Tanania, Fabio

Subtle Characteristic Classes and Hermitian Forms

Doc. Math. 24, 2493-2523 (2019)
DOI: 10.25537/dm.2019v24.2493-2523
Communicated by Nikita Karpenko

Summary

Following \textit{A. Smirnov} and \textit{A. Vishik} [``Subtle characteristic classes'', Preprint, \url{arXiv:1401.6661v1}], we compute the motivic cohomology ring of the Nisnevich classifying space of the unitary group associated to the standard split hermitian form of a quadratic extension. This provides us with subtle characteristic classes which take value in the motivic cohomology of the Čech simplicial scheme associated to a hermitian form. Comparing these new classes with subtle Stiefel-Whitney classes arising in the orthogonal case, we obtain relations among the latter ones holding in the motivic cohomology of the Čech simplicial scheme associated to a quadratic form divisible by a 1-fold Pfister form. Moreover, we present a description of the motive of the torsor corresponding to a hermitian form in terms of its subtle characteristic classes.

Mathematics Subject Classification

11E39, 14F42, 20G15, 55R40

Keywords/Phrases

Hermitian forms, motivic cohomology, Nisnevich classifying space, characteristic classes

References

  • 1. T. Bachmann, On the invertibility of motives of affine quadrics, Doc. Math. 22 (2017), 363-395. https://www.elibm.org/article/10000443; zbl 1420.14011; MR3628786.
  • 2. T. Bachmann, A. Vishik, Motivic equivalence of affine quadrics, Math. Ann. 371 (2018), no. 1-2, 741-751. DOI 10.1007/s00208-018-1641-8; zbl 1391.14039; MR3788865; arxiv 1707.08087.
  • 3. P. Deligne, Théorie de Hodge. III, Publications Mathématiques de l'IHÉS, no. 44 (1974), 5-77. DOI 10.1007/BF02685881; zbl 0237.14003; MR0498552.
  • 4. S. I. Gelfand, Y. I. Manin, Methods of homological algebra, Translated from the 1988 Russian original. Springer-Verlag, Berlin, 1996. zbl 0855.18001; MR1438306.
  • 5. P. G. Goerss, J. F. Jardine, Simplicial homotopy theory, Progress in Mathematics, 174. Birkhäuser Verlag, Basel, 1999. zbl 0949.55001; MR1711612.
  • 6. N. A. Karpenko, Unitary Grassmannians, J. Pure Appl. Algebra 216 (2012), no. 12, 2586-2600. DOI 10.1016/j.jpaa.2012.03.024; zbl 1298.14048; MR2943741; arxiv 1204.0379.
  • 7. M.-A. Knus, A. Merkurjev, M. Rost, J.-P. Tignol, The book of involutions, American Mathematical Society Colloquium Publications, 44. American Mathematical Society, Providence, RI, 1998. zbl 0955.16001; MR1632779.
  • 8. C. Mazza, V. Voevodsky, C. Weibel, Lecture notes on motivic cohomology, Clay Mathematics Monographs, 2. American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2006. zbl 1115.14010; MR2242284.
  • 9. F. Morel, V. Voevodsky, \(A^1\)-homotopy theory of schemes, Publications Mathématiques de l'IHÉS, no. 90 (1999), 45-143. DOI 10.1007/BF02698831; zbl 0983.14007; MR1813224.
  • 10. M. Ojanguren, I. Panin, Rationally trivial Hermitian spaces are locally trivial, Math. Z. 237 (2001), no. 1, 181-198. DOI 10.1007/PL00004859; zbl 1042.11024; MR1836777.
  • 11. D. Orlov, A. Vishik, V. Voevodsky, An exact sequence for \(K^M_* /2\) with applications to quadratic forms, Ann. Math. 165 (2007), 1-13. DOI 10.4007/annals.2007.165.1; zbl 1124.14017; MR2276765.
  • 12. M. Rost, The motive of a Pfister form, Preprint, 1998. Available at: https://www.math.uni-bielefeld.de/\( \sim\) rost/motive.html.
  • 13. W. Scharlau, Quadratic and Hermitian forms, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 270. Springer-Verlag, Berlin, 1985. zbl 0584.10010; MR0770063.
  • 14. A. Smirnov, A. Vishik, Subtle Characteristic Classes, Preprint, 2014. arxiv 1401.6661.
  • 15. A. Vishik, On the Chow groups of quadratic Grassmannians, Doc. Math. 10 (2005), 111-130. https://www.elibm.org/article/10000061; zbl 1115.14002; MR2148072.
  • 16. V. Voevodsky, Motives over simplicial schemes, J. K-Theory 5 (2010), no. 1, 1-38. DOI 10.1017/is010001030jkt107; zbl 1194.14029; MR2600283; arxiv 0805.4431.
  • 17. V. Voevodsky, Motivic cohomology with Z/2-coefficients, Publications Mathématiques de l'IHÉS, no. 98, pp. 59-104, 2003. DOI 10.1007/s10240-003-0010-6; zbl 1057.14028; MR2031199.
  • 18. V. Voevodsky, Triangulated categories of motives over a field, Cycles, transfers, and motivic homology theories, 188-238, Ann. of Math. Stud., 143, Princeton Univ. Press, Princeton, NJ, 2000. zbl 1019.14009; MR1764202.
  • 19. N. Yagita, Applications of Atiah-Hirzebruch spectral sequences for motivic cobordisms, Proc. London Math. Soc. 90 (2005), 783-816. DOI 10.1112/S0024611504015084; zbl 1086.55005; MR2137831.

Affiliation

Tanania, Fabio
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK

Downloads