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Selecta Mathematica

, Volume 24, Issue 2, pp 1121–1146 | Cite as

The derived functors of unramified cohomology

  • Bruno Kahn
  • R. Sujatha
Article

Abstract

We study the first “derived functors of unramified cohomology” in the sense of Kahn and Sujatha (IMRN 2016. doi: 10.1093/imrn/rnw184), applied to the sheaves \(\mathbb {G}_m\) and \(\mathcal {K}_2\). We find interesting connections with classical cycle-theoretic invariants of smooth projective varieties, involving notably a version of the Griffiths group and the group of indecomposable (2, 1)-cycles.

Mathematics Subject Classification

19E15 14E99 

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References

  1. 1.
    Ayoub, J., Barbieri-Viale, L.: 1-Motivic sheaves and the Albanese functor. J. Pure Appl. Algebra 213(5), 809–839 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barbieri-Viale, L., Kahn, B.: On the derived category of 1-motives. Astérisque 381, 1–254 (2016)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Beilinson, A., Bernstein, J., Deligne, P.: Faisceaux pervers. Astérisque 100, 5–171 (1984)Google Scholar
  4. 4.
    Bloch, S., Esnault, H.: The coniveau filtration and non-divisibility for algebraic cycles. Math. Ann. 304, 303–314 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Clemens, H.: Homological equivalence, modulo algebraic equivalence, is not finitely generated. Publ. Math. IHÉS 58, 19–38 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Colliot-Thélène, J.-L.: Unramified cohomology, birational invariants and the Gersten conjecture. Proc. Sympos. Pure Math. vol. 58, Part 1. American Mathematical Society, Providence (1995)Google Scholar
  7. 7.
    Colliot-Thélène, J.-L., Raskind, W.: \(\cal{K}_2\)-cohomology and the second Chow group. Math. Ann. 270, 165–199 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Coombes, K.: The arithmetic of zero cycles on surfaces with geometric genus and irregularity zero. Math. Ann. 291, 429–452 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gabber, O.: Sur la torsion dans la cohomologie \(l\)-adique d’une variété. C. R. Acad. Sci. Paris 297, 179–182 (1983)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Griffiths, P.A.: On the periods of certain rational integrals. I, II. Ann. Math. 90, 460–495, 496–541 (1969)Google Scholar
  11. 11.
    Huber, A., Kahn, B.: The slice filtration and mixed Tate motives. Compos. Math. 142, 907–936 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kahn, B.: Motivic cohomology of smooth geometrically cellular varieties. Proc. Symp. Pure Math. 67, 149–174 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kahn, B.: The Brauer group and indecomposable (2, 1)-cycles. Compos. Math. 152, 1041–1051 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kahn, B., Sujatha, R.: Birational motives, I: pure birational motives. Ann. K-Theory 1, 379–440 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kahn, B., Sujatha, R.: Birational motives, II: triangulated birational motives. IMRN (2016). doi: 10.1093/imrn/rnw184 zbMATHGoogle Scholar
  16. 16.
    Kahn, B., Yamazaki, T.: Voevodsky’s motives and Weil reciprocity. Duke Math. J. 162(14), 2751–2796 (2013). (erratum: 164, 2093–2098, 2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lieberman, D.: Numerical and homological equivalence of algebraic cycles on Hodge manifolds. Am. J. Math. 90, 366–374 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mazza, C., Voevodsky, V., Weibel, C.: Lecture Notes on Motivic Cohomology. Clay Mathematics Monographs, vol. 2. AMS, Providence (2006)zbMATHGoogle Scholar
  19. 19.
    Milne, J.: Abelian varieties. In: Cornell, G., Silverman, J. (eds.) Arithmetic Geometry, rev. printing, pp. 103–150. Springer, Berlin (1998)Google Scholar
  20. 20.
    Murre, J.P.: Un résultat en théorie des cycles algébriques de codimension deux. C. R. Acad. Sci. Paris 296, 981–984 (1983)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Riou, J.: Théorie homotopique des \(S\)-schémas, mémoire de DEA, Paris 7 (2002). http://www.math.u-psud.fr/~riou/dea/dea.pdf
  22. 22.
    Schoen, C.: Complex varieties for which the Chow group mod \(n\) is not finite. J. Algebr. Geom. 11, 41–100 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Spieß, M., Szamuely, T.: On the Albanese map for smooth quasi-projective varieties. Math. Ann. 325, 1–17 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Voevodsky, V.: Cohomological theory of presheaves with transfers. In: Friedlander, E., Suslin, A., Voevodsky, V. (eds.) Cycles, Transfers and Motivic Cohomology Theories, Annals of Mathematics Studies, vol. 143, pp. 187–188. Princeton University Press, Princeton (2000)Google Scholar
  25. 25.
    Voevodsky, V.: Triangulated categories of motives over a field. In: Friedlander, E., Suslin, A., Voevodsky, V. (eds.) Cycles, Transfers and Motivic Cohomology Theories, Annals of Mathematics Studies, vol. 143, pp. 188–238. Princeton University Press, Princeton (2000)Google Scholar
  26. 26.
    Voevodsky, V.: Cancellation theorem, Doc. Math. 2010, Extra volume: Andrei A. Suslin sixtieth birthday, pp. 671–685Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.IMJ-PRGParis Cedex 05France
  2. 2.Department of MathematicsUniversity of British ColumbiaVancouverCanada

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