Homotopical localizations of module spectra

Casacuberta, Carles; Gutiérrez, Javier

doi:10.1090/S0002-9947-04-03552-4

Homotopical localizations of module spectra
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by Carles Casacuberta and Javier J. Gutiérrez PDF

Trans. Amer. Math. Soc. 357 (2005), 2753-2770 Request permission

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–Mac Lane 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.

References

J. F. Adams, Stable homotopy and generalised homology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill.-London, 1974. MR 0402720
D. W. Anderson and Luke Hodgkin, The $K$-theory of Eilenberg-MacLane complexes, Topology 7 (1968), 317–329. MR 231369, DOI 10.1016/0040-9383(68)90009-8
Bernard Badzioch, Recognition principle for generalized Eilenberg-Mac Lane spaces, Cohomological methods in homotopy theory (Bellaterra, 1998) Progr. Math., vol. 196, Birkhäuser, Basel, 2001, pp. 21–26. MR 1851244
Francis Borceux, Handbook of categorical algebra. 1, Encyclopedia of Mathematics and its Applications, vol. 50, Cambridge University Press, Cambridge, 1994. Basic category theory. MR 1291599
A. K. Bousfield, Constructions of factorization systems in categories, J. Pure Appl. Algebra 9 (1976/77), no. 2, 207–220. MR 478159, DOI 10.1016/0022-4049(77)90067-6
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 homology equivalences and homological localizations of spaces, Amer. J. Math. 104 (1982), no. 5, 1025–1042. MR 675308, DOI 10.2307/2374082
A. K. Bousfield, Unstable localization and periodicity, Algebraic topology: new trends in localization and periodicity (Sant Feliu de Guíxols, 1994) Progr. Math., vol. 136, Birkhäuser, Basel, 1996, pp. 33–50. MR 1397720
A. K. Bousfield, On $K(n)$-equivalences of spaces, Homotopy invariant algebraic structures (Baltimore, MD, 1998) Contemp. Math., vol. 239, Amer. Math. Soc., Providence, RI, 1999, pp. 85–89. MR 1718077, DOI 10.1090/conm/239/03598
A. K. Bousfield and E. M. Friedlander, Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977) Lecture Notes in Math., vol. 658, Springer, Berlin, 1978, pp. 80–130. MR 513569
A. K. Bousfield and D. M. Kan, The core of a ring, J. Pure Appl. Algebra 2 (1972), 73–81. MR 308107, DOI 10.1016/0022-4049(72)90023-0
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
Carles Casacuberta, On structures preserved by idempotent transformations of groups and homotopy types, Crystallographic groups and their generalizations (Kortrijk, 1999) Contemp. Math., vol. 262, Amer. Math. Soc., Providence, RI, 2000, pp. 39–68. MR 1796125, DOI 10.1090/conm/262/04167
Carles Casacuberta and José L. Rodríguez, On towers approximating homological localizations, J. London Math. Soc. (2) 56 (1997), no. 3, 645–656. MR 1610420, DOI 10.1112/S0024610797005607
C. Casacuberta, J. L. Rodríguez, and J.-Y. Tai, Localizations of abelian Eilenberg–Mac Lane spaces of finite type, preprint, 2000.
Emmanuel Dror Farjoun, Cellular spaces, null spaces and homotopy localization, Lecture Notes in Mathematics, vol. 1622, Springer-Verlag, Berlin, 1996. MR 1392221, DOI 10.1007/BFb0094429
Manfred Dugas, Adolf Mader, and Charles Vinsonhaler, Large $E$-rings exist, J. Algebra 108 (1987), no. 1, 88–101. MR 887193, DOI 10.1016/0021-8693(87)90123-2
William G. Dwyer and John H. Palmieri, Ohkawa’s theorem: there is a set of Bousfield classes, Proc. Amer. Math. Soc. 129 (2001), no. 3, 881–886. MR 1712921, DOI 10.1090/S0002-9939-00-05669-0
A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs, vol. 47, American Mathematical Society, Providence, RI, 1997. With an appendix by M. Cole. MR 1417719, DOI 10.1090/surv/047
Peter Gabriel and Friedrich Ulmer, Lokal präsentierbare Kategorien, Lecture Notes in Mathematics, Vol. 221, Springer-Verlag, Berlin-New York, 1971 (German). MR 0327863, DOI 10.1007/BFb0059396
Paul G. Goerss and John F. Jardine, Simplicial homotopy theory, Progress in Mathematics, vol. 174, Birkhäuser Verlag, Basel, 1999. MR 1711612, DOI 10.1007/978-3-0348-8707-6
J. J. Gutiérrez, Strict modules and homotopy modules in stable homotopy, preprint, 2003.
Philip S. Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs, vol. 99, American Mathematical Society, Providence, RI, 2003. MR 1944041, DOI 10.1090/surv/099
Mark Hovey, Cohomological Bousfield classes, J. Pure Appl. Algebra 103 (1995), no. 1, 45–59. MR 1354066, DOI 10.1016/0022-4049(94)00096-2
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
Mark Hovey, Brooke Shipley, and Jeff Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), no. 1, 149–208. MR 1695653, DOI 10.1090/S0894-0347-99-00320-3
Irving Kaplansky, Infinite abelian groups, Revised edition, University of Michigan Press, Ann Arbor, Mich., 1969. MR 0233887
Amnon Neeman, Triangulated categories, Annals of Mathematics Studies, vol. 148, Princeton University Press, Princeton, NJ, 2001. MR 1812507, DOI 10.1515/9781400837212
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
Yuli B. Rudyak, On Thom spectra, orientability, and cobordism, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. With a foreword by Haynes Miller. MR 1627486
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.