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The Balmer spectrum of the equivariant homotopy category of a finite abelian group. (English) Zbl 1417.55016
In this paper, the authors identify the Balmer spectrum of the subcategory of compact objects in the homotopy category of genuine \(G\)-equivariant spectra for any finite abelian group \(G\).
To explain this in more detail, we recall that following P. Balmer [J. Reine Angew. Math. 588, 149–168 (2005; Zbl 1080.18007)], a prime ideal in a tensor triangulated category \(\mathcal T\) is a proper full subcategory \(\mathfrak p\) that is closed under completing triangles, retracts, and tensoring with arbitrary objects and that has the property that whenever \(A \otimes B \in \mathfrak p\), then \(A\) or \(B\) is in \(\mathfrak p\). The set of prime ideals supporting the objects in \(\mathcal T\) generate a topology on the set \(\mathrm{Spc}(\mathcal T)\) of prime ideals in \(\mathcal T\). The paper under review is concerned with the case where the triangulated category in question is the category \(\mathrm{Sp}_G^{\omega}\) of compact objects in the homotopy category of genuine \(G\)-spectra for a finite abelian group \(G\). P. Balmer and B. Sanders [Invent. Math. 208, No. 1, 283–326 (2017; Zbl 1373.18016)] have identified the underlying set of the spectrum of \(\mathrm{Sp}_G^{\omega}\) by describing it in terms of the images of the geometric fixed point functors \(\Phi^H : \mathrm{Sp}_G^{\omega}\to \mathrm{Sp}^{\omega}\) for varying subgroups \(H \subseteq G\) and the known structure of the spectrum of the non-equivariant stable homotopy category. Using results about blue-shift in Tate cohomology by Hovey-Sadofsky and Kuhn, Balmer-Sanders [loc. cit.] also completely described the topology of \(\mathrm{Spc}(\mathrm{Sp}_G^{\omega})\) in the case where the order of \(G\) is square-free. Moreover, they reduced the description of the topology for general finite abelian \(G\) to a question about the inclusions between certain prime ideals \(\mathcal P(H,p,n)\) in the case where \(G\) is a \(p\)-group, \(H \subseteq G\) is a subgroup, and \(1 \leq n \leq \infty\).
In the paper under review, the authors determine the inclusion relations between these prime ideals and thereby complete the description of the topology of \(\mathrm{Spc}(\mathrm{Sp}_G^{\omega})\) for a finite abelian group \(G\). More specifically, they introduce the “\(n\)th blue-shift number” associated with \( 1 \leq n \leq \infty\) and a pair of subgroups \(K \subseteq H\) of a finite \(p\)-group \(G\). These blue-shift numbers determine the above mentioned inclusions of prime ideals and thus the topology of \(\mathrm{Spc}(\mathrm{Sp}_G^{\omega})\). The authors show that they are given by the \(p\)-ranks of the quotients \(H/K\) when \(G\) is an abelian \(p\)-group and \( 1 \leq n < \infty\). This identification agrees with what Balmer-Sanders conjectured when \(G\) is an elementary abelian \(p\)-group, and differs otherwise.
The strategy of proof is to establish an upper bound for the blue shift numbers by proving a blue-shift theorem for Lubin-Tate spectra, to establish a lower bound for the blue shift numbers using certain equivariant finite complexes constructed using partition complexes, and to show that these upper and lower bounds coincide.

MSC:
55P91 Equivariant homotopy theory in algebraic topology
55P42 Stable homotopy theory, spectra
18E30 Derived categories, triangulated categories (MSC2010)
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[1] Arone, G.; Dwyer, WG; Lesh, K., Bredon homology of partition complexes, Doc. Math., 21, 1227-1268, (2016) · Zbl 1360.55005
[2] Arone, G., Lesh, K.: Fixed Points of Coisotropic Subgroups of \(\Gamma _{k}\) on Decomposition Spaces (2017). http://front.math.ucdavis.edu/1701.06070
[3] Arone, G.; Mahowald, M., The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres, Invent. Math., 135, 743-788, (1999) · Zbl 0997.55016
[4] Ando, M.; Morava, J.; Sadofsky, H., Completions of \({\mathbf{Z}}/(p)\)-Tate cohomology of periodic spectra, Geom. Topol., 2, 145-174, (1998) · Zbl 0907.55006
[5] Arone, G., Iterates of the suspension map and Mitchell’s finite spectra with \(A_k\)-free cohomology, Math. Res. Lett., 5, 485-496, (1998) · Zbl 0930.55004
[6] Balmer, P., Presheaves of triangulated categories and reconstruction of schemes, Math. Ann., 324, 557-580, (2002) · Zbl 1011.18007
[7] Balmer, P., The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew. Math., 588, 149-168, (2005) · Zbl 1080.18007
[8] Balmer, P., Spectra, spectra, spectra-tensor triangular spectra versus Zariski spectra of endomorphism rings, Algebr. Geom. Topol., 10, 1521-1563, (2010) · Zbl 1204.18005
[9] Benson, DJ; Carlson, JF; Rickard, J., Thick subcategories of the stable module category, Fund. Math., 153, 59-80, (1997) · Zbl 0886.20007
[10] Benson, DJ; Iyengar, SB; Krause, H., Stratifying modular representations of finite groups, Ann. Math. (2), 174, 1643-1684, (2011) · Zbl 1261.20057
[11] Balmer, P.; Sanders, B., The spectrum of the equivariant stable homotopy category of a finite group, Invent. Math., 208, 283-326, (2017) · Zbl 1373.18016
[12] Douglas, C.L., Francis, J., Henriques, A.G., Hill, M.A. (eds.): Topological Modular Forms, Volume 201 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2014) · Zbl 1304.55002
[13] Goerss, P.; Henn, H-W; Mahowald, M.; Rezk, C., A resolution of the \(K(2)\)-local sphere at the prime 3, Ann. Math. (2), 162, 777-822, (2005) · Zbl 1108.55009
[14] Greenlees, J.P.C., Peter May, J.: Completions in algebra and topology. In: James, I.M. (ed.) Handbook of Algebraic Topology, pp. 255-276. North-Holland, Amsterdam (1995) · Zbl 0869.55007
[15] Greenlees, J.P.C., May, J.P.: Equivariant stable homotopy theory. In: James, I.M. (ed.) Handbook of Algebraic Topology, pp. 277-323. North-Holland, Amsterdam (1995) · Zbl 0866.55013
[16] Greenlees, JPC; May, JP, Generalized Tate cohomology, Mem. Am. Math. Soc., 113, viii+178, (1995) · Zbl 0876.55003
[17] Greenlees, JPC; Sadofsky, H., The Tate spectrum of \(v_n\)-periodic complex oriented theories, Math. Z., 222, 391-405, (1996)
[18] Hahn, J.: On the Bousfield Classes of \(H_\infty \)-Ring Spectra (2016). http://front.math.ucdavis.edu/1612.041386
[19] Hill, MA; Hopkins, MJ; Ravenel, DC, On the nonexistence of elements of Kervaire invariant one, Ann. Math. (2), 184, 1-262, (2016) · Zbl 1366.55007
[20] Hopkins, MJ; Kuhn, NJ; Ravenel, DC, Generalized group characters and complex oriented cohomology theories, J. Am. Math. Soc., 13, 553-594, (2000) · Zbl 1007.55004
[21] Hopkins, M.J., Lurie, J.: Ambidexterity in \(K(n)\)-Local Stable Homotopy Theory (2013). http://www.math.harvard.edu/ lurie/papers/Ambidexterity.pdf
[22] Hopkins, M.J.: Global methods in homotopy theory. In: Homotopy Theory-Proceedings Durham Symposium 1985. Cambridge University Press, Cambridge (1987)
[23] Hovey, M.; Sadofsky, H., Tate cohomology lowers chromatic Bousfield classes, Proc. Am. Math. Soc., 124, 3579-3585, (1996) · Zbl 0866.55011
[24] Hopkins, MJ; Smith, JH, Nilpotence and stable homotopy theory. II, Ann. Math. (2), 148, 1-49, (1998) · Zbl 0924.55010
[25] Joachimi, R.: Thick Ideals in Equivariant and Motivic Stable Homotopy Categories (2015). http://front.math.ucdavis.edu/1503.08456
[26] Kuhn, NJ, Tate cohomology and periodic localization of polynomial functors, Invent. Math., 157, 345-370, (2004) · Zbl 1069.55007
[27] Laumon, G., Moret-Bailly, L.: Champs algébriques, volume 39 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer, Berlin (2000)
[28] Lewis, L.G., May, J.P., Steinberger, M.: Equivariant Stable Homotopy Theory, Volume 1213 of Lecture Notes in Mathematics. Springer, Berlin (1986)
[29] Lurie, J.: Chromatic Homotopy Theory (2010). http://www.math.harvard.edu/ lurie/252x.html
[30] Lurie, J.: Spectral Schemes (2011). http://www.math.harvard.edu/ lurie/papers/DAG-VII.pdf
[31] Lurie, J.: Spectral Algebraic Geometry (2017). http://www.math.harvard.edu/ lurie/papers/SAG-rootfile.pdf
[32] Mitchell, SA, Finite complexes with \(A(n)\)-free cohomology, Topology, 24, 227-246, (1985) · Zbl 0568.55021
[33] Mathew, A.; Meier, L., Affineness and chromatic homotopy theory, J. Topol., 8, 476-528, (2015) · Zbl 1325.55004
[34] Mathew, A., Naumann, N., Noel, J.: Derived Induction and Restriction Theory (2015). http://front.math.ucdavis.edu/1507.06867
[35] Mathew, A.; Naumann, N.; Noel, J., Nilpotence and descent in equivariant stable homotopy theory, Adv. Math., 305, 994-1084, (2017) · Zbl 1420.55024
[36] Neeman, A., The chromatic tower for \(D(R)\), Topology, 31, 519-532, (1992) · Zbl 0793.18008
[37] Neeman, A: Triangulated Categories, Volume 148 of Annals of Mathematics Studies. Princeton University Press, Princeton (2001) · Zbl 0974.18008
[38] Ravenel, D.C.: Nilpotence and Periodicity in Stable Homotopy Theory, Volume 128 of Annals of Mathematics Studies. Princeton University Press, Princeton (1992). Appendix C by Jeff Smith · Zbl 0774.55001
[39] Rezk, C.: Notes on the Hopkins-Miller theorem. In: Homotopy Theory via Algebraic Geometry and Group Representations (Evanston, IL, 1997), Volume 220 of Contemporary Mathematics, pp. 313-366. American Mathematical Society (1997) · Zbl 0910.55004
[40] Stapleton, N., Transchromatic generalized character maps, Algebr. Geom. Topol., 13, 171-203, (2013) · Zbl 1300.55011
[41] Strickland, NP, Finite subgroups of formal groups, J. Pure Appl. Algebra, 121, 161-208, (1997) · Zbl 0916.14025
[42] Stroilova, O.: The Generalized Tate Construction (2012). http://web.mit.edu/stroilo/www/main_no_cover.pdf
[43] Tate, J.T.: \(p\)-Divisible groups. In: Proceedings of the Conference on Local Fields (Driebergen, 1966), pp. 158-183. Springer, Berlin (1967)
[44] Thomason, RW, The classification of triangulated subcategories, Compos. Math., 105, 1-27, (1997) · Zbl 0873.18003
[45] Verdier, J.-L.: Des catégories dérivées des catégories abéliennes. Astérisque (239), xii+253 (1996). With a preface by Luc Illusie, Edited and with a note by Georges Maltsiniotis (1997)
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