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Periodicity of hermitian K-groups

Published online by Cambridge University Press:  16 May 2011

A. J. Berrick ,
M. Karoubi  and
P. A. Østvær
A. J. Berrick
Affiliation:
Department of Mathematics, National University of Singapore, Singaporeberrick@math.nus.edu.sg
M. Karoubi
Affiliation:
UFR de Mathématiques, Université Paris 7, Francemax.karoubi@gmail.com
P. A. Østvær
Affiliation:
Department of Mathematics, University of Oslo, Norwaypaularne@math.uio.no
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Abstract

Bott periodicity for the unitary and symplectic groups is fundamental to topological K-theory. Analogous to unitary topological K-theory, for algebraic K-groups with finite coefficients, similar results are consequences of the Milnor and Bloch-Kato conjectures, affirmed by Voevodsky, Rost and others. More generally, we prove that periodicity of the algebraic K-groups for any ring implies periodicity for the hermitian K-groups, analogous to orthogonal and symplectic topological K-theory.

The proofs use in an essential way higher KSC-theories, extending those of Anderson and Green. They also provide an upper bound for the higher hermitian K-groups in terms of higher algebraic K-groups.

We also relate periodicity to étale hermitian K-groups by proving a hermitian version of Thomason's étale descent theorem. The results are illustrated in detail for local fields, rings of integers in number fields, smooth complex algebraic varieties, rings of continuous functions on compact spaces, and group rings.

Keywords

Bott periodicityhigher hermitian K-groupsKSC-theoriesétale cohomology
Type
Research Article
Information
Journal of K-Theory , Volume 7 , Issue 3 , June 2011 , pp. 429 - 493
Copyright
Copyright © ISOPP 2011

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References

1.Adams, J. F.. On the group J(X) IV. Topology 5:21–71;1966.CrossRefGoogle Scholar
2.Anderson, D. W.. The real K-theory of classifying spaces. Proc. Nat. Acad. Sci. 51:634–636, 1964.CrossRefGoogle ScholarPubMed
3.Araki, S. and Toda, H.. Multiplicative structures in mod q cohomology theories, I, II. Osaka J. Math. 2, 3:71–115, 81–120, 1965, 1966.Google Scholar
4.Bass, H.. Algebraic K-theory. W. A. Benjamin, Inc., New York-Amsterdam, 1968.Google Scholar
5.Berrick, A. J. and Karoubi, M.. Hermitian K-theory of the integers. Amer. J. Math. 127(4):785–823, 2005.CrossRefGoogle Scholar
6.Berrick, A. J., Karoubi, M., and Østvær, P. A.. Hermitian K-theory and 2-regularity for totally real number fields. Math. Annalen 349, 117–159, 2011.CrossRefGoogle Scholar
7.Berrick, A. J., Karoubi, M., Schlichting, M., Østvær, P. A.. The homotopy limit problem and étale hermitian K-theory, in preparation.Google Scholar
8.Boardman, J. M.. Conditionally convergent spectral sequences. In Homotopy invariant algebraic structures (Baltimore, MD, 1998). Contemp. Math. 239, 49–84. Amer. Math. Soc. Providence, RI, 1999.CrossRefGoogle Scholar
9.Bökstedt, M.. The rational homotopy type of ΩWhDiff(∗). Algebraic topology, Aarhus 1982 (Aarhus, 1982). Lecture Notes in Math. 1051, 25–37. Springer, Berlin, 1984.CrossRefGoogle Scholar
10.Borel, A.. Stable real cohomology of arithmetic groups. Ann. Sci. École Norm. Sup. 7(4):235–272 (1975), 1974.CrossRefGoogle Scholar
11.Bott, R.. The stable homotopy of the classical groups. Ann. of Math. 70(2):313–337, 1959.CrossRefGoogle Scholar
12.Bousfield, A. K.. The localization of spectra with respect to homology. Topology 18(4):257–281, 1979.CrossRefGoogle Scholar
13.Connes, A.. Noncommutative differential geometry. Academic Press, San Diego (1994).Google Scholar
14.Crabb, M. C. and Knapp, K.. Adams periodicity in stable homotopy. Topology 24(4):475–486, 1985.CrossRefGoogle Scholar
15.Dwyer, W. G. and Friedlander, E. M.. Algebraic and etale K-theory. Trans. Amer. Math. Soc. 292(1):247–280, 1985.Google Scholar
16.Dwyer, W. G., Friedlander, E. M., Snaith, V., and Thomason, R. W.. Algebraic K-theory eventually surjects onto topological K-theory. Invent. Math. 66(3):481–491, 1982.CrossRefGoogle Scholar
17.Feit, W., Thompson, J. G.. Solvability of groups of odd order. Pacific. J. Math. 13:775–1029, 1963.CrossRefGoogle Scholar
18.Fischer, T.. K-theory of function rings. J. Pure Appl. Algebra 69(1):33–50, 1990.CrossRefGoogle Scholar
19.Friedlander, E. M.. Computations of K-theories of finite fields. Topology 15(1):87–109, 1976.CrossRefGoogle Scholar
20.Green, P. S.. A cohomology theory based upon self-conjugacies of complex vector bundles. Bull. Amer. Math. Soc. 70:522–524, 1964.CrossRefGoogle Scholar
21.Haesemeyer, C. and Weibel, C.. Norm varieties and the chain lemma (after Markus Rost). Algebraic Topology, The Abel Symposium, 2007 4, 95–130. Springer, Berlin, 2009.CrossRefGoogle Scholar
22.Hiller, H. L.. Karoubi theory of finite fields. J. Pure Appl. Algebra 11(1-3):271–278, 1977/1978.CrossRefGoogle Scholar
23.Hodgkin, L. and Østvær, P. A.. The homotopy type of two-regular K-theory. In Categorical decomposition techniques in algebraic topology (Isle of Skye, 2001), Progr. Math. 215, 167–178. Birkhäuser, Basel, 2004.Google Scholar
24.Hornbostel, J.. Constructions and dévissage in Hermitian K-theory. K-Theory 26(2):139–170, 2002.CrossRefGoogle Scholar
25.Hu, P., Kriz, I., Ormsby, K.. Equivariant and motivic stable homotopy theory K-Preprint.Google Scholar
26.Jannsen, U.. Continuous étale cohomology. Math. Ann. 280(2):207–245, 1988.CrossRefGoogle Scholar
27.Jardine, J. F.. A rigidity theorem for L-theory. Preprint, 1983.Google Scholar
28.Jardine, J. F.. Supercoherence. J. Pure Appl. Algebra 75(2):103–194, 1991.CrossRefGoogle Scholar
29.Jardine, J. F.. Generalized étale cohomology theories, Progress in Mathematics 146, Birkhäuser Verlag, Basel, 1997.CrossRefGoogle Scholar
30.Karoubi, M.. Foncteurs dérivés et K-théorie. In Séminaire Heidelberg-Saarbrücken-Strasbourg sur la K-théorie (1967/68), Lecture Notes in Mathematics 136, 107–186. Springer, Berlin, 1970.CrossRefGoogle Scholar
31.Karoubi, M.. La périodicité de Bott en K-théorie générale. Ann. Sci. École Norm. Sup. 4(4):63–95, 1971.CrossRefGoogle Scholar
32.Karoubi, M.. Théorie de Quillen et homologie du groupe orthogonal Ann. of Math. (2), 112(2):207–257, 1980.CrossRefGoogle Scholar
33.Karoubi, M.. Le théorème fondamental de la K-théorie hermitienne. Ann. of Math. (2), 112(2):259–282, 1980.CrossRefGoogle Scholar
34.Karoubi, M.. Relations between algebraic K-theory and Hermitian K-theory. In Proceedings of the Luminy conference on algebraic K-theory (Luminy, 1983) 34, 259–263, 1984.Google Scholar
35.Karoubi, M.. Homologie cyclique et K-théorie. Astérisque 149:147, 1987.Google Scholar
36.Karoubi, M.. Periodicity of Hermitian K-theory and Milnor's K-groups. In Algebraic and arithmetic theory of quadratic forms, Contemp. Math. 344, 197–206. Amer. Math. Soc., Providence, RI, 2004.CrossRefGoogle Scholar
37.Karoubi, M.. Bott periodicity in topological, algebraic and Hermitian K-theory. In Handbook of K-theory 1, 111–137. Springer, Berlin, 2005.CrossRefGoogle Scholar
38.Karoubi, M.. Stabilization of the Witt group. C. R. Math. Acad. Sci. Paris 342(3):165–168, 2006.Google Scholar
39.Mitchell, S. A.. Hypercohomology spectra and Thomason's descent theorem Algebraic K-theory (Toronto, ON, 1996), Fields Inst. Commun. 16, 221–277. Amer. Math. Soc., Providence, RI, 1997.Google Scholar
40.Mitchell, S. A.. K-theory hypercohomology spectra of number rings at the prime 2. In Une dégustation topologique: homotopy theory in the Swiss Alps (Arolla, 1999), Contemp. Math. 265, 129–157. Amer. Math. Soc., Providence, RI, 2000.CrossRefGoogle Scholar
41.Neukirch, J., Schmidt, A., and Wingberg, K.. Cohomology of number fields, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 323, Springer-Verlag, Berlin, 2000.Google Scholar
42.Østvær, P. A.. Calculation of two-primary algebraic K-theory of some group rings. K-Theory 16(4):391–397, 1999.CrossRefGoogle Scholar
43.Østvær, P. A.. Étale descent for real number fields. Topology 42(1):197–225, 2003.CrossRefGoogle Scholar
44.Pedrini, C. and Weibel, C.. The higher K-theory of complex varieties. K-Theory 21(4):367–385, 2000. Special issues dedicated to Daniel Quillen on the occasion of his sixtieth birthday, Part V.CrossRefGoogle Scholar
45.Pedrini, C. and Weibel, C.. The higher K-theory of a complex surface. Compositio Math. 129(3):239–271, 2001.CrossRefGoogle Scholar
46.Prasolov, A. V.. Algebraic K-theory of Banach algebras. Dokl. Akad. Nauk BSSR 28(8):677–679, 1984.Google Scholar
47.Quillen, D.. On the cohomology and K-theory of the general linear groups over a finite field. Ann. of Math. 96(2):552–586, 1972.CrossRefGoogle Scholar
48.Rognes, J. and Østvær, P. A.. Two-primary algebraic K-theory of two-regular number fields. Math. Z. 233(2):251–263, 2000.CrossRefGoogle Scholar
49.Rognes, J. and Weibel, C.. Two-primary algebraic K-theory of rings of integers in number fields. J. Amer. Math. Soc. 13(1):1–54, 2000. Appendix A by Manfred Kolster.CrossRefGoogle Scholar
50.Rosenberg, J.. Comparison between algebraic and topological K-theory for Banach algebras and C*-algebras. In Handbook of K-theory 2, 843–874. Springer, Berlin, 2005.CrossRefGoogle Scholar
51.Rosenschon, A. and Østvær, P. A.. K-theory of curves over number fields. J. Pure Appl. Algebra 178(3):307–333, 2003.CrossRefGoogle Scholar
52.Rosenschon, A. and Østvær, P. A.. The homotopy limit problem for two-primary algebraic K-theory. Topology 44(6):1159–1179, 2005.CrossRefGoogle Scholar
53.Rosenschon, A. and Østvær, P. A.. Descent for K-theories. J. Pure Appl. Algebra 206(1-2):141–152, 2006.CrossRefGoogle Scholar
54.Rost, M.. Chain lemma for splitting fields of symbols. Preprint, 1998, www.math.uni-bielefeld.de/~rost/chain-lemma.html.Google Scholar
55.Rost, M.. Construction of splitting varieties. Preprint, 1998, http://www.math.uni-bielefeld.de/~rost/chain-lemma.html.Google Scholar
56.Schlichting, M.. Hermitian K-theory, derived equivalences and Karoubi's fundamental theorem. In preparation.Google Scholar
57.Snaith, V.. A descent theorem for Hermitian K-theory. Canad. J. Math. 39(4):835–847, 1987.CrossRefGoogle Scholar
58.Suslin, A.. Algebraic K-theory and motivic cohomology. In Proceedings of the International Congress of Mathematicians 1, 2, (Zürich, 1994), 342–351, Basel, 1995. Birkhäuser.CrossRefGoogle Scholar
59.Suslin, A. and Joukhovitski, S.. Norm varieties. J. Pure Appl. Algebra 206(1-2):245–276, 2006.CrossRefGoogle Scholar
60.Thomason, R. W.. Algebraic K-theory and étale cohomology. Ann. Sci. École Norm. Sup. (4), 18(3):437–552, 1985.CrossRefGoogle Scholar
61.Thomason, R. W. and Trobaugh, T.. Higher algebraic K-theory of schemes and of derived categories. In The Grothendieck Festschrift, Vol. III, Progr. Math. 88, 247–435. Birkhäuser Boston, Boston, MA, 1990.CrossRefGoogle Scholar
62.Voevodsky, V.. Motivic cohomology with Z/2-coefficients. Publ. Math. Inst. Hautes Études Sci. 98:59–104, 2003.CrossRefGoogle Scholar
63.Voevodsky, V.. Reduced power operations in motivic cohomology. Publ. Math. Inst. Hautes Études Sci. 98:1–57, 2003.CrossRefGoogle Scholar
64.Weibel, C.. The norm residue isomorphism theorem. J. Topol. 2:346–372, 2009.CrossRefGoogle Scholar
65.Weibel, C.. The 2-torsion in the K-theory of the integers. C. R. Acad. Sci. Paris Sér. I Math. 324(6):615–620, 1997.CrossRefGoogle Scholar
66.Weibel, C.. Bott periodicity for groups rings. Appendix to this paper. J. K-Theory 7, …, 2011.CrossRefGoogle Scholar