Delta-discrete G–spectra and iterated homotopy fixed points

Davis, Daniel G¹

¹Department of Mathematics, University of Louisiana at Lafayette, Lafayette LA 70504, USA

Algebraic and Geometric Topology, Tome 11 (2011) no. 5, pp. 2775-2814

Résumé

Let G be a profinite group with finite virtual cohomological dimension and let X be a discrete G–spectrum. If H and K are closed subgroups of G, with H ◃ K, then, in general, the K∕H–spectrum XhH is not known to be a continuous K∕H–spectrum, so that it is not known (in general) how to define the iterated homotopy fixed point spectrum (XhH)hK∕H. To address this situation, we define homotopy fixed points for delta-discrete G–spectra and show that the setting of delta-discrete G–spectra gives a good framework within which to work. In particular, we show that by using delta-discrete K∕H–spectra, there is always an iterated homotopy fixed point spectrum, denoted (XhH)hδK∕H, and it is just XhK.

Additionally, we show that for any delta-discrete G–spectrum Y , there is an equivalence Y hδHhδK∕H ≃ Y hδK. Furthermore, if G is an arbitrary profinite group, there is a delta-discrete G–spectrum Xδ that is equivalent to X and, though XhH is not even known in general to have a K∕H–action, there is always an equivalence ((Xδ)hδH)hδK∕H ≃ (X δ)hδK. Therefore, delta-discrete L–spectra, by letting L equal H,K, and K∕H, give a way of resolving undesired deficiencies in our understanding of homotopy fixed points for discrete G–spectra.

Détail

DOI : 10.2140/agt.2011.11.2775

Classification : 55P42, 55P91
Keywords: homotopy fixed point spectrum, discrete $G$–spectrum, iterated homotopy fixed point spectrum

Bibliographie

[1] M Behrens, D G Davis, The homotopy fixed point spectra of profinite Galois extensions, Trans. Amer. Math. Soc. 362 (2010) 4983

[2] J E Bergner, Homotopy limits of model categories and more general homotopy theories

[3] F Borceux, Handbook of categorical algebra 3: Categories of sheaves, Encyclopedia of Mathematics and its Applications 52, Cambridge University Press (1994)

[4] C Casacuberta, B Chorny, The orthogonal subcategory problem in homotopy theory, from: "An alpine anthology of homotopy theory", Contemp. Math. 399, Amer. Math. Soc. (2006) 41

[5] D G Davis, Homotopy fixed points for $L_{K(n)}(E_n\wedge X)$ using the continuous action, J. Pure Appl. Algebra 206 (2006) 322

[6] D G Davis, Explicit fibrant replacement for discrete $G$–spectra, Homology, Homotopy Appl. 10 (2008) 137

[7] D G Davis, Iterated homotopy fixed points for the Lubin–Tate spectrum, Topology Appl. 156 (2009) 2881

[8] E S Devinatz, A Lyndon–Hochschild–Serre spectral sequence for certain homotopy fixed point spectra, Trans. Amer. Math. Soc. 357 (2005) 129

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

[10] D Dikranjan, Topological characterization of $p$–adic numbers and an application to minimal Galois extensions, Annuaire Univ. Sofia Fac. Math. Méc. 73 (1979)

[11] D Dugger, Combinatorial model categories have presentations, Adv. Math. 164 (2001) 177

[12] W G Dwyer, P S Hirschhorn, D M Kan, J H Smith, Homotopy limit functors on model categories and homotopical categories, Mathematical Surveys and Monographs 113, American Mathematical Society (2004)

[13] W Dwyer, H Miller, J Neisendorfer, Fibrewise completion and unstable Adams spectral sequences, Israel J. Math. 66 (1989) 160

[14] W G Dwyer, C W Wilkerson, Homotopy fixed-point methods for Lie groups and finite loop spaces, Ann. of Math. $(2)$ 139 (1994) 395

[15] H Fausk, Equivariant homotopy theory for pro-spectra, Geom. Topol. 12 (2008) 103

[16] P G Goerss, M J Hopkins, Moduli spaces of commutative ring spectra, from: "Structured ring spectra", London Math. Soc. Lecture Note Ser. 315, Cambridge Univ. Press (2004) 151

[17] P G Goerss, J F Jardine, Simplicial homotopy theory, Progress in Mathematics 174, Birkhäuser Verlag (1999)

[18] R I Grigorchuk, Just infinite branch groups, from: "New horizons in pro–$p$ groups", Progr. Math. 184, Birkhäuser (2000) 121

[19] K Hess, A general framework for homotopic descent and codescent

[20] P S Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs 99, American Mathematical Society (2003)

[21] M Hovey, Spectra and symmetric spectra in general model categories, J. Pure Appl. Algebra 165 (2001) 63

[22] J F Jardine, Generalized étale cohomology theories, Progress in Mathematics 146, Birkhäuser Verlag (1997)

[23] J F Jardine, Representability theorems for presheaves of spectra, J. Pure Appl. Algebra 215 (2011) 77

[24] J Lurie, Higher topos theory, Annals of Mathematics Studies 170, Princeton University Press (2009)

[25] S A Mitchell, Hypercohomology spectra and Thomason's descent theorem, from: "Algebraic $K$–theory (Toronto, ON, 1996)", Fields Inst. Commun. 16, Amer. Math. Soc. (1997) 221

[26] S A Morris, S Oates-Williams, H B Thompson, Locally compact groups with every closed subgroup of finite index, Bull. London Math. Soc. 22 (1990) 359

[27] J Rognes, Galois extensions of structured ring spectra. Stably dualizable groups, Mem. Amer. Math. Soc. 192 (2008)

[28] J Rosický, Generalized Brown representability in homotopy categories, Theory Appl. Categ. 14 (2005) 451

[29] R W Thomason, Algebraic $K$–theory and étale cohomology, Ann. Sci. École Norm. Sup. $(4)$ 18 (1985) 437

[30] B Toën, G Vezzosi, Homotopical algebraic geometry II: Geometric stacks and applications, Mem. Amer. Math. Soc. 193 (2008)

[31] J S Wilson, On just infinite abstract and profinite groups, from: "New horizons in pro–$p$ groups", Progr. Math. 184, Birkhäuser (2000) 181