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The Balmer spectrum of the equivariant homotopy category of a finite abelian group

Inventiones mathematicae volume 216, pages 215–240 (2019)Cite this article

Abstract

For a finite abelian group A, we determine the Balmer spectrum of \({\mathrm {Sp}}_A^{\omega }\), the compact objects in genuine A-spectra. This generalizes the case \(A={\mathbb {Z}}/p{\mathbb {Z}}\) due to Balmer and Sanders (Invent Math 208(1):283–326, 2017), by establishing (a corrected version of) their \(\hbox {log}_p\)-conjecture for abelian groups. We also work out the consequences for the chromatic type of fixed-points and establish a generalization of Kuhn’s blue-shift theorem for Tate-constructions (Kuhn in Invent Math 157(2):345–370, 2004).

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Notes

  1. The consequence of this technical assumption is that every thick \(\otimes \)-ideal of \(\mathcal {T} \) is radical, see [7, Rem. 4.3 and Prop. 4.4].

  2. Note that when the tensor unit \(\mathbf {1}\in \mathcal {T} \) generates \(\mathcal {T} \) as a thick subcategory, then every thick subcategory of \(\mathcal {T} \) is also a thick \(\otimes \)-ideal. So there is no distinction between thick \(\otimes \)-ideals and thick subcategories of \({\mathrm {Sp}}^\omega \).

  3. The fact that determining these inclusions is equivalent to knowing the topology follows from the second sentence of [11, Cor. 8.19].

  4. For comparison with [26], observe that \(\varphi ^A(X)=\Phi ^A({\tilde{E}}A\wedge F(EA_+, i_*X))\) are the geometric fixed points of the Tate-construction of the A-spectrum \(i_*X\), where \(i_* X\) is the inflation of X.

  5. Since we are now working p-locally, we will omit the second entry in the notation \({{\mathcal {P}}}(A',q,n)\) from the introduction, since q will always be p.

  6. The construction of \({\mathcal O}_{{\mathrm {Sp}}\acute{\mathrm{e}}{\mathrm {t}}(R)}\) will partially be recalled during the proof of Lemma 3.11 below.

  7. More is true: This is part of an equivalence between p-divisible commutative formal Lie groups over \(\pi _0(E)\) and connected p-divisible groups over \(\pi _0(E)\) [43, Prop. 1].

  8. Poincaré duality for K(n)-cohomology makes it clear that the types of a finite complex and its dual agree.

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Acknowledgements

Markus Hausmann thanks Gregory Arone for helpful conversations on the subject of this paper. Niko Naumann thanks Neil Strickland for making unpublished notes of his available. We thank Paul Balmer, Beren Sanders, and the anonymous referee for pointing out inaccuracies in a preliminary draft of this paper. This work first began while Tobias Barthel, Thomas Nikolaus, and Nathaniel Stapleton were in Bonn and they thank the MPIM for its hospitality.

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Authors and Affiliations

  1. University of Copenhagen, Copenhagen, Denmark

    Tobias Barthel & Markus Hausmann

  2. NWF I - Mathematik, University of Regensburg, Regensburg, Germany

    Niko Naumann

  3. University of Münster, Münster, Germany

    Thomas Nikolaus

  4. Booking.com, Amsterdam, Netherlands

    Justin Noel

  5. University of Kentucky, Lexington, KY, USA

    Nathaniel Stapleton

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  1. Tobias Barthel
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Correspondence to Niko Naumann.

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Justin Noel was partially supported by the DFG Grants: NO 1175/1-1 and SFB 1085—Higher Invariants, Regensburg. Niko Naumann and Nathaniel Stapleton were also partially supported by the SFB 1085—Higher Invariants, Regensburg. Tobias Barthel and Markus Hausmann were supported by the DNRF92.

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Barthel, T., Hausmann, M., Naumann, N. et al. The Balmer spectrum of the equivariant homotopy category of a finite abelian group. Invent. math. 216, 215–240 (2019). https://doi.org/10.1007/s00222-018-0846-5

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