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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.

55P91 Equivariant homotopy theory in algebraic topology
55P42 Stable homotopy theory, spectra
18E30 Derived categories, triangulated categories (MSC2010)
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