Harmonic localization of algebraic 𝐾-theory spectra

Mitchell, Stephen A.

doi:10.1090/S0002-9947-1992-1069739-1

Harmonic localization of algebraic $K$-theory spectra
HTML articles powered by AMS MathViewer

by Stephen A. Mitchell PDF

Trans. Amer. Math. Soc. 332 (1992), 823-837 Request permission

Abstract:

The Lichtenbaum-Quillen conjectures hold for the harmonic localization of the $K$-theory spectrum of a nice scheme. Various consequences of this fact are explored; for example, the harmonic localization of the $K$-theory of the integers at a regular prime is explicitly identified.

References

Armand Borel, Stable real cohomology of arithmetic groups, Ann. Sci. École Norm. Sup. (4) 7 (1974), 235–272 (1975). MR 387496, DOI 10.24033/asens.1269
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, Uniqueness of infinite deloopings for $K$-theoretic spaces, Pacific J. Math. 129 (1987), no. 1, 1–31. MR 901254, DOI 10.2140/pjm.1987.129.1
William G. Dwyer and Eric M. Friedlander, Algebraic and etale $K$-theory, Trans. Amer. Math. Soc. 292 (1985), no. 1, 247–280. MR 805962, DOI 10.1090/S0002-9947-1985-0805962-2
W. G. Dwyer and E. M. Friedlander, Conjectural calculations of general linear group homology, Applications of algebraic $K$-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983) Contemp. Math., vol. 55, Amer. Math. Soc., Providence, RI, 1986, pp. 135–147. MR 862634, DOI 10.1090/conm/055.1/862634

Topological models for arithmetic

W. G. Dwyer, E. M. Friedlander, and S. A. Mitchell, The generalized Burnside ring and the $K$-theory of a ring with roots of unity, $K$-Theory 6 (1992), no. 4, 285–300. MR 1193146, DOI 10.1007/BF00966114
William Dwyer, Eric Friedlander, Victor Snaith, and Robert Thomason, Algebraic $K$-theory eventually surjects onto topological $K$-theory, Invent. Math. 66 (1982), no. 3, 481–491. MR 662604, DOI 10.1007/BF01389225
Ethan S. Devinatz, Michael J. Hopkins, and Jeffrey H. Smith, Nilpotence and stable homotopy theory. I, Ann. of Math. (2) 128 (1988), no. 2, 207–241. MR 960945, DOI 10.2307/1971440
B. Harris and G. Segal, $K_{i}$ groups of rings of algebraic integers, Ann. of Math. (2) 101 (1975), 20–33. MR 387379, DOI 10.2307/1970984

Nilpotence in stable homotopy theory

Peter S. Landweber, Annihilator ideals and primitive elements in complex bordism, Illinois J. Math. 17 (1973), 273–284. MR 322874
J. P. May, The spectra associated to permutative categories, Topology 17 (1978), no. 3, 225–228. MR 508886, DOI 10.1016/0040-9383(78)90027-7
Haynes R. Miller, On relations between Adams spectral sequences, with an application to the stable homotopy of a Moore space, J. Pure Appl. Algebra 20 (1981), no. 3, 287–312. MR 604321, DOI 10.1016/0022-4049(81)90064-5
S. A. Mitchell, The Morava $K$-theory of algebraic $K$-theory spectra, $K$-Theory 3 (1990), no. 6, 607–626. MR 1071898, DOI 10.1007/BF01054453
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
Douglas C. Ravenel, The structure of Morava stabilizer algebras, Invent. Math. 37 (1976), no. 2, 109–120. MR 420619, DOI 10.1007/BF01418965
Victor Snaith, Unitary $K$-homology and the Lichtenbaum-Quillen conjecture on the algebraic $K$-theory of schemes, Algebraic topology, Aarhus 1982 (Aarhus, 1982) Lecture Notes in Math., vol. 1051, Springer, Berlin, 1984, pp. 128–155. MR 764577, DOI 10.1007/BFb0075565
R. W. Thomason, Algebraic $K$-theory and étale cohomology, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 3, 437–552. MR 826102, DOI 10.24033/asens.1495
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