Centralizers in good groups are good

Barthel, Tobias¹ ; Stapleton, Nathaniel¹

¹Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany

Algebraic and Geometric Topology, Tome 16 (2016) no. 3, pp. 1453-1472

Résumé

We modify transchromatic character maps of the second author to land in a faithfully flat extension of Morava E–theory. Our construction makes use of the interaction between topological and algebraic localization and completion. As an application we prove that centralizers of tuples of commuting prime-power order elements in good groups are good and we compute a new example.

Détail

DOI : 10.2140/agt.2016.16.1453

Classification : 55N20
Keywords: Morava E-theory, character theory, chromatic homotopy theory, good groups

Bibliographie

[1] J F Adams, A variant of E H Brown’s representability theorem, Topology 10 (1971) 185

[2] T Barthel, M Frankland, Completed power operations for Morava E–theory, Algebr. Geom. Topol. 15 (2015) 2065

[3] M Basterra, M A Mandell, The multiplication on BP, J. Topol. 6 (2013) 285

[4] A K Bousfield, The localization of spectra with respect to homology, Topology 18 (1979) 257

[5] M Demazure, Lectures on p–divisible groups, 302, Springer (1986)

[6] A D Elmendorf, I Kriz, M A Mandell, J P May, Rings, modules, and algebras in stable homotopy theory, 47, Amer. Math. Soc. (1997)

[7] J P C Greenlees, J P May, Derived functors of I–adic completion and local homology, J. Algebra 149 (1992) 438

[8] J P C Greenlees, J P May, Completions in algebra and topology, from: "Handbook of algebraic topology" (editor I M James), North-Holland (1995) 255

[9] M Hopkins, Complex oriented cohomology theories and the language of stacks, course notes (1999)

[10] M J Hopkins, N J Kuhn, D C Ravenel, Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (2000) 553

[11] M A Hovey, vn–elements in ring spectra and applications to bordism theory, Duke Math. J. 88 (1997) 327

[12] M Hovey, Some spectral sequences in Morava E–theory (2004)

[13] M Hovey, J H Palmieri, N P Strickland, Axiomatic stable homotopy theory, 610, Amer. Math. Soc. (1997)

[14] M Hovey, N P Strickland, Morava K–theories and localisation, 666, Amer. Math. Soc. (1999)

[15] I Kriz, Morava K–theory of classifying spaces : some calculations, Topology 36 (1997) 1247

[16] I Kriz, K P Lee, Odd-degree elements in the Morava K(n) cohomology of finite groups, Topology Appl. 103 (2000) 229

[17] J Lurie, DAG, XII: Proper morphisms, completions, and the Grothendieck existence theorem (2011)

[18] N Minami, From K(n + 1)∗(X) to K(n)∗(X), Proc. Amer. Math. Soc. 130 (2002) 1557

[19] D C Ravenel, Localization with respect to certain periodic homology theories, Amer. J. Math. 106 (1984) 351

[20] C Rezk, Analytic completion (2013)

[21] T M Schlank, N Stapleton, A transchromatic proof of Strickland’s theorem, Adv. Math. 285 (2015) 1415

[22] B Schuster, Morava K–theory of classifying spaces, Habilitationsschrift, Wuppertal (2006)

[23] B Schuster, Morava K–theory of groups of order 32, Algebr. Geom. Topol. 11 (2011) 503

[24] B Schuster, On the Morava K–theory of M12, Proc. Amer. Math. Soc. 142 (2014) 1437

[25] N Stapleton, Transchromatic generalized character maps, Algebr. Geom. Topol. 13 (2013) 171

[26] M Tanabe, On Morava K–theories of Chevalley groups, Amer. J. Math. 117 (1995) 263

[27] T S P Authors, The Stacks Project (2014)