Hostname: page-component-77c89778f8-vpsfw Total loading time: 0 Render date: 2024-07-17T17:59:11.742Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on the Witt group and the KO-theory of complex Grassmannians

Published online by Cambridge University Press:  22 March 2011

Nobuaki Yagita
Nobuaki Yagita
Affiliation:
Department of Mathematics, Faculty of Education, Ibaraki University Mito, Ibaraki, Japan, yagita@mx.ibaraki.ac.jp
Get access

Abstract

For a complex Grassmannian X, there is an isomorphism

between Balmer's Witt group and the quotient of topological K-theories.

Keywords

GrassmannianWitt groupK-theory
Type
Research Article
Information
Journal of K-Theory , Volume 9 , Issue 1 , February 2012 , pp. 161 - 175
Copyright
Copyright © ISOPP 2011

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

At.Atiyah, M.. K-theory and reality. Quart. J. Oxford 17 (1966) 367–386.Google Scholar
At-Bo-Sh.Atiyah, M., Bott, R. and Shapiro, A.. Clifford modules. Topology 3 (1964) 3–38.CrossRefGoogle Scholar
Ba.Balmer, P.. Witt groups. Handbook of K-theory, 539–576, Springer, Berlin 2005.Google Scholar
Ba-Ca1.Balmer, P. and Calmès, B.Geometric description of the connecting homomorphism for Witt groups. Doc. Math. 14 (2009) 525–550.CrossRefGoogle Scholar
Ba-Ca2.Balmer, P. and Calmès, B.. Witt groups of Grassmann varieties. www.math.uiuc.edu/K-theory/903 (2008).Google Scholar
Ba-Wa.Balmer, P. and Walter, C.. A Gersten-Witt spectral sequence for regular schemes. Ann.Scient.Éc.Norm.Sup. 35 (2002) 127–152.CrossRefGoogle Scholar
Br.Brosnan, P.. Steenrod operations in Chow theory. Trans. of Amer. Math. Soc. 355 (2003) 1869–1903.CrossRefGoogle Scholar
Ca-Ho.Calmès, B. and Hornbostel, J.. Witt motives, transfers and devissage. preprint. (2006).Google Scholar
Fj.Fujii, M.KO-group of projective spaces. Osaka J, Math. 4 (1967) 141–149.Google Scholar
Fu.Fulton, W.. Intersection theory, second ed. Ergebnisse der Mathematic und ihrer Grenzgebiete. 3 Springer-Verlag, Berlin. (1998).Google Scholar
Gi.Gille, S.. A graded Gersten-Witt complex for schemes with a dualizing complex and the Chow group. J. Pure and Apllied Algebra 208 (2007) 391–419.CrossRefGoogle Scholar
Ha.Hara, S.. Note on KO-theory of BO(n) and BU(n). J. Math. Kyoto Univ. 31 (1991) 487–493.Google Scholar
Ho.Hornbostel, J.. -representability of hermitian K-theory and Witt group. Topology. 44 (2005) 661–687.CrossRefGoogle Scholar
Ko-Ha.Kono, A. and Hara, S.. KO-theory of complex Grassmannian. J. Math. Kyoto Univ. 31 (1991) 827–833.Google Scholar
La.Laksov, D.. Algebraic cycles on Grassmann varieties. Adv. in Math. 9 (1972) 267–295.CrossRefGoogle Scholar
Ne.Nenashev, A.. Gysin maps in Balmer-Witt theory. J. Pure and Appl. Algebra. 211 (2007) 203–221.Google Scholar
Or-Vi-Vo.Orlov, D., Vishik, A. and Voevodsky, V.. An exact sequence for Milnor's K-theory with applications to quadratic forms. Ann of Math. 165 (2007) 1–13.CrossRefGoogle Scholar
Pa.Pardon, W.. The filtered Gersten-Witt complex for regular schemes. www.math.uiuc.edu/K-theory/0419 (2000).Google Scholar
To.Totaro, B.. Non-injectivity of the map from the Witt group of a variety to the Witt group of its function field. J. Inst. Math. Jussieu 2 (2003) 483–493.CrossRefGoogle Scholar
Vo1.Voevodsky, V.. The Milnor conjecture. www.math.uiuc.edu/K-theory/0170 (1996).Google Scholar
Vo2.Voevodsky, V.. Motivic cohomology are isomorphic to higher Chow groups. www.math.uiuc.edu/K-theory/378 (1999).Google Scholar
Vo3.Voevodsky, V.. Reduced power operations in motivic cohomology. Publ.Math. IHES. 98 (2003) 1–54.Google Scholar
Ya.Yagita, N.. Motivic cohomology of quadrics and the coniveau spectral sequence. J. K-theory 6 (2010) 547–589.CrossRefGoogle Scholar