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

Published online by Cambridge University Press:  22 March 2011

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

For a complex Grassmannian X, there is an isomorphism

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

Type
Research Article
Copyright
Copyright © ISOPP 2011

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References

At.Atiyah, M.. K-theory and reality. Quart. J. Oxford 17 (1966) 367386.Google Scholar
At-Bo-Sh.Atiyah, M., Bott, R. and Shapiro, A.. Clifford modules. Topology 3 (1964) 338.CrossRefGoogle Scholar
Ba.Balmer, P.. Witt groups. Handbook of K-theory, 539576, 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) 525550.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) 127152.CrossRefGoogle Scholar
Br.Brosnan, P.. Steenrod operations in Chow theory. Trans. of Amer. Math. Soc. 355 (2003) 18691903.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) 141149.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) 391419.CrossRefGoogle Scholar
Ha.Hara, S.. Note on KO-theory of BO(n) and BU(n). J. Math. Kyoto Univ. 31 (1991) 487493.Google Scholar
Ho.Hornbostel, J.. -representability of hermitian K-theory and Witt group. Topology. 44 (2005) 661687.CrossRefGoogle Scholar
Ko-Ha.Kono, A. and Hara, S.. KO-theory of complex Grassmannian. J. Math. Kyoto Univ. 31 (1991) 827833.Google Scholar
La.Laksov, D.. Algebraic cycles on Grassmann varieties. Adv. in Math. 9 (1972) 267295.CrossRefGoogle Scholar
Ne.Nenashev, A.. Gysin maps in Balmer-Witt theory. J. Pure and Appl. Algebra. 211 (2007) 203221.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) 113.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) 483493.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) 154.Google Scholar
Ya.Yagita, N.. Motivic cohomology of quadrics and the coniveau spectral sequence. J. K-theory 6 (2010) 547589.CrossRefGoogle Scholar