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Transchromatic generalized character maps. (English) Zbl 1300.55011
Given a finite group $$G$$ one of the most classical results in representation theory (and thus $$K$$-theory) says that associating to a complex representation of the group its character induces an isomorphism $$L \otimes R(G) \to Cl(G,L)$$; here $$L$$ denotes a minimal field of characteristic zero containing all roots of unity, $$R(G)$$ denotes the complex representation ring, and $$Cl(G,L)$$ denotes the class functions with values in $$L$$. On the other hand there is the classical Atiyah-Segal completion theorem stating that the Borel $$K$$-cohomology group $$K^0(EG\times_G X)$$ is isomorphic to $$K_G^0(X)\otimes_{R(G)} R(G)\widehat{_I}$$, the tensor product of the equivariant $$K$$-cohomology of $$X$$ and the completion of the represention ring with respect to the augmentation ideal. The two statements in fact can be combined, and in [J. Am. Math. Soc. 13, No. 3, 553–594 (2000; Zbl 1007.55004)], M. J. Hopkins et al. proved a generalization of the combined statement for a large class of complex oriented cohomology theories $$E$$, e.g. like $$K$$-theory, Morava $$E$$-theories $$E_n$$ or the Lubin-Tate-theories $$E\widehat{_n}$$ (see Theorem C, ibid.), namely that there is an isomorphism $$L(E^{*}) \otimes_{E^{*}} E^{*}(EG \times_G X) \cong Cl(G,X; L(E^*))$$ where $$L(E^*)$$ denotes a suitable ring constructed from the cohomology theory $$E^{*}$$, and $$Cl(G,X; L(E^*))$$ is a ring of generalized class functions with values in $$L(E^*)$$ built from $$X$$. The result can be viewed as a generalization of the classical statements for $$K$$-theory, which is a complex oriented cohomology theory of height 1, to corresponding statements for complex oriented cohomology theories of higher height.
In his paper Stapleton now constructs for Morava $$E_n$$-theory and any $$0\leq t <n$$ a complex oriented cohomology theory $$C_{n,t}^{*}$$ of height $$t$$ together with an according character isomorphism $$C_{n,t} \otimes_{E_n^{*}} E_{n}^{*}(EG \times_G X) \cong C_{n,t}^*(X)$$ where $$C_{n,t}$$ is a suitable ring over $$E_n^*$$. And indeed, the isomorphism of Kuhn, Hopkins and Ravenel coincides with the special case of $$t=0$$. One major input for getting this result is an investigation of the formal group $$\mathbb{G}$$ obtained from the formal group $$\mathbb{G}_{E_{n}}$$ of $$E_n$$ by extending scalars along the homomorphism $$\pi_0 E_n \to \pi_0 L_{K(t)}E_n$$ induced by the Bousfield localization with respect to Morava $$K(t)$$-theory. The author shows that $$\mathbb{G}$$ fits into a short exact sequence of $$p$$-divisible groups $$\mathbb{G}_0 \to \mathbb{G} \to \mathbb{G}_{\acute{e}t}$$, where $$\mathbb{G}_0$$ is formal and $$\mathbb{G}_{\acute{e}t}$$ is étale, and that the short exact sequence splits into a formal group of height $$t$$ and a constant height $$n-t$$ étale $$p$$-divisible group after extending scalars from $$E_{n}^{*}$$ to $$C_{n,t}$$. In fact, $$C_{n,t}$$ is essentially characterized by the property that it is initial with that property. After the splitting has been established the author uses decomposition techniques for $$G$$-spaces similar to those used in [ibid.] to finally obtain his desired generalized character isomorphism.

##### MSC:
 55N91 Equivariant homology and cohomology in algebraic topology 55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology 55N22 Bordism and cobordism theories and formal group laws in algebraic topology
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##### References:
 [1] M Ando, Isogenies of formal group laws and power operations in the cohomology theories $$E_n$$, Duke Math. J. 79 (1995) 423 · Zbl 0862.55004 · doi:10.1215/S0012-7094-95-07911-3 [2] M F Atiyah, Characters and cohomology of finite groups, Inst. Hautes Études Sci. Publ. Math. 9 (1961) 247 · Zbl 0107.02303 · doi:10.1007/BF02698718 · numdam:PMIHES_1961__9__23_0 · eudml:103826 [3] A K Bousfield, D M Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics 304, Springer (1972) · Zbl 0259.55004 [4] M Demazure, Lectures on $$p$$-divisible groups, Lecture Notes in Mathematics 302, Springer (1972) · Zbl 0247.14010 [5] M Demazure, P Gabriel, Groupes algébriques, Tome I: Géométrie algébrique, généralités, groupes commutatifs, Masson (1970) · Zbl 0203.23401 [6] D Dugger, A primer on homotopy colimits (2008) · pages.uoregon.edu [7] D Eisenbud, Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics 150, Springer (1995) · Zbl 0819.13001 · doi:10.1007/978-1-4612-5350-1 [8] P Gabriel, M Zisman, Calculus of fractions and homotopy theory, Ergeb. Math. Grenzgeb. 35, Springer (1967) · Zbl 0186.56802 [9] M J Hopkins, N J Kuhn, D C Ravenel, Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (2000) 553 · Zbl 1007.55004 · doi:10.1090/S0894-0347-00-00332-5 [10] M A Hovey, $$v_n$$-elements in ring spectra and applications to bordism theory, Duke Math. J. 88 (1997) 327 · Zbl 0880.55006 · doi:10.1215/S0012-7094-97-08813-X [11] N M Katz, B Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies 108, Princeton Univ. Press (1985) · Zbl 0576.14026 [12] N J Kuhn, Character rings in algebraic topology (editors S M Salamon, B Steer, W A Sutherland), London Math. Soc. Lecture Note Ser. 139, Cambridge Univ. Press (1989) 111 · Zbl 0693.55014 [13] H Matsumura, Commutative algebra, Mathematics Lecture Note Series 56, Benjamin/Cummings (1980) · Zbl 0441.13001 [14] J P May, The geometry of iterated loop spaces, Lectures Notes in Mathematics 271, Springer (1972) · Zbl 0244.55009 [15] J S Milne, Étale cohomology, Princeton Mathematical Series 33, Princeton University Press (1980) · Zbl 0433.14012 [16] F Oort, Commutative group schemes, Lecture Notes in Mathematics 15, Springer (1966) · Zbl 0216.05603 · doi:10.1007/BFb0097479 [17] D C Ravenel, Nilpotence and periodicity in stable homotopy theory, Annals of Mathematics Studies 128, Princeton Univ. Press (1992) · Zbl 0774.55001 [18] D C Ravenel, W S Wilson, The Morava $$K$$-theories of Eilenberg-Mac Lane spaces and the Conner-Floyd conjecture, Amer. J. Math. 102 (1980) 691 · Zbl 0466.55007 · doi:10.2307/2374093 [19] C Rezk, Notes on the Hopkins-Miller theorem (editors M Mahowald, S Priddy), Contemp. Math. 220, Amer. Math. Soc. (1998) 313 · Zbl 0910.55004 · doi:10.1090/conm/220/03107 [20] C Rezk, The units of a ring spectrum and a logarithmic cohomology operation, J. Amer. Math. Soc. 19 (2006) 969 · Zbl 1106.55002 · doi:10.1090/S0894-0347-06-00521-2 [21] N P Strickland, Finite subgroups of formal groups, J. Pure Appl. Algebra 121 (1997) 161 · Zbl 0916.14025 · doi:10.1016/S0022-4049(96)00113-2
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