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Stratification and duality for homotopical groups. (English) Zbl 1426.55017
Let \(G\) be a finite group. Some of the most celebrated results in group cohomology are the \(F\)-isomorphism theorem of Quillen and the study of the spectrum of the cohomology ring of \(G\) with coefficients in a field \(k\) whose characteristic divides the order of \(G\). There have been generalizations of these results to other instances like \(p\)-compact groups and \(p\)-local finite groups. The main objective of this paper is to generalize these (and many related) results to \(p\)-local compact groups. Much of the work is devoted to establishing the technical tools to achieve these results. We provide some examples. A \(p\)-local compact group \(\mathcal{G}=(S,\mathcal{F})\) satisfies Choinard’s theorem if induction and coinduction along the morphism induced by restriction \[ C^{*}(B\mathcal{G},k)\to \prod_{\mathcal{E}(\mathcal{G})} C^{*}(BE,k), \] is conservative, where \(\mathcal{E}(\mathcal{G})\) denotes a set of representatives of \(\mathcal{F}\)-conjugacy classes of elementary abelian subgroups of \(S\) and \(C^{*}(B(-),k)\) is the spectrum of \(k\)-valued cochains on \(B(-)\). The authors prove that connected \(p\)-local groups satisfy Choinard’s theorem. They also prove the generalization of Quillen’s \(F\)-isomorphism theorem to these groups:
Theorem. Let \(\mathcal{G}=(S,\mathcal{F})\) be a \(p\)-local compact group, the the restriction to elementary abelian subgroups of \(S\) induces an \(F\)-isomorphism \[ H^{*}(B\mathcal{G},\mathbf{F}_p)\to \lim_{\mathcal{F}^{e}} H^{*}(BE,\mathbb{F}_p), \] where \(\mathbb{F}^{e}\) denotes the full subcategory of \(\mathbb{F}\) consisting elementary abelian \(p\)-groups.
They also prove that \(p\)-local compact groups satisfy Quillen’s stratification.

MSC:
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology
20J05 Homological methods in group theory
13D45 Local cohomology and commutative rings
55P42 Stable homotopy theory, spectra
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References:
[1] Aguadé, J., Constructing modular classifying spaces, Israel J. Math., 66, 1-3, 23-40, (1989) · Zbl 0697.55002
[2] Andersen, K. K.S.; Grodal, J., The classification of 2-compact groups, J. Amer. Math. Soc., 22, 2, 387-436, (2009) · Zbl 1360.55014
[3] Andersen, K. K.S.; Grodal, J.; Møller, J. M.; Viruel, A., The classification of p-compact groups for p odd, Ann. of Math. (2), 167, 1, 95-210, (2008) · Zbl 1149.55011
[4] Balmer, P.; Dell’Ambrogio, I.; Sanders, B., Grothendieck-Neeman duality and the Wirthmüller isomorphism, Compos. Math., 152, 8, 1740-1776, (2016) · Zbl 1408.18026
[5] Barthel, T.; Heard, D.; Valenzuela, G., Local duality for structured ring spectra, J. Pure Appl. Algebra, (2017)
[6] Barthel, T.; Heard, D.; Valenzuela, G., Local duality in algebra and topology, Adv. Math., 335, 563-663, (2018) · Zbl 1403.55008
[7] Bauer, T., p-compact groups as framed manifolds, Topology, 43, 3, 569-597, (2004) · Zbl 1052.55019
[8] Benson, D., Modules with injective cohomology, and local duality for a finite group, New York J. Math., 7, 201-215, (2001), (electronic) · Zbl 0994.20009
[9] Benson, D.; Greenlees, J., Stratifying the derived category of cochains on BG for G a compact Lie group, J. Pure Appl. Algebra, 218, 4, 642-650, (2014) · Zbl 1291.18014
[10] Benson, D.; Krause, H., Pure injectives and the spectrum of the cohomology ring of a finite group, J. Reine Angew. Math., 542, 23-51, (2002) · Zbl 0987.20026
[11] Benson, D.; Iyengar, S. B.; Krause, H., Local cohomology and support for triangulated categories, Ann. Sci. Éc. Norm. Supér. (4), 41, 4, 573-619, (2008)
[12] Benson, D.; Iyengar, S. B.; Krause, H., Stratifying triangulated categories, J. Topol., 4, 3, 641-666, (2011) · Zbl 1239.18013
[13] Benson, D. J.; Greenlees, J. P.C., Commutative algebra for cohomology rings of classifying spaces of compact Lie groups, J. Pure Appl. Algebra, 122, 1-2, 41-53, (1997) · Zbl 0886.57022
[14] Benson, D. J.; Greenlees, J. P.C., Localization and duality in topology and modular representation theory, J. Pure Appl. Algebra, 212, 7, 1716-1743, (2008) · Zbl 1161.20005
[15] Benson, D. J.; Krause, H., Complexes of injective kG-modules, Algebra Number Theory, 2, 1, 1-30, (2008)
[16] Benson, D. J.; Carlson, J. F.; Rickard, J., Thick subcategories of the stable module category, Fund. Math., 153, 1, 59-80, (1997) · Zbl 0886.20007
[17] Benson, D. J.; Iyengar, S. B.; Krause, H., Stratifying modular representations of finite groups, Ann. of Math. (2), 174, 3, 1643-1684, (2011) · Zbl 1261.20057
[18] Benson, D. J.; Iyengar, S. B.; Krause, H., Colocalizing subcategories and cosupport, J. Reine Angew. Math., 673, 161-207, (2012) · Zbl 1271.18012
[19] Benson, D. J.; Greenlees, J. P.C.; Shamir, S., Complete intersections and mod p cochains, Algebr. Geom. Topol., 13, 1, 61-114, (2013) · Zbl 1261.13007
[20] Bousfield, A. K., The localization of spectra with respect to homology, Topology, 18, 4, 257-281, (1979) · Zbl 0417.55007
[21] Bousfield, A. K.; Kan, D. M., Homotopy Limits, Completions and Localizations, Lecture Notes in Mathematics, vol. 304, (1972), Springer-Verlag: Springer-Verlag Berlin-New York · Zbl 0259.55004
[22] Broto, C.; Zarati, S., Nil-localization of unstable algebras over the Steenrod algebra, Math. Z., 199, 4, 525-537, (1988) · Zbl 0639.55012
[23] Broto, C.; Levi, R.; Oliver, B., The homotopy theory of fusion systems, J. Amer. Math. Soc., 16, 4, 779-856, (2003) · Zbl 1033.55010
[24] Broto, C.; Levi, R.; Oliver, B., Discrete models for the p-local homotopy theory of compact Lie groups and p-compact groups, Geom. Topol., 11, 315-427, (2007) · Zbl 1135.55008
[25] Broto, C.; Levi, R.; Oliver, B., An algebraic model for finite loop spaces, Algebr. Geom. Topol., 14, 5, 2915-2981, (2014) · Zbl 1306.55008
[26] Brown, E. H.; Comenetz, M., Pontrjagin duality for generalized homology and cohomology theories, Amer. J. Math., 98, 1, 1-27, (1976) · Zbl 0325.55008
[27] Cantarero, J.; Castellana, N., Unitary embeddings of finite loop spaces, Forum Math., 29, 2, 287-311, (2017) · Zbl 1362.55011
[28] Castellana, N., On the p-compact groups corresponding to the p-adic reflection groups \(G(q, r, n)\), Trans. Amer. Math. Soc., 358, 7, 2799-2819, (2006) · Zbl 1145.55014
[29] Chermak, A., Fusion systems and localities, Acta Math., 211, 1, 47-139, (2013) · Zbl 1295.20021
[30] Chouinard, L. G., Projectivity and relative projectivity over group rings, J. Pure Appl. Algebra, 7, 3, 287-302, (1976) · Zbl 0327.20020
[31] Dave, B., Idempotent kG-modules with injective cohomology, J. Pure Appl. Algebra, 212, 7, 1744-1746, (2008) · Zbl 1156.20042
[32] Díaz, A.; Ruiz, A.; Viruel, A., All p-local finite groups of rank two for odd prime p, Trans. Amer. Math. Soc., 359, 4, 1725-1764, (2007) · Zbl 1113.55010
[33] Dwyer, W.; Zabrodsky, A., Maps between classifying spaces, (Algebraic Topology. Algebraic Topology, Barcelona, 1986. Algebraic Topology. Algebraic Topology, Barcelona, 1986, Lecture Notes in Math., vol. 1298, (1987), Springer: Springer Berlin), 106-119
[34] Dwyer, W. G.; Wilkerson, C. W., A new finite loop space at the prime two, J. Amer. Math. Soc., 6, 1, 37-64, (1993) · Zbl 0769.55007
[35] Dwyer, W. G.; Wilkerson, C. W., Homotopy fixed-point methods for Lie groups and finite loop spaces, Ann. of Math. (2), 139, 2, 395-442, (1994) · Zbl 0801.55007
[36] Dwyer, W. G.; Wilkerson, C. W., The fundamental group of a p-compact group, Bull. Lond. Math. Soc., 41, 3, 385-395, (2009) · Zbl 1170.55006
[37] Dwyer, W. G.; Miller, H. R.; Wilkerson, C. W., The homotopic uniqueness of \(B S^3\), (Algebraic Topology. Algebraic Topology, Barcelona, 1986. Algebraic Topology. Algebraic Topology, Barcelona, 1986, Lecture Notes in Math., vol. 1298, (1987), Springer: Springer Berlin), 90-105
[38] Elmendorf, A. D.; Kriz, I.; Mandell, M. A.; May, J. P., Rings, Modules, and Algebras in Stable Homotopy Theory, Mathematical Surveys and Monographs, vol. 47, (1997), American Mathematical Society: American Mathematical Society Providence, RI, With an appendix by M. Cole · Zbl 0894.55001
[39] Evens, L., The cohomology ring of a finite group, Trans. Amer. Math. Soc., 101, 224-239, (1961) · Zbl 0104.25101
[40] Gonzalez, A., Finite approximations of p-local compact groups, Geom. Topol., 20, 5, 2923-2995, (2016) · Zbl 1405.55015
[41] Goto, S.; Yamagishi, K., Finite generation of Noetherian graded rings, Proc. Amer. Math. Soc., 89, 1, 41-44, (1983) · Zbl 0528.13015
[42] Greenlees, J., Homotopy invariant commutative algebra over fields, (Building Bridges Between Algebra and Topology. Building Bridges Between Algebra and Topology, Adv. Courses Math. CRM Barcelona, (2018), Birkhäuser/Springer: Birkhäuser/Springer Cham), 103-169 · Zbl 1399.13016
[43] Greenlees, J. P.C.; Lyubeznik, G., Rings with a local cohomology theorem and applications to cohomology rings of groups, J. Pure Appl. Algebra, 149, 3, 267-285, (2000) · Zbl 0965.13012
[44] Hovey, M.; Palmieri, J. H.; Strickland, N. P., Axiomatic stable homotopy theory, Mem. Amer. Math. Soc., 128, 610, (1997), x+114 · Zbl 0881.55001
[45] Hovey, M.; Shipley, B.; Smith, J., Symmetric spectra, J. Amer. Math. Soc., 13, 1, 149-208, (2000) · Zbl 0931.55006
[46] Lannes, J., Sur les espaces fonctionnels dont la source est le classifiant d’un p-groupe abélien élémentaire, Publ. Math. Inst. Hautes Études Sci., 75, 135-244, (1992), With an appendix by Michel Zisman · Zbl 0857.55011
[47] Levi, R.; Libman, A., Existence and uniqueness of classifying spaces for fusion systems over discrete p-toral groups, J. Lond. Math. Soc. (2), 91, 1, 47-70, (2015) · Zbl 1346.55015
[48] Levi, R.; Oliver, B., Construction of 2-local finite groups of a type studied by Solomon and Benson, Geom. Topol., 6, 917-990, (2002) · Zbl 1021.55010
[49] Linckelmann, M., Quillen’s stratification for fusion systems, Comm. Algebra, 45, 12, 5227-5229, (2017) · Zbl 1375.20056
[50] Lurie, J., Higher algebra, (2017), Draft available from author’s website as
[51] Margolis, H. R., Modules over the Steenrod algebra and the stable homotopy category, (Spectra and the Steenrod Algebra. Spectra and the Steenrod Algebra, North-Holland Mathematical Library, vol. 29, (1983), North-Holland Publishing Co.: North-Holland Publishing Co. Amsterdam) · Zbl 0552.55002
[52] Notbohm, D., Topological realization of a family of pseudoreflection groups, Fund. Math., 155, 1, 1-31, (1998) · Zbl 0896.55013
[53] Notbohm, D., On the 2-compact group \(\operatorname{DI}(4)\), J. Reine Angew. Math., 555, 163-185, (2003) · Zbl 1011.55009
[54] Oliver, B., Existence and uniqueness of linking systems: Chermak’s proof via obstruction theory, Acta Math., 211, 1, 141-175, (2013) · Zbl 1292.55007
[55] Puig, L., Frobenius categories, J. Algebra, 303, 1, 309-357, (2006) · Zbl 1110.20011
[56] Quillen, D., The spectrum of an equivariant cohomology ring. I, II, Ann. of Math. (2). Ann. of Math. (2), Ann. of Math. (2), 94, 573-602, (1971) · Zbl 0247.57013
[57] Ragnarsson, K., Classifying spectra of saturated fusion systems, Algebr. Geom. Topol., 6, 195-252, (2006) · Zbl 1098.55012
[58] Ragnarsson, K., Retractive transfers and p-local finite groups, Proc. Edinb. Math. Soc. (2), 51, 2, 465-487, (2008) · Zbl 1158.55019
[59] Ravenel, D. C., Nilpotence and Periodicity in Stable Homotopy Theory, Annals of Mathematics Studies, vol. 128, (1992), Princeton University Press: Princeton University Press Princeton, NJ, Appendix C by Jeff Smith · Zbl 0774.55001
[60] Rector, D. L., Loop Structures on the Homotopy Type of \(S \operatorname{\Sigma}^\infty 3\), Lecture Notes in Math., vol. 249, 99-105, (1971)
[61] Rector, D. L., Noetherian cohomology rings and finite loop spaces with torsion, J. Pure Appl. Algebra, 32, 2, 191-217, (1984) · Zbl 0543.57030
[62] Rognes, J., Galois extensions of structured ring spectra. Stably dualizable groups, Mem. Amer. Math. Soc., 192, 898, (2008), viii+137 · Zbl 1166.55001
[63] Schwartz, L., Unstable Modules Over the Steenrod Algebra and Sullivan’s Fixed Point Set Conjecture, Chicago Lectures in Mathematics, (1994), University of Chicago Press: University of Chicago Press Chicago, IL · Zbl 0871.55001
[64] Shamir, S., Stratifying derived categories of cochains on certain spaces, Math. Z., 272, 3-4, 839-868, (2012) · Zbl 1271.55010
[65] Venkov, B. B., Cohomology algebras for some classifying spaces, Dokl. Akad. Nauk SSSR, 127, 943-944, (1959) · Zbl 0099.38802
[66] Ziemiański, K., A faithful unitary representation of the 2-compact group \(\operatorname{DI}(4)\), J. Pure Appl. Algebra, 213, 7, 1239-1253, (2009) · Zbl 1169.55009
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