Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T13:59:19.439Z Has data issue: false hasContentIssue false
  • English
  • Français

RATIONAL LOCAL SYSTEMS AND CONNECTED FINITE LOOP SPACES

Part of: Homotopy theory Spectral sequences

Published online by Cambridge University Press:  14 January 2021

DREW HEARD
DREW HEARD*
Affiliation:
Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway e-mail:[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Greenlees has conjectured that the rational stable equivariant homotopy category of a compact Lie group always has an algebraic model. Based on this idea, we show that the category of rational local systems on a connected finite loop space always has a simple algebraic model. When the loop space arises from a connected compact Lie group, this recovers a special case of a result of Pol and Williamson about rational cofree G-spectra. More generally, we show that if K is a closed subgroup of a compact Lie group G such that the Weyl group WGK is connected, then a certain category of rational G-spectra “at K” has an algebraic model. For example, when K is the trivial group, this is just the category of rational cofree G-spectra, and this recovers the aforementioned result. Throughout, we pay careful attention to the role of torsion and complete categories.

MSC classification

Primary: 55P91: Equivariant homotopy theory
Secondary: 55P35: Loop spaces 55T15: Adams spectral sequences
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 64 , Issue 1 , January 2022 , pp. 136 - 164
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

References

REFERENCES

Kasper, K. Andersen, S., Tilman, Bauer, Jesper Grodal, and Erik Kjaer Pedersen, A finite loop space not rationally equivalent to a compact Lie group, Invent. Math. 157 (2004), no. 1, 1–10. MR 2135183Google Scholar
David, Ayala, Aaron, Mazel-Gee, and Nick, Rozenblyum, Stratified noncommutative geometry, arXiv:1910.14602.Google Scholar
David, Barnes, Rational O(2)-equivariant spectra, Homology Homotopy Appl. 19 (2017), no. 1, 225–252. MR 3633720Google Scholar
Tobias, Barthel, Natàlia, Castellana, Drew, Heard, and Gabriel, Valenzuela, Stratification and duality for homotopical groups, Adv. Math. 354 (2019), 106733, 61. MR 3989930Google Scholar
Tibor, Beke, Sheafifiable homotopy model categories, Math. Proc. Cambridge Philos. Soc. 129 (2000), no. 3, 447–475. MR 1780498Google Scholar
Tobias, Barthel and Martin, Frankland, Completed power operations for Morava E-theory, Algebr. Geom. Topol. 15 (2015), no. 4, 2065–2131. MR 3402336Google Scholar
Scott, Balchin and Greenlees, J. P. C., Adelic models of tensor-triangulated categories, Advances in Mathematics 375 (2020).10.1016/j.aim.2020.107339CrossRefGoogle Scholar
Tobias, Barthel, Drew, Heard, and Gabriel, Valenzuela, Local duality for structured ring spectra, J. Pure Appl. Algebra 222 (2018), no. 2, 433–463. MR 3694463Google Scholar
Tobias, Barthel, Drew, Heard, and Gabriel, Valenzuela, Local duality in algebra and topology, Adv. Math. 335 (2018), 563–663. MR 383667410.1016/j.aim.2018.07.017CrossRefGoogle Scholar
Tobias, Barthel, Drew, Heard, and Gabriel, Valenzuela, Derived completion for comodules, Manuscripta Math. 161 (2020), no. 3-4, 409–438. MR 4060488Google Scholar
Tilman, Bauer, Nitu, Kitchloo, Dietrich, Notbohm, and Erik Kjæ, r Pedersen, Finite loop spaces are manifolds, Acta Math. 192 (2004), no. 1, 5–31. MR 2079597Google Scholar
Tobias, Barthel, May, J. P., and Emily, Riehl, Six model structures for DG-modules over DGAs: model category theory in homological action, New York J. Math. 20 (2014), 1077–1159. MR 3291613Google Scholar
Brodmann, M. P. and Sharp, R. Y., Local cohomology, second ed., Cambridge Studies in Advanced Mathematics, vol. 136, Cambridge University Press, Cambridge, 2013, An algebraic introduction with geometric applications. MR 3014449Google Scholar
Dwyer, W. G. and Greenlees, J. P. C., Complete modules and torsion modules, Amer. J. Math. 124 (2002), no. 1, 199–220. MR 1879003Google Scholar
Dwyer, W. G., Greenlees, J. P. C., and Iyengar, S., Duality in algebra and topology, Adv. Math. 200 (2006), no. 2, 357–402. MR 2200850Google Scholar
Dwyer, W. G., Strong convergence of the Eilenberg-Moore spectral sequence, Topology 13 (1974), 255–265. MR 39466310.1016/0040-9383(74)90018-4CrossRefGoogle Scholar
Elmendorf, A. D., Kriz, I., Mandell, M. A., and May, J. P., Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs, vol. 47, American Mathematical Society, Providence, RI, 1997, With an appendix by M. Cole. MR 1417719Google Scholar
Greenlees, J. P. C. and May, J. P., Derived functors of I-adic completion and local homology, J. Algebra 149 (1992), no. 2, 438–453. MR 1172439Google Scholar
Greenlees, J. P. C. and May, J. P., Generalized Tate cohomology, Mem. Amer. Math. Soc. 113 (1995), no. 543, viii+178. MR 1230773Google Scholar
Greenlees, J. P. C., Tate cohomology in axiomatic stable homotopy theory, Cohomological methods in homotopy theory (Bellaterra, 1998), Progr. Math., vol. 196, Birkhäuser, Basel, 2001, pp. 149–176. MR 185125310.1007/978-3-0348-8312-2_12CrossRefGoogle Scholar
Greenlees, J. P. C., Triangulated categories of rational equivariant cohomology theories, Oberwolfach Reports 8/2006 (2006), 480–488.Google Scholar
Greenlees, J. P. C., The Balmer spectrum of rational equivariant cohomology theories, J. Pure Appl. Algebra 223 (2019), no. 7, 2845–2871. MR 3912951Google Scholar
Greenlees, J. P. C. and Shipley, B., An algebraic model for free rational G-spectra for connected compact Lie groups G, Math. Z. 269 (2011), no. 1-2, 373–400. MR 2836075Google Scholar
Greenlees, J. P. C. and Shipley, B., The cellularization principle for Quillen adjunctions, Homology Homotopy Appl. 15 (2013), no. 2, 173–184. MR 3138375Google Scholar
Greenlees, J. P. C. and Shipley, B., An algebraic model for free rational G-spectra, Bull. Lond. Math. Soc. 46 (2014), no. 1, 133–142. MR 3161769Google Scholar
Greenlees, J. P. C. and Shipley, B., An algebraic model for rational torus-equivariant spectra, J. Topol. 11 (2018), no. 3, 666–719. MR 3830880Google Scholar
Hill, M. A., Hopkins, M. J., and Ravenel, D. C., On the nonexistence of elements of Kervaire invariant one, Ann. of Math. (2) 184 (2016), no. 1, 1–262. MR 3505179Google Scholar
Vladimir, Hinich, Dwyer-Kan localization revisited, Homology Homotopy Appl. 18 (2016), no. 1, 27–48. MR 3460765Google Scholar
Kathryn, Hess, Magdalena, Kędziorek, Emily, Riehl, and Brooke, Shipley, A necessary and sufficient condition for induced model structures, J. Topol. 10 (2017), no. 2, 324–369. MR 3653314Google Scholar
Michael, Hopkins and Jacob, Lurie, Ambidexterity in K(n)-local stable homotopy theory, Available at https://www.math.ias.edu/˜lurie/papers/Ambidexterity.pdf.Google Scholar
Mark, Hovey, Some spectral sequences in Morava E-theory, Available at https://hopf.math.purdue.edu/Hovey/morava-E-SS.pdf.Google Scholar
Mark, Hovey, Palmieri, John H., and Strickland, Neil P., Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128 (1997), no. 610, x+114. MR 1388895Google Scholar
Mark, Hovey and Strickland, Neil P., Morava K-theories and localisation, Mem. Amer. Math. Soc. 139 (1999), no. 666, viii+100. MR 1601906Google Scholar
Magdalena, Kędziorek, An algebraic model for rational SO(3)-spectra, Algebr. Geom. Topol. 17 (2017), no. 5, 3095–3136. MR 3704254Google Scholar
Lewis, L. G., , J. P.Jr. May, , Steinberger, M., and McClure, J. E., Equivariant stable homotopy theory, Lecture Notes in Mathematics, vol. 1213, Springer-Verlag, Berlin, 1986, With contributions by J. E. McClure. MR 866482CrossRefGoogle Scholar
Wolfgang, Lück, Survey on classifying spaces for families of subgroups, Infinite groups: geometric, combinatorial and dynamical aspects, Progr. Math., vol. 248, Birkhäuser, Basel, 2005, pp. 269–322. MR 2195456Google Scholar
Jacob, Lurie, Higher topos theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, Princeton, NJ, 2009. MR 252265910.1515/9781400830558CrossRefGoogle Scholar
Jacob, Lurie, Higher Algebra, 2017, Draft available from author’s website as https://www.math.ias.edu/˜lurie/papers/HA.pdf.Google Scholar
Jacob, Lurie, Spectral Algebraic Geometry, 2018, Draft available from author’s website as https://www.math.ias.edu/˜lurie/papers/SAG-rootfile.pdf.Google Scholar
Mandell, M. A. and May, J. P., Equivariant orthogonal spectra and S-modules, Mem. Amer. Math. Soc. 159 (2002), no. 755, x+108. MR 1922205Google Scholar
Akhil, Mathew, Niko, Naumann, and Justin, Noel, Nilpotence and descent in equivariant stable homotopy theory, Adv. Math. 305 (2017), 994–1084. MR 3570153Google Scholar
Amnon, Neeman, Grothendieck duality made simple, K-theory in algebra, analysis and topology, Contemp. Math., vol. 749, Amer. Math. Soc., Providence, RI, 2020, pp. 279–325. MR 408764310.1090/conm/749/15076CrossRefGoogle Scholar
Thomas, Nikolaus and Peter, Scholze, On topological cyclic homology, Acta Math. 221 (2018), no. 2, 203–409. MR 3904731Google Scholar
Leonid, Positselski, Dedualizing complexes and MGM duality, J. Pure Appl. Algebra 220 (2016), no. 12, 3866–3909. MR 3517561Google Scholar
Marco, Porta, Liran, Shaul, and Amnon, Yekutieli, On the homology of completion and torsion, Algebr. Represent. Theory 17 (2014), no. 1, 31–67. MR 3160712Google Scholar
Luca, Pol and Jordan, Williamson, The left localization principle, completions, and cofree G-spectra, Journal of Pure and Applied Algebra 224 (2020), no. 11, 106408.Google Scholar
Quigley, J. D. and Jay, Shah, On the parametrized Tate construction and two theories of real p-cyclotomic spectra, arXiv:1909.03920.Google Scholar
Charles, Rezk, Analytic completion, Available at https://faculty.math.illinois.edu/˜rezk/analytic-paper.pdf.Google Scholar
Stefan, Schwede, Lecture notes on equivariant stable homotopy theory, Available at http://www.math.uni-bonn.de/people/schwede/equivariant.pdf.Google Scholar
Stefan, Schwede, Global homotopy theory, New Mathematical Monographs, vol. 34, Cambridge University Press, Cambridge, 2018. MR 3838307Google Scholar
Brooke, Shipley, Hℤ-algebra spectra are differential graded algebras, Amer. J. Math. 129 (2007), no. 2, 351–379. MR 2306038Google Scholar
The Stacks project authors, The stacks project, https://stacks.math.columbia.edu, 2020.Google Scholar