Dieudonné rings associated with 𝐾(𝑛)_{∗}\underline{𝑘(𝑛)}_{∗}

Saramago, Rui

doi:10.1090/S0002-9939-08-09235-6

Dieudonné rings associated with $K(n)_\ast \underline {k(n)}_{ \ast }$
HTML articles powered by AMS MathViewer

by Rui Miguel Saramago

Proc. Amer. Math. Soc. 136 (2008), 2699-2709

DOI: https://doi.org/10.1090/S0002-9939-08-09235-6

Published electronically: April 10, 2008

PDF | Request permission

Abstract:

We use Dieudonné theory for periodically graded Hopf rings to determine the Dieudonné ring structure of the $\mathbb {Z}/2(p^n - 1)$-graded Morava $K$-theory $\overline {K(n)}_\ast (-)$, with $p$ an odd prime, when applied to the $\Omega$-spectrum $\underline {k(n)}_{ \ast }$ (and to $\underline {K(n)}_{ \ast }$). We also expand these results in order to accomodate the case of the full Morava $K$-theory $K(n)_\ast (-)$.

References

J. Michael Boardman, David Copeland Johnson, and W. Stephen Wilson, Unstable operations in generalized cohomology, Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 687–828. MR 1361900, DOI 10.1016/B978-044481779-2/50016-X
J. Michael Boardman, Richard L. Kramer, and W. Stephen Wilson, The periodic Hopf ring of connective Morava $K$-theory, Forum Math. 11 (1999), no. 6, 761–767. MR 1725596, DOI 10.1515/form.1999.024
A. K. Bousfield, On $\lambda$-rings and the $K$-theory of infinite loop spaces, $K$-Theory 10 (1996), no. 1, 1–30. MR 1373816, DOI 10.1007/BF00534886
A. K. Bousfield, On $p$-adic $\lambda$-rings and the $K$-theory of $H$-spaces, Math. Z. 223 (1996), no. 3, 483–519. MR 1417857, DOI 10.1007/PL00004271
Victor Buchstaber and Andrey Lazarev, Dieudonné modules and $p$-divisible groups associated with Morava $K$-theory of Eilenberg-Mac Lane spaces, Algebr. Geom. Topol. 7 (2007), 529–564. MR 2308956, DOI 10.2140/agt.2007.7.529
Michel Demazure, Lectures on $p$-divisible groups, Lecture Notes in Mathematics, Vol. 302, Springer-Verlag, Berlin-New York, 1972. MR 0344261
Michel Demazure and Pierre Gabriel, Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, Masson & Cie, Éditeurs, Paris; North-Holland Publishing Co., Amsterdam, 1970 (French). Avec un appendice Corps de classes local par Michiel Hazewinkel. MR 0302656
P. Goerss, J. Lannes, and F. Morel, Hopf algebras, Witt vectors, and Brown-Gitler spectra, Algebraic topology (Oaxtepec, 1991) Contemp. Math., vol. 146, Amer. Math. Soc., Providence, RI, 1993, pp. 111–128. MR 1224910, DOI 10.1090/conm/146/01218
Paul G. Goerss, Hopf rings, Dieudonné modules, and $E_*\Omega ^2S^3$, Homotopy invariant algebraic structures (Baltimore, MD, 1998) Contemp. Math., vol. 239, Amer. Math. Soc., Providence, RI, 1999, pp. 115–174. MR 1718079, DOI 10.1090/conm/239/03600
Michael J. Hopkins, Douglas C. Ravenel, and W. Stephen Wilson, Morava Hopf algebras and spaces $K(n)$ equivalent to finite Postnikov systems, Stable and unstable homotopy (Toronto, ON, 1996) Fields Inst. Commun., vol. 19, Amer. Math. Soc., Providence, RI, 1998, pp. 137–163. MR 1622344
John R. Hunton and Paul R. Turner, Coalgebraic algebra, J. Pure Appl. Algebra 129 (1998), no. 3, 297–313. MR 1631257, DOI 10.1016/S0022-4049(97)00076-5
David Copeland Johnson and W. Stephen Wilson, $BP$ operations and Morava’s extraordinary $K$-theories, Math. Z. 144 (1975), no. 1, 55–75. MR 377856, DOI 10.1007/BF01214408
Irving Kaplansky, Bialgebras, Lecture Notes in Mathematics, University of Chicago, Department of Mathematics, Chicago, Ill., 1975. MR 0435126
R. L. Kramer, “The periodic Hopf ring of connective Morava $K$-theory,” Ph.D. Thesis, The Johns Hopkins University, 1990.
John W. Milnor and John C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211–264. MR 174052, DOI 10.2307/1970615
Douglas C. Ravenel, Dieudonné modules for abelian Hopf algebras, Conference on homotopy theory (Evanston, Ill., 1974) Notas Mat. Simpos., vol. 1, Soc. Mat. Mexicana, México, 1975, pp. 177–183. MR 761728
Douglas C. Ravenel and W. Stephen Wilson, The Hopf ring for complex cobordism, J. Pure Appl. Algebra 9 (1976/77), no. 3, 241–280. MR 448337, DOI 10.1016/0022-4049(77)90070-6
Douglas C. Ravenel and W. Stephen Wilson, The Morava $K$-theories of Eilenberg-Mac Lane spaces and the Conner-Floyd conjecture, Amer. J. Math. 102 (1980), no. 4, 691–748. MR 584466, DOI 10.2307/2374093
Hal Sadofsky and W. Stephen Wilson, Commutative Morava homology Hopf algebras, Homotopy theory via algebraic geometry and group representations (Evanston, IL, 1997) Contemp. Math., vol. 220, Amer. Math. Soc., Providence, RI, 1998, pp. 367–373. MR 1642903, DOI 10.1090/conm/220/03108
R. M. Saramago, “Dieudonné Theory for Ungraded and Periodically Graded Hopf Rings,” Ph.D. Thesis, The Johns Hopkins University, 2000.
R. M. Saramago, “Dieudonné Module Structures for Ungraded and Periodically Graded Hopf Rings,” to appear in Algebras and Representation Theory.
R. M. Saramago, “Connected Hopf Algebra Maps,” preprint.
Colette Schoeller, Étude de la catégorie des algèbres de Hopf commutatives connexes sur un corps, Manuscripta Math. 3 (1970), 133–155 (French, with English summary). MR 281709, DOI 10.1007/BF01273307
W. Stephen Wilson, The Hopf ring for Morava $K$-theory, Publ. Res. Inst. Math. Sci. 20 (1984), no. 5, 1025–1036. MR 764345, DOI 10.2977/prims/1195180879