Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-28T02:17:19.506Z Has data issue: false hasContentIssue false
  • English
  • Français

Twisted Witt Groups of Flag Varieties

Published online by Cambridge University Press:  17 April 2014

Marcus Zibrowius
Marcus Zibrowius*
Affiliation:
Bergische Universität Wuppertal, Gaußstraße 20, 42119 Wuppertal, [email protected]
Get access

Abstract

Calmès and Fasel have shown that the twisted Witt groups of split flag varieties vanish in a large number of cases. For flag varieties over algebraically closed fields, we sharpen their result to an if-and-only-if statement. In particular, we show that the twisted Witt groups vanish in many previously unknown cases. In the non-zero cases, we find that the twisted total Witt group forms a free module of rank one over the untwisted total Witt group, up to a difference in grading.

Our proof relies on an identification of the Witt groups of flag varieties with the Tate cohomology groups of their K-groups, whereby the verification of all assertions is eventually reduced to the computation of the (twisted) Tate cohomology of the representation ring of a parabolic subgroup.

Keywords

Witt groupsflag varietieshomogeneous spacesTate cohomologyparabolic subgroupsPrimary: 18F25Secondary: 19G99, 20G05
Type
Research Article
Information
Journal of K-Theory , Volume 14 , Issue 1 , August 2014 , pp. 139 - 184
Copyright
Copyright © ISOPP 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1.Adams, J. Frank, Lectures on Lie groups, W. A. Benjamin, Inc., New York-Amsterdam, 1969.Google Scholar
2.Balmer, Paul, Triangular Witt groups. I. The 12-term localization exact sequence, K-Theory 19(4) (2000), 311–363.Google Scholar
3.Balmer, Paul, Triangular Witt groups. II. From usual to derived, Math. Z. 236(2) (2001), 351–382.Google Scholar
4.Balmer, Paul, Calmès, Baptiste, Witt groups of Grassmann varieties, J. Algebraic Geom. 21(4) (2012), 601–642.Google Scholar
5.Balmer, Paul, Calmès, Baptiste, Bases of total Witt groups and lax-similitude, J. Algebra Appl. 11(3) (2012), 1250045, 24.CrossRefGoogle Scholar
6.Balmer, Paul, Walter, Charles, A Gersten-Witt spectral sequence for regular schemes, Ann. Sci. École Norm. Sup. (4) 35(1) (2002), 127–152.Google Scholar
7.Balmer, Paul, Gille, Stefen, Koszul complexes and symmetric forms over the punctured affine space, Proc. London Math. Soc. (3) 91(2) (2005), 273–299.Google Scholar
8.Bourbaki, Nikolas, Lie groups and Lie algebras. Chapters 4–6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by Pressley, Andrew.Google Scholar
9.Bousfield, Aldridge K., A classification of K-local spectra, J. Pure Appl. Algebra 66(2) (1990), 121–163.CrossRefGoogle Scholar
10.Bousfield, Aldridge K., On the 2-primary vi-periodic homotopy groups of spaces, Topology 44(2) (2005), 381–413.Google Scholar
11.Calmès, Baptiste, Fasel, Jean, Trivial Witt groups of flag varieties, J. Pure Appl. Algebra 216(2) (2012), 404–406.Google Scholar
12.Calmès, Baptiste, Hornbostel, Jens, Witt motives, transfers and reductive groups, (2004), available at http://www.mathematik.uni-bielefeld.de/LAG/man/143.html.Google Scholar
13.Cartan, Henri, Eilenberg, Samuel, Homological algebra, Princeton University Press, Princeton, N. J., 1956.Google Scholar
14.Fulton, William, Harris, Joe, Representation theory, Graduate Texts in Mathematics 129, Springer-Verlag, New York, 1991.Google Scholar
15.Gille, Stefen, Nenashev, Alexander, Pairings in triangular Witt theory, J. Algebra 261(2), (2003), 292–309.Google Scholar
16.Hodgkin, Luke, The equivariant Künneth theorem in K-theory, Topics in K-theory. Two independent contributions, pp. 1–101. Lecture Notes in Math. 496, Springer, Berlin, 1975.Google Scholar
17.Jantzen, Jens Carsten, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs 107, American Mathematical Society, Providence, RI, 2003.Google Scholar
18.Karoubi, Max, Le théorème fondamental de la K-théorie hermitienne, Ann. of Math. (2) 112(2), (1980), 259–282.Google Scholar
19.Köck, Bernhard, Chow motif and higher Chow theory of G/P, Manuscripta Math. 70(4), (1991), 363–372.Google Scholar
20.Merkurjev, Alexander S., Tignol, Jean-Pierre, The multipliers of similitudes and the Brauer group of homogeneous varieties, J. Reine Angew. Math. 461 (1995), 13–47.Google Scholar
21.Nenashev, Alexander, Gysin maps in Balmer-Witt theory, J. Pure Appl. Algebra 211(1) (2007), 203–221.Google Scholar
22.Panin, Ivan A., On the algebraic K-theory of twisted flag varieties, K-Theory 8(6) (1994), 541–585.Google Scholar
23.Quebbemann, Heinz-Georg, Scharlau, Winfried, Schulte, Manfred, Quadratic and Hermitian forms in additive and abelian categories, J. Algebra 59(2) (1979), 264–289.Google Scholar
24.Serre, Jean-Pierre, Groupes de Grothendieck des schémas en groupes réductifs déployés, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 37–52.Google Scholar
25.Théorie des intersections et théorème de Riemann-Roch, Lecture Notes in Mathematics 225, Springer-Verlag, Berlin, 1971. Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6); Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre.Google Scholar
26.Steinberg, Robert, On a theorem of Pittie, Topology 14 (1975), 173–177.Google Scholar
27.Walter, Charles, Grothendieck-Witt groups of triangulated categories (2003), available at http://www.math.uiuc.edu/K-theory/0643/.Google Scholar
28.Walter, Charles, Grothendieck-Witt groups ofprojective bundles (2003), available at http://www.math.uiuc.edu/K-theory/0644/.Google Scholar
29.Yagita, Nobuaki, Witt groups of algebraic groups (2011), available at http://www.mathematik.uni-bielefeld.de/LAG/man/430.html.Google Scholar
30.Zibrowius, Marcus, Witt groups of complex cellular varieties, Documenta Math. 16 (2011), 465–511.Google Scholar
31.Zibrowius, Marcus, KO-Rings of Full Flag Varieties (2012), available at arXiv:1208.1497, To appear in Trans. Amer. Math. Soc.Google Scholar