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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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