Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Iwasawa theory for $ K(1)$-local spectra

Authors: Rebekah Hahn and Stephen Mitchell
Journal: Trans. Amer. Math. Soc. 359 (2007), 5207-5238
MSC (2000): Primary 55N15, 19L20
Published electronically: June 4, 2007
MathSciNet review: 2327028
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Iwasawa algebra $ \Lambda$ is a power series ring in one variable over the $ p$-adic integers. It has long been studied by number theorists in the context of $ \mathbb{Z}_p$-extensions of number fields. It also arises, however, as a ring of operations in $ p$-adic topological $ K$-theory. In this paper we study $ K(1)$-local stable homotopy theory using the structure theory of modules over the Iwasawa algebra. In particular, for $ p$ odd we classify $ K(1)$-local spectra up to pseudo-equivalence (the analogue of pseudo-isomorphism for $ \lambda$-modules) and give an Iwasawa-theoretic classification of the thick subcategories of the weakly dualizable spectra.

References [Enhancements On Off] (What's this?)

  • 1. Adams, J.F., Stable Homotopy Theory Lecture, Lecture Notes in Mathematics, Vol. 3, Springer, New York, 1964. MR 0185597 (32:3061)
  • 2. Bousfield, A.K., The localization of spectra with respect to homology, Topology 18 (1979), 257-281. MR 0551009 (80m:55006)
  • 3. Bousfield, A.K., On the homotopy theory of K-local spectra at an odd prime, Amer. J. Math. 107 (1985), 895-932. MR 0796907 (87c:55010)
  • 4. Bousfield, A.K., A classification of K-local spectra, J. Pure Appl. Algebra 66 (1990), 120-163. MR 1075335 (92d:55003)
  • 5. Bousfield, A.K., and Kan, D., Homotopy limits, completions and localizations, Springer Lecture Notes in Math. Vol. 304 (2nd corrected printing), Springer-Verlag, Berlin-Heidelberg-New York, 1987. MR 0365573 (51:1825)
  • 6. Devinatz, E., Morava modules and Brown-Comenetz duality, Amer. J. Math. 119 (1997), 741-770. MR 1465068 (98i:55008)
  • 7. Franke, J., Uniqueness theorems for certain triangulated categories possessing an Adams spectral sequence, preprint,
  • 8. Hahn, R., University of Washington Ph.D. thesis, 2003.
  • 9. Hopkins, M., Global methods in homotopy theory, Proceedings of the 1985 LMS Symposium on Homotopy Theory, J.D.S. Jones and E. Rees, editors, pp. 73-96, 1987. MR 0932260 (89g:55022)
  • 10. Hopkins, M., Mahowald, M., and Sadofsky, H., Constructions of elements in Picard groups, Topology and Representation Theory (E. Friedlander and M. Mahowald, eds.), Contemp. Math. 158 (1994), 89-126. MR 1263713 (95a:55020)
  • 11. Hovey, M., Some spectral sequences in Morava E-theory, preprint 2004.
  • 12. Hovey, M., and Strickland, N., Morava K-theories and localisation, Mem. Amer. Math. Soc. 139 (1999), no. 666, x+100. MR 1601906 (99b:55017)
  • 13. Hovey, M., Palmieri, J., and Strickland, N., Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128 (1997), no. 610, x+114. MR 1388895 (98a:55017)
  • 14. Madsen, I., Snaith, V., and Tornehave, J., Infinite loop maps in geometric topology, Math. Proc. Camb. Phil. Soc. 81 (1977), 399-430. MR 0494076 (58:13007)
  • 15. Mitchell, S.A., On p-adic topological K-theory, in Algebraic K-Theory and Algebraic Topology, P.G. Goerss and J.F. Jardine, editors, Kluwer Academic Publishers 1993, 197-204. MR 1367298 (96h:19006)
  • 16. Mitchell, S.A., $ K(1)$-local homotopy theory, Iwasawa theory and algebraic K-theory, Handbook of Algebraic K-theory, edited by E. Friedlander and D. Grayson, vol. 2, pp. 955-1010 Springer, 2005. MR 2181837 (2006g:11220)
  • 17. Neukirch, J., Schmidt, A., and Wingberg, K., Cohomology of Number Fields, Springer, 2000. MR 1737196 (2000j:11168)
  • 18. Ravenel, D.C., Localization with respect to certain periodic homology theories, Amer. J. Math. 106 (1984), 351-414. MR 0737778 (85k:55009)
  • 19. Serre, J.P., Cohomologie des groupes discrets, in Prospects in Mathematics, Annals of Math. Studies 70 (1971), 77-169. MR 0385006 (52:5876)
  • 20. Serre, J.P, Linear Representations of Finite Groups, Springer-Verlag, 1977. MR 0450380 (56:8675)
  • 21. Washington, L., Introduction to Cyclotomic Fields, Springer-Verlag, l982. MR 0718674 (85g:11001)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 55N15, 19L20

Retrieve articles in all journals with MSC (2000): 55N15, 19L20

Additional Information

Rebekah Hahn
Affiliation: 6805 Windhaven Parkway, S126, The Colony, Texas 75056

Stephen Mitchell
Affiliation: Department of Mathematics, University of Washington, P.O. Box 354350, Seattle, Washington 98195-0001

Keywords: K-theory, homotopy theory, Iwasawa algebra
Received by editor(s): June 14, 2005
Published electronically: June 4, 2007
Additional Notes: The second author was supported by a grant from the National Science Foundation
Article copyright: © Copyright 2007 American Mathematical Society

American Mathematical Society