Gross–Hopkins duals of higher real K–theory spectra
HTML articles powered by AMS MathViewer
- by Tobias Barthel, Agnès Beaudry and Vesna Stojanoska PDF
- Trans. Amer. Math. Soc. 372 (2019), 3347-3368 Request permission
Abstract:
We determine the Gross–Hopkins duals of certain higher real $K$–theory spectra. More specifically, let $p$ be an odd prime, and consider the Morava $E$–theory spectrum of height $n=p-1$. It is known, in expert circles, that for certain finite subgroups $G$ of the Morava stabilizer group, the homotopy fixed point spectra $E_n^{hG}$ are Gross–Hopkins self-dual up to a shift. In this paper, we determine the shift for those finite subgroups $G$ which contain $p$–torsion. This generalizes previous results for $n=2$ and $p=3$.References
- Gert Almkvist and Robert Fossum, Decomposition of exterior and symmetric powers of indecomposable $\textbf {Z}/p\textbf {Z}$-modules in characteristic $p$ and relations to invariants, Séminaire d’Algèbre Paul Dubreil, 30ème année (Paris, 1976–1977), Lecture Notes in Math., vol. 641, Springer, Berlin, 1978, pp. 1–111. MR 499459
- D. W. Anderson, Universal coefficient theorems for $K$-theory, mimeographed notes, Univ. of California, Berkeley (1969).
- T. Barthel, A. Beaudry, P. G. Goerss, and V. Stojanoska, Constructing the determinant sphere using a Tate twist, arXiv e-prints, 2018.
- Mark Behrens, A modular description of the $K(2)$-local sphere at the prime 3, Topology 45 (2006), no. 2, 343–402. MR 2193339, DOI 10.1016/j.top.2005.08.005
- Mark Behrens, The Goodwillie tower and the EHP sequence, Mem. Amer. Math. Soc. 218 (2012), no. 1026, xii+90. MR 2976788, DOI 10.1090/S0065-9266-2011-00645-3
- I. Bobkova and P. G. Goerss, Topological resolutions in K(2)-local homotopy theory at the prime 2, Journal of Topology 11 (2018), no. 4, 918–957.
- Paul G. Goerss and Hans-Werner Henn, The Brown-Comenetz dual of the $K(2)$-local sphere at the prime 3, Adv. Math. 288 (2016), 648–678. MR 3436395, DOI 10.1016/j.aim.2015.08.024
- P. Goerss, H.-W. Henn, M. Mahowald, and C. Rezk, A resolution of the $K(2)$-local sphere at the prime 3, Ann. of Math. (2) 162 (2005), no. 2, 777–822. MR 2183282, DOI 10.4007/annals.2005.162.777
- J. P. C. Greenlees and J. P. May, Generalized Tate cohomology, Mem. Amer. Math. Soc. 113 (1995), no. 543, viii+178. MR 1230773, DOI 10.1090/memo/0543
- J. P. C. Greenlees and Hal Sadofsky, The Tate spectrum of $v_n$-periodic complex oriented theories, Math. Z. 222 (1996), no. 3, 391–405. MR 1400199, DOI 10.1007/PL00004264
- D. Heard, The Tate spectrum of the higher real $K$-theories at height $n=p-1$, arXiv e-prints, 2015.
- Hans-Werner Henn, On finite resolutions of $K(n)$-local spheres, Elliptic cohomology, London Math. Soc. Lecture Note Ser., vol. 342, Cambridge Univ. Press, Cambridge, 2007, pp. 122–169. MR 2330511, DOI 10.1017/CBO9780511721489.008
- M. J. Hopkins and B. H. Gross, Equivariant vector bundles on the Lubin-Tate moduli space, Topology and representation theory (Evanston, IL, 1992) Contemp. Math., vol. 158, Amer. Math. Soc., Providence, RI, 1994, pp. 23–88. MR 1263712, DOI 10.1090/conm/158/01453
- M. J. Hopkins and B. H. Gross, The rigid analytic period mapping, Lubin-Tate space, and stable homotopy theory, Bull. Amer. Math. Soc. (N.S.) 30 (1994), no. 1, 76–86. MR 1217353, DOI 10.1090/S0273-0979-1994-00438-0
- Michael J. Hopkins, Mark Mahowald, and Hal Sadofsky, Constructions of elements in Picard groups, Topology and representation theory (Evanston, IL, 1992) Contemp. Math., vol. 158, Amer. Math. Soc., Providence, RI, 1994, pp. 89–126. MR 1263713, DOI 10.1090/conm/158/01454
- Drew Heard, Akhil Mathew, and Vesna Stojanoska, Picard groups of higher real $K$-theory spectra at height $p-1$, Compos. Math. 153 (2017), no. 9, 1820–1854. MR 3705278, DOI 10.1112/S0010437X17007242
- 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
- Drew Heard and Vesna Stojanoska, $K$-theory, reality, and duality, J. K-Theory 14 (2014), no. 3, 526–555. MR 3349325, DOI 10.1017/is014007001jkt275
- Mark Mahowald and Charles Rezk, Brown-Comenetz duality and the Adams spectral sequence, Amer. J. Math. 121 (1999), no. 6, 1153–1177. MR 1719751
- Lee S. Nave, The Smith-Toda complex $V((p+1)/2)$ does not exist, Ann. of Math. (2) 171 (2010), no. 1, 491–509. MR 2630045, DOI 10.4007/annals.2010.171.491
- 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, Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics, vol. 121, Academic Press, Inc., Orlando, FL, 1986. MR 860042
- Vesna Stojanoska, Duality for topological modular forms, Doc. Math. 17 (2012), 271–311. MR 2946825
- N. P. Strickland, Gross-Hopkins duality, Topology 39 (2000), no. 5, 1021–1033. MR 1763961, DOI 10.1016/S0040-9383(99)00049-X
- Peter Symonds, The Tate-Farrell cohomology of the Morava stabilizer group $S_{p-1}$ with coefficients in $E_{p-1}$, Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic $K$-theory, Contemp. Math., vol. 346, Amer. Math. Soc., Providence, RI, 2004, pp. 485–492. MR 2066512, DOI 10.1090/conm/346/06301
Additional Information
- Tobias Barthel
- Affiliation: Department of Mathematics, University of Copenhagen, DK-2100 Copenhagen, Denmark
- MR Author ID: 1015635
- Agnès Beaudry
- Affiliation: Department of Mathematics, University of Colorado Boulder, Boulder, Colorado 80309
- Vesna Stojanoska
- Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
- MR Author ID: 857759
- Received by editor(s): May 19, 2017
- Received by editor(s) in revised form: July 19, 2018, and October 17, 2018
- Published electronically: May 30, 2019
- Additional Notes: This material is based upon work supported by the National Science Foundation under Grant No. DMS-1606479 and Grant No. DMS-1612020/1725563.
The first-named author was partially supported by the DNRF92. - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 3347-3368
- MSC (2010): Primary 55M05, 55P42, 20J06, 55Q91, 55Q51, 55P60
- DOI: https://doi.org/10.1090/tran/7730
- MathSciNet review: 3988613