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Picard groups of higher real $$K$$-theory spectra at height $$p-1$$. (English) Zbl 1374.14006
We consider the formal group law $$F_n(X, Y) \in \mathbb{F}_{p^n}[[X, Y]]$$ with $$p$$-series $$[p]_{F_n}(X)=X^{p^n}$$ where we suppose that $$p$$ is an odd prime and $$n=p-1$$. Denote by $$\mathbb{S}_n$$ the group of automorphisms of $$F_n$$ given by power series $$f \in \mathbb{F}_{p^n}[[X]]$$ such that $$f(0)=0$$, $$f'(0)\neq 0$$ and $$F_n(f(X), f(Y))=f(F(X, Y))$$. Then the assignment $$f \mapsto f'(0)$$ induces a homomorphism $$\mathbb{S}_n \to \mathbb{F}_{p^n}^\times$$ which is surjective and whose kernel is a pro-$$p$$-group. This allows us to define a semidirect product of $$\mathbb{S}_n$$ and $$\text{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)$$, which is written $$\mathbb{G}_n=\mathbb{S}_n \rtimes \text{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)$$. Let $$E_n$$ be the $$n$$th Morava $$E$$-theory, i.e. an $${\mathbf E}_\infty$$-ring spectrum with $$\pi_*E_n=W(\mathbb{F}_{p^n})[[u_1, \ldots, u_{n-1}]][u^{\pm{1}}]$$ where $$W(\mathbb{F}_{p^n})$$ denotes the ring of Witt vectors over $$\mathbb{F}_{p^n}$$. Then $$\mathbb{G}_n$$ acts on $$E_n$$ through $${\mathbf E}_\infty$$-ring maps. If we denote by $$E_n^{hG}$$ the homotopy fixed points of $$E_n$$ for a finite subgroup $$G \subset \mathbb{G}_n$$, then the main theorem of this paper (Theorem 4.1) states that the Picard group $$\text{Pic}(E_n^{hG})$$ is a cyclic group generated by the suspension $$\Sigma E_n^{hG}$$. In order to prove this, the authors first check that the extension $$E_n^{hG} \to E_n$$ is a faithful $$G$$-Galois extension of ring spectra, thereby obtaining the descent spectral sequence for Picard groups. The proof is done by analyzing this spectral sequence, using the technique developed in the second part of the paper. Incidentally, this paper consists of two parts I and II, both having three sections. We then refer to a remark on the example of the $$C_2$$-Galois extension $$KO \to KU$$. It states that the theorem above is also true when $$p=2$$, $$n=1$$; as a result we have $$\text{Pic}(KO)=\mathbb{Z}/8$$, generated by $$\Sigma KO$$.

##### MSC:
 14C22 Picard groups 55P43 Spectra with additional structure ($$E_\infty$$, $$A_\infty$$, ring spectra, etc.) 55Q51 $$v_n$$-periodicity 55S99 Operations and obstructions in algebraic topology
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##### References:
 [1] Almkvist, G. and Fossum, R., Decomposition of exterior and symmetric powers of indecomposable Z/pZ -modules in characteristic p and relations to invariants, in Séminaire d’Algèbre Paul Dubreil, 30ème année (Paris 1976-1977), (Springer, Berlin, 1978), 1-111; (81b:14024). [2] Ando, M., Blumberg, A. J., Gepner, D., Hopkins, M. J. and Rezk, C., Units of ring spectra, orientations and Thom spectra via rigid infinite loop space theory, J. Topol.7 (2014), 1077-1117; . doi:10.1112/jtopol/jtu009 · Zbl 1316.55005 [3] Baker, A. and Richter, B., Invertible modules for commutative S-algebras with residue fields, Manuscripta Math.118 (2005), 99-119; (2006f:55009). doi:10.1007/s00229-005-0582-1 · Zbl 1092.55007 [4] Baker, A. and Richter, B., Realizability of algebraic Galois extensions by strictly commutative ring spectra, Trans. Amer. Math. Soc.359 (2007), 827-857 (electronic); (2007m:55007). doi:10.1090/S0002-9947-06-04201-2 · Zbl 1111.55009 [5] Beaudry, A., The algebraic duality resolution at p = 2, Algebr. Geom. Topol., 15, 3653-3705, (2015) · Zbl 1350.55019 [6] Behrens, M., The homotopy groups of S_E (2) at p⩾5 revisited, Adv. Math., 230, 458-492, (2012) · Zbl 1256.55004 [7] Bobkova, I., Resolutions in the$$K(2)$$-local category at the prime 2, PhD thesis, Northwestern University (ProQuest LLC, Ann Arbor, MI 2014); . [8] Bujard, C., Finite subgroups of extended Morava stabilizer groups, Preprint (2012), arXiv:1206.1951. [9] Cohen, F. R., Lada, T. J. and May, J. P., The homology of iterated loop spaces, (Springer, New York, 1976); (55 #9096). doi:10.1007/BFb0080464 · Zbl 0334.55009 [10] Devinatz, E. S. and Hopkins, M. J., The action of the Morava stabilizer group on the Lubin-Tate moduli space of lifts, Amer. J. Math.117 (1995), 669-710; (97a:55007). doi:10.2307/2375086 · Zbl 0842.14034 [11] Devinatz, E. S. and Hopkins, M. J., Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups, Topology43 (2004), 1-47; (2004i:55012). doi:10.1016/S0040-9383(03)00029-6 · Zbl 1047.55004 [12] Fröhlich, A., Formal groups, (Springer, Berlin, 1968); (39 #4164). doi:10.1007/BFb0074373 · Zbl 0177.04801 [13] Gepner, D. and Lawson, T., Brauer groups and Galois cohomology of commutative ring spectra, Preprint (2016), arXiv:1607.01118. [14] Goerss, P., Henn, H.-W. and Mahowald, M., The homotopy of L_2V (1) for the prime 3, in Categorical decomposition techniques in algebraic topology (Isle of Skye 2001), (Birkhäuser, Basel, 2004), 125-151; (2004k:55010). · Zbl 1052.55010 [15] Goerss, P., Henn, H.-W., Mahowald, M. and Rezk, C., A resolution of the K (2)-local sphere at the prime 3, Ann. of Math. (2)162 (2005), 777-822; (2006j:55016). doi:10.4007/annals.2005.162.777 · Zbl 1108.55009 [16] Goerss, P., Henn, H.-W., Mahowald, M. and Rezk, C., On Hopkins’ Picard groups for the prime 3 and chromatic level 2, J. Topol.8 (2015), 267-294; . doi:10.1112/jtopol/jtu024 · Zbl 1314.55006 [17] Goerss, P. G. and Hopkins, M. J., Moduli spaces of commutative ring spectra, in Structured ring spectra, (Cambridge University Press, Cambridge, 2004), 151-200; (2006b:55010). doi:10.1017/CBO9780511529955.009 · Zbl 1086.55006 [18] Gorbounov, V., Mahowald, M. and Symonds, P., Infinite subgroups of the Morava stabilizer groups, Topology37 (1998), 1371-1379; (99m:16032). doi:10.1016/S0040-9383(97)00100-6 · Zbl 0937.16026 [19] Heard, D., The Tate spectrum of the higher real$$K$$-theories at height$$n=p-1$$, Preprint (2015), arXiv:1501.07759. [20] Henn, H.-W., On finite resolutions of K (n)-local spheres, in Elliptic cohomology, (Cambridge University Press, Cambridge, 2007), 122-169; (2009e:55014). doi:10.1017/CBO9780511721489.008 · Zbl 1236.55015 [21] Hewett, T., Finite subgroups of division algebras over local fields, J. Algebra173 (1995), 518-548; (96b:16012). doi:10.1006/jabr.1995.1101 · Zbl 0829.16023 [22] Hill, M., Computational methods for higher real$$K$$-theory with applications to TMF, PhD thesis, Massachusetts Institute of Technology (2006). [23] Hill, M. and Meier, L., The$$C_{2}$$-spectrum$$Tmf_{1}(3)$$and its invertible modules, Algebr. Geom. Topol., to appear, arXiv:1507.08115. [24] Hopkins, M. J., Mahowald, M. and Sadofsky, H., Constructions of elements in Picard groups, in Topology and representation theory (Evanston, IL 1992), (American Mathematical Society, Providence, RI, 1994), 89-126; (95a:55020). doi:10.1090/conm/158/01454 · Zbl 0799.55005 [25] Karamanov, N., On Hopkins’ Picard group Pic_2 at the prime 3, Algebr. Geom. Topol.10 (2010), 275-292; (2011c:55013). doi:10.2140/agt.2010.10.275 · Zbl 1185.55010 [26] Kraines, D., Massey higher products, Trans. Amer. Math. Soc.124 (1966), 431-449; (34 #2010). doi:10.1090/S0002-9947-1966-0202136-1 · Zbl 0146.19201 [27] Lader, O., Une résolution projective pour le second groupe de Morava pour$$p\geqslant 5$$et applications, PhD thesis, Institut de Recherche Mathmatiques Avance (July 2013). [28] Lurie, J., Derived algebraic geometry XI: descent theorems, available at http://www.math.harvard.edu/∼lurie/papers/DAG-XI.pdf. [29] Mathew, A., A thick subcategory theorem for modules over certain ring spectra, Geom. Topol., 19, 2359-2392, (2015) · Zbl 1405.55009 [30] Mathew, A., The Galois group of a stable homotopy theory, Adv. Math., 291, 403-541, (2016) · Zbl 1338.55009 [31] Mathew, A. and Meier, L., Affineness and chromatic homotopy theory, J. Topol.8 (2015), 476-528; . doi:10.1112/jtopol/jtv005 · Zbl 1325.55004 [32] Mathew, A., Naumann, N. and Noel, J., Derived induction and restriction theory, Preprint (2015), arXiv:1507.06867. [33] Mathew, A. and Stojanoska, V., The Picard group of topological modular forms via descent theory, Geom. Topol.20 (2016), 3133-3217. doi:10.2140/gt.2016.20.3133 · Zbl 1373.14008 [34] Mumford, D., Lectures on curves on an algebraic surface, With a section by G. M. Bergman, (Princeton University Press, Princeton, NJ, 1966); (35 #187). · Zbl 0187.42701 [35] Nave, L. S., The cohomology of finite subgroups of Morava stabilizer groups and Smith-Toda complexes, PhD thesis, University of Washington (ProQuest LLC, Ann Arbor, MI 1999); . [36] Nave, L. S., The Smith-Toda complex V ((p + 1)/2) does not exist, Ann. of Math. (2)171 (2010), 491-509; (2011m:55012). doi:10.4007/annals.2010.171.491 · Zbl 1194.55017 [37] Priddy, S., Mod-p right derived functor algebras of the symmetric algebra functor, J. Pure Appl. Algebra3 (1973), 337-356; (49 #7338). doi:10.1016/0022-4049(73)90036-4 · Zbl 0284.55026 [38] Ravenel, D. C., The non-existence of odd primary Arf invariant elements in stable homotopy, Math. Proc. Cambridge Philos. Soc.83 (1978), 429-443; (57 #13938). doi:10.1017/S0305004100054712 · Zbl 0374.55023 [39] Rezk, C., Notes on the Hopkins-Miller theorem, in Homotopy theory via algebraic geometry and group representations (Evanston, IL 1997), (American Mathematical Society, Providence, RI, 1998), 313-366; (2000i:55023). doi:10.1090/conm/220/03107 · Zbl 0910.55004 [40] Rognes, J., Galois extensions of structured ring spectra. Stably dualizable groups, Mem. Amer. Math. Soc.192 (2008), viii + 137; (2009c:55007). · Zbl 1166.55001 [41] Serre, J.-P., Local fields, (Springer, New York, 1979). Translated from the French by Marvin Jay Greenberg; (82e:12016). doi:10.1007/978-1-4757-5673-9 [42] Grothendieck, A. and Raynaud, M., Revêtements étales et groupe fondamental (SGA 1), in Séminaire de géométrie algébrique du Bois Marie 1960-61, (Société Mathématique de France, Paris, 2003), updated and annotated reprint of the 1971 original; (2004g:14017). [43] Toda, H., An important relation in homotopy groups of spheres, Proc. Japan Acad.43 (1967), 839-842; (37 #5872). doi:10.3792/pja/1195521423 · Zbl 0169.25603 [44] Toda, H., Extended pth powers of complexes and applications to homotopy theory, Proc. Japan Acad.44 (1968), 198-203; (37 #5873). doi:10.3792/pja/1195521243 · Zbl 0181.26405
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