Obtaining intermediate rings of a local profinite Galois extension without localization

2010/06/16 by Daniel G. Davis, Davis, Daniel G.
Mathematics · #55N20 #55P43 #55P60 #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AT #msc:55N20 #msc:55P43 #msc:55P60

paper · pdf · doi:10.48550/arxiv.1006.3288

Updated the Acknowledgements and References. This version is accepted for publication in Journal of Homotopy and Related Structures

openalex publication_date 2010/06/16 · arxiv created 2010/08/29 · arxiv updated 2010/08/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let E_n be the Lubin-Tate spectrum and let G_n be the nth extended Morava stabilizer group. Then there is a discrete G_n-spectrum F_n, with L_K(n)(F_n) ≃ E_n, that has the property that (F_n)^hU ≃ E_n^hU, for every open subgroup U of G_n. In particular, (F_n)^hG_n ≃ L_K(n)(S⁰). More generally, for any closed subgroup H of G_n, there is a discrete H-spectrum Z_{n, H}, such that (Z_{n, H})^hH ≃ E_n^hH. These conclusions are obtained from results about consistent k-local profinite G-Galois extensions E of finite vcd, where L_k(-) is L_M(L_T(-)), with M a finite spectrum and T smashing. For example, we show that L_k(E^hH) ≃ E^hH, for every open subgroup H of G.

Related

• The homotopy fixed point spectra of profinite Galois extensions
• Every K(n)-local spectrum is the homotopy fixed points of its Morava module
• Galois extensions of Lubin-Tate spectra
• Some properties of Lubin-Tate cohomology for classifying spaces of finite groups
• Iterated homotopy fixed points for the Lubin-Tate spectrum, with an Appendix: An example of a discrete G-spectrum that is not hyperfibrant