, 2, New York: Springer-Verlag, 2005, pp. 193-234. · Zbl 1113.14017 [11] T. Geisser, ”The affine part of the Picard scheme,” Compos. Math., vol. 145, iss. 2, pp. 415-422, 2009. · Zbl 1163.14025 [12] T. Geisser and M. Levine, ”The $$K$$-theory of fields in characteristic $$p$$,” Invent. Math., vol. 139, iss. 3, pp. 459-493, 2000. · Zbl 0957.19003 [13] T. Geisser and M. Levine, ”The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky,” J. Reine Angew. Math., vol. 530, pp. 55-103, 2001. · Zbl 1023.14003 [14] M. Gros and N. Suwa, ”La conjecture de Gersten pour les faisceaux de Hodge-Witt logarithmique,” Duke Math. J., vol. 57, iss. 2, pp. 615-628, 1988. · Zbl 0715.14011 [15] A. Grothendieck, Technique de descente et theoremes d’existence en geometrie algebrique. VI. Les schemas de Picard. Proprietes generales. · Zbl 0238.14015 [16] U. Jannsen, ”Hasse principles for higher-dimensional fields,” Universität Regensburg, preprint 18/2004 , 2004. · Zbl 1346.14057 [17] 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 [18] U. Jannsen, S. Saito, and K. Sato, ”Etale Duality for Constructible Sheaves on Arithmetic Schemes,” Universität Regensburg, preprint 8/2004. · Zbl 1299.14026 [19] K. Kato, ”Duality theories for the $$p$$-primary étale cohomology. I,” in Algebraic and Topological Theories, Tokyo: Kinokuniya, 1986, pp. 127-148. · Zbl 0800.14009 [20] K. Kato, ”A Hasse principle for two-dimensional global fields,” J. Reine Angew. Math., vol. 366, pp. 142-183, 1986. · Zbl 0576.12012 [21] K. Kato and T. Kuzumaki, ”The dimension of fields and algebraic $$K$$-theory,” J. Number Theory, vol. 24, iss. 2, pp. 229-244, 1986. · Zbl 0608.12029 [22] N. M. Katz and S. Lang, ”Finiteness theorems in geometric classfield theory,” Enseign. Math., vol. 27, iss. 3-4, pp. 285-319 (1982), 1981. · Zbl 0495.14011 [23] A. Krishna and V. Srinivas, ”Zero-cycles and $$K$$-theory on normal surfaces,” Ann. of Math., vol. 156, iss. 1, pp. 155-195, 2002. · Zbl 1060.14015 [24] S. L. Kleiman, ”The Picard scheme,” in Fundamental Algebraic Geometry, Providence, RI: Amer. Math. Soc., 2005, vol. 123, pp. 235-321. · Zbl 1085.14001 [25] M. Levine, ”Torsion zero-cycles on singular varieties,” Amer. J. Math., vol. 107, iss. 3, pp. 737-757, 1985. · Zbl 0579.14007 [26] M. Levine, ”Techniques of localization in the theory of algebraic cycles,” J. Algebraic Geom., vol. 10, iss. 2, pp. 299-363, 2001. · Zbl 1077.14509 [27] M. Levine, ”Motivic cohomology and K-theory of schemes. K-theory,” , preprint archives 336. · Zbl 0883.19001 [28] M. Levine and C. Weibel, ”Zero cycles and complete intersections on singular varieties,” J. Reine Angew. Math., vol. 359, pp. 106-120, 1985. · Zbl 0555.14004 [29] B. Mazur, ”Notes on étale cohomology of number fields,” Ann. Sci. École Norm. Sup., vol. 6, pp. 521-552 (1974), 1973. · Zbl 0282.14004 [30] J. S. Milne, ”Zero cycles on algebraic varieties in nonzero characteristic: Rojtman’s theorem,” Compositio Math., vol. 47, iss. 3, pp. 271-287, 1982. · Zbl 0506.14006 [31] J. S. Milne, Étale Cohomology, Princeton, NJ: Princeton Univ. Press, 1980, vol. 33. · Zbl 0433.14012 [32] J. S. Milne, Arithmetic Duality Theorems, Boston, MA: Academic Press, 1986, vol. 1. · Zbl 0613.14019 [33] 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 [34] T. Moser, ”A duality theorem for étale $$p$$-torsion sheaves on complete varieties over a finite field,” Compositio Math., vol. 117, iss. 2, pp. 123-152, 1999. · Zbl 0954.14012 [35] E. Nart, ”The Bloch complex in codimension one and arithmetic duality,” J. Number Theory, vol. 32, iss. 3, pp. 321-331, 1989. · Zbl 0728.14002 [36] A. A. Rojtman, ”The torsion of the group of $$0$$-cycles modulo rational equivalence,” Ann. of Math., vol. 111, iss. 3, pp. 553-569, 1980. · Zbl 0504.14006 [37] J. Roos, ”Sur les foncteurs dérivés de $$\underleftarrow\lim$$. Applications,” C. R. Acad. Sci. Paris, vol. 252, pp. 3702-3704, 1961. · Zbl 0102.02501 [38] S. Saito, ”Torsion zero-cycles and étale homology of singular schemes,” Duke Math. J., vol. 64, iss. 1, pp. 71-83, 1991. · Zbl 0764.14011 [39] A. Seidenberg, ”The hyperplane sections of normal varieties,” Trans. Amer. Math. Soc., vol. 69, pp. 357-386, 1950. · Zbl 0040.23501 [40] ”Théorie des topos et cohomologie étale des schémas. Tome 3,” in Séminair de Géométrie Algébrique du Bois-Marie 1963-1964 (SGA 4), Artin, M., Grothendieck, A., and Verdier, J. L., Eds., New York: Springer-Verlag, 1973, vol. 305. · Zbl 0245.00002 [41] ”Cohomologie $$l$$-adique et fonctions $$L$$. (French),” in Séminair de Géométrie Algébrique du Bois-Marie 1965-1965 (SGA 5), Illusie, L., Ed., New York: Springer-Verlag, 1977, vol. 589. · Zbl 0345.00011 [42] J. P. Serre, Morphismes universels et variétès d’Albanese. · Zbl 0123.13903 [43] N. Spaltenstein, ”Resolutions of unbounded complexes,” Compositio Math., vol. 65, iss. 2, pp. 121-154, 1988. · Zbl 0636.18006 [44] M. Spieß, ”Artin-Verdier duality for arithmetic surfaces,” Math. Ann., vol. 305, iss. 4, pp. 705-792, 1996. · Zbl 0887.14008 [45] A. A. Suslin, ”Higher Chow groups and etale cohomology,” in Cycles, Transfers, and Motivic Homology Theories, Princeton, NJ: Princeton Univ. Press, 2000, vol. 143, pp. 239-254. · Zbl 1019.14001 [46] 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 [47] R. W. Thomason, ”Algebraic $$K$$-theory and étale cohomology,” Ann. Sci. École Norm. Sup., vol. 18, iss. 3, pp. 437-552, 1985. · Zbl 0596.14012 [48] V. Voevodsky, ”Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic,” Int. Math. Res. Not., vol. 2002, iss. 7, pp. 351-355, 2002. · Zbl 1057.14026
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