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Homotopical localizations of module spectra

Authors: Carles Casacuberta and Javier J. Gutiérrez
Journal: Trans. Amer. Math. Soc. 357 (2005), 2753-2770
MSC (2000): Primary 55P42, 55P43, 55P60
Published electronically: September 23, 2004
MathSciNet review: 2139526
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Abstract: We prove that stable $f$-localizations (where $f$ is any map of spectra) preserve ring spectrum structures and module spectrum structures, under suitable hypotheses, and we use this fact to describe all possible localizations of the integral Eilenberg-MacLane spectrum $H{\mathbb{Z} }$. As a consequence of this study, we infer that localizations of stable GEMs are stable GEMs, and it also follows that there is a proper class of nonequivalent stable localizations.

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  • [Ada74] J. F. Adams,
    Stable Homotopy and Generalised Homology,
    University of Chicago Press, Chicago, 1974. MR 53:6534
  • [AH68] D. W. Anderson and L. Hodgkin,
    The $K$-theory of Eilenberg-MacLane complexes,
    Topology 7 (1968), pp. 317-329. MR 37:6924
  • [Bad01] B. Badzioch,
    Recognition principle for generalized Eilenberg-MacLane spaces,
    In: Cohomological Methods in Homotopy Theory (Barcelona, 1998),
    Progress in Math., vol. 196, Birkhäuser, Basel, 2001, pp. 21-26. MR 2002f:55019
  • [Bor94] F. Borceux,
    Handbook of Categorical Algebra II: Categories and Structures,
    Encyclopedia of Mathematics and its Applications, vol. 51,
    Cambridge University Press, Cambridge, 1994. MR 96g:18001b
  • [Bou77] A. K. Bousfield,
    Constructions of factorization systems in categories,
    J. Pure Appl. Algebra 9 (1976/77), no. 2, pp. 207-220. MR 57:17648
  • [Bou79] A. K. Bousfield,
    The localization of spectra with respect to homology,
    Topology 18 (1979), no. 4, pp. 257-281. MR 80m:55006
  • [Bou82] A. K. Bousfield,
    On homology equivalences and homological localizations of spaces,
    Amer. J. Math. 104 (1982), no. 5, pp. 1025-1042. MR 84g:55014
  • [Bou96] A. K. Bousfield,
    Unstable localization and periodicity,
    In: Algebraic Topology: New Trends in Localization and Periodicity (Sant Feliu, 1994),
    Progress in Math., vol. 136, Birkhäuser, Basel, 1996, pp. 33-50. MR 98c:55014
  • [Bou99] A. K. Bousfield,
    On $K(n)$-equivalences of spaces,
    In: Homotopy Invariant Algebraic Structures (Baltimore, 1998),
    Contemp. Math., vol. 239, Amer. Math. Soc., Providence, 1999, pp. 85-89. MR 2001k:55013
  • [BF78] A. K. Bousfield and E. M. Friedlander,
    Homotopy theory of $\Gamma $-spaces, spectra, and bisimplicial sets,
    In: Geometric Applications of Homotopy Theory (Evanston, 1977), II, Lecture Notes in Math., vol. 658, Springer-Verlag, Berlin, Heidelberg, New York, 1978, pp. 80-130. MR 80e:55021
  • [BK72a] A. K. Bousfield and D. M. Kan,
    The core of a ring,
    J. Pure Appl. Algebra 2 (1972), pp. 73-81. MR 46:7222
  • [BK72b] A. K. Bousfield and D. M. Kan,
    Homotopy Limits, Completions and Localizations,
    Lecture Notes in Math., vol. 304,
    Springer-Verlag, Berlin Heidelberg New York, 1972. MR 51:1825
  • [Cas00] C. Casacuberta,
    On structures preserved by idempotent transformations of groups and homotopy types,
    In: Crystallographic Groups and Their Generalizations (Kortrijk, 1999),
    Contemp. Math., vol. 262, Amer. Math. Soc., Providence, 2000, pp. 39-68. MR 2001i:55012
  • [CR97] C. Casacuberta and J. L. Rodríguez,
    On towers approximating homological localizations,
    J. London Math. Soc. 56 (1997), pp. 645-656. MR 99b:55016
  • [CRT00] C. Casacuberta, J. L. Rodríguez, and J.-Y. Tai,
    Localizations of abelian Eilenberg-Mac Lane spaces of finite type,
    preprint, 2000.
  • [Dro96] E. Dror Farjoun,
    Cellular Spaces, Null Spaces and Homotopy Localization, Lecture Notes in Math., vol. 1622,
    Springer-Verlag, Berlin Heidelberg New York, 1996. MR 98f:55010
  • [DMV87] M. Dugas, A. Mader, and C. Vinsonhaler,
    Large $E$-rings exist,
    J. Algebra 108 (1987), pp. 88-101. MR 88e:16047
  • [DP01] W. G. Dwyer, J. H. Palmieri,
    Ohkawa's theorem: there is a set of Bousfield classes,
    Proc. Amer. Math. Soc. 129 (2001), no. 3, pp. 881-886. MR 2001f:55015
  • [EKMM97] A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May,
    Rings, Modules, and Algebras in Stable Homotopy Theory,
    Math. Surveys Monographs, vol. 47,
    Amer. Math. Soc., Providence, 1997. MR 97h:55006
  • [GU71] P. Gabriel and F. Ulmer,
    Lokal Präsentierbare Kategorien,
    Lecture Notes in Math., vol. 221,
    Springer-Verlag, Berlin Heidelberg New York, 1971. MR 48:6205
  • [GJ99] P. G. Goerss and J. F. Jardine,
    Simplicial Homotopy Theory,
    Progress in Math., vol. 174,
    Birkhäuser, Basel, 1999. MR 2001d:55012
  • [Gut03] J. J. Gutiérrez,
    Strict modules and homotopy modules in stable homotopy,
    preprint, 2003.
  • [Hir03] P. S. Hirschhorn,
    Model Categories and Their Localizations,
    Math. Surveys Monographs, vol. 99,
    Amer. Math. Soc., Providence, 2003. MR 2003j:18018
  • [Hov95] M. Hovey,
    Cohomological Bousfield classes,
    J. Pure Appl. Algebra 103 (1995), pp. 45-59. MR 96g:55008
  • [HPS97] M. Hovey, J. H. Palmieri, and N. P. Strickland,
    Axiomatic Stable Homotopy Theory,
    Mem. Amer. Math. Soc. 128 (1997), no. 610. MR 98a:55017
  • [HSS00] M. Hovey, B. Shipley, and J. Smith,
    Symmetric spectra,
    J. Amer. Math. Soc. 13 (2000), no. 1, pp. 149-208. MR 2000h:55016
  • [Kap69] I. Kaplansky,
    Infinite Abelian Groups,
    The Univ. of Michigan Press,
    Ann Arbor, 1969. MR 38:2208
  • [Nee01] A. Neeman,
    Triangulated Categories,
    Ann. of Math. Studies, vol. 148, Princeton Univ. Press, Princeton, 2001. MR 2001k:18010
  • [Rav84] D. C. Ravenel,
    Localization with respect to certain periodic homology theories,
    Amer. J. Math. 106 (1984), pp. 351-414. MR 85k:55009
  • [Rud98] Y. B. Rudyak,
    On Thom Spectra, Orientability, and Cobordism,
    Springer Monographs in Math.,
    Springer-Verlag, Berlin Heidelberg New York, 1998. MR 99f:55001
  • [Str04] N. P. Strickland,
    Axiomatic stable homotopy--a survey,
    In: Axiomatic, Enriched and Motivic Homotopy Theory (Cambridge, 2002), NATO Science Series II: Mathematics, Physics and Chemistry, vol. 131, Kluwer Academic Publishers, Dordrecht, 2004, pp. 69-98.

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Additional Information

Carles Casacuberta
Affiliation: Departament d’Àlgebra i Geometria, Universitat de Barcelona, Gran Via, 585, 08007 Barcelona, Spain

Javier J. Gutiérrez
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra, Spain
Address at time of publication: Departament d’Àlgebra i Geometria, Universitat de Barcelona, Gran Via, 585, 08007 Barcelona, Spain

Keywords: Localization, ring spectrum, module spectrum, stable GEM
Received by editor(s): May 1, 2002
Received by editor(s) in revised form: November 3, 2003
Published electronically: September 23, 2004
Additional Notes: The authors were supported by MCyT grants PB97-0202, BFM2001-2031, and FP98 16587447
Article copyright: © Copyright 2004 American Mathematical Society

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