Algebraic & Geometric Topology

Delta-discrete $G$–spectra and iterated homotopy fixed points

Daniel G Davis

Full-text: Open access

Abstract

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 HK, then, in general, the KH–spectrum XhH is not known to be a continuous KH–spectrum, so that it is not known (in general) how to define the iterated homotopy fixed point spectrum (XhH)hKH. 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 KH–spectra, there is always an iterated homotopy fixed point spectrum, denoted (XhH)hδKH, and it is just XhK.

Additionally, we show that for any delta-discrete G–spectrum Y, there is an equivalence YhδHhδKHYhδK. 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 KH–action, there is always an equivalence ((Xδ)hδH)hδKH(Xδ)hδK. Therefore, delta-discrete L–spectra, by letting L equal H,K, and KH, give a way of resolving undesired deficiencies in our understanding of homotopy fixed points for discrete G–spectra.

Article information

Source
Algebr. Geom. Topol., Volume 11, Number 5 (2011), 2775-2814.

Dates
Received: 13 June 2010
Accepted: 27 September 2010
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715305

Digital Object Identifier
doi:10.2140/agt.2011.11.2775

Mathematical Reviews number (MathSciNet)
MR2846911

Zentralblatt MATH identifier
1230.55006

Subjects
Primary: 55P42: Stable homotopy theory, spectra 55P91: Equivariant homotopy theory [See also 19L47]

Keywords
homotopy fixed point spectrum discrete $G$–spectrum iterated homotopy fixed point spectrum

Citation

Davis, Daniel G. Delta-discrete $G$–spectra and iterated homotopy fixed points. Algebr. Geom. Topol. 11 (2011), no. 5, 2775--2814. doi:10.2140/agt.2011.11.2775. https://projecteuclid.org/euclid.agt/1513715305


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References

  • M Behrens, D G Davis, The homotopy fixed point spectra of profinite Galois extensions, Trans. Amer. Math. Soc. 362 (2010) 4983–5042
  • J E Bergner, Homotopy limits of model categories and more general homotopy theories to appear in Bull. Lond. Math. Soc.
  • F Borceux, Handbook of categorical algebra 3: Categories of sheaves, Encyclopedia of Mathematics and its Applications 52, Cambridge University Press, Cambridge (1994)
  • C Casacuberta, B Chorny, The orthogonal subcategory problem in homotopy theory, from: “An alpine anthology of homotopy theory”, Contemp. Math. 399, Amer. Math. Soc., Providence, RI (2006) 41–53
  • D G Davis, Homotopy fixed points for $L_{K(n)}(E_n\wedge X)$ using the continuous action, J. Pure Appl. Algebra 206 (2006) 322–354
  • D G Davis, Explicit fibrant replacement for discrete $G$–spectra, Homology, Homotopy Appl. 10 (2008) 137–150
  • D G Davis, Iterated homotopy fixed points for the Lubin–Tate spectrum, Topology Appl. 156 (2009) 2881–2898 With an appendix by Daniel G. Davis and Ben Wieland
  • E S Devinatz, A Lyndon–Hochschild–Serre spectral sequence for certain homotopy fixed point spectra, Trans. Amer. Math. Soc. 357 (2005) 129–150
  • E S Devinatz, M J Hopkins, Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups, Topology 43 (2004) 1–47
  • D Dikranjan, Topological characterization of $p$–adic numbers and an application to minimal Galois extensions, Annuaire Univ. Sofia Fac. Math. Méc. 73 (1979) 103–110 (1986)
  • D Dugger, Combinatorial model categories have presentations, Adv. Math. 164 (2001) 177–201
  • 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, Providence, RI (2004)
  • W Dwyer, H Miller, J Neisendorfer, Fibrewise completion and unstable Adams spectral sequences, Israel J. Math. 66 (1989) 160–178
  • W G Dwyer, C W Wilkerson, Homotopy fixed-point methods for Lie groups and finite loop spaces, Ann. of Math. $(2)$ 139 (1994) 395–442
  • H Fausk, Equivariant homotopy theory for pro-spectra, Geom. Topol. 12 (2008) 103–176
  • 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, Cambridge (2004) 151–200
  • P G Goerss, J F Jardine, Simplicial homotopy theory, Progress in Mathematics 174, Birkhäuser Verlag, Basel (1999)
  • R I Grigorchuk, Just infinite branch groups, from: “New horizons in pro–$p$ groups”, Progr. Math. 184, Birkhäuser, Boston (2000) 121–179
  • K Hess, A general framework for homotopic descent and codescent
  • P S Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs 99, American Mathematical Society, Providence, RI (2003)
  • M Hovey, Spectra and symmetric spectra in general model categories, J. Pure Appl. Algebra 165 (2001) 63–127
  • J F Jardine, Generalized étale cohomology theories, Progress in Mathematics 146, Birkhäuser Verlag, Basel (1997)
  • J F Jardine, Representability theorems for presheaves of spectra, J. Pure Appl. Algebra 215 (2011) 77–88
  • J Lurie, Higher topos theory, Annals of Mathematics Studies 170, Princeton University Press, Princeton, NJ (2009)
  • S A Mitchell, Hypercohomology spectra and Thomason's descent theorem, from: “Algebraic $K$–theory (Toronto, ON, 1996)”, Fields Inst. Commun. 16, Amer. Math. Soc., Providence, RI (1997) 221–277
  • 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) 359–361
  • J Rognes, Galois extensions of structured ring spectra. Stably dualizable groups, Mem. Amer. Math. Soc. 192 (2008) viii+137
  • J Rosický, Generalized Brown representability in homotopy categories, Theory Appl. Categ. 14 (2005) no. 19, 451–479
  • R W Thomason, Algebraic $K$–theory and étale cohomology, Ann. Sci. École Norm. Sup. $(4)$ 18 (1985) 437–552
  • B Toën, G Vezzosi, Homotopical algebraic geometry II: Geometric stacks and applications, Mem. Amer. Math. Soc. 193 (2008) x+224
  • J S Wilson, On just infinite abstract and profinite groups, from: “New horizons in pro–$p$ groups”, Progr. Math. 184, Birkhäuser, Boston (2000) 181–203