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ÉTALE MOTIVIC COHOMOLOGY AND ALGEBRAIC CYCLES

Part of: $K$-theory in geometry

Published online by Cambridge University Press:  24 November 2014

Andreas Rosenschon  and
V. Srinivas
Andreas Rosenschon
Affiliation:
Mathematisches Institut, Ludwigs Maximilians Universität, München, Germany (axr@math.lmu.de)
V. Srinivas
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India (srinivas@math.tifr.res.in)
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Abstract

We consider étale motivic or Lichtenbaum cohomology and its relation to algebraic cycles. We give an geometric interpretation of Lichtenbaum cohomology and use it to show that the usual integral cycle maps extend to maps on integral Lichtenbaum cohomology. We also show that Lichtenbaum cohomology, in contrast to the usual motivic cohomology, compares well with integral cohomology theories. For example, we formulate integral étale versions of the Hodge and the Tate conjecture, and show that these are equivalent to the usual rational conjectures.

Keywords

motivic cohomologyalgebraic cycles

MSC classification

Primary: 19E15: Algebraic cycles and motivic cohomology
Secondary: 19E20: Relations with cohomology theories
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 15 , Issue 3 , July 2016 , pp. 511 - 537
Copyright
© Cambridge University Press 2014 

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References

Artin, M. and Mazur, B., Etale Homotopy, Lecture Notes in Mathematics, Volume 100 (Springer, Berlin, 1986).Google Scholar
Atiyah, M. F. and Hirzebruch, F., Analytic cycles on complex manifolds, Topology 1 (1962), 25–45.Google Scholar
Bloch, S. and Ogus, A., Gersten’s conjecture and the homology of schemes, Ann. Sci. Éc. Norm. Supér (4) 7 (1974), 181–201.Google Scholar
Bloch, S., Algebraic K-theory and crystalline cohomology, Publ. Math. Inst. Hautes Études Sci. 47 (1977), 187–268.Google Scholar
Bloch, S., Torsion algebraic cycles and a theorem of Roitman, Compositio Math. 39 (1979), 107–127.Google Scholar
Bloch, S., Algebraic cycles and higher K-theory, Adv. Math. 61 (1986), 267–304.Google Scholar
Bloch, S., Algebraic cycles and the Beĭlinson conjectures, in The Lefschetz Centennial Conference, Part I (Mexico City, 1984), Contemparary Mathematics, Volume 58, pp. 65–79 (American Mathematical Society, Providence, RI, 1986).Google Scholar
Bloch, S., 1994. Some notes on elementary properties of higher Chow groups (unpublished note).Google Scholar
Bloch, S., The moving lemma for higher Chow groups, J. Algebraic Geom. 3 (1994), 537–568.Google Scholar
Brown, K. S. and Gersten, S. M., Algebraic K-theory as generalized sheaf cohomology, in Algebraic K-Theory, I: Higher K-Theories (Proc. Conf., Battelle Memorial Inst., Seattle, WA, 1972), Lecture Notes in Mathematics, Volume 341, pp. 266–292 (Springer, Berlin, 1973).Google Scholar
Colliot-Thélène, J.-L. and Kahn, B., Cycles de codimension 2 et H 3 non ramifié pour les variétés sur les corps finis, J. K-Theory 11 (2013), 1–53.Google Scholar
Colliot-Thélène, J.-L., Sansuc, J.-J. and Soulé, C., Torsion dans le groupe de Chow de codimension deux, Duke Math. J. 50 (1983), 763–801.Google Scholar
Colliot-Thélène, J.-L. and Szamuely, T., Autour de la conjecture de Tate à coefficients Zℓ pour les variétés sur les corps finis, in The Geometry of Algebraic Cycles, Clay Mathematics Proceedings, Volume 9, pp. 83–98 (American Mathematical Society, 2010).Google Scholar
Colliot-Thélène, J.-L. and Voisin, C., Cohomologie non ramifée conjecture de Hodge entière, Duke Math. J. 161 (2012), 735–801.CrossRefGoogle Scholar
Esnault, H. and Viehweg, E., Deligne–Beĭlinson cohomology, in Beĭlinson’s Conjectures on special values of L-functions, Perspect Math., Volume 4, pp. 43–91 (Academic Press, Boston, MA, 1988).Google Scholar
Geisser, T., Tate’s conjecture, algebraic cycles and rational K-theory, K-Theory 13 (1998), 109–128.Google Scholar
Geisser, T. and Levine, M., The K-theory of fields in characteristic p, Invent. Math. 139 (2000), 459–493.Google Scholar
Geisser, T. and Levine, M., The Bloch–Kato conjecture and a theorem of Suslin–Voevodsky, J. Reine Angew. Math. 530 (2001), 55–103.Google Scholar
Grothendieck, A., Hodge’s general conjecture is false for trivial reasons, Topology 8 (1969), 299–303.Google Scholar
Hodge, W. V. D., The topological invariants of algebraic varieties, in Proceedings of the International Congress of Mathematicians, Cambridge, MA, 1950, Volume 1, pp. 182–192 (American Mathematical Society, Providence, RI, 1952).Google Scholar
Illusie, L., Complexes de de Rham–Witt et cohomologie cristalline, Ann. Sci. Éc. Norm. Supér 12 (1979), 501–661.Google Scholar
Jannsen, U., Continuous étale cohomology, Math. Ann. 280 (1988), 207–245.Google Scholar
Jannsen, U., On the l-adic cohomology of varieties over number fields and its Galois cohomology, in Galois Groups Over Q (Berkeley, CA, 1987), Mathematical Sciences Research Institute Publications, Volume 16, pp. 315–360 (Springer, New York, 1989).CrossRefGoogle Scholar
Kahn, B., Applications of weight-two motivic cohomology, Doc. Math. 1 (1996), 395–416.Google Scholar
Kahn, B., The Geisser–Levine method revisited and algebraic cycles over a finite field, Math. Ann. 324 (2002), 581–617.Google Scholar
Kahn, B., Classes de cycles motiviques étales, Algebra Number Theory 6 (2012), 1369–1407.CrossRefGoogle Scholar
Kollar, J., Trento examples, in Classification of Irregular Varieties (Trento, 1990), Lecture Notes in Mathematics, Volume 1515, pp. 134–139 (Springer, Berlin, 1992).Google Scholar
Levine, M., Chow’s moving lemma and the homotopy coniveau tower, K-Theory 37 (2006), 129–209.Google Scholar
Levine, M., Mixed Motives, Math. Surveys and Monographs, Volume 57 (American Mathematical Society, Providence, RI, 1992).Google Scholar
Lichtenbaum, S., The construction of weight-two arithmetic cohomology, Invent. Math. 88 (1987), 183–215.CrossRefGoogle Scholar
Lichtenbaum, S., New results on weight-two motivic cohomology, in The Grothendieck Festschrift, Vol. III, Progress in Mathematics, Volume 88, pp. 35–55 (Birkhäuser, Boston, MA, 1990).Google Scholar
Merkurjev, A. S. and Suslin, A. A., K-cohomology of Severi–Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), 1011–1046, 1135–1136.Google Scholar
Merkurjev, A. S. and Suslin, A. A., Norm residue homomorphism of degree three, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), 339–356.Google Scholar
Milne, J. S., Motivic cohomology and values of zeta functions, Compositio Math. 68 (1988), 59–100.Google Scholar
Nori, M. V., Algebraic cycles and Hodge-theoretic connectivity, Invent. Math. 111 (1993), 349–373.Google Scholar
Rosenschon, A. and Srinivas, V., The Griffiths group of the generic abelian 3-fold, in Proc. Int. Colloq. on Cycles, Motives and Shimura Varieties, Mumbai, TIFR Studies in Mathematics, Volume 21, pp. 449–467 (Narosa Publishing, 2010).Google Scholar
Schoen, C., Cyclic covers of Pv branched along v + 2 hyperplanes and the generalized Hodge conjecture for certain abelian varieties, in Arithmetic of Complex Manifolds (Erlangen, 1988), Lecture Notes in Mathematics, Volume 1399, pp. 137–154 (Springer, Berlin, 1989).Google Scholar
Schoen, C., An integral analog of the Tate conjecture for one-dimensional cycles on varieties over finite fields, Math. Ann. 311 (1998), 493–500.Google Scholar
Schoen, C., The Chow group modulo ℓ for the triple product of a general elliptic curve, Asian J. Math. 4 (2000), 987–996.Google Scholar
Schoen, C., On certain exterior product maps of Chow groups, Math. Res. Lett. 7 (2000), 177–194.Google Scholar
Schoen, C., Complex varieties for which the Chow group mod n is not finite, J. Algebraic Geom. 11 (2002), 41–100.Google Scholar
Suslin, A., Higher Chow groups and etale cohomology, in Cycles, Transfers, and Motivic Homology Theories, Annals of Mathematical Studies, Volume 143, pp. 239–254 (Princeton University Press, 2002).Google Scholar
Soulé, C. and Voisin, C., Torsion cohomology classes and algebraic cycles on complex projective manifolds, Adv. Math. 198 (2005), 107–127.Google Scholar
Suslin, A. and Voevodsky, V., Singular homology of abstract algebraic varieties, Invent. Math. 123 (1996), 61–94.Google Scholar
Suslin, A. and Voevodsky, V., Bloch–Kato conjecture and motivic cohomology with finite coefficients, in The Arithmetic and Geometry of Algebraic Cycles (Banff, AB, 1998), NATO Science Series C: Mathematics, Physics and Science, Volume 548, pp. 117–189.Google Scholar
Tate, J., On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, in Séminaire Bourbaki, 9, Exp. No. 306, pp. 415–440 (Soc. Math. France, 1965).Google Scholar
Tate, J., Relations between K 2 and Galois cohomology, Invent. Math. 36 (1976), 257–274.Google Scholar
Tate, J., Conjectures on algebraic cycle in ℓ-adic cohomology, in Motives, Seattle, WA, 1991, Proceedings of Symposia in Pure Mathematics, Volume 55, pp. 71–83 (American Mathematical Society).Google Scholar
Thomason, R. W., A finiteness condition equivalent to the Tate conjecture over Fq, in Algebraic K-Theory and Algebraic Number Theory, Honululu, HI, 1987, Contemporary Mathematics, Volume 83, pp. 385–392.Google Scholar
Totaro, B., Torsion algebraic cycles and complex cobordism, J. Amer. Math. Soc. 10 (1997), 467–493.CrossRefGoogle Scholar
Voevodsky, V., Motivic cohomology with Z/2-coefficients, Publ. Math. Inst. Hautes Études Sci. 98 (2003), 59–104.Google Scholar
Voevodsky, V., On motivic cohomology with Z/l-coefficients, Ann. of Math. (2) 174 (2011), 401–438.Google Scholar
Voisin, C., On integral Hodge classes on uniruled and Calabi–Yau manifolds, in Moduli Spaces and Arithmetic Geometry, Advanced Studies in Pure Mathematics, Volume 45, pp. 43–73. (2006).Google Scholar
Voisin, C., Remarks on curve classes on rationally connected varieties, in A Celebration of Algebraic Geometry, Clay Math. Proc. 18, pp. 591–599 (American Mathematical Society, Providence, RI, 2013).Google Scholar
Voisin, C., Degree 4 unramified cohomology with finite coefficients and torsion codimension 3 cycles, in Geometry and Arithmetic, Schiermonnikoog, EMS Congress Report, pp. 347–368 (European Mathematical Society, 2010).Google Scholar
Weibel, C., The norm residue isomorphism theorem, J. Topol. 2 (2009), 346–372.Google Scholar