Iwasawa theory for 𝐾(1)-local spectra

Hahn, Rebekah; Mitchell, Stephen

doi:10.1090/S0002-9947-07-04204-3

Iwasawa theory for $K(1)$-local spectra
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by Rebekah Hahn and Stephen Mitchell PDF

Trans. Amer. Math. Soc. 359 (2007), 5207-5238 Request permission

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

J. Frank Adams, Stable homotopy theory, Springer-Verlag, Berlin-Göttingen-Heidelberg-New York, 1964. Lectures delivered at the University of California at Berkeley, 1961; Notes by A. T. Vasquez. MR 0185597
A. K. Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), no. 4, 257–281. MR 551009, DOI 10.1016/0040-9383(79)90018-1
A. K. Bousfield, On the homotopy theory of $K$-local spectra at an odd prime, Amer. J. Math. 107 (1985), no. 4, 895–932. MR 796907, DOI 10.2307/2374361
A. K. Bousfield, A classification of $K$-local spectra, J. Pure Appl. Algebra 66 (1990), no. 2, 121–163. MR 1075335, DOI 10.1016/0022-4049(90)90082-S
A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR 0365573
Ethan S. Devinatz, Morava modules and Brown-Comenetz duality, Amer. J. Math. 119 (1997), no. 4, 741–770. MR 1465068
Franke, J., Uniqueness theorems for certain triangulated categories possessing an Adams spectral sequence, preprint, http://www.math.uiuc.edu/K–theory/0139/
Hahn, R., University of Washington Ph.D. thesis, 2003.
Michael J. Hopkins, Global methods in homotopy theory, Homotopy theory (Durham, 1985) London Math. Soc. Lecture Note Ser., vol. 117, Cambridge Univ. Press, Cambridge, 1987, pp. 73–96. MR 932260
Michael J. Hopkins, Mark Mahowald, and Hal Sadofsky, Constructions of elements in Picard groups, Topology and representation theory (Evanston, IL, 1992) Contemp. Math., vol. 158, Amer. Math. Soc., Providence, RI, 1994, pp. 89–126. MR 1263713, DOI 10.1090/conm/158/01454
Hovey, M., Some spectral sequences in Morava E-theory, preprint 2004.
Mark Hovey and Neil P. Strickland, Morava $K$-theories and localisation, Mem. Amer. Math. Soc. 139 (1999), no. 666, viii+100. MR 1601906, DOI 10.1090/memo/0666
Mark Hovey, John H. Palmieri, and Neil P. Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128 (1997), no. 610, x+114. MR 1388895, DOI 10.1090/memo/0610
I. Madsen, V. Snaith, and J. Tornehave, Infinite loop maps in geometric topology, Math. Proc. Cambridge Philos. Soc. 81 (1977), no. 3, 399–430. MR 494076, DOI 10.1017/S0305004100053482
Stephen A. Mitchell, On $p$-adic topological $K$-theory, Algebraic $K$-theory and algebraic topology (Lake Louise, AB, 1991) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 407, Kluwer Acad. Publ., Dordrecht, 1993, pp. 197–204. MR 1367298, DOI 10.1007/978-94-017-0695-7_{9}
Stephen A. Mitchell, $K(1)$-local homotopy theory, Iwasawa theory and algebraic $K$-theory, Handbook of $K$-theory. Vol. 1, 2, Springer, Berlin, 2005, pp. 955–1010. MR 2181837, DOI 10.1007/978-3-540-27855-9_{1}9
Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg, Cohomology of number fields, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323, Springer-Verlag, Berlin, 2000. MR 1737196
Douglas C. Ravenel, Localization with respect to certain periodic homology theories, Amer. J. Math. 106 (1984), no. 2, 351–414. MR 737778, DOI 10.2307/2374308
Jean-Pierre Serre, Cohomologie des groupes discrets, Prospects in mathematics (Proc. Sympos., Princeton Univ., Princeton, N.J., 1970) Ann. of Math. Studies, No. 70, Princeton Univ. Press, Princeton, N.J., 1971, pp. 77–169 (French). MR 0385006
Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott. MR 0450380
Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR 718674, DOI 10.1007/978-1-4684-0133-2