Transchromatic generalized character maps. (English) Zbl 1300.55011

Stapleton, Nathaniel

Algebr. Geom. Topol. 13, No. 1, 171-203 (2013).

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.

Reviewer: Michael Joachim (Münster)

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

Keywords:

Borel cohomology; generalized character theory; Morava \(E\)-theory; formal groups; \(p\)-divisible groups

Citations:

Zbl 1007.55004

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
[2]	M F Atiyah, Characters and cohomology of finite groups, Inst. Hautes Études Sci. Publ. Math. 9 (1961) 247 · Zbl 0107.02303
[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)
[7]	D Eisenbud, Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics 150, Springer (1995) · Zbl 0819.13001
[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
[10]	M A Hovey, \(v_n\)-elements in ring spectra and applications to bordism theory, Duke Math. J. 88 (1997) 327 · Zbl 0880.55006
[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
[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
[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
[20]	C Rezk, The units of a ring spectrum and a logarithmic cohomology operation, J. Amer. Math. Soc. 19 (2006) 969 · Zbl 1106.55002
[21]	N P Strickland, Finite subgroups of formal groups, J. Pure Appl. Algebra 121 (1997) 161 · Zbl 0916.14025