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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)].
55P91 Equivariant homotopy theory in algebraic topology
55P42 Stable homotopy theory, spectra
18G80 Derived categories, triangulated categories
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