Morava 𝐸-theory of filtered colimits

Hovey, Mark

doi:10.1090/S0002-9947-07-04298-5

Morava $E$-theory of filtered colimits
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Trans. Amer. Math. Soc. 360 (2008), 369-382 Request permission

Abstract:

Morava $E$-theory $E_{n*}^{\vee }(-)$ is a much-studied theory in algebraic topology, but it is not a homology theory in the usual sense, because it fails to preserve coproducts (resp. filtered homotopy colimits). The object of this paper is to construct a spectral sequence to compute the Morava $E$-theory of a coproduct (resp. filtered homotopy colimit). The $E_{2}$-term of this spectral sequence involves the derived functors of direct sum (resp. filtered colimit) in an appropriate abelian category. We show that there are at most $n-1$ (resp. $n$) of these derived functors. When $n=1$, we recover the known result that homotopy commutes with an appropriate version of direct sum in the $K(1)$-local stable homotopy category.

References

William G. Dwyer, Philip S. Hirschhorn, Daniel M. Kan, and Jeffrey H. Smith, Homotopy limit functors on model categories and homotopical categories, Mathematical Surveys and Monographs, vol. 113, American Mathematical Society, Providence, RI, 2004. MR 2102294, DOI 10.1090/surv/113
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
P. G. Goerss and M. J. Hopkins, Moduli spaces of commutative ring spectra, Structured ring spectra, London Math. Soc. Lecture Note Ser., vol. 315, Cambridge Univ. Press, Cambridge, 2004, pp. 151–200. MR 2125040, DOI 10.1017/CBO9780511529955.009
J. P. C. Greenlees and J. P. May, Derived functors of $I$-adic completion and local homology, J. Algebra 149 (1992), no. 2, 438–453. MR 1172439, DOI 10.1016/0021-8693(92)90026-I
J. P. C. Greenlees and J. P. May, Completions in algebra and topology, Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 255–276. MR 1361892, DOI 10.1016/B978-044481779-2/50008-0
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, Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999. MR 1650134
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
Michael J. Hopkins and Jeffrey H. Smith, Nilpotence and stable homotopy theory. II, Ann. of Math. (2) 148 (1998), no. 1, 1–49. MR 1652975, DOI 10.2307/120991
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, 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
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
Stefan Schwede and Brooke E. Shipley, Algebras and modules in monoidal model categories, Proc. London Math. Soc. (3) 80 (2000), no. 2, 491–511. MR 1734325, DOI 10.1112/S002461150001220X