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 blue-shift phenomenon for Tate cohomology. Finally, we draw conclusions on the classification of thick tensor ideals.

34 pages, to appear in Invent. Math

References in corpus (1)

• Grothendieck-Neeman duality and the Wirthmüller isomorphism

Cited by in corpus (25)

• Nilpotence theorems via homological residue fields
• G-symmetric monoidal categories of modules over equivariant commutative ring spectra
• Stratification and the comparison between homological and tensor triangular support
• On the Balmer spectrum for compact Lie groups
• Thick tensor ideals of right bounded derived categories
• Computing homological residue fields in algebra and topology
• The compactness locus of a geometric functor and the formal construction of the Adams isomorphism
• A characterization of finite étale morphisms in tensor triangular geometry
• Primes and fields in stable motivic homotopy theory
• The Balmer spectrum of the equivariant homotopy category of a finite abelian group
• Balmer spectra and Drinfeld centers
• The Spectrum of the Burnside Tambara Functor of a Cyclic Group
• Tate blueshift and vanishing for Real oriented cohomology
• The spectrum of equivariant Kasparov theory for cyclic groups of prime order
• On some adjunctions in equivariant stable homotopy theory
• The spectrum of excisive functors
• The spectrum of the Burnside Tambara functor
• On the Tambara Affine Line
• An $\mathbb{R}$-motivic $v_1-$self-map of periodicity $1$
• Derived Mackey functors and $C_{p^n}$-equivariant cohomology
• The reduced ring of the $RO(C_2)$-graded $C_2$-equivariant stable stems
• Equivariant $v_{1,\vec{0}}$-self maps
• Invariant prime ideals in equivariant Lazard rings
• Homological stratification and descent
• Algebraicity in monochromatic homotopy theory