An unstable change of rings for Morava E–theory

Thompson, Robert¹

¹Department of Mathematics and Statistics, Hunter College and the Graduate Center, CUNY, New York, NY, United States

Algebraic and Geometric Topology, Tome 20 (2020) no. 5, pp. 2145-2176

Résumé

The Bousfield–Kan (or unstable Adams) spectral sequence can be constructed for various homology theories, such as Brown–Peterson homology theory BP, Johnson–Wilson theory E(n) or Morava E–theory En. For nice spaces the E2–term is given by Ext in a category of unstable comodules. We establish an unstable Morava change of rings isomorphism between Ext𝒰Γ B(B,M) and Ext𝒰E n∗En∕In(En∗∕In,En∗⊗BP∗M), where (B,ΓB) denotes the Hopf algebroid (vn−1BP∗∕In,vn−1BP∗BP∕In). We show that the latter groups can be interpreted as Ext in the category of continuous modules over the profinite monoid of endomorphisms of the Honda formal group law. By comparing this with the cohomology of the Morava stabilizer group we obtain an unstable Morava vanishing theorem when p − 1 ∤ n. 

Détail

DOI : 10.2140/agt.2020.20.2145

Classification : 55N20, 55Q51, 55T15
Keywords: unstable Adams spectral sequence, Morava changes of rings theorem

Bibliographie

[1] M Bendersky, Some calculations in the unstable Adams–Novikov spectral sequence, Publ. Res. Inst. Math. Sci. 16 (1980) 739 | DOI

[2] M Bendersky, E B Curtis, H R Miller, The unstable Adams spectral sequence for generalized homology, Topology 17 (1978) 229 | DOI

[3] M Bendersky, E B Curtis, D Ravenel, EHP sequences in BP theory, Topology 21 (1982) 373 | DOI

[4] M Bendersky, J R Hunton, On the coalgebraic ring and Bousfield–Kan spectral sequence for a Landweber exact spectrum, Proc. Edinb. Math. Soc. 47 (2004) 513 | DOI

[5] M Bendersky, R D Thompson, The Bousfield–Kan spectral sequence for periodic homology theories, Amer. J. Math. 122 (2000) 599 | DOI

[6] T P Bisson, A Joyal, Q–rings and the homology of the symmetric groups, from: "Operads : Proceedings of Renaissance Conferences" (editors J L Loday, J D Stasheff, A A Voronov), Contemp. Math. 202, Amer. Math. Soc. (1997) 235 | DOI

[7] M. Boardman, Two more bases of QBP∗(BP∗), preprint (1984)

[8] A K Bousfield, The K–theory localizations and v1–periodic homotopy groups of H–spaces, Topology 38 (1999) 1239 | DOI

[9] A K Bousfield, Cosimplicial resolutions and homotopy spectral sequences in model categories, Geom. Topol. 7 (2003) 1001 | DOI

[10] A K Bousfield, D M Kan, The homotopy spectral sequence of a space with coefficients in a ring, Topology 11 (1972) 79 | DOI

[11] E S Devinatz, Morava’s change of rings theorem, from: "The Čech centennial" (editors M Cenkl, H Miller), Contemp. Math. 181, Amer. Math. Soc. (1995) 83 | DOI

[12] E S Devinatz, M J Hopkins, Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups, Topology 43 (2004) 1 | DOI

[13] M Hopkins, Hopf-algebroids and a new proof of the Morava–Miller–Ravenel change of rings theorem, preprint (1995)

[14] M Hovey, Morita theory for Hopf algebroids and presheaves of groupoids, Amer. J. Math. 124 (2002) 1289 | DOI

[15] M Hovey, Homotopy theory of comodules over a Hopf algebroid, from: "Homotopy theory : relations with algebraic geometry, group cohomology, and algebraic –theory" (editors P Goerss, S Priddy), Contemp. Math. 346, Amer. Math. Soc. (2004) 261 | DOI

[16] M Hovey, H Sadofsky, Invertible spectra in the E(n)–local stable homotopy category, J. London Math. Soc. 60 (1999) 284 | DOI

[17] H R Miller, D C Ravenel, Morava stabilizer algebras and the localization of Novikov’s E2–term, Duke Math. J. 44 (1977) 433 | DOI

[18] J Morava, Noetherian localisations of categories of cobordism comodules, Ann. of Math. 121 (1985) 1 | DOI

[19] D Mulcahey, An unstable variant of the Morava change of rings theorem for K(n) theory, PhD thesis, City University of New York (2011)

[20] G M L Powell, Unstable modules over the Steenrod algebra revisited, from: "Proceedings of the school and conference in algebraic topology" (editors J Hubbuck, N H V Hu’ng, L Schwartz), Geom. Topol. Monogr. 11, Geom. Topol. Publ. (2007) 245 | DOI

[21] D C Ravenel, Complex cobordism and stable homotopy groups of spheres, 121, Academic (1986)

[22] D C Ravenel, W S Wilson, The Hopf ring for complex cobordism, J. Pure Appl. Algebra 9 (1977) 241 | DOI

[23] D C Ravenel, W S Wilson, The Hopf ring for P(n), Canad. J. Math. 48 (1996) 1044 | DOI