# zbMATH — the first resource for mathematics

On the Balmer spectrum for compact Lie groups. (English) Zbl 1431.55012
Given a compact Lie group $$G$$, the homotopy category of finite genuine $$G$$-spectra $$\mathcal{SH}^c_G$$ is a tensor triangulated category. A lot of information about the global structure of such a category is captured by its Balmer spectrum, see [P. Balmer, J. Reine Angew. Math. 588, 149–168 (2005; Zbl 1080.18007)]. The Balmer spectrum of $$\mathcal{SH}^c_G$$ is a topological space $$\mathrm{Spc}(\mathcal{SH}^c_G)$$ whose points are the prime ideals in $$\mathcal{SH}^c_G$$, that is, the proper full subcategories $$I$$ that are closed under completing triangles, retracts, and tensoring with arbitrary objects and that have the property that whenever $$A \otimes B \in I$$, then $$A$$ or $$B$$ is in $$I$$.
The first main result of the paper under review is a full description of the underlying set of $$\mathrm{Spc}(\mathcal{SH}^c_G)$$ in terms of geometric fixed points functors for closed subgroups of $$G$$ and the known structure of prime ideals in the homotopy category of $$p$$-local finite spectra. More specifically, the authors show that every prime ideal in the homotopy category of finite $$p$$-local genuine $$G$$-spectra $$\mathcal{SH}^c_{G,(p)}$$ is of the form $P_G(H,n) = \{ X \in\mathcal{SH}^c_{G,(p)}\; | \; K(n-1)_* (\Phi^H(X)) = 0 \}$ where $$H$$ is a closed subgroup of $$G$$, $$\Phi^H$$ is the $$H$$-geometric fixed points functor, $$n$$ is a non-negative integer or infinity, and $$K(n-1)$$ is the Morava $$K$$-theory spectrum of height $$n-1$$ at the prime $$p$$. The second main result provides a description of the topology of $$\mathrm{Spc}(\mathcal{SH}^c_G)$$ under the additional assumption that $$G$$ is an abelian compact Lie group.
The results in this paper significantly extend earlier work in the finite group case by P. Balmer and B. Sanders [Invent. Math. 208, No. 1, 283–326 (2017; Zbl 1373.18016)] and T. Barthel et al. [ibid. 216, No. 1, 215–240 (2019; Zbl 1417.55016)].
##### MSC:
 55P91 Equivariant homotopy theory in algebraic topology 55P42 Stable homotopy theory, spectra 18G80 Derived categories, triangulated categories
Full Text:
##### References:
 [1] G.Arone, Iterates of the suspension map and Mitchell’s finite spectra with A_k-free cohomology, Math. Res. Lett.5 (1998), 485-496; MR 1653316. · Zbl 0930.55004 [2] G.Arone and K.Lesh, Fixed points of coisotropic subgroups of $$\unicode[STIX]{x1D6E4}_k$$ on decomposition spaces, Preprint (2017), arXiv:1701.06070. [3] G.Arone and M.Mahowald, The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres, Invent. Math.135 (1999), 743-788; MR 1669268. · Zbl 0997.55016 [4] P.Balmer, The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew. Math.588 (2005), 149-168; MR 2196732. · Zbl 1080.18007 [5] P.Balmer, Spectra, spectra, spectra – tensor triangular spectra versus Zariski spectra of endomorphism rings, Algebr. Geom. Topol.10 (2010), 1521-1563; MR 2661535. · Zbl 1204.18005 [6] P.Balmer and B.Sanders, The spectrum of the equivariant stable homotopy category of a finite group, Invent. Math.208 (2017), 283-326; MR 3621837. · Zbl 1373.18016 [7] T.Barthel, M.Hausmann, N.Naumann, T.Nikolaus, J.Noel and N.Stapleton, The Balmer spectrum of the equivariant homotopy category of a finite abelian group, Invent. Math.216 (2019), 215-240; MR 3935041. · Zbl 1417.55016 [8] A. K.Bousfield, On K (n)-equivalences of spaces, in Homotopy invariant algebraic structures (Baltimore, MD, 1998), Contemporary Mathematics, vol. 239 (American Mathematical Society, Providence, RI, 1999), 85-89; MR 1718077. · Zbl 0939.55004 [9] G.Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, vol. 46 (Academic Press, New York-London, 1972); MR 0413144. · Zbl 0246.57017 [10] T.Bröcker and T.tom Dieck, Representations of compact Lie groups, Graduate Texts in Mathematics, vol. 98 (Springer, New York, 1985); MR 781344. [11] E. S.Devinatz, M. J.Hopkins and J. H.Smith, Nilpotence and stable homotopy theory. I, Ann. of Math. (2)128 (1988), 207-241; MR 960945. · Zbl 0673.55008 [12] T.tom Dieck, Bordism of G-manifolds and integrality theorems, Topology9 (1970), 345-358; MR 0266241. · Zbl 0209.27504 [13] T.tom Dieck, The Burnside ring and equivariant stable homotopy, Lecture Notes by Michael C. Bix (Department of Mathematics, University of Chicago, Chicago, IL, 1975); MR 0423389. · Zbl 0313.57030 [14] T.tom Dieck, The Burnside ring of a compact Lie group. I, Math. Ann.215 (1975), 235-250; MR 0394711. · Zbl 0313.57030 [15] T.tom Dieck, A finiteness theorem for the Burnside ring of a compact Lie group, Compositio Math.35 (1977), 91-97; MR 0474344. · Zbl 0354.57007 [16] T.tom Dieck, Transformation groups and representation theory, Lecture Notes in Mathematics, vol. 766 (Springer, Berlin, 1979); MR 551743. · Zbl 0445.57023 [17] A.Dress, A characterisation of solvable groups, Math. Z.110 (1969), 213-217; MR 0248239. · Zbl 0174.30806 [18] H.Fausk, Survey on the Burnside ring of compact Lie groups, J. Lie Theory18 (2008), 351-368; MR 2431120. · Zbl 1143.19001 [19] H.Fausk and B.Oliver, Continuity of 𝜋-perfection for compact Lie groups, Bull. Lond. Math. Soc.37 (2005), 135-140; MR 2106728. · Zbl 1066.22005 [20] J. P. C.Greenlees and J. P.May, Equivariant stable homotopy theory, in Handbook of algebraic topology (North-Holland, Amsterdam, 1995), 277-323; MR 1361893. · Zbl 0866.55013 [21] J. P. C.Greenlees and J. P.May, Generalized Tate cohomology, Mem. Amer. Math. Soc.113 (1995), MR 1230773 (96e:55006). [22] J. P. C.Greenlees, The Balmer spectrum of rational equivariant cohomology theories, J. Pure Appl. Algebra223 (2019), 2845-2871; MR 3912951. · Zbl 1412.55008 [23] J. P. C.Greenlees and H.Sadofsky, The Tate spectrum of v_n-periodic complex oriented theories, Math. Z.222 (1996), 391-405; MR 1400199. · Zbl 0849.55005 [24] M. J.Hopkins and J. H.Smith, Nilpotence and stable homotopy theory. II, Ann. of Math. (2)148 (1998), 1-49; MR 1652975. · Zbl 0927.55015 [25] M.Hovey, J. H.Palmieri and N. P.Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc.128 (1997), MR 1388895. · Zbl 0881.55001 [26] G.Kelly, Basic concepts of enriched category theory, London Mathematical Society Lecture Note Series, vol. 64 (Cambridge University Press, Cambridge-New York, 1982); MR 651714. · Zbl 0478.18005 [27] P.Landweber, Conjugations on complex manifolds and equivariant homotopy of MU, Bull. Amer. Math. Soc.74 (1968), 271-274; MR 0222890. · Zbl 0181.26801 [28] L. G.LewisJr., J. P.May, M.Steinberger and J. E.McClure, Equivariant stable homotopy theory, Lecture Notes in Mathematics, vol. 1213 (Springer, Berlin, 1986), with contributions by J. E. McClure; MR 866482. [29] J. P.May, Equivariant homotopy and cohomology theory, CBMS Regional Conference Series in Mathematics, vol. 91 (American Mathematical Society, Providence, RI, 1996); MR 1413302. [30] M. A.Mandell and J. P.May, Equivariant orthogonal spectra and S-modules, Mem. Amer. Math. Soc.159 (2002), MR 1922205. · Zbl 1025.55002 [31] A.Mathew, N.Naumann and J.Noel, Nilpotence and descent in equivariant stable homotopy theory, Adv. Math.305 (2017), 994-1084; MR 3570153. · Zbl 1420.55024 [32] S.Mitchell, Finite complexes with A (n)-free cohomology, Topology24 (1985), 227-246; MR 793186 (86k:55007). · Zbl 0568.55021 [33] D.Montgomery and L.Zippin, A theorem on Lie groups, Bull. Amer. Math. Soc.48 (1942), 448-452; MR 0006545. · Zbl 0063.04079 [34] P.Orlik and H.Terao, Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 300 (Springer, Berlin), 1992; MR 1217488. · Zbl 0757.55001 [35] D.Ravenel, Localization with respect to certain periodic homology theories, Amer. J. Math.106 (1984), 351-414; MR 737778. · Zbl 0586.55003 [36] S.Schwede, Global homotopy theory, New Mathematical Monographs, vol. 34 (Cambridge University Press, Cambridge, 2018); MR 3838307. · Zbl 06928448 [37] D. P.Sinha, Computations of complex equivariant bordism rings, Amer. J. Math.123 (2001), 577-605; MR 1844571. · Zbl 0997.55008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.