Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-12-01T00:14:02.608Z Has data issue: false hasContentIssue false
  • English
  • Français

Motivic cohomology of quadrics and the coniveau spectral sequence

Published online by Cambridge University Press:  05 November 2010

Nobuaki Yagita
Nobuaki Yagita
Affiliation:
Department of Mathematics, Faculty of Education, Ibaraki University, Mito, Ibaraki, Japan. [email protected]
Get access

Abstract

We study the coniveau spectral sequence for quadrics defined by Pfister forms. In particular, we explicitly compute the motivic cohomology of anisotropic quadrics over ℝ, by showing that their coniveau spectral sequences collapse from the -term

Keywords

real quadratic formRost's motivemotivic cohomologyconiveau spectral sequence
Type
Research Article
Information
Journal of K-Theory , Volume 6 , Issue 3 , December 2010 , pp. 547 - 589
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

Ba-Wa.Balmer, P. and Walter, C.. A Gersten-Witt spectral sequence for regular schemes. Ann. Sci. École. Norm. Sup. 35 (2002) 127–152.CrossRefGoogle Scholar
Bs-Ta.Bass, H. and Tate, J.. The Milnor ring of a global field. Lecture Notes in Math. 342 (1973) 349–446.Google Scholar
Bl-Og.Bloch, S. and Ogus, A.. Gersten's conjecture and the homology of schemes. Ann. Sci. École. Norm. Sup. 7 (1974) 181–202.CrossRefGoogle Scholar
CoTh-Sj.Colliot-Théléne, J. and Sujatha, R.. Unramified Witt groups of real anisotropic quadrics. K-theory and algebraic geometry: connections with quadratic forms and division algebras, Proc. Sympos. Pure Math. 58 (1995), 127–147.Google Scholar
Co.Cox, D.. The etale homotopy type of variety over ℝ. Proc. Amer. Math. Soc. 76 (1979), 17–22.Google Scholar
El-La.Elman, R. and Lam, T.. Pfister forms and K-theory of fields. J.Algebra. 23 (1972), 181–213.CrossRefGoogle Scholar
Ho.Hoffmann, D.. Isotropy of quadric forms over the function field of a quadric. Math. Z. 220 (1995), 461–476.CrossRefGoogle Scholar
Hu-Kr.Hu, P. and Kriz, I.. Some remarks on real and algebraic cobordism. K-theory. 22 (2001), 335–366.CrossRefGoogle Scholar
Ka1.Kahn, B.. La conjecture de Milnor (d'aprés V.Voevodsky). Séminaire Bourbaki, Vol. 1996/97. Astérisque 834 379–418.Google Scholar
Ka2.Kahn, B.. Motivic cohomology of smooth geometrically cellular varieties. Proc. Sympos. Pure Math. “Algebraic K-theory” (1997:University of Washington, Seattle) 67 (1999), 149–174.CrossRefGoogle Scholar
Ka-Ro-Sj.Kahn, B., Rost, M. and Sujatha, R.. Unramified cohomology of quadrics I. Amer.J.Math. 120 (1998), 841–891.CrossRefGoogle Scholar
Ka-Sj.Kahn, B. and Sujatha, R.. Motivic cohomology and unramified cohomology of quadrics. J. Eur. Math. Soc. 2 (2000), 145–177.CrossRefGoogle Scholar
Me-Su.Merkurjev, A. and Suslin, A.. K-cohomology of Severi-Brauer varieties and the norm residue homomorphism. Izv. Acad. Nauk. SSSR Ser. Math. 46 (1982), 1011–1046.Google Scholar
Mi.Milnor, J.. On the cobordism ring Ω and a complex analogue. I. Amer. J. Math. 82 (1960), 505–521.CrossRefGoogle Scholar
Or-Vi-Vo.Orlov, D., Vishik, A. and Voevodsky, V.. An exact sequence for Milnor's K-theory with applications to quadric forms. Ann of Math. 165 (2007), 1–13.CrossRefGoogle Scholar
Pa.Paranjape, K.. Some spectral sequences for filtered complexes and applications. J. Algebra 186 (1996) 793–806.CrossRefGoogle Scholar
Pd.Pardon, W.. The filtered Gersten-Witt complex for regular schemes. www.math.uiuc.edu/K-theory/0419 (2000).Google Scholar
Ro1.Rost, M.. Some new results on the Chow groups of quadrics. Preprint (1990).Google Scholar
Ro2.Rost, M.. The motive of a Pfister form. Preprint (1998).Google Scholar
Sj.Sujatha, R.. Unramified cohomology and Witt groups of anisotropic Pfister quadrics. Trans. Amer. Math. Soc. 349 (1997), 2341–2358.CrossRefGoogle Scholar
Su-Jo.Suslin, A. and Joukhovitski, S.. Norm Variety. J. Pure Appl. Algebra 206 (2006), 245–276.CrossRefGoogle Scholar
To1.Totaro, B.. The Chow ring of classifying spaces. Proc. Sympos. Pure Math. “Algebraic K-theory” (1997: University of Washington, Seattle) 67 (1999), 248–281.Google Scholar
To2.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
Vi.Vishik, A.. Motives of quadrics with applications to the theory of quadratic forms. Geometric methods in algebraic theory of quadratic forms, by Izhboldin, Kahn, Karpenko and Vishik. Lecture Notes in Math. 1835 (2004), 25–101.Google Scholar
Vi-Ya.Vishik, A. and Yagita, N.. Algebraic cobordisms of a Pfister quadric. J. London Math. Soc. 76 (2007), 586–604.CrossRefGoogle Scholar
Vo1.Voevodsky, V.. The Milnor conjecture. www.math.uiuc.edu/K-theory/0170 (1996).Google Scholar
Vo2.Voevodsky, V.. Voevodsky's Seattle lectures : K-theory and motivic cohomology. Notes by C. Weibel. Proc. Sympos. Pure Math. “Algebraic K-theory” (1997: University of Washington, Seattle) 67 (1999), 283–303.CrossRefGoogle Scholar
Vo3.Voevodsky, V.. Reduced power operations in motivic cohomology. Publ. Math. Inst. Hautes Études Sci. 98 (2003), 1–54.CrossRefGoogle Scholar
Vo4.Voevodsky, V.. Motivic cohomology with ℤ/2-coefficients. Publ. Math. Inst. Hautes Études Sci. 98 (2003), 59–104.CrossRefGoogle Scholar
Vo5.Voevodsky, V.. On motivic cohomology with ℤ/ℓ-coefficient. www.math.uiuc.edu/K-theory/0639 (2003).Google Scholar
Vo6.Voevodsky, V.. Motivic Eilenberg-MacLane spaces. www.math.uiuc.edu/Ktheory/0864 (2007).Google Scholar
Ya1.Yagita, N.. Examples for the mod p motivic cohomology of classifying spaces. Trans. Amer. Math. Soc. 355 (2003), 4427–4450.CrossRefGoogle Scholar
Ya2.Yagita, N.. Applications of Atiyah-Hirzebruch spectral sequences for motivic cobordism. Proc. London Math. Soc. 90 (2005), 783–816.CrossRefGoogle Scholar