The spectrum of the equivariant stable homotopy category of a finite group

Abstract

We study the spectrum of prime ideals in the tensor-triangulated category of compact equivariant spectra over a finite group. We completely describe this spectrum as a set for all finite groups. We also make significant progress in determining its topology and obtain a complete answer for groups of square-free order. For general finite groups, we describe the topology up to an unresolved indeterminacy, which we reduce to the case of p-groups. We then translate the remaining unresolved question into a new chromatic blueshift phenomenon for Tate cohomology. Finally, we draw conclusions on the classification of thick tensor ideals.

Equivariant chromatic localizations and commutativity

“…The Balmer-Sanders' Classification. The equivariant thick subcategories of Sp G c have been classified by Balmer-Sanders using Balmer's notion of the spectrum of a tensor triangulated category [3], where again Sp G c is the full subcategory of compact objects in G-spectra. The Balmer spectrum of a tensor triangulated category should be thought of as an extension of the classical Zariski spectrum to a context which formally looks like the derived category of modules over a ring [2].…”

On some adjunctions in equivariant stable homotopy theory

“…Thus, in this case, precisely the functors f (−1) , f (2) , f (−2) exist, leading to a chain of a total of 7 adjoints. In fact, these functors can be described geometrically by taking successive left adjoints of (7): We have…”

“…Kriz also acknowledges the support of a Simons Collaboration Grant. under suitable assumptions (see Section 2 below), only three possibilities arise, namely a chain of 3, 5 or infinitely many adjoints on both sides. Geometric fixed points, in the case of a complete universe, satisfy the assumptions of [1], and in this context (see also [2]), it seemed interesting to look at this example more closely.…”

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

“…It has been much studied in recent years, especially after its crucial role in the solution of the Kervaire invariant one problem [HHR16]. Building on unpublished work of Strickland and Joachimi [Joa15], Balmer and Sanders [BS17] determine the underlying set of Spc(Sp ω G ) and show that the topology of Spc(Sp ω G ) is closely related to the blue-shift phenomenon in generalized Tate cohomology discovered by Greenlees, Hovey, and Sadofsky [GS96,HS96]. More precisely, they show that determining the topology on Spc(Sp ω G ) is equivalent to computing the blue-shift numbers n (G; −, −) (defined below in Definition 1.2) of G, which broadly speaking measure how equivariant homotopy theory interacts with chromatic homotopy theory.…”

“…In [BS17], Balmer and Sanders study, for a finite group G, the Balmer spectrum Spc(Sp ω G ) of the homotopy category Sp ω G of compact genuine G-spectra [LMS86]. We recommend the introduction of [BS17] for a thorough overview of this problem. The results depend on the thick subcategory theorem of Hopkins and Smith [HS98,Rav92], which we will now recall (see [Bal10,Sec.…”