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On the push-forwards for motivic cohomology theories with invertible stable Hopf element

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Abstract

We present a geometric construction of push-forward maps along projective morphisms for cohomology theories representable in the stable motivic homotopy category assuming that the element corresponding to the stable Hopf map is inverted in the coefficient ring of the theory. The construction is parallel to the one given by Nenashev for derived Witt groups. Along the way we introduce cohomology groups twisted by a formal difference of vector bundles as cohomology groups of a certain Thom space and compute twisted cohomology groups of projective spaces.

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References

  1. Ananyevskiy A.: The special linear version of the projective bundle theorem. Compos. Math. 151(3), 461–501 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ananyevskiy, A., Levine, M., Panin, I.: Witt sheaves and the η-inverted sphere spectrum. arXiv:1504.04860

  3. Andrews, M., Miller, H.: Inverting the Hopf map. http://www-math.mit.edu/~hrm/papers/andrews-miller-jun14

  4. Balmer P.: Derived Witt groups of a scheme. J. Pure Appl. Algebra 141, 101–129 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Barge J, Morel F.: Groupes de Chow des cycles orientés et classes d’Euler des fibrés vectoriels. C. R. Math. Acad. Sci. 330, 287–290 (2000)

    MathSciNet  Google Scholar 

  6. Calmès B., Hornbostel J.: Push-forwards for Witt groups of schemes. Comment. Math. Helv. 86(2), 437–468 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gille S.: A note on the Witt group of \({\mathbb{P}^n}\). Math. Z. 237, 601–619 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Jardine J.F.: Motivic symmetric spectra. Doc. Math. 5, 445–552 (2000)

    MathSciNet  MATH  Google Scholar 

  9. Jouanolou, J. P.: Un suite exacte de Mayer–Vietoris en K-theorie algebrique. In: Algebraic K-theory I, Proceedings of the 1972 Battelle Institute Conference, Lecture Notes in Mathematics, No. 341, Springer, Berlin, pp. 293–316 (1973)

  10. Hu P.: On the Picard group of the stable \({\mathbb{A}^1}\)-homotopy category. Topology 44(3), 609–640 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Levine M., Morel F.: Algebraic Cobordism. Springer, Berlin (2007)

    MATH  Google Scholar 

  12. Morel, F.: An introduction to \({\mathbb{A}^1}\)-homotopy theory. In: Karoubi, M., Kuku, A.O., Pedrini, C. (eds.) Contemporary Developments in Algebraic K-Theory, ICTP Lecture Notes, vol. XV, pp. 357–441. Abdus Salam International Center for Theoretical Physics, Trieste (2004)

  13. Morel F., Voevodsky V.: \({\mathbb{A}^1}\)-homotopy theory of schemes. Publ. Math. IHES 90, 45–143 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Nenashev A.: Gysin maps in Balmer–Witt theory. J. Pure Appl. Algebra 211, 203–221 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Nenashev A.: Projective push-forwards in the Witt theory of algebraic varieties. Adv. Math. 220(6), 1923–1944 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Nenashev A.: On the Witt groups of projective bundles and split quadrics: geometric reasoning. J. K-Theory 3(3), 533–546 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Nenashev A., Zainoulline K.: Oriented cohomology and motivic decompositions of relative cellular spaces. J. Pure Appl. Algebra 205(2), 323–340 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Panin I.: Oriented cohomology theories of algebraic varieties. K-Theory 30, 265–314 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Panin, I.: Riemann–Roch theorems for oriented cohomology. In: Axiomatic, Enriched and Motivic Homotopy Theory, Kluwer Academic Publishers, Dordrecht, pp. 261—333 (2004)

  20. Panin I.: Oriented cohomology theories of algebraic varieties II. Homol. Homotopy Appl. 11(1), 349–405 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Panin, I., Walter, C.: Quaternionic Grassmannians and Pontryagin classes in algebraic geometry. arXiv:1011.0649

  22. Panin, I., Walter, C.: On the motivic commutative spectrum BO. arXiv:1011.0650

  23. Panin, I., Walter, C.: On the algebraic cobordism spectra MSL and MSp. arXiv:1011.0651

  24. Riou J.: Dualité de Spanier–Whitehead en géométrie algébrique. Comptes Rendus Math. 340(6), 431–436 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Schlichting M.: Hermitian K-theory of exact categories. J. K-theory 5(1), 105–165 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Smirnov A.: Orientations and transfers in cohomology of algebraic varieties. St. Petersb. Math. J. 18(2), 305–346 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Smirnov A.: RiemannRoch theorem for operations in cohomology of algebraic varieties. St. Petersb. Math. J. 18(5), 837–856 (2007)

    Article  MATH  Google Scholar 

  28. Voevodsky, V.: \({\mathbb{A}^1}\)-homotopy theory. Doc. Math. Extra vol. I, pp. 579–604 (1998)

  29. Walter, C.: Grothendieck-Witt groups of projective bundles. K-theory Preprint Archives. http://www.math.uiuc.edu/K-theory/0644 (2003). Accessed 9 Dec 2014

  30. Weibel C.: Homotopy algebraic K-theory. Contemp. Math. 83, 461–488 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zibrowius, M.: Witt groups of complex cellular varieties. Doc. Math. 16, 465–511 (2011)

    MathSciNet  MATH  Google Scholar 

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  1. Chebyshev Laboratory, St. Petersburg State University, 14th Line, 29b, Saint Petersburg, 199178, Russia

    Alexey Ananyevskiy

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  1. Alexey Ananyevskiy
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Ananyevskiy, A. On the push-forwards for motivic cohomology theories with invertible stable Hopf element. manuscripta math. 150, 21–44 (2016). https://doi.org/10.1007/s00229-015-0799-6

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  • DOI: https://doi.org/10.1007/s00229-015-0799-6

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