Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-23T03:22:24.775Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Witt groups of projective bundles and split quadrics: geometric reasoning

Published online by Cambridge University Press:  23 July 2008

Alexander Nenashev
Alexander Nenashev
Affiliation:
Department of Mathematics, York University – Glendon College, 2275 Bayview Av., Toronto, ON, Canada M4N 3M6, [email protected].
Get access

Abstract

Formulas for the derived Witt groups of projective bundles are obtained. We deduce them from general properties of Witt theory, with the help of twisted Thom isomorphisms, avoiding explicit use of triangulated categories. Witt groups of completely split quadrics are also considered.

Keywords

Triangulated and derived Witt groupsprojective bundleThom isomorphism
Type
Research Article
Information
Journal of K-Theory , Volume 3 , Issue 3 , June 2009 , pp. 533 - 546
Copyright
Copyright © ISOPP 2009

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

1.Arason, J.: Der Wittring projektiver Räume Math. Ann. 253 (1980), 205–212CrossRefGoogle Scholar
2.Balmer, P.: Triangular Witt groups. I: The 12-term exact sequence K-Theory 19 (2000), 311–363CrossRefGoogle Scholar
3.Balmer, P.: Triangular Witt groups. II: From usual to derived Math. Z. 236 (2001), 351–382CrossRefGoogle Scholar
4.Balmer, P.: Witt cohomology, Mayer-Vietoris, homotopy invariance and the Gersten conjecture K-Theory 23 (2001), 15–30CrossRefGoogle Scholar
5.Balmer, P.: Products of degenerate quadratic forms Compositio Mathematica 141 (2005), 1374–1404Google Scholar
6.Balmer, P. and Gille, S.: Koszul complexes and symmetric forms over the punctured affine space Proc. London Math. Soc. 91 (2005), 273–299CrossRefGoogle Scholar
7.Baum, P., Fulton, W., and MacPherson, R.: Riemann-Roch and topological K-theory for singular varieties Acta Math. 143 (1979), 155–192CrossRefGoogle Scholar
8.Bernstein, J., Gelfand, I., and Gelfand, S.: Algebraic vector bundles on and problems of linear algebra Functional Anal. Appl. 12 (1978), 212–214CrossRefGoogle Scholar
9.Fulton, W.: Intersection Theory Springer-Verlag 1984CrossRefGoogle Scholar
10.Gille, S.: A note on the Witt group of Math. Z. 237 (2001), 601–619CrossRefGoogle Scholar
11.Gille, S.: A transfer morphism for Witt groups J. reine angew. Math. 564 (2003), 215–233Google Scholar
12.Gille, S.: On Witt groups with support Math. Ann. 322 (2002), 103–137CrossRefGoogle Scholar
13.Gille, S.: Homotopy invariance of coherent Witt groups Math. Zeitschrift 244 (2003), 211–233CrossRefGoogle Scholar
14.Gille, S.: The general dévissage theorem for Witt groups of schemes Archiv Math. 88 (2007), 333–343CrossRefGoogle Scholar
15.Gille, S. and Nenashev, A.: Pairings in triangular Witt theory J. Algebra 261 (2003), 292–309CrossRefGoogle Scholar
16.Karpenko, N.: Algebro-geometric invariants of quadratic forms Leningrad (St. Pet.) Math. J. 2 issue 1 (1991), 119–138Google Scholar
17.Nenashev, A.: Projective bundle theorem in homology theories with Chern structure Documenta Math. 9 (2004), 487–497CrossRefGoogle Scholar
18.Nenashev, A.: Gysin maps in oriented theories J. of Algebra 302 (2006), 200–213CrossRefGoogle Scholar
19.Nenashev, A.: Gysin maps in Balmer-Witt theory J. Pure Appl. Algebra 211(2007), 203–221CrossRefGoogle Scholar
20.Nenashev, A. and Zainoulline, K.: Oriented theories of relative cellular spaces J. Pure Appl. Algebra 205 issue 2 (2006), 323–340CrossRefGoogle Scholar
21.Panin, I. (after I. Panin and A. Smirnov): Oriented cohomology theories of algebraic varieties K-Theory 30(2003), 263–312CrossRefGoogle Scholar
22.Pimenov, K.: Traces in oriented homology theories, II preprint at http://www.math.uiuc.edu/K-theory/0724/ (2005)Google Scholar
23.Walter, C.: Grothendieck-Witt groups of projective bundles, preprint at http://www.math.uiuc.edu/K-theory/0645/ (2003)Google Scholar