The chromatic convergence theorem and a tower in algebraic 𝐾-theory

McClure, J. E.; Staffeldt, R. E.

doi:10.1090/S0002-9939-1993-1164148-5

The chromatic convergence theorem and a tower in algebraic $K$-theory
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by J. E. McClure and R. E. Staffeldt PDF

Proc. Amer. Math. Soc. 118 (1993), 1005-1012 Request permission

Abstract:

In this note we show how the chromatic convergence theorem of Hopkins and Ravenel implies that a tower of relative algebraic $K$-theories constructed by Waldhausen converges to the $p$-local part of the algebraic $K$-theory of the one-point space relative to the $K$-theory of the integers. The notion of convergence used here is made precise using the language of pro-homotopy theory.

References

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 and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR 0365573, DOI 10.1007/978-3-540-38117-4

A proof of the smash product conjecture

Jean-Louis Loday, $K$-théorie algébrique et représentations de groupes, Ann. Sci. École Norm. Sup. (4) 9 (1976), no. 3, 309–377 (French). MR 447373, DOI 10.24033/asens.1312
J. Peter May, Simplicial objects in algebraic topology, Van Nostrand Mathematical Studies, No. 11, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0222892
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
Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
Friedhelm Waldhausen, Algebraic $K$-theory of spaces, localization, and the chromatic filtration of stable homotopy, Algebraic topology, Aarhus 1982 (Aarhus, 1982) Lecture Notes in Math., vol. 1051, Springer, Berlin, 1984, pp. 173–195. MR 764579, DOI 10.1007/BFb0075567
Douglas C. Ravenel, Life after the telescope conjecture, 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. 205–222. MR 1367299, DOI 10.1007/978-94-017-0695-7_{1}0