THE SPECTRUM OF THE EQUIVARIANT STABLE HOMOTOPY CATEGORY OF A FINITE GROUP

PAUL BALMER AND BEREN SANDERS

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.

Contents

1. Introduction 1 2. Background on equivariant stable homotopy theory 6 3. The spectrum of the Burnside ring 10 4. The set Spc(SH(G)^{c}) 11 5. Tensor idempotents and the Tate construction 14 6. Inclusions of primes and the comparison map 19 7. The topology of Spc(SH(C_{p})^{c}) 21 8. The topology for general G and the log_{p}-Conjecture 23 9. Tate re-interpretation of the log_{p}-Conjecture 29 10. The classification of tt-ideals in SH(G)^{c} 31 References 33

1. Introduction

Let G be a finite group and let SH(G) denote the G-equivariant stable homotopy category, with respect to a complete G-universe. It is a tensor-triangulated category. We want to determine the tensor-triangular spectrum Spc(SH(G)^{c}) of its subcategory of compact objects SH(G)^{c}.

As we explain below, this question is the equivariant analogue of the celebrated chromatic filtration in classical stable homotopy theory, due to Devinatz-Hopkins-Smith [DHS88, HS98]. Important work on this topic has been done by Strickland [Str12]; see also Joachimi [Joa15, Chap. 3]. However, the question remained open even for the group of order two. We return to a discussion of earlier literature at the end of the Introduction, after stating our results.

Date: August 1, 2016.

First-named author supported by NSF grant DMS-1303073.

Second-named author supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92).

For a tensor-triangulated category 𝒦, like the above SH(G)^{c}, the topological space Spc(𝒦) := { 𝒫 ⊊ 𝒦 | 𝒫 is prime } consists of all proper tt-ideals 𝒫 (thick triangulated ⊗-ideal subcategories of 𝒦) which are prime (x ⊗ y ∈ 𝒫 implies x ∈ 𝒫 or y ∈ 𝒫); the support of an object x ∈ 𝒦 is the closed subset supp(x) := { 𝒫 ∈ Spc(𝒦) | x ∉ 𝒫 } and these closed subsets form a closed basis for the topology of Spc(𝒦). Conceptually, this spectrum is the universal space which carries a decent notion of support for objects of 𝒦. (See [Bal05, Bal10b] for more details.)

Computing the topological space Spc(𝒦) is essentially equivalent to classifying all tt-ideals of 𝒦, and the latter may be thought of as classifying the objects of 𝒦 up to the naturally available tensor-triangulated structure. More precisely, a classification of tt-ideals describes when two objects can be built from each other by taking suspensions, direct sums, direct summands, cofiber sequences, and tensor products. It is a major development of the last twenty years that such classification theorems can be obtained for highly non-trivial categories arising in stable homotopy theory, algebraic geometry, and modular representation theory [HS98, Tho97, BCR97, FP07].

Up until now, such classification theorems were obtained before the determination of the corresponding tensor-triangular spectrum. The category 𝒦 = SH(G)^{c} is a first case where the computation of Spc(𝒦) can be achieved beforehand, and the classification of tt-ideals deduced as a corollary. Indeed, the fact that we attack the spectrum directly via methods of tensor-triangular geometry is one of the novel aspects of our approach.

The subject of classifying tt-ideals was born in non-equivariant stable homotopy theory. In that case, the ⊗-unit (the sphere spectrum) generates the category, hence every thick subcategory is automatically a ⊗-ideal. It then follows from the Thick Subcategory Theorem of Hopkins and Smith [HS98] that the corresponding Spc(SH^{c}) is the space depicted at the top of the following picture (see [Bal10a, §9]):

(1.1)  [figure: 𝒞_{2,∞}; 𝒞_{3,∞}; ⋯; 𝒞_{p,∞}; ⋯; Spc(SH^{c}) =; ρ_{SH}^{c}; ⋮; ⋮; ⋮; 𝒞_{2,n+1}; 𝒞_{3,n+1}; ⋯; 𝒞_{p,n+1}; ⋯; 𝒞_{2,n}; 𝒞_{3,n}; ⋯; 𝒞_{p,n}; ⋯; ⋮; ⋮; ⋮; 𝒞_{2,2}; 𝒞_{3,2}; ⋯; 𝒞_{p,2}; ⋯; 𝒞_{0,1}; Spec(ℤ) =; 2ℤ; 3ℤ; ⋯; pℤ; ⋯; (0)]

Here 𝒞_{p,n} is the kernel in SH^{c} of the p-local (n − 1)-th Morava K-theory (composed with localization SH^{c} → SH_{(p)}^{c} at p). In particular, 𝒞_{p,1} = SH^{c,tor} =: 𝒞_{0,1} is the subcategory of torsion finite spectra, independently of p, while 𝒞_{p,∞} = ∩_{n≥1}𝒞_{p,n} = Ker(SH^{c} → SH_{(p)}^{c}) is the subcategory of p-acyclic finite spectra. The closure of each point consists of the points above it in the picture (related by an upwards line), so that each 𝒞_{p,∞} is a closed point, while 𝒞_{0,1} is a dense point. A general closed subset is the closure of finitely many points.

For an arbitrary ⊗-triangulated category 𝒦, there is a so-called comparison map

(1.2)

ρ_{𝒦} : Spc(𝒦) ⟶  Spec(End_{𝒦} (𝟙))
𝒫  ⟼ { f : 𝟙 → 𝟙 | cone(f) ∉ 𝒫 }

from the tensor-triangular spectrum of 𝒦 to the Zariski spectrum of the endomorphism ring of the ⊗-unit object 𝟙. For 𝒦 = SH^{c}, we have End_{𝒦} (𝟙) = ℤ and ρ_{SH}^{c} is the projection 𝒞_{p,∗} ↦ pℤ displayed in picture (1.1). This viewpoint highlights how the spectrum of the non-equivariant stable homotopy category is a chromatic refinement of Spec(ℤ): The Morava K-theories provide an interpolation between H ℚ and H 𝔽_{p}, or in other words, between the prime (0) and the prime pℤ.

For our tensor-triangulated category 𝒦 = SH(G)^{c}, the ring End_{𝒦} (𝟙) is isomorphic to the Burnside ring A(G), whose Zariski spectrum was determined by Dress [Dre69], and the comparison map ρ_{SH(G)^{c}} : Spc(SH(G)^{c}) → Spec(A(G)) will similarly exhibit Spc(SH(G)^{c}) as a kind of “chromatic refinement” of Spec(A(G)).

We review Dress’s result in Section 3. In short, Spec(A(G)) is covered by copies of Spec(ℤ) coming via the “H-fixed points” ring homomorphisms

f^{H} : A(G) ⟶ A(1) = ℤ
[X] ⟼ |X^{H}|  (for all finite G-sets X)

for each subgroup H ≤ G (up to conjugacy). These copies of Spec(ℤ) overlap at some primes p dividing the order of G; namely, for two subgroups H, K ≤ G, the primes 𝔭(H, p) := (f^{H})^{−1}(pℤ) and 𝔭(K, p) = (f^{K})^{−1}(pℤ) coincide in Spec(A(G)) when O^{p}(H) and O^{p}(K) are conjugate in G; here O^{p}(H) denotes the smallest normal subgroup of H of index a power of p (see details in Definition 3.1).

For every subgroup H ≤ G, the above H-fixed-points homomorphism f^{H} : A(G) = End_{SH(G)^{c}}(𝟙) → End_{SH}^{c}(𝟙) = A(1) is precisely the map induced on endomorphism rings by the geometric H-fixed points functor

Φ^{H} : SH(G)^{c} → SH^{c}

recalled in Section 2, Part (H) below. Moreover, since Φ^{H} is a tensor-triangulated functor it provides a means for pulling back the non-equivariant primes 𝒞_{p,n} to obtain “G-equivariant primes”, i.e., primes in SH(G)^{c}, as follows:

𝒫(H, p, n) := (Φ^{H})^{−1}(𝒞_{p,n}) ∈ Spc(SH(G)^{c}).

Our first main result states that every prime of SH(G)^{c} is of this form, and that the overlap is minimal.

Theorem (Theorems 4.9 and 4.14). All G-equivariant primes are obtained by pulling back non-equivariant primes via geometric fixed point functors with respect to the various subgroups H ≤ G. Moreover, there is no redundancy, in the sense that the primes 𝒫(H, p, n) and 𝒫(H′, p′, n′) coincide only if the subgroups H and H′ are conjugate in G and the chromatic primes 𝒞_{p,n} and 𝒞_{p′,n′} coincide in SH^{c}.

We thus obtain a complete description of Spc(SH(G)^{c}) as a set, for every finite group G. Interestingly, this already shows that the height one collisions observed by Dress in the Zariski spectrum of the Burnside ring do not occur in the tensortriangular spectrum of SH(G)^{c}. Thus, Spc(SH(G)^{c}) is not only a chromatic refinement of Spec(A(G)) but it also remembers some group-theoretic information which was lost in Spec(A(G)). Another interesting consequence of these results is an equivariant version of the Nilpotence Theorem (Theorem 4.15). The latter was obtained by Strickland [Str12] by different methods.

There are two main ingredients in the proofs of Theorems 4.9 and 4.14. The first is the exploitation of the fact that the geometric fixed point functor Φ^{G} : SH(G) → SH is a finite Bousfield localization. Although this property already appears in [LMS86], it does not seem to have been given the attention it deserves in the literature. We will make a systematic use of finite localizations on the category of G-equivariant spectra — both in the equivariant and chromatic directions. However, we will need to go beyond such “classical” localizations, towards the tensor-triangular analogue of the étale topology. In this vein, the second crucial ingredient is a result of [BDS15] which states that the restriction functor res_{H}^{G} : SH(G) → SH(H) is a separable extension (a. k. a. a finite étale extension). We remind the reader in (D) below. This fact enables us to apply results from [Bal13] on the tensor-triangular geometry of separable extensions. These techniques give us complete control of the geometric fixed point functors Φ^{H} at the level of tensortriangular spectra, as each Φ^{H} can be understood as a finite étale extension followed by a finite localization.

With this knowledge of the underlying set of Spc(SH(G)^{c}), we focus on its topology from Section 6 onwards. This question reduces (cf. Proposition 6.1) to understanding the inclusions between equivariant primes: 𝒫(K, q, n) ⊂ 𝒫(H, p, m). For primes corresponding to the same conjugacy class (K ∼_{G} H), these inclusions precisely match the inclusions in the original non-equivariant case, i.e., the inclusions displayed in diagram (1.1). On the other hand, the comparison map ρ_{SH(G)^{c}} : Spc(SH(G)^{c}) → Spec(A(G)) greatly restricts the possible inclusions among primes associated to different conjugacy classes of subgroups. Ultimately, the determination of the topology reduces to the question of understanding possible inclusions 𝒫(K, p, n) ⊂ 𝒫(H, p, m) for the same prime p dividing the order of G and for “p-subnormal subgroups” K ≤ H in the sense of 3.2.

We first attack this question for G = C_{p}, where we are able to give a complete answer by utilizing results of Hovey-Sadofsky [HS96] and Kuhn [Kuh04] on blueshift phenomena in Tate cohomology. It turns out that there is a shift by one: 𝒫(1, p, n) ⊂ 𝒫(C_{p}, p, m) iff n ≥ m + 1. Here is a complete picture of the spectrum in this case, together with the comparison map to the Zariski spectrum of A(C_{p}):

(1.3) [figure: ρ_{SH(Cp)^{c}}; Spc(SH(C_{p})^{c}) =; Spec(A(C_{p})) =; 𝒫(1,p,∞); 𝒫(1,p,2); 𝒫(1,p,3); 𝒫(1,p,4); 𝒫(1,p,5); •; ∘; •; ∘; •; ∘; •; ∘; ⋮; •; ∘; 𝒫(C_{p},p,∞); 𝒫(C_{p},p,2); 𝒫(C_{p},p,3); 𝒫(C_{p},p,4); 𝒫(C_{p},p,5); •; ∘; •; ∘; •; ∘; •; ∘; ⋮; •; ∘; 𝒫(1,q,n) ...; (q≠p, n≥2); •; ∘; •; ∘; •; ∘; •; ∘; ⋮; •; ∘; •; ∘; •; ∘; •; ∘; •; ∘; ⋮; •; ∘; •; ∘; •; ∘; •; ∘; •; ∘; ⋮; •; ∘; •; ∘; •; ∘; •; ∘; •; ∘; ⋮; •; ∘; …; …; …; …; …; 𝒫(C_{p},q,n) ...; (q≠p, n≥2); •; ∘; •; ∘; •; ∘; •; ∘; ⋮; •; ∘; •; ∘; •; ∘; •; ∘; •; ∘; ⋮; •; ∘; •; ∘; •; ∘; •; ∘; •; ∘; ⋮; •; ∘; •; ∘; •; ∘; •; ∘; •; ∘; ⋮; •; ∘; …; …; …; …; …; •; ∘; •; ∘; 𝒫(1,0,1); 𝒫(C_{p},0,1); ; 𝔭(1,p)=𝔭(C_{p},p); ◖ ◗; ◯; 𝔭(1,q) ...; (q≠p); •; ∘; •; ∘; •; ∘; •; ∘; …; 𝔭(C_{p},q) ...; (q≠p); •; ∘; •; ∘; •; ∘; •; ∘; …; •; ∘; •; ∘; 𝔭(1,0); 𝔭(C_{p},0)]

Looking only at the primes involving the subgroup C_{p} itself (the red dots), we see a copy of the chromatic picture in (1.1); similarly with the trivial subgroup (the green dots). Note how the collision 𝔭(1, p) = 𝔭(C_{p}, p) in the spectrum of the Burnside ring A(C_{p}) disappears in the spectrum of the C_{p}-equivariant stable homotopy category SH(C_{p})^{c}, in which the fiber over that green-red point is a connected pair of “chromatic towers”. We provide further comments in Remark 8.5.

More generally, we can fully determine the topology of Spc(SH(G)^{c}) for groups whose order is square-free (Theorem 8.12). For instance, the spectrum of the symmetric group G = S_{3} is discussed in Example 8.14.

For arbitrary finite groups, we can completely describe the topology of our spectrum up to an unresolved indeterminacy in the case of p-groups. This indeterminacy can be explained as follows. For a p-group G of order p^{r} and for n > r, we can prove that 𝒫(1, p, n) ⊂ 𝒫(G, p, n − r) but it might be possible a priori that this is not the best such inclusion: We could have an inclusion into a smaller prime, i.e., 𝒫(1, p, n) ⊂ 𝒫(G, p, n − r + 1). Our current guess is that the inclusion 𝒫(1, p, n) ⊂ 𝒫(G, p, n − r) is the sharpest one, and we introduce this as Conjecture 8.7, mainly to fix the ideas. If the answer turns out to be different than the one predicted by Conjecture 8.7, the complete determination of the topology of Spc(SH(G)^{c}) will nevertheless follow, mutatis mutandis. Here is a summary of what we know about the topology, for G an arbitrary finite group:

1.4. Theorem (Corollary 8.19). Every closed subset of Spc(SH(G)^{c}) is a finite union of irreducible closed subsets, and every irreducible closed set is of the form \overline{\{𝒫\}} = \{ 𝒬 ∈ Spc(SH(G)^{c}) | 𝒬 ⊆ 𝒫 \} for a unique prime 𝒫 ∈ Spc(SH(G)^{c}). The only inclusions 𝒬 ⊆ 𝒫 between equivariant primes happen when 𝒬 = 𝒫(K, p, n) and 𝒫 = 𝒫(H, p, m) with K conjugate in G to a p-subnormal subgroup of H (see 3.2) and when n is greater or equal to a certain chromatic integer n_{min}, which depends on p, H, K and m. We have that n_{min} ≤ m + log_{p}(|H|/|K|). Conjecture 8.7 is equivalent to n_{min} = m + log_{p}(|H|/|K|). This holds if G has square-free order.

In Section 9 we translate our conjecture into a new chromatic blue-shift phenomenon for the Tate construction, which would enhance and clarify known results in the subject. Existing blue-shift results in the literature usually consider the functor t_{G}(triv(−))^{G} : SH → SH from non-equivariant spectra to itself and roughly state that this functor lowers chromatic degree (at p) by one for all finite groups G whose order is divisible by p. (See Theorem 5.21 for a precise statement.) In particular, the chromatic degree only goes down by one even if p^{2} divides the order of G, or p^{3}, etc. In contrast, we conjecture that the functor Φ^{G}(t_{G}(triv(−))) : SH → SH, obtained by replacing categorical fixed points by geometric fixed points, reduces chromatic degree by log_{p}(|G|) for any p-group G, and more generally for Tate cohomology with respect to suitable families of subgroups. The precise statements can be found in Section 9. Specifically, we prove in Theorem 9.1 that this conjectural property of Tate cohomology is equivalent to Conjecture 8.7 and would therefore complete the determination of the topology of Spc(SH(G)^{c}) for all finite groups. Again, if future research isolates a different behavior of Tate cohomology than the shift by log_{p}(|G|), the methods we provide in Section 9 will still allow the determination of the topology of Spc(SH(G)^{c}), which is the ultimate goal.

Finally, Section 10 contains the translation of the computation of Spc(SH(G)^{c}) into the classification of tt-ideals. See Corollary 10.6.

Let us conclude our introduction with a word about the existing literature. Beyond of course the non-equivariant case [HS98] already mentioned, the problem of classifying tt-ideals in SH(G)^{c} is seriously considered only in Strickland’s unpublished notes [Str12], parts of which have been written up in Joachimi [Joa15, Chap. 3]. Although we do not rely on any result from this source, we are extremely grateful to Neil Strickland for sharing this material with us while we were working on the project. The inspiration for Proposition 7.1 clearly came from his work. It is our understanding that [Str12] does not contain a complete classification of tt-ideals, even for the cyclic group G = C_{p}; in contrast, we do provide a complete answer in this case, among others. Nevertheless, [Str12] remains a highly valuable source, discussing several other topics. For instance, Strickland proves the equivariant version of the Nilpotence Theorem, which is also our already mentioned Theorem 4.15.

Acknowledgements : We are very grateful to Neil Strickland, for the reasons explained above. We also thank John Greenlees and Mike Hill for several stimulating discussions.

2. Background on equivariant stable homotopy theory

Everywhere, “tt” is short for “tensor-triangulated” or “tensor-triangular”.

In this preliminary section we isolate some basic features of equivariant stable homotopy theory, seen from the angle of tt-geometry. All results we need concern the stable homotopy category SH(G), not any given model, and we will remain agnostic about the choice of such model-theoretic foundations. All of the following facts can be checked using, e. g., the Lewis-May approach to G-spectra [LMS86], but more recent foundations such as equivariant orthogonal spectra [MM02] would serve equally well. Our goal is not to be exhaustive, nor brilliantly pedagogical, but simply to exhibit key features of the tt-category SH(G) which enable our proofs to work. One motivation is that our techniques may have some success in computing the spectrum of other equivariant tt-categories which exhibit similar features (e. g., have a nice enough set of generators and well-behaved analogues of the geometric fixed point functors). In any case, the reader will find a modern detailed discussion of the fundamental features of equivariant stable homotopy theory in the Appendices of [HHR16], while the triangulated category aspects are discussed in [HPS97]. Before we begin, let us get the following notation and terminology out of the way:

2.1. Notation. We write H ∼_{G} H^{g} = { g^{−1}h g | h ∈ H } to indicate conjugation. We say that H is subconjugate to K if H^{g} ≤ K for some g ∈ G and write H ≤_{G} K.

2.2. Terminology. We use standard terminology from the theory of tensor triangulated categories (such as thick subcategory, localizing subcategory, ⊗-ideal, compact, rigid) all of which may be found in [BF11, §§1-2]. Note that the topology literature sometimes uses “small” or “finite” as a synonym for “compact”, and “strongly dualizable” for “rigid”. The compatibility axioms for our tt-categories are those of [HPS97, Appendix A]. Finally, by a tt-functor we mean a triangulated functor which is also a strong tensor functor.

The equivariant stable homotopy category.

(A) For any finite group G, the equivariant stable homotopy category of genuine G-spectra SH(G) is a rigidly-compactly generated tt-category; i.e., it is a compactly generated triangulated category with a compatible closed symmetric monoidal structure having the property that the compact objects coincide with the rigid (a. k. a. strongly dualizable) objects. Every pointed G-space X defines a suspension G-spectrum Σ_{G}^{∞}X in SH(G) and the collection of Σ_{G}^{∞}G/H_{+} for all subgroups H ≤ G provides a set of compact-rigid generators (see e. g. [HPS97, §9.4]). We denote the tt-category of compactrigid objects by SH(G)^{c} ⊂ SH(G).

(B) Any group homomorphism α : H → G induces a tt-functor α^{∗} : SH(G) → SH(H) which necessarily preserves compact objects (since compact=rigid in these categories and any strong tensor functor preserves rigid objects). For example, inclusion of a subgroup H ↪ G provides the restriction functor res_{H}^{G} : SH(G) → SH(H), while a quotient G ↠ G/N provides the inflation functor infl_{G/N}^{G} : SH(G/N) → SH(G).

(C) Restriction admits adjoints on both sides (induction and coinduction) which are isomorphic: ind_{H}^{G} ≃ coind_{H}^{G}. On the other hand, the right adjoint to inflation is the “categorical” fixed-point functor (−)^{N} : SH(G) → SH(G/N). This does not recover usual fixed-points on suspension spectra, nor does it preserve compact objects (cf. the tom Dieck splitting theorem) and therefore inflation does not have a left adjoint (by [BDS16]). Although (−)^{N} ∘ infl_{G/N}^{G} ≄ Id_{SH(G/N)}, we do have a projection formula:

(infl_{G/N}^{G}(x) ⊗ y)^{N} ≃ x ⊗ y^{N}

for every x ∈ SH(G/N) and y ∈ SH(G).

Restriction as a separable-extension.

(D) For every subgroup H ≤ G, the G-spectrum A_{H}^{G} := G/H_{+} (we often drop the Σ_{G}^{∞} for readability) is a separable commutative ring object (=a tt-ring, or a finite étale ring) in the tt-category SH(G). (See [Bal11] and [Bal13] for the definition of separable and for a discussion of how such rings provide tt-geometry with an analogue of the étale topology.) By [BDS15], there is a tt-equivalence

A_{H}^{G} - Mod_{SH(G)} ≅ SH(H)

such that extension-of-scalars SH(G) ⟶ A_{H}^{G} - Mod_{SH(G)} along A_{H}^{G} is isomorphic to the restriction functor res_{H}^{G} : SH(G) ⟶ SH(H). Since A_{H}^{G} is compact, the same relation holds for the compact objects : SH(H)^{c} ≅ A_{H}^{G} - Mod_{SH(G)^{c}}.

(E) By [Bal13], for any tt-category 𝒦 and any tt-ring A in 𝒦, the map induced on spectra Spc(A - Mod_{𝒦} ) → Spc(𝒦) has image exactly supp(A). Hence, in view of (D), we have the following equality of subsets of Spc(SH(G)^{c}):

Im ( Spc(res_{H}^{G})) = supp(G/H_{+}).

(F) Also by [Bal13], if a tt-ring A has finite degree (in the sense of [Bal14]) then φ_{A} : Spc(A - Mod_{𝒦} ) → Spc(𝒦) satisfies the “Going-Up Theorem”: For every 𝒬 ∈ Spc(A - Mod_{𝒦} ) and every 𝒫′ ∈ \overline{\{φ_{A}(𝒬)\}} there exists 𝒬′ ∈ \overline{\{𝒬\}} such that φ_{A}(𝒬′) = 𝒫′. It follows from [Bal14, Corollary 4.8] and the fact that the geometric fixed point functors (see below) are jointly conservative that every tt-ring in SH(G)^{c} has finite degree. So, in particular, the Going-Up Theorem holds for A = G/H_{+}, that is, for res_{H}^{G}.

Geometric fixed points.

(G) To quickly motivate the construction below – in light of the incompatibility between categorical fixed-points and suspension spectra – note that for subgroups H, K ≤ G, the H-fixed points of the G-set G/K is the set N_{G}(H, K)/K where N_{G}(H, K) := { g ∈ G | H^{g} ⊆ K }. For instance, (G/K)^{H} is empty if N_{G}(H, K) = ∅, that is, if H ≰_{G} K. In particular, when H = N ⊴ G is normal, the N-fixed points of the G-set G/K is either empty when N ≰ K, or G/K when N ≤ K. So, adding base-points, we expect N-fixed points to “kill” G/K_{+} (i.e., map it to ∅_{+} = 0) when N ≰ K. This “killing” is nothing but a localization, as we formalize below.

(H) Let N ⊴ G be a normal subgroup and let

ℒ_{N} := Loc_{⊗}(G/K_{+} | K ≤ G, N ≰ K)

denote the localizing ⊗-ideal of SH(G) generated by the compact objects { G/K_{+} | K ≤ G, N ≰ K }; see (G). Then inflation SH(G/N) → SH(G) composed with the (Bousfield) localization of SH(G) with respect to ℒ_{N}

\xymatrix{SH(G/N) \ar[r]^{^{infl{G \atop G/N}}} \ar[rr]_{≅} & SH(G) \ar@{->>}[r] & SH(G)/ℒ_{N}}

is an equivalence of tt-categories (by [LMS86, Cor. II.9.6]). Then, the composite

(2.3)  SH(G) ↠ SH(G)/ℒ_{N} ⟶^{≅} SH(G/N)

is the so-called geometric N-fixed point functor Φ̃^{N} : SH(G) ↠ SH(G/N). By construction, inflation splits the geometric fixed point functor:

(2.4)  Φ̃^{N} ∘ infl_{G/N}^{G} ≅ Id_{SH(G/N)} .

Since ℒ_{N} is compactly generated, (2.3) presents SH(G/N) as a finite localization of SH(G), with Φ̃^{N} as the localization functor. It follows from Neeman [Nee92, Thm. 2.1] that the same is also true on the subcategories of compact objects, up to an idempotent completion (denoted ^{♮}):

SH(G)^{c} ↠ SH(G)^{c}/𝒥_{N} ↪ (SH(G)^{c}/𝒥_{N})^{♮} ⟶^{≅} SH(G/N)^{c}

where 𝒥_{N} := thick_{⊗}(G/K_{+}|N ≰ K). (We will discuss such finite localizations in more detail in Section 5; see especially Remark 5.19.) Note that in this special case, the idempotent completion (−)^{♮} is not necessary, since Φ̃^{N} is split by infl_{G/N}^{G}, which preserves compact objects.

(I) For an arbitrary subgroup H ≤ G, we define the (absolute) geometric H-fixed point functor Φ^{H} : SH(G) → SH as the composite

Φ^{H} : SH(G) ⟶^{res_{H}^{G}} SH(H) \underrightarrow{Φ̃^{H}} SH .

Note that this functor is the composition of an “étale” extension in the sense of (D) with a finite localization in the sense of (H). When necessary to distinguish the ambient group, we shall write Φ^{H,G} : SH(G) → SH. Since the functor Φ^{H,G} : SH(G) → SH is a tt-functor which preserves compact objects, it induces a continuous map on tt-spectra, that we denote

φ^{H,G} : Spc(SH^{c}) → Spc(SH(G)^{c}).

It is defined by 𝒫 ↦ (Φ^{H,G})^{−1}(𝒫) = { x ∈ SH(G)^{c} | Φ^{H,G}(x) ∈ 𝒫 }.

(J) For any N ≤ N′ ⊴ G, with N ⊴ G, we have Φ̃^{N′/N} ∘ Φ̃^{N} ≅ Φ̃^{N′} from SH(G) to SH(G/N′), as localization can be performed in two “nested” steps. In fact, for any N ≤ H ≤ G, with N ⊴ G, we also have Φ^{H/N} ∘ Φ̃^{N} ≅ Φ^{H} from SH(G) to SH. For example, Φ^{G/N} ∘ Φ̃^{N} ≅ Φ^{G} and res_{1}^{G/N}∘ Φ̃^{N} ≅ Φ^{N} for any N ⊴ G.

(K) Let α : G → G′ be a group homomorphism. Consider H ≤ G and its image α(H) ≤ G′. Then we have an isomorphism Φ^{H,G} ∘ α^{∗} = Φ^{α(H),G′}. Indeed, the following diagram commutes up to isomorphism, for N := ker(α) ∩ H:

\xymatrix{SH(G′) \ar[r]^{res} \ar[dd]_{α^{∗}} \ar[drrr]^{Φ^{α(H),G′}} & SH(α(H)) \ar[dr]^{id} \ar[dd]_{α_{|H}^{∗}=infl_{α(H)}^{H}} \\  &  & SH(α(H)) \ar@{->>}[r]^{Φ̃^{α(H)}} & SH . \\ SH(G) \ar[r]_{res} \ar[urrr]_{Φ^{H,G}} & SH(H) \ar@{->>}[ur]^{Φ̃^{N}} \ar@{->>}[urr]_{Φ̃^{H}}}

It uses (2.4) for the middle triangle and (J) on the right, plus the definitions. Hence the induced maps from Spc(SH^{c}) to Spc(SH(G′)^{c}) coincide:

Spc(α^{∗}) ∘ φ^{H,G} = φ^{α(H),G′}.

(L) For any g ∈ G, the isomorphism (−)^{g} : H ⟶^{∼} H^{g} induces an equivalence SH(H^{g}) ⟶^{≅} SH(H) such that the diagram

\xymatrix{ & SH(H^{g}) \ar[dr]^{Φ̃^{H^{g}}} \ar[dd]^{≅} \\ SH(G) \ar[dr]_{_{res{G \atop H}}} \ar[ur]^{^{res{G \atop H^{g}}}} &  & SH \\  & SH(H) \ar[ur]_{Φ̃^{H}}}

commutes up to isomorphism. (The equivalence SH(G) ⟶^{≅} SH(G) induced by the inner automorphism (−)^{g} : G ⟶^{∼} G is naturally isomorphic to the identity functor.) Hence Φ^{H,G} ≅ Φ^{H^{g},G} and φ^{H,G} = φ^{H^{g},G}.

(M) The trivial homomorphism G → 1 induces a tt-functor infl_{1}^{G} abbreviated

triv : SH → SH(G)

which sends a spectrum to the “trivial” G-spectrum. It splits the (absolute) H-fixed point functor Φ^{H} : SH(G) → SH for every H ≤ G. Hence Spc(triv) retracts φ^{H,G} : Spc(SH^{c}) → Spc(SH(G)^{c}) for every H ≤ G, which shows that the maps φ^{H,G} are all injective.

(N) The geometric fixed point functors have the nice property that they commute with suspension spectra (see e. g. [LMS86, Cor. II.9.9]). In particular, for a finite G-set X and for any subgroup H ≤ G, we have Φ^{H}(Σ_{G}^{∞}X_{+}) ≅ Σ^{∞}X_{+}^{H} in SH^{c}.

3. The spectrum of the Burnside ring

The purpose of this section is to briefly recall Dress’s description of the spectrum of the Burnside ring [Dre69] and related ideas concerning p-perfect subgroups.

3.1. Definition. A group is said to be p-perfect if it admits no non-trivial homomorphism to a p-group, or equivalently if the only normal subgroup of index a power of p is the group itself. Every finite group G has a unique normal p-perfect subgroup O^{p}(G) such that the quotient G/ O^{p}(G) is a p-group; O^{p}(G) is the intersection of all normal subgroups of index a power of p. So, O^{p}(G) contains all p-perfect subgroups H of G (since the composite H → G → G/ O^{p}(G) must be trivial) and O^{p}(G) is contained in every normal subgroup of G of index a power of p.

3.2. Terminology. A p-subnormal tower H = H_{r} ⊲_{p} H_{r−1} ⊲_{p} ⋯ ⊲_{p} H_{1} ⊲_{p} H_{0} = G is a subnormal tower H_{i+1} ⊲ H_{i} all of whose subquotients H_{i}/H_{i+1} have order p.

3.3. Lemma. Let H ≤ G be a subgroup. The following are equivalent: (1) O^{p}(H) = O^{p}(G);

(2) H ⊇ O^{p}(G);

(3) There exists a p-subnormal tower from H to G. In this case, we say that H is a p-subnormal subgroup of G.

Proof. Since O^{p}(G) contains all p-perfect subgroups of G, we have O^{p}(H) ⊆ O^{p}(G). The equivalence (1) ⇔ (2) follows. To prove (1) ⇒ (3), assume (1), let N := O^{p}(H) = O^{p}(G) and consider H/N ≤ G/N in the p-group G/N. Every subgroup of a p-group admits a p-subnormal tower to the ambient group (by an easy inductive argument using the non-trivial center, for instance) and a p-subnormal tower from H/N to G/N lifts to a p-subnormal tower from H to G. Finally, (3) ⇒ (1) follows inductively from the observation that if N ⊲_{p} G then N ⊇ O^{p}(G). □

3.4. Remark. Recall that the Burnside ring, A(G), is the Grothendieck ring of the category of finite G-sets with respect to disjoint union, and equipped with Cartesian product as multiplication. For each subgroup H ≤ G, there is a ring homomorphism f^{H} : A(G) → ℤ which sends (the class of) a finite G-set X to the number of H-fixed points |X^{H}|. By pulling back the prime ideals of ℤ, we get prime ideals

(3.5)  𝔭(H, p) := (f^{H})^{−1}(pℤ)  and  𝔭(H, 0) := (f^{H})^{−1}((0))

of A(G). Dress’s result [Dre69] can be summarized as follows:

3.6. Theorem (Dress). Every prime ideal of A(G) is of the form 𝔭(H, p) or 𝔭(H, 0) for some H ≤ G. Moreover, for any subgroups H, K ≤ G and any primes p, q:

(a) 𝔭(H, 0) ⊆ 𝔭(K, 0) iff 𝔭(H, 0) = 𝔭(K, 0) iff H ∼_{G} K.

(b) 𝔭(H, p) ⊆ 𝔭(K, q) implies p = q and 𝔭(H, p) = 𝔭(K, p).

(c) 𝔭(H, 0) ⊆ 𝔭(K, p) iff 𝔭(H, p) = 𝔭(K, p) iff O^{p}(H) ∼_{G} O^{p}(K).

(d) 𝔭(H, 0) ⊂ 𝔭(H, p) and 𝔭(H, p) ⊈ 𝔭(K, 0).

3.7. Remark. In other words, the ring A(G) has Krull dimension 1 and Spec(A(G)) is covered by copies of Spec(ℤ) indexed by conjugacy classes of subgroups H ≤ G. The copy of Spec(ℤ) in Spec(A(G)) corresponding to H ≤ G is an irreducible closed subset with generic point 𝔭(H, 0). In the closure of 𝔭(H, 0) are height 1 closed points 𝔭(H, p) for all prime numbers p. The only overlap between two such copies, say for two subgroups H and K, happens at closed points, corresponding to the same prime p in each copy, exactly when O^{p}(H) and O^{p}(K) are conjugate in G. For given H and K, this can happen at several primes p. However, note that if p does not divide the order of G then O^{p}(H) = H for all H ≤ G. Hence there can only be non-trivial collisions for p dividing the order of G. See the bottom of (1.3) for a picture of the G = C_{p} case (which is representative of the situation for all p-groups) and see Example 8.14 for a picture of the G = S_{3} case.

3.8. Remark. The ring homomorphism A(G) ≅ End_{SH(G)^{c}}(𝟙) → End_{SH}^{c}(𝟙) ≅ ℤ induced by the geometric fixed point functor Φ^{H} : SH(G)^{c} → SH^{c} is exactly the map f^{H} : A(G) → ℤ used above (see the proof of Prop. 6.7, if necessary). From this point of view, it is very natural to attempt a categorified version of what Dress accomplished for the Burnside ring, as announced in the Introduction. Furthermore, this compatibility between Φ^{H} and f^{H} will allow (cf. Proposition 6.7) for an easy description of the comparison map ρ_{SH(G)^{c}} : Spc(SH(G)^{c}) → Spec(A(G)) once we know what the primes in Spc(SH(G)^{c}) actually are.

4. The set Spc(SH(G)^{c})

Recall the Hopkins-Smith non-equivariant primes 𝒞_{p,n} ∈ Spc(SH^{c}) from (1.1). We now consider their image under the map φ^{H,G} : Spc(SH^{c}) → Spc(SH(G)^{c}) induced by geometric H-fixed points Φ^{H,G} : SH(G) → SH, as explained in (I).

4.1. Definition. For every subgroup H ≤ G, every prime number p and every “chromatic integer” 1 ≤ n ≤ ∞, we have a tt-prime in SH(G)^{c} given by

𝒫(H, p, n) := φ^{H,G}(𝒞_{p,n}) = { x ∈ SH(G)^{c} | Φ^{H}(x) ∈ 𝒞_{p,n} in SH^{c} }.

By (L), we have that 𝒫(H, p, n) = 𝒫(H^{g}, p, n) for every g ∈ G. So 𝒫(H, p, n) only depends on the conjugacy class of H in G. Also, since 𝒞_{p,1} = 𝒞_{q,1} for all primes p, q, the same is true for 𝒫(H, p, 1) = 𝒫(H, q, 1) and we could replace p by 0 in the notation, if it avoids putting the wrong emphasis on a particular p:

𝒫(H, 0, 1) := 𝒫(H, p, 1) (for any p) = { x ∈ SH(G)^{c} | Φ^{H}(x) is torsion }.

Finally, when necessary, we will write 𝒫_{G}(H, p, n) to indicate the ambient group G.

4.2. Remark. Since 𝒞_{p,∞} = ∩_{n≥1}𝒞_{p,n} we get 𝒫(H, p, ∞) = ∩_{n≥1}𝒫(H, p, n).

Let us say a word about the functoriality of these primes 𝒫_{G}(H, p, n) in the group G. We recall that for every tt-functor F : 𝒦 → ℒ the induced map Spc(F) : Spc(ℒ) → Spc(𝒦) sends 𝒬 to F^{−1}(𝒬) and is inclusion-preserving.

4.3. Proposition. Let α : G → G′ be a group homomorphism, and α^{∗} : SH(G′) → SH(G) the induced tt-functor. Then the induced map Spc(α^{∗}) : Spc(SH(G)^{c}) → Spc(SH(G′)^{c}) maps 𝒫_{G}(H, p, n) to 𝒫_{G′}(α(H), p, n) for every H ≤ G.

Proof. This follows directly from (K) and Definition 4.1. □

4.4. Corollary. Consider H ≤ G and the tt-functor res_{H}^{G} : SH(G) → SH(H). The induced map Spc(res_{H}^{G}) : Spc(SH(H)^{c}) → Spc(SH(G)^{c}) maps 𝒫_{H}(K, p, n) to 𝒫_{G}(K, p, n) for any K ≤ H. □

4.5. Corollary. Consider N ⊴ G and the tt-functor infl_{G/N}^{G} : SH(G/N) → SH(G). The induced map Spc(infl_{G/N}^{G}) : Spc(SH(G)^{c}) ↠ Spc(SH(G/N)^{c}) maps 𝒫_{G}(H, p, n) to 𝒫_{G/N}(HN/N, p, n) for any H ≤ G. □

The special case N = G shows that we can, in some sense, drop the grouptheoretic information by projecting onto the chromatic information; see also (M):

4.6. Corollary. The tt-functor triv = infl_{1}^{G} : SH → SH(G) induces an inclusionpreserving map Spec(triv) : Spc(SH(G)^{c}) → Spc(SH^{c}) which sends the equivariant prime 𝒫(H, p, n) to the non-equivariant prime 𝒞_{p,n}. □

4.7. Proposition. Consider N ⊴ G and the tt-functor Φ̃^{N} : SH(G) ↠ SH(G/N). The induced map Spc(Φ̃^{N}) : Spc(SH(G/N)^{c}) ↪ Spc(SH(G)^{c}) is injective and maps 𝒫_{G/N}(H/N, p, n) to 𝒫_{G}(H, p, n) for any N ≤ H ≤ G. The image of Spc(Φ̃^{N}) is exactly { 𝒫 ∈ Spc(SH(G)^{c}) | G/H_{+} ∈ 𝒫 for all H s.t. N ≰ H }.

Proof. In view of (H), Φ̃^{N} : SH(G)^{c} ↠ SH(G/N)^{c} is the localization with respect to 𝒥_{N} := thick_{⊗}(G/K_{+}|N ≰ K). It is a general fact that Spc(𝒦/𝒥) ≅ { 𝒫 ∈ Spc(𝒦) | 𝒥 ⊆ 𝒫 }, hence the injectivity of the induced map on spectra and the description of its image. The description of its value on 𝒫_{G/N}(H/N, p, n) comes from the functoriality of Spc(−) and the relation Φ^{H/N} ∘ Φ̃^{N} ≅ Φ^{H} in (J). □

4.8. Remark. Contrary to what happens in Proposition 4.7, the apparently harmless map of Corollary 4.4 is not injective in general. There is some fusion phenomenon happening, since 𝒫_{H}(K, p, n) depends on the conjugacy class of K in H, whereas 𝒫_{G}(K, p, n) only depends on the conjugacy class of K in the larger group G.

4.9. Theorem. Every prime 𝒫 ∈ Spc(SH(G)^{c}) is of the form 𝒫(H, p, n) for some H ≤ G, some prime p and some 1 ≤ n ≤ ∞.

Proof. We need to show that Spc(SH(G)^{c}) = ⋃_{H≤G} Im(φ^{H,G}). We proceed by induction on |G|. The result is clear for G trivial. So, let us suppose the result known for all proper subgroups of G. By Proposition 4.7 for N = G, we can then identify the image of φ^{G,G} = Spc(Φ^{G}) as follows:

Im(φ^{G,G}) = { 𝒫 ∈ Spc(SH(G)^{c}) | 𝒫 ∋ G/H_{+} for all H ⪇ G }.

Since { 𝒫 | 𝒫 ∌ G/H_{+} } = supp(G/H_{+}) by definition of the support, we have

Spc(SH(G)^{c}) = Im(φ^{G,G}) ∪ ⋃_{H⪇G} supp(G/H_{+}) .

By (E), we have supp(G/H_{+}) = Im(Spc(res_{H}^{G})). By the induction hypothesis applied to H ⪇ G, we know that Spc(SH(H)^{c}) is covered by the images of φ^{K,H} for K ≤ H and we need to know what happens to those images Im(φ^{K,H}) under Spc(res_{H}^{G}). By (K), we know that Spc(res_{H}^{G}) will map Im(φ^{K,H}) into Im(φ^{K,G}) and this completes the proof. □

We now want to describe the support of some basic objects in SH(G)^{c}.

4.10. Lemma. For two subgroups H, K ≤ G, we have in SH that

Φ^{H}(G/K_{+}) = \{ \begin{matrix}𝟙^{⊕ℓ} & for some ℓ > 0 if H ≤_{G} K (Notation 2.1), \\ 0 & if H ≰_{G} K. \end{matrix}

Proof. By (N), it suffices to compute the H-fixed subset of the finite G-set G/K. The result holds with ℓ = |N_{G}(H, K)/K| by the discussion in (G). □

4.11. Proposition. Let H, K ≤ G be two subgroups, p a prime and 1 ≤ n ≤ ∞. Then G/K_{+} ∈ 𝒫(H, p, n) if and only if H ≰_{G} K.

Proof. This is immediate from the definition of 𝒫(H, p, n) as (Φ^{H})^{−1}(𝒞_{p,n}) and Lemma 4.10 since 𝟙 ∉ 𝒫 and 0 ∈ 𝒫 for every prime 𝒫 = 𝒞_{p,n} in Spc(SH^{c}). □

4.12. Corollary. If 𝒫(K, q, n) ⊆ 𝒫(H, p, m) then K ≤_{G} H.

Proof. Use (the contrapositive of) Proposition 4.11 twice: As H ≤_{G} H, we have G/H_{+} ∉ 𝒫(H, p, m) and therefore G/H_{+} ∉ 𝒫(K, q, n), which implies K ≤_{G} H. □

4.13. Corollary. Let K ≤ G be a subgroup. We have

supp(G/K_{+}) = { 𝒫(H, p, n) | H ≤_{G} K } = Im(Spc(res_{K}^{G})).

Proof. By definition, supp(G/K_{+}) = { 𝒫 | G/K_{+} ∉ 𝒫 } and by Theorem 4.9, every prime 𝒫 is of the form 𝒫(H, p, n). So, the first equality results from Proposition 4.11. The other equality supp(G/K_{+}) = Im(Spc(res_{K}^{G})) is (E) again. □

4.14. Theorem. The primes 𝒫(H, p, n) are uniquely characterized by the conjugacy class of H, the prime p and the chromatic integer 1 ≤ n ≤ ∞. More precisely, if 𝒫(H, p, m) = 𝒫(K, q, n) then H^{g} = K for some g ∈ G and 𝒞_{p,m} = 𝒞_{q,n} in SH^{c} (meaning m = n, and, as long as m = n > 1 then also p = q).

Proof. Suppose that 𝒫(H, p, m) = 𝒫(K, q, n). By Corollary 4.12, we have H ≤_{G} K and K ≤_{G} H, so H ∼_{G} K. Finally Corollary 4.6 gives 𝒞_{p,m} = 𝒞_{q,n} as desired. □

At this stage, we have a complete description of the set Spc(SH(G)^{c}). We also have a complete description of the maps Spc(α^{∗}) induced by group homomorphisms α : G → G′. Before moving on to the topology of Spc(SH(G)^{c}), we can prove the following result, which is also obtained in [Str12]. It will not be used in the rest of the paper.

4.15. Theorem (Equivariant Nilpotence Theorem). The collection of functors

K(p, n)_{∗}(Φ^{H}(−)) : SH(G)^{c} → K(p, n)_{∗} - Mod_{∗}

detects ⊗-nilpotence. That is, a morphism f : x → y between compact G-spectra x, y ∈ SH(G)^{c} is ⊗-nilpotent in SH(G)^{c} (i.e., f^{⊗ℓ} = 0 : x^{⊗ℓ} → y^{⊗ℓ} for some ℓ ≥ 1) if and only if K(p, n)_{∗}(Φ^{H}(f)) = 0 for all H ≤ G, all primes p and all 1 ≤ n ≤ ∞.

Proof. By the non-equivariant Nilpotence Theorem, up to replacing f by some ⊗- power, we can assume that Φ^{H}(f) = 0 for all of the finitely many subgroups H ≤ G. By induction on the order of the group, applying the result to proper subgroups H ⪇ G, together with the fact that Φ^{K,G} = Φ^{K,H} ∘res_{H}^{G} for all K ≤ H ≤ G, we can similarly assume that res_{H}^{G}(f) = 0 for all H ⪇ G. Consequently, f ⊗ G/H_{+} = 0 for all H ⪇ G. Since the subcategory “on which f is ⊗-nilpotent”, { z ∈ SH(G)^{c} | f^{⊗ℓ}⊗ z = 0 for some ℓ ≥ 1 }, is a tt-ideal (see [Bal10a, Prop. 2.12]) we deduce that for every z ∈ 𝒥 := thick_{⊗} { G/H_{+} | H ⪇ G } there exists ℓ = ℓ(z) ≥ 1 such that f^{⊗ℓ} ⊗ z = 0. Finally, from Φ^{G}(f) = 0 and the description of Φ^{G} as a localization (see (H) for N = G), we see that f ↦ 0 under SH(G)^{c} ↠ SH(G)^{c}/𝒥. Therefore, by a general fact on Verdier localization, f factors via some object z ∈ 𝒥. Combining the two results, we see that f^{⊗(ℓ+1)} factors via f^{⊗ℓ} ⊗ z = 0 for ℓ = ℓ(z) as above. □

5. Tensor idempotents and the Tate construction

In the second part of the paper, about the topology of Spc(SH(G)^{c}), we shall need additional tools relating to generalized Rickard idempotents in the sense of [BF11] and generalized Tate cohomology in the sense of Greenlees [Gre01].

Let 𝒯 be a rigidly-compactly generated tt-category, like SH(G). We shall use [−, −] to denote the internal hom functor, which restricts to 𝒯^{c} by assumption, and we shall sometimes write x^{∨} := [x, 𝟙] for the dual of an object x ∈ 𝒯.

5.1. Remark. Let 𝒦 be an essentially small and rigid tt-category, like 𝒦 = 𝒯^{c}. Recall that a Thomason subset of Spc(𝒦) is a set which can be written as a (possibly infinite) union of closed subsets each having a quasi-compact complement. By [Bal05, Thm. 4.10], the tt-ideals of 𝒦 correspond to the Thomason subsets of Spc(𝒦) via 𝒞 ↦ supp(𝒞) := ⋃_{x∈𝒞} supp(x), with inverse given by Y ↦ 𝒦_{Y} := { x ∈ 𝒦 | supp(x) ⊆ Y }. This is the classification of tt-ideals resulting from a description of the space Spc(𝒦).

5.2. Notation. For any Thomason subset Y ⊆ Spc(𝒯^{c}), recall from [BF11] that

Δ_{Y} := ( e_{Y} → 𝟙 → f_{Y} → Σe_{Y} )

denotes the idempotent triangle in 𝒯 for the finite localization associated to the tt-ideal (𝒯^{c})_{Y} = { x ∈ 𝒯^{c} | supp(x) ⊆ Y }. The localizing subcategory 𝒯_{Y} of acyclic objects is by definition the one generated by (𝒯^{c})_{Y} (and we have (𝒯_{Y})^{c} = (𝒯^{c})_{Y}), and the idempotents e_{Y}^{⊗2} ≅ e_{Y} and f_{Y}^{⊗2} ≅ f_{Y} satisfy 𝒯_{Y} = e_{Y} ⊗ 𝒯 = Ker(f_{Y} ⊗ −) and (𝒯_{Y})^{⊥} = f_{Y} ⊗ 𝒯 = Ker(e_{Y} ⊗ −) = Ker([e_{Y} , −]).

5.3. Remark. For Y ⊆ Spc(𝒯^{c}) a Thomason subset and V := Spc(𝒯^{c}) ∖ Y its complement, we have the following diagram of adjunctions (a. k. a. recollement)

[figure: ^{⊥}𝒯(V) = e_{Y} ⊗ 𝒯 = 𝒯_{Y}; incl; [e_{Y} ,−]; [e_{Y} ,−]; ≅; [e_{Y} , 𝒯] = 𝒯(V)^{⊥} = (𝒯_{Y})^{⊥⊥}; e_{Y} ⊗−; incl; e_{Y} ⊗−; 𝒯; e_{Y} ⊗−; [e_{Y} ,−]; f_{Y} ⊗−; [f_{Y} ,−]; 𝒯(V) := f_{Y} ⊗ 𝒯 = [f_{Y} , 𝒯] = (𝒯_{Y})^{⊥}; incl]

in which each of the six “vertical” sequences • ↣ 𝒯 ↠ • is a (Bousfield) localization sequence of triangulated categories. In particular, the localization f_{Y} ⊗ − : 𝒯 ↠ 𝒯(V) = f_{Y} ⊗ 𝒯 is a tt-functor, the tensor on 𝒯(V) being that of 𝒯. Moreover, 𝒯(V) is also a rigidly-compactly generated tt-category whose subcategory of compact objects is, by Neeman’s Theorem [Nee92, Thm. 2.1], the idempotent completion of the image of the compact objects of 𝒯:

(f_{Y} ⊗ 𝒯^{c})^{♮} ≅ 𝒯(V)^{c}.

The spectrum of this tt-category 𝒯(V)^{c} identifies with V = Spc(𝒯^{c})∖Y,

(5.4)  Spc(𝒯(V)^{c}) ≅ V,

via Spc(f_{Y} ⊗ − : 𝒯^{c} → 𝒯(V)^{c}); see [Bal10b, §2.2]. The internal hom [−, −] on 𝒯 restricts to an internal hom on the local category 𝒯(V) = f_{Y} ⊗ 𝒯. However, localization f_{Y} ⊗ − : 𝒯 → 𝒯(V) is not closed monoidal. Indeed, one can show that for every x, y ∈ 𝒯, the following triangle is exact in 𝒯(V):

f_{Y} ⊗ [x, e_{Y} ⊗ y] → f_{Y} ⊗ [x, y] → [f_{Y} ⊗ x, f_{Y} ⊗ y] → ·

The second object is the image of the internal hom and the third is the internal hom of the images. The first object describes the difference.

5.5. Remark. For every x, y ∈ 𝒯, it follows from [e_{Y} ⊗ 𝒯, f_{Y} ⊗ 𝒯] = 0 and from the exact triangle x ⊗ Δ_{Y} that we have [x, f_{Y} ⊗ y] ≅ [f_{Y} ⊗ x, f_{Y} ⊗ y]. Similarly, [e_{Y} ⊗ x, y] ≅ [e_{Y} ⊗ x, e_{Y} ⊗ y].

5.6. Lemma. Let e_{Z} → 𝟙 → f_{Z} → Σe_{Z} be the idempotent triangle in a rigidlycompactly generated tt-category 𝒯 associated to a Thomason closed subset Z ⊆ Spc(𝒯^{c}). If there is a non-trivial decomposition e_{Z} = e_{1} ⊕ e_{2} with e_{1} ⊗ e_{2} = 0 then Z is disconnected, Z = Z_{1} ⊔ Z_{2}, for closed Z_{i} such that e_{i} = e_{Z_{i}} for i = 1, 2.

Proof. Let 𝒥 := 𝒯_{Z}^{c} denote the compact part generating the idempotent triangle; that is, e_{Z} ⊗ 𝒯 = Loc(𝒥) = Ker(f_{Z} ⊗ −). Since Z is Thomason closed, there is (by [Bal05, Prop. 2.14]) some w ∈ 𝒯^{c} such that supp(w) = Z and 𝒥 = thick_{⊗}(w) (the thick tensor-ideal of 𝒯^{c} generated by w). The object w belongs to 𝒥 ⊂ Loc(𝒥), hence e_{Z} ⊗ w ≅ w. So, we obtain a decomposition w ≅ w_{1} ⊕ w_{2} with w_{1} ⊗ w_{2} = 0 by setting w_{1} := e_{1} ⊗ w and w_{2} := e_{2} ⊗ w. Moreover, it is simple to check that 𝒥_{1} := e_{1} ⊗ 𝒥 = thick_{⊗}(w_{1}) and 𝒥_{2} := e_{2} ⊗ 𝒥 = thick_{⊗}(w_{2}). Finally, these thick tensor-ideals 𝒥_{i} are nonzero. For example, if e_{1} ⊗ 𝒥 = 0 then Loc(𝒥) is contained in Ker(e_{1} ⊗ −) and, since e_{Z} ∈ Loc(𝒥), we would have 0 = e_{1} ⊗ e_{Z} ≅ e_{1} contradicting the assumption that e_{1} ≠ 0. In conclusion, Z = supp(w) = supp(w_{1}) ⊔ supp(w_{2}) is a disjoint union of non-empty closed sets and the rest follows easily. □

5.7. Definition (Greenlees [Gre01]). Let Y ⊆ Spc(𝒯^{c}) be a Thomason subset. We define the Tate functor with respect to Y to be

t_{Y} := [f_{Y} , Σe_{Y} ⊗ −] : 𝒯 → 𝒯.

5.8. Remark. Many classical results about the Tate spectrum [GM95] hold for this more general construction. For example, we have

(5.9)  t_{Y} = [f_{Y} , Σe_{Y} ⊗ −] ≅ f_{Y} ⊗ [e_{Y} , −]

(cf. [Gre01, Corollary 2.5]). Furthermore, the Tate functor t_{Y} is 𝒯^{c}-linear, i.e.,

(5.10)  t_{Y} (x) ⊗ y ≅ t_{Y} (x ⊗ y)

for all x ∈ 𝒯 and y ∈ 𝒯^{c}. Indeed, every compact object y ∈ 𝒯^{c} is rigid, hence [−, −] ⊗ y ≅ [−, − ⊗ y]. Finally, although t_{Y} is not a monoidal functor, it is lax-monoidal (from the “lax-monoidality” of [−, −], the fact that e_{Y} and f_{Y} are ⊗-idempotents, and description (5.9) of t_{Y}). Hence t_{Y} (−) preserves ring objects and modules. In fact, if R is a ring object in 𝒯 then not only is t_{Y} (R) also a ring object, but also the natural map R → t_{Y} (R) is a ring homomorphism. If M is an R-module then t_{Y} (M) inherits the structure of a t_{Y} (R)-module and hence can also be regarded as an R-module.

5.11. Proposition. Let F : 𝒯 → 𝒯′ be a tt-functor preserving coproducts. Let 𝒦 = 𝒯^{c} and 𝒦′ = (𝒯′)^{c} and let φ : Spc(𝒦′) → Spc(𝒦) be the induced map. Let Y ⊆ Spc(𝒦) be a Thomason subset and set Y′ := φ^{−1}(Y) ⊆ Spc(𝒦′). Then 𝒦′_{Y′} = thick_{⊗}(F (𝒦_{Y})) and there is a unique isomorphism of idempotent triangles

F (e_{Y} → 𝟙 → f_{Y} → Σe_{Y} ) ≅ (e_{Y′} → 𝟙 → f_{Y′} → Σe_{Y′})

in 𝒯′. If moreover F is closed monoidal then F ∘ t_{Y} ≅ t_{Y′} ∘ F.

Proof. For any collection of objects ℰ ⊂ 𝒦, the tt-ideal thick_{⊗}(ℰ) corresponds to the Thomason subset ⋃_{x∈ℰ} supp(x). The equality of tt-ideals 𝒦′_{Y′} = thick_{⊗}(F (𝒦_{Y})) then follows from the following equality of Thomason subsets of Spc(𝒦′):

Y′ = φ^{−1}(Y) = φ^{−1}( ⋃_{x∈𝒦_{Y}} supp(x)) = ⋃_{x∈𝒦_{Y}} φ^{−1}(supp(x)) = ⋃_{x∈𝒦_{Y}} supp(F x).

Then 𝒯′_{Y′} = Loc(𝒦′_{Y′}) = Loc(F (𝒦_{Y})) and it follows that F (f_{Y})⊗𝒯′_{Y′} = 0. Since F preserves coproducts, F (e_{Y}) ∈ F (Loc(𝒦_{Y})) ⊆ Loc(F (𝒦_{Y})) = 𝒯′_{Y′}. It then follows from the uniqueness of idempotent triangles (cf. [BF11, Theorem 3.5]) that the triangle F (e_{Y}) → 𝟙 → F (f_{Y}) → ΣF (e_{Y}) is isomorphic to e_{Y′} → 𝟙 → f_{Y′} → Σe_{Y′}. The statement about the Tate construction then follows from the definitions. □

5.12. Example. Let e_{p,n} → 𝟙 → f_{p,n} → Σe_{p,n} be the idempotent triangle in SH associated to the tt-ideal 𝒞_{p,n} ⊆ SH^{c} =: 𝒦. Here, 𝒞_{p,n} = 𝒦_{Y_{p,n}} for Y_{p,n} = supp(𝒞_{p,n}) = { 𝒬 | 𝒞_{p,n} ⊈ 𝒬 } = { 𝒞_{p,m} | m ≥ n + 1 } ∪ { 𝒞_{q,m} | q ≠ p, m > 1 } in Spc(SH^{c}). The right idempotent f_{p,n} is sometimes denoted L_{n−1}^{f}S^{0} in the literature (the exponent “f” referring to “finite” localization and the index “n − 1” referring to the Morava K-theory in use).

It follows from Proposition 5.11 that Y_{p,n}^{G} := Spc(triv)^{−1}(Y_{p,n}) = {𝒫(H, p, m) | H ≤ G, m ≥ n + 1} ∪ { 𝒫(H, q, m) | H ≤ G, q ≠ p, m > 1 } is a Thomason subset of Spc(SH(G)^{c}) with associated idempotent triangle

(5.13)  Δ_{Y_{p,n}^{G}} = ( triv(e_{p,n}) → 𝟙 → triv(f_{p,n}) → Σ triv(e_{p,n}))

in SH(G). The corresponding localized category

𝒮ℋ(G)_{p,n} := triv(f_{p,n}) ⊗ SH(G)

will be very useful in our discussion of Spc(SH(G)^{c}). It can be called the truncation of SH(G) below p-chromatic level n.

Note that the n = ∞ case is just localization at p. In particular, localization with respect to f_{p,n} in SH, or triv(f_{p,n}) in SH(G), already localizes at p for free.

5.14. Example. For a family ℱ of subgroups of G, always assumed below to be stable under conjugation and under passing to subgroups, we can consider the tt-ideal 𝒞_{ℱ} := thick_{⊗}(G/L_{+} | L ∈ ℱ) of SH(G)^{c}. (By the Mackey formula, we can drop ⊗ in the notation thick_{⊗}(...) but this is a pedantic detail.) This tt-ideal 𝒞_{ℱ} corresponds to the Thomason closed subset Y_{ℱ} := ⋃_{L∈ℱ} supp(G/L_{+}). By Corollary 4.13, we have Y_{ℱ} = { 𝒫(H, p, n) | H ∈ ℱ, all p, n }. In this example, we write

Δ_{ℱ} = ( e_{ℱ} → 𝟙 → f_{ℱ} → Σe_{ℱ} )

for the associated idempotent triangle Δ_{Y_{ℱ}} in SH(G). We also use the notation

t_{ℱ} := [f_{ℱ} , Σe_{ℱ} ⊗ −]

for the associated Tate functor. In more usual notation, e_{ℱ} = Σ_{G}^{∞}(Eℱ_{+}) and f_{ℱ} = Σ_{G}^{∞}(Ẽℱ).

5.15. Remark. The trivial families are ℱ = ∅ and F = {all subgroups} whose idempotent triangles are 0 → 𝟙 → 𝟙 → 0 and 𝟙 → 𝟙 → 0 → Σ𝟙 respectively. Their Tate functors are zero.

5.16. Remark. The extreme non-trivial examples are the family ℱ = {1} consisting only of the trivial subgroup, and the family ℱ_{proper} consisting of all proper subgroups. For the former, the notation e_{{1}} = Σ_{G}^{∞}EG_{+} and f_{{1}} = Σ_{G}^{∞}ẼG is probably more familiar to some readers. The Tate functor t_{G} := t_{{1}} is the original construction studied in [GM95]. We shall also use the notation e_{G} := e_{{1}} and f_{G} := f_{{1}} to keep some semblance to the classical notation.

5.17. Example. For every group G, one can define a family of subgroups by bounding the order of the subgroups, or equivalently their index in G. We shall denote by ℱ_{≤ℓ} := { H ≤ G | |H| ≤ ℓ } the family of subgroups of order at most ℓ. Such families will play an important role in Section 9.

5.18. Example. Let ℱ be a family of subgroups of G as in Example 5.14. Let H ≤ G be a subgroup and set ℱ ∩ H := { K ∈ ℱ | K ≤ H }. Then we can apply Proposition 5.11 to the tt-functor res_{H}^{G} : SH(G) → SH(H), which is closed monoidal. Combining this with Example 5.14 and the fact that res_{H}^{G}(G/L_{+}) ≃ ⊕_{[g]∈H\G/L}(H/H ∩^{g}L)_{+}, it is easy to see that res_{H}^{G} ∘t_{ℱ} ≅ t_{ℱ∩H} ∘ res_{H}^{G}.

5.19. Remark. We have already encountered in (H) the localization of SH(G) with respect to the family ℱ[≱N] := { K ≤ G | K ⊉ N } associated to a normal subgroup N ⊴ G. The thick ⊗-ideal here is thick_{⊗} { G/K_{+} | K ⊉ N }. We repeat the crucial fact that the corresponding localization f_{ℱ[≱N]} ⊗ SH(G) ≅ SH(G/N) yields the equivariant stable homotopy category for the quotient group G/N, in such a way that Φ̃^{N} : SH(G) → SH(G/N) becomes the localization functor.

5.20. Remark. According to [HS96] and [Kuh04], the original Tate construction t_{G} (Rem. 5.16) lowers chromatic degree by one. The following theorem states this result in the form we will need.

5.21. Theorem (Hovey-Sadofsky, Kuhn). Let p be a prime dividing the order of a finite group G, and let f_{p,n} denote the right idempotent for the localization of SH associated to 𝒞_{p,n} (see Example 5.12). For any n ≥ 2, the kernel of the functor t_{G}(triv(f_{p,n}))^{G} ⊗ − : SH^{c} → SH is precisely 𝒞_{p,n−1}.

Proof. Note that if x ∈ SH^{c} then t_{G}(triv(f_{p,n}))^{G} ⊗ x = t_{G}(triv(f_{p,n} ⊗ x))^{G}. Define 𝒯_{G}(−) := t_{G}(triv(−))^{G} for convenience. The main results of [Kuh04, Theorem 1.5 and Lemma 4.1] imply that 𝒯_{G}(f_{p,n} ⊗ x) = 0 for all x ∈ 𝒞_{p,n−1}. Just observe that f_{p,n} = L_{n−1}^{f}S^{0} and that f_{p,n}⊗x ≃ L_{n−1}^{f}(x) ≃ Tel(x; v_{n−1}) for any x ∈ 𝒞_{p,n−1}∖𝒞_{p,n}.

Conversely, L_{m}(S^{0}) is an L_{m}^{f}(S^{0})-module, hence 𝒯_{G}(L_{m}(S^{0})) is a 𝒯_{G}(L_{m}^{f}(S^{0}))- module, so that 𝒯_{G}(L_{m}(S^{0})) is 𝒯_{G}(L_{m}^{f}(S^{0}))-local. It follows that if x ∈ SH^{c} is a finite spectrum such that 𝒯_{G}(L_{m}^{f}(S^{0})) ⊗ x = 0 then

(5.22)  𝒯_{G}(L_{m}(S^{0})) ⊗ x^{∨} = [x, 𝒯_{G}(L_{m}(S^{0}))] = 0.

However, the main result of [HS96] establishes that 𝒯_{G}(L_{m}(S^{0})) and L_{m−1}(S^{0}) have the same Bousfield class provided that p divides the order of G. Hence (5.22) implies that L_{m−1}(S^{0})⊗x^{∨} = L_{m−1}(x^{∨}) = 0 so that x^{∨} (and hence x) is contained in 𝒞_{p,m}. Hence, plugging m = n − 1 and using f_{p,n} = L_{n−1}^{f}S^{0} again, we see that 𝒯_{G}(f_{p,n} ⊗ x) = 0 for a finite spectrum x ∈ SH^{c} implies that x ∈ 𝒞_{p,n−1}. □

5.23. Remark. If G = C_{p} then the family of proper subgroups coincides with the trivial family {1}. In light of Remark 5.19, we see that in this case, t_{G}(−)^{G} ≅ Φ^{G}t_{G}(−). This is something very special about the case of the cyclic group C_{p} and is certainly not true for arbitrary groups. This remark lies at the heart of our application of Theorem 5.21 and our conjectured blue-shift results in Section 9.

We conclude this section by describing how chromatic localization (Example 5.12) affects the tt-spectrum of SH(G)^{c}.

5.24. Proposition. Let p be a prime and let 1 ≤ n ≤ ∞ be a chromatic integer. Consider the tt-category 𝒮ℋ(G)_{p,n} = triv(f_{p,n}) ⊗ SH(G). Then the localization functor SH(G)^{c} → 𝒮ℋ(G)_{p,n}^{c} induces a homeomorphism between Spc(𝒮ℋ(G)_{p,n}^{c}) and the following subset V_{p,n} of Spc(SH(G)^{c}):

V_{p,n} := { 𝒫(H, p, n) | H ≤ G, 1 ≤ m ≤ n }.

Proof. The statement is a special case of (5.4) in Remark 5.3, for the big tt-category 𝒯 = SH(G) and for the Thomason subset Y = Y_{p,n}^{G} of its spectrum, where Y_{p,n}^{G} is given in Example 5.12. Its complement Spc(SH(G)^{c})∖Y_{p,n}^{G} is exactly our V_{p,n}. □

5.25. Corollary. For every prime p, the spectrum of SH(G)_{(p)}^{c} is homeomorphic to the subset V_{p,∞} := { 𝒫(H, p, m) | H ≤ G, 1 ≤ m ≤ ∞ } of Spc(SH(G)^{c}). □

5.26. Remark. As is customary, we shall use the same notation 𝒫(H, p, m) for the prime in SH(G)^{c} and the corresponding prime in 𝒮ℋ(G)_{p,n}^{c}.

6. Inclusions of primes and the comparison map

Our goal is to determine the topology of Spc(SH(G)^{c}). We begin by reducing the problem to the task of understanding all inclusions among the equivariant primes:

6.1. Proposition. Every closed subset of Spc(SH(G)^{c}) is a finite union of irreducible closed subsets. Every (non-empty) irreducible closed subset is of the form \overline{\{𝒫\}} = \{ 𝒬 ∈ Spc(SH(G)^{c}) | 𝒬 ⊆ 𝒫 \} for some unique point 𝒫.

Proof. It is basic tt-geometry [Bal05] that every irreducible closed subset of Spc(𝒦) has a unique generic point and that the closure of a point 𝒫 is exactly { 𝒬 ∈ Spc(𝒦) | 𝒬 ⊆ 𝒫 }; beware the reversal of the inclusion compared to the Zariski topology. For the non-equivariant case, the proposition was proved in [Bal10a, Corollary 9.5(e)]. By Theorem 4.9, the space Spc(SH(G)^{c}) is covered by the images of φ^{H,G} = Spc(Φ^{H}) : Spc(SH^{c}) → Spc(SH(G)^{c}) for subgroups H ≤ G. Then consider any closed subset Z ⊆ Spc(SH(G)^{c}). By what we have just said, Z = ∪_{H≤G} φ^{H,G}((φ^{H,G})^{−1}(Z)). For any H ≤ G, (φ^{H})^{−1}(Z) is a closed subset of Spc(SH^{c}). So, by the non-equivariant result it is a finite union of irreducible closed sets \overline{\{𝒞_{p,n}\}} for some primes p and integers 1 ≤ n ≤ ∞. It follows that Z is the union of the closures of the points 𝒫(H, p, n) corresponding to the generic points 𝒞_{p,n} of the irreducible components of (φ^{H})^{−1}(Z), for all H ≤ G. As there are only finitely many subgroups H ≤ G, and finitely many generic points for (φ^{H})^{−1}(Z), this is a finite union. □

6.2. Proposition. For every subgroup H ≤ G, primes p, q, and chromatic integers 1 ≤ m, n ≤ ∞, the following are equivalent:

(1) 𝒫(H, q, n) ⊆ 𝒫(H, p, m);

(2) 𝒞_{q,n} ⊆ 𝒞_{p,m};

(3) n ≥ m and, either m = 1 or q = p.

Proof. The inclusions among the 𝒞_{p,n} manifest as inclusions among the 𝒫(H, p, n), which are their preimages under Φ^{H}. That these are the only inclusions among the primes 𝒫(H, p, m) for fixed H follows from Corollary 4.6. □

6.3. Remark. In other words, the inclusions among the primes 𝒫(H, p, m) – for a fixed H – precisely match the inclusions among the non-equivariant primes 𝒞_{p,m}. Thus, Spc(SH(G)^{c}) consists of a number of copies of the non-equivariant spectrum Spc(SH^{c}) displayed in (1.1) – one such copy for each conjugacy class of subgroups of G. The question is what the interaction is among the different copies; that is, what are the inclusions 𝒫(K, q, n) ⊆ 𝒫(H, p, m) for K ≁_{G} H?

We can start by projecting onto the chromatic information, via Corollary 4.6:

6.4. Corollary. Suppose 𝒫(K, q, n) ⊆ 𝒫(H, p, m) for two subgroups H, K ≤ G, primes p, q and chromatic integers 1 ≤ n, m ≤ ∞. Then 𝒞_{q,n} ⊆ 𝒞_{p,m} in SH^{c} and in particular n ≥ m. If furthermore m > 1, then p = q. □

6.5. Remark. If m = 1 in the previous statement, we cannot conclude that p = q but this is slightly artificial because then 𝒫(H, p, 1) = 𝒫(H, 0, 1) = 𝒫(H, q, 1) and p is irrelevant anyway. In that case, we are equivalently discussing the inclusion 𝒫(K, q, n) ⊆ 𝒫(H, q, m). In other words, when studying inclusions 𝒫(K, q, n) ⊆ 𝒫(H, p, m) between primes in SH(G)^{c}, we can just as well assume p = q.

6.6. Remark. Recall the homeomorphism Spc(𝒮ℋ(G)_{p,n}^{c}) ≃ V_{p,n} of Proposition 5.24. This homeomorphism respects and detects inclusions among primes. So, whenever we have to decide whether 𝒫(K, p, n) ⊆ 𝒫(H, p, m) holds, we can just as well localize in 𝒮ℋ(G)_{p,n} = triv(f_{p,n}) ⊗ SH(G) or in 𝒮ℋ(G)_{p,∞} = SH(G)_{(p)}.

Next we can project onto the group-theoretic information:

6.7. Proposition. The comparison map ρ_{SH(G)^{c}} : Spc(SH(G)^{c}) → Spec(A(G)) is inclusion-reversing. It sends 𝒫(H, p, 1) = 𝒫(H, 0, 1) to 𝔭(H, 0) and it sends 𝒫(H, p, n) to 𝔭(H, p) for n > 1.

Proof. Recall that the isomorphism A(G) ≅ End_{SH(G)}(𝟙) sends the class [G/K] to the composite 𝟙 → G/K_{+} → 𝟙 where 𝟙 → G/K_{+} is the stable transfer map (i.e., the unit for the ring structure on the G-spectrum G/K_{+}) and G/K_{+} → 𝟙 is the stable map induced by the projection G/K_{+} → S^{0}. Applying Φ^{H} this composite becomes 𝟙 → 𝟙^{⊕ℓ} → 𝟙, a sum of ℓ copies of the identity map, where ℓ = |(G/K)^{H}| (cf. (N) and Lemma 4.10). It follows that the ring homomorphism A(G) ≅ End_{SH(G)}(𝟙) → End_{SH}(𝟙) ≅ ℤ induced by the geometric fixed point functor Φ^{H} : SH(G) → SH is precisely the ring homomorphism f^{H} from Section 3 and therefore the following diagram

\xymatrix{Spc(SH^{c}) \ar[d]_{ρ_{SH}^{c}} \ar[r]^{Spc(Φ^{H})} & Spc(SH(G)^{c}) \ar[d]^{ρ_{SH(G)^{c}}} \\ Spec(ℤ) \ar[r]^{Spec(f^{H})} & Spec(A(G))}

commutes (by naturality of the ρ_{𝒦} construction, see [Bal10a]). The result then follows from the definitions and the description of ρ_{𝒦} recalled in (1.2). □

6.8. Example. If 𝒫(K, q, 1) ⊆ 𝒫(H, p, m) then m = 1 by Corollary 6.4 and 𝔭(K, 0) = 𝔭(H, 0) by Proposition 6.7. The latter forces H ∼_{G} K by Dress’s Theorem 3.6. So we have equality 𝒫(K, q, 1) = 𝒫(K, 0, 1) = 𝒫(H, 0, 1) = 𝒫(H, p, m).

6.9. Proposition. Suppose 𝒫_{G}(K, p, n) ⊆ 𝒫_{G}(H, p, m) for subgroups H, K ≤ G, a prime p and chromatic integers 1 ≤ n, m ≤ ∞. Then the following hold:

(a) K is G-conjugate to a p-subnormal subgroup of H; i.e., K^{g} ≤ H and O^{p}(K^{g}) = O^{p}(H) for some g ∈ G (see 3.2). Moreover, we can find g ∈ G such that 𝒫_{H}(K^{g}, p, n) ⊆ 𝒫_{H}(H, p, m) in SH(H)^{c}.

(b) If n = 1 then m = 1 and K ∼_{G} H (so the two tt-primes were equal).

Proof. By Corollary 4.12, we must have K ≤_{G} H. So, replacing K by a conjugate, we can assume that K ≤ H. Then Going-Up (F) applied to res_{H}^{G} implies that there is a prime 𝒫_{H}(K′, p′, n′) ∈ Spc(SH(H)^{c}) such that 𝒫_{H}(K′, p′, n′) ⊆ 𝒫_{H}(H, p, m) and such that 𝒫_{G}(K′, p′, n′) = 𝒫_{G}(K, p, n) (cf. Cor. 4.4). The latter equality implies by Theorem 4.14 that K ∼_{G} K′ and 𝒞_{p,n} = 𝒞_{p′,n′}. Hence we have an inclusion 𝒫_{H}(K′, p, n) = 𝒫_{H}(K′, p′, n′) ⊆ 𝒫_{H}(H, p, m). Applying Proposition 6.7 yields an inclusion of primes in the Burnside ring. By Dress’s description (cf. Theorem 3.6) of the inclusions of prime ideals of A(H), we must have O^{p}(K′) ∼_{H} O^{p}(H). However, O^{p}(H) ⊴ H, so in fact O^{p}(K′) = O^{p}(H). (Lemma 3.3 explains how this means that K′ is a p-subnormal subgroup of H.) Part (b) is Example 6.8. □

6.10. Remark. By Remark 6.5 and Proposition 6.9, we thus need only understand the inclusions 𝒫_{G}(K, p, n) ⊆ 𝒫_{G}(H, p, m) for the same prime p and for p-subnormal subgroups K of H. In particular, we need only consider those primes p dividing the order of G. (Otherwise, no subgroup of G has any proper p-subnormal subgroup.)

Finally, we can further reduce our problem to the case where G is a p-group:

6.11. Proposition. Let H, K ≤ G be subgroups. Then for any 1 ≤ n, m ≤ ∞, we have an inclusion 𝒫_{G}(K, p, n) ⊆ 𝒫_{G}(H, p, m) if and only if K is G-conjugate to a p-subnormal subgroup K^{g} ≤ H such that 𝒫_{H/N}(K^{g}/N, p, n) ⊆ 𝒫_{H/N}(H/N, p, m) where N := O^{p}(H) = O^{p}(K^{g}). Note that the subquotient H/N of G is a p-group.

Proof. By Proposition 6.9 (a), there exists a G-conjugate K^{g} of K such that K^{g} ≤ H, such that 𝒫_{H}(K^{g}, p, n) ⊆ 𝒫_{H}(H, p, m), and such that O^{p}(H) = O^{p}(K^{g}). Let N := O^{p}(H) = O^{p}(K^{g}) be this normal subgroup of H. Under the map induced by inflation infl_{G/N}^{G} (Cor. 4.5), our inclusion becomes 𝒫_{H/N}(K^{g}/N, p, n) ⊆ 𝒫_{H/N}(H/N, p, m). Conversely, the latter inclusion of primes in SH(H/N) gets mapped by Spc(Φ̃^{N}) to the inclusion 𝒫_{H}(K^{g}, p, n) ⊆ 𝒫_{H}(H, p, m) by Prop. 4.7, which then becomes 𝒫_{G}(K^{g}, p, n) ⊆ 𝒫_{G}(H, p, m) under restriction (Cor. 4.4). □

7. The topology of Spc(SH(C_{p})^{c})

Let us consider G = C_{p} cyclic of prime order. Since there are two subgroups, there are only two classes of primes in SH(G)^{c}:

𝒫(C_{p}, q, n) = { x ∈ SH(G)^{c} | Φ^{G}(x) ∈ 𝒞_{q,n} }

and

𝒫(1, q, n) = { x ∈ SH(G)^{c} | res_{1}^{G}(x) ∈ 𝒞_{q,n} } .

By Remark 6.10, we only need to understand when there is an inclusion 𝒫(1, p, n) ⊆ 𝒫(C_{p}, p, m), at the prime q = p.

7.1. Proposition. We have 𝒫(1, p, n) ⊆ 𝒫(C_{p}, p, n − 1) for every n ≥ 2.

Proof. Let x ∈ SH(G)^{c} be a compact G-spectrum such that res_{1}^{G}(x) ∈ 𝒞_{p,n}. This means that 0 = f_{p,n} ⊗ res_{1}^{G}(x) ≅ res_{1}^{G}(triv(f_{p,n}) ⊗ x) where f_{p,n} is, as always, the idempotent associated to 𝒞_{p,n} (cf. Ex. 5.12). We deduce that G_{+}⊗triv(f_{p,n})⊗x = 0. Therefore Loc_{⊗}(G_{+}) ⊗ triv(f_{p,n}) ⊗ x = 0; in particular e_{G} ⊗ triv(f_{p,n}) ⊗ x = 0, where e_{G} → 𝟙 → f_{G} → Σe_{G} is the idempotent triangle in SH(G) associated to the smashing subcategory Loc_{⊗}(G_{+}) (cf. Remark 5.16). It follows that

t_{G}(triv(f_{p,n}) ⊗ x) = [f_{G}, Σe_{G} ⊗ triv(f_{p,n}) ⊗ x] = 0.

Applying geometric fixed points we obtain

0 = Φ^{G}(t_{G}(triv(f_{p,n}) ⊗ x)) = Φ^{G}(t_{G}(triv(f_{p,n})) ⊗ x) = Φ^{G}(t_{G}(triv(f_{p,n}))) ⊗ Φ^{G}(x).

Note that Φ^{G}(t_{G}(triv(f_{p,n}))) = t_{G}(triv(f_{p,n}))^{G} since G = C_{p} (cf. Remark 5.23) hence Theorem 5.21 implies that f_{p,n−1} ⊗Φ^{G}(x) = 0. That is, Φ^{G}(x) ∈ 𝒞_{p,n−1}. □

7.2. Corollary. We have 𝒫(1, p, ∞) ⊆ 𝒫(C_{p}, p, ∞).

Proof. This is immediate from the above proposition, since 𝒞_{p,∞} = ⋂_{n≥1} 𝒞_{p,n}. □

7.3. Remark. We know that 𝒫(1, p, n) ⊈ 𝒫(C_{p}, p, n + 1), e. g. by Cor. 6.4. So, the only remaining question is whether or not 𝒫(1, p, n) can be included in 𝒫(C_{p}, p, n).

Although not necessary in the formal arguments for the hyper-brainy readers, there is a way to build intuition from geometric pictures which can help average readers, down around 160 I.Q. points. Localizing at p, we have by Corollary 5.25 the following (left-hand side) towers of primes in SH(C_{p})^{c}:

[figure: •; 𝒫(1,p,1); •; 𝒫(1,p,2); •; 𝒫(1,p,3); ⋮; •; 𝒫(1,p,n−1); •; 𝒫(1,p,n); ⋮; •; 𝒫(1,p,∞); •; 𝒫(C_{p},p,1); •; 𝒫(C_{p},p,2); •; 𝒫(C_{p},p,3); ⋮; •; 𝒫(C_{p},p,n−1); •; 𝒫(C_{p},p,n); ⋮; •; 𝒫(C_{p},p,∞); Spc(SH(C_{p})_{(p)}^{c})]  ⊃  [figure: •; •; •; ⋮; •; •; 𝒫(1,p,n); •; •; •; ⋮; •; •; 𝒫(C_{p},p,n); Spc(𝒮ℋ(C_{p})_{p,n}^{c})]

The dashed lines −− indicate that the inclusions 𝒫(1, p, n) ⊂^{?} 𝒫(C_{p}, p, n) have not been ruled out yet. As in Proposition 5.24, we can further localize below chromatic level n, via SH(G) ↠ triv(f_{p,n}) ⊗ SH(G) = 𝒮ℋ(G)_{p,n}. This yields the right-hand side tt-spectrum in the above picture. The advantage of doing so is that we see the two primes in question, 𝒫(1, p, n) and 𝒫(C_{p}, p, n), at the top of the picture. Consider now the closed subset consisting of those two primes, diagrammatically depicted as

[figure: ∘; ∘; ∘; ⋮; ∘; •; 𝒫(1,p,n); ∘; ∘; ∘; ⋮; ∘; •; 𝒫(C_{p},p,n)]

where the two • indicate the two points that we include in the closed subset and the ∘ mark the primes we exclude. The inclusion marked by the top dashed line • − −• holds if and only if this closed subset is connected (and is the closure of the right-hand prime, corresponding to 𝒫(C_{p}, p, n)). To show that this inclusion does not hold, we are going to show that this closed set is disconnected. As explained in Section 5, this can be done by carefully analyzing idempotent triangles and this is why we proceed with the following computation.

7.4. Lemma. Let G = C_{p}. For any n ≥ 2, we have a decomposition

triv(e_{p,n−1} ⊗ f_{p,n}) ≃ (e_{G} ⊗ triv(e_{p,n−1} ⊗ f_{p,n})) ⊕ (f_{G} ⊗ triv(e_{p,n−1} ⊗ f_{p,n}))

in SH(G), where e_{p,n} and f_{p,n} are the chromatic idempotents (Ex. 5.12) and e_{G} and f_{G} are the group-theoretic ones (Ex. 5.14 and Rem. 5.16).

Proof. More precisely, we prove that the triangle Δ_{G} ⊗ triv(e_{p,n−1} ⊗ f_{p,n}) splits, by showing that the last map in this triangle (the vertical one below) is zero:

\xymatrix{e_{G} ⊗ triv(e_{p,n−1} ⊗ f_{p,n}) \ar[r] & triv(e_{p,n−1} ⊗ f_{p,n}) \ar[r] & f_{G} ⊗ triv(e_{p,n−1} ⊗ f_{p,n}) \ar[d] \\  &  & Σe_{G} ⊗ triv(e_{p,n−1} ⊗ f_{p,n}).}

It suffices to show that Hom(f_{G} ⊗triv(e_{p,n−1} ⊗f_{p,n}), Σe_{G} ⊗triv(e_{p,n−1} ⊗f_{p,n})) = 0, which in turn follows from [f_{G} ⊗ triv(e_{p,n−1} ⊗ f_{p,n}), Σe_{G} ⊗ triv(e_{p,n−1} ⊗ f_{p,n})] = 0. Let us temporarily stop writing “triv” for readability. Using Remark 5.5, we can remove the factor e_{p,n−1} on the right and the factor f_{p,n} on the left. This reads:

[f_{G} ⊗ e_{p,n−1} ⊗ f_{p,n}, Σe_{G} ⊗ e_{p,n−1} ⊗ f_{p,n}] = [f_{G} ⊗ e_{p,n−1}, Σe_{G} ⊗ f_{p,n}].

So we want to prove that [f_{G} ⊗ e_{p,n−1}, Σe_{G} ⊗ f_{p,n}] = 0.

To this end, we start from Theorem 5.21, which gives us t_{G}(f_{p,n})^{G} ⊗ 𝒞_{p,n−1} = 0. Thus, t_{G}(f_{p,n} ⊗ x^{∨})^{G} = 0 for all x ∈ 𝒞_{p,n−1}. Moreover, since G = C_{p}, this actually gives an equality t_{G}(f_{p,n} ⊗ x^{∨}) = 0 in SH(G) because t_{G}(−) takes f_{G}-local values and here, f_{G} = f_{proper}. (See Rem. 5.23.) Thus [f_{G} ⊗ triv(x), Σe_{G} ⊗ f_{p,n}] = 0. But now, the kernel of the functor [f_{G} ⊗ triv(−), Σe_{G} ⊗ f_{p,n}] is a localizing subcategory of SH. Since we just proved that this kernel contains 𝒞_{p,n−1}, it contains e_{p,n−1} ∈ Loc(𝒞_{p,n−1}). Hence [f_{G} ⊗ e_{p,n−1}, Σe_{G} ⊗ f_{p,n}] = 0 as desired. □

7.5. Proposition. Let G = C_{p} and n ≥ 1. Then 𝒫(1, p, n) ⊈ 𝒫(C_{p}, p, n).

Proof. Example 6.8 deals with the n = 1 case, so assume n ≥ 2. Let 𝒮ℋ(G)_{p,n} = triv(f_{p,n}) ⊗ SH(G) be the truncation of SH(G) below p-chromatic level n; see Ex. 5.12. As explained in Remark 7.3, the tt-spectrum Spc(𝒮ℋ(G)_{p,n}^{c}) is homeomorphic to the open subset of Spc(SH(G)^{c}) consisting of those 𝒫_{G}(H, p, m) – same p – for all H ∈ {1, G} and 1 ≤ m ≤ n. Moreover, the image in 𝒮ℋ(G)_{p,n} of the (n−1)-th triangle e_{p,n−1} → 𝟙 → f_{p,n−1} → Σe_{p,n−1} is the idempotent triangle in 𝒮ℋ(G)_{p,n} associated with the closed set {𝒫(1, p, n), 𝒫(C_{p}, p, n)}; see Example 5.12. By Lemma 7.4, the left idempotent e_{p,n−1} ⊗ f_{p,n} of this triangle splits in 𝒮ℋ(G)_{p,n}. That is, we have e_{p,n−1} ⊗ f_{p,n} = (e_{p,n−1} ⊗ f_{p,n} ⊗ e_{G}) ⊕ (e_{p,n−1} ⊗ f_{p,n} ⊗ f_{G}). By applying Φ^{G} and res_{1}^{G} respectively we see that the two direct summands are non-zero; hence it is a non-trivial decomposition. Invoking Lemma 5.6, we conclude that the closed subset {𝒫(1, p, n), 𝒫(C_{p}, p, n)} of Spc(𝒮ℋ(G)_{p,n}^{c}) consisting of precisely two points is disconnected. The claim follows (see Remark 7.3 if necessary). □

7.6. Remark. We thus have a complete description of the topology for G = C_{p}. See (1.3) for a picture of the spectrum. As a consequence we obtain a complete classification of the thick ⊗-ideals in SH(C_{p})^{c}. More on this topic in Section 10.

8. The topology for general G and the log_{p}-Conjecture

Next we consider the problem of determining the topology of Spc(SH(G)^{c}) for an arbitrary finite group G. As explained in Proposition 6.11, we have reduced the problem to understanding the possible inclusions 𝒫_{G}(K, p, n) ⊆ 𝒫_{G}(H, p, m) for p a prime number and G a p-group.

We obtain an immediate generalization of Proposition 7.5:

8.1. Proposition. Let G be a finite group, H, K ≤ G subgroups, p a prime and 1 ≤ n < ∞. Then 𝒫_{G}(K, p, n) ⊈ 𝒫_{G}(H, p, n) – for the same n – unless H ∼_{G} K (in which case the two primes coincide).

Proof. Suppose ab absurdo that 𝒫_{G}(K, p, n) ⊆ 𝒫_{G}(H, p, n) for some n ≥ 1 and H, K ≤ G with H ≁_{G} K. Proposition 6.11 allows us to assume that G is a p-group and that H = G. Our (absurd) situation is now: 𝒫_{G}(K, p, n) ⊆ 𝒫_{G}(G, p, n), where G is a p-group and K < G. Then there exists a maximal proper subgroup N ⊲ G containing K. Since G is a p-group, N must have index p and G/N ≃ C_{p}. Finally, apply Corollary 4.5 to 𝒫_{G}(K, p, n) ⊆ 𝒫_{G}(G, p, n) to obtain the absurd conclusion 𝒫_{C_{p}}(1, p, n) ⊆ 𝒫_{C_{p}}(C_{p}, p, n), contradicting Proposition 7.5. □

We can also generalize Proposition 7.1 to arbitrary finite groups:

8.2. Theorem. Let G be a finite group, p a prime, and K ≤ H ≤ G subgroups such that K admits a p-subnormal tower to H of length s (see Terminology 3.2). Then 𝒫_{G}(K, p, n) ⊆ 𝒫_{G}(H, p, n − s) for all n > s.

Proof. By Corollary 4.4, we can assume that H = G. We then proceed by induction on s. For the s = 1 base case, Proposition 4.7 applied to N = K turns the inclusion 𝒫_{C_{p}}(1, p, n) ⊆ 𝒫_{C_{p}}(C_{p}, p, n − 1) of Prop. 7.1 into 𝒫_{G}(K, p, n) ⊆ 𝒫_{G}(G, p, n − 1). For the general case, let K ≤ K′ ⊴ G with [G : K′] = p and K ≤ K′ a p-subnormal subgroup of index p^{s−1}. By induction hypothesis, 𝒫_{K′}(K, p, n) ⊆ 𝒫_{K′}(K′, p, n − s + 1) for all n > s − 1. Applying Corollary 4.4, we get 𝒫_{G}(K, p, n) ⊆ 𝒫_{G}(K′, p, n − s + 1). The base case s = 1 gives the last bit: 𝒫_{G}(K′, p, n − s + 1) ⊆ 𝒫_{G}(G, p, n − s). □

8.3. Corollary. Let G be a p-group and K ≤ H ≤ G subgroups with [H : K] = p^{s}. Then 𝒫_{G}(K, p, n) ⊆ 𝒫_{G}(H, p, n − s) for all n > s. □

Finally, Corollary 7.2 (and its proof) passes to arbitrary finite groups:

8.4. Corollary. Let G be a finite group, p a prime, and K ≤ H ≤ G subgroups such that K admits a p-subnormal tower to H. Then 𝒫_{G}(K, p, ∞) ⊆ 𝒫_{G}(H, p, ∞). □

8.5. Remark. We saw in Proposition 6.7 how Spc(SH(G)^{c}) provides a “chromatic refinement” of Spec(A(G)), in the same way that Spc(SH^{c}) refines Spec(ℤ). However, Theorem 8.2 goes beyond this chromatic refinement. Namely, the collision 𝔭(H, p) = 𝔭(K, p) when O^{p}(H) ∼_{G} O^{p}(K) for non-conjugate subgroups H and K (Thm. 3.6) does not happen in Spc(SH(G)^{c}). We have instead the inclusions of Theorem 8.2, which under the comparison map ρ_{SH(G)^{c}} : Spc(SH(G)^{c}) → Spec(A(G)) yield inclusions between maximal primes, i.e., necessarily equalities 𝔭(H, p) = 𝔭(K, p) in A(G). These are exactly the equalities observed by Dress.

8.6. Remark. By the above results, the question we have in the general case is similar to the one we had for G = C_{p}. Theorem 8.2 shows that we have a shift by s when K admits a p-subnormal tower of length s to H. The problem is whether this is optimal. Proposition 8.1 shows that we do not have inclusion 𝒫_{G}(K, p, n) ⊆ 𝒫_{G}(H, p, n), but for s > 1 there is still the possibility that we could have a more efficient inclusion than 𝒫_{G}(K, p, n) ⊆ 𝒫_{G}(H, p, n−s). Moreover, by Proposition 6.11 the question of this indeterminacy is equivalent to the question for a p-group. To be clear, if G is a p-group and H ≤ G is a subgroup of index p^{s} then we have 𝒫_{G}(H, p, n) ⊆ 𝒫_{G}(G, p, n − s) for all n > s but the question is whether we could have 𝒫_{G}(H, p, n) ⊆ 𝒫_{G}(G, p, n − s + 1). We conjecture that this does not happen. In fact, this conjecture reduces to the H = 1 case, which we state below:

8.7. Conjecture (log_{p}-Conjecture). We say that a group G of order p^{r} satisfies the log_{p}-Conjecture if, for all n ≥ r, we have 𝒫_{G}(1, p, n) ⊈ 𝒫_{G}(G, p, n − r + 1).

8.8. Example. Proposition 7.5 says that C_{p} satisfies the log_{p}-Conjecture.

8.9. Remark. As we have seen in Remark 6.6, inclusions 𝒫(K, p, n) ⊆ 𝒫(H, p, m)

– for the same p – can be tested p-locally. In other words, the log_{p}-Conjecture is truly a conjecture about SH(G)_{(p)}; see Corollary 5.25.

In the next section we will connect this conjecture to Tate cohomology but for now we would like to demonstrate its consequences for the complete determination of the topology of Spc(SH(G)^{c}), in conjunction with Proposition 6.1.

8.10. Theorem. Let G be a p-group and let H ≤ G. Suppose that G satisfies the log_{p}-Conjecture. Then 𝒫(H, p, n) ⊆ 𝒫(G, p, m) if and only if n ≥ m + log_{p}[G : H].

Proof. One direction is Theorem 8.2. Conversely suppose 𝒫(H, p, n) ⊆ 𝒫(G, p, m) and let r = log_{p} |G| and s = log_{p} |H|. By Theorem 8.2 again, we know that 𝒫(1, p, n + s) ⊆ 𝒫(H, p, n). Combining these inclusions, we obtain 𝒫(1, p, n + s) ⊆ 𝒫(G, p, m). By hypothesis, G satisfies the log_{p}-Conjecture, hence n+s ≥ m+r. □

More generally, we get:

8.11. Theorem. Let G be a finite group and p a prime. Let H, K ≤ G be arbitrary subgroups and 1 ≤ n, m ≤ ∞. Suppose the p-group H/ O^{p}(H), a subquotient of G, satisfies the log_{p}-Conjecture 8.7. Then we have an inclusion of equivariant primes 𝒫(K, p, n) ⊆ 𝒫(H, p, m) if and only if K is G-conjugate to a p-subnormal subgroup of H and n ≥ m + log_{p}(|H|/|K|).

Proof. In view of Proposition 6.11, we can reduce the question to the announced p-group H/ O^{p}(H); note that |H|/|K| does not change under this reduction. So we are down to the special case of p-groups established in Theorem 8.10. □

From the results of Section 7, we can deduce a conjecture-free statement:

8.12. Theorem. Let G be a finite group of square-free order. For any prime p dividing |G|, any subgroups H, K ≤ G with H ≁_{G} K and any chromatic integers 1 ≤ m, n ≤ ∞, the following are equivalent:

(1) 𝒫(K, p, n) ⊂ 𝒫(H, p, m).

(2) n ≥ m + 1 and K is G-conjugate to a normal subgroup of index p in H.

Proof. We can apply the conclusion of Theorem 8.11 unconditionally since the only p-subgroup which appears as a subquotient of G is C_{p}, which satisfies the log_{p}- Conjecture by Proposition 7.5. Also because |G| is square-free, we see that there is no p-subnormal tower beyond length s = 1. This gives the statement. □

8.13. Remark. In the situation of Theorem 8.12, the prime p cannot divide |K|. So K is p-perfect, O^{p}(K) = K ∼_{G} O^{p}(H) and the latter has index p in H.

8.14. Example. Consider G = S_{3} and p a prime. Let us draw a picture of the space Spc(SH(G)^{c}) at the prime p, that is, above Spec(A(G)_{(p)}), or equivalently let us draw the picture of Spc(SH(G)_{(p)}^{c}); see Rem. 8.9. This depends on the prime p, with three cases to discuss: p = 2, p = 3, and p not dividing |G|.

[figure: Spc(SH(S_{3})_{(p)}^{c}) =; Spec(A(S_{3})_{(p)}) =; ρ; •; •; •; •; ⋮; •; •; •; •; •; ⋮; •; •; •; •; •; ⋮; •; •; •; •; •; ⋮; •; •; •; •; •; •; •; {1}; C_{2}; A_{3}; S_{3}; if p = 2; •; •; •; •; ⋮; •; •; •; •; •; ⋮; •; •; •; •; •; ⋮; •; •; •; •; •; ⋮; •; •; •; •; •; •; •; •; {1}; C_{2}; A_{3}; S_{3}; if p = 3; •; •; •; •; ⋮; •; •; •; •; •; ⋮; •; •; •; •; •; ⋮; •; •; •; •; •; ⋮; •; •; •; •; •; •; •; •; •; {1}; C_{2}; A_{3}; S_{3}; if p ≠ 2, 3]

At the very right, when p ≠ 2, 3, the spectrum of the Burnside ring localized at p is a disjoint union of four copies of Spec(ℤ_{(p)}) = {(0), (p)}, one for each conjugacy class of subgroup H ≤ G. Accordingly, above each copy of Spec(ℤ_{(p)}), we see a chromatic tower with infinitely many primes above (p) and only one above (0). At p = 2, Dress’s collision happens between the trivial subgroup and any of the conjugate cyclic subgroups of order 2, as well as between A_{3} = C_{3} and G. Each of those closed points now admits a chromatic refinement for each H, i.e., here there are two towers of primes in SH(G)^{c} projecting down to the closed point (p). Moreover, the two towers are not disconnected but the one for the smaller subgroup is in some sense “in the closure” of the one for the bigger subgroup. A similar phenomenon happens at p = 3 for the subgroups {1} and A_{3} = C_{3}. Note however that although (any) H = C_{2} has index p = 3 in G, it is not p-subnormal simply because it is not normal. We see here that Dress’s result already distinguishes 𝔭(H, p) = 𝔭(C_{2}, 3) from 𝔭(G, p) = 𝔭(S_{3}, 3), which shows that the corresponding chromatic towers { 𝒫(C_{2}, 3, n) | n ≥ 1 } and { 𝒫(S_{3}, 3, n) | n ≥ 1 } are disconnected.

Let us give a re-interpretation of Conjecture 8.7 in more geometric terms:

8.15. Proposition. Let G be a group of order p^{r}. The following are equivalent: (A) Conjecture 8.7 holds: 𝒫(1, p, n) ⊈ 𝒫(G, p, n − r + 1), whenever r ≤ n < ∞. (B) For every N > r, the following subset of Spc(SH(G)_{(p)}^{c}) is closed:

Z_{p,N} := { 𝒫(H, p, ℓ) ∈ Spc(SH(G)_{(p)}^{c}) | H ≤ G, N − log_{p} |H| ≤ ℓ ≤ ∞ } .

(C) For every N > r, there exists x ∈ SH(G)_{(p)}^{c} with supp(x) = Z_{p,N}.

When these conditions hold, the closed set Z_{p,N} is irreducible and has 𝒫(G, p, N − r) as generic point.

Proof. (A)⇒(B): Since N > r, Corollary 8.3 implies that Z_{p,N} is contained in the closed set \{ 𝒬 ∈ Spc(SH(G)_{(p)}) | 𝒬 ⊆ 𝒫(G, p, N − r) \} = \overline{\{𝒫(G, p, N − r)\}}. If we prove the reverse inclusion, \overline{\{𝒫(G, p, N − r)\}} ⊆ Z_{p,N}, then Z_{p,N} is closed irreducible with the announced generic point. To prove this, suppose ab absurdo that there exists a point 𝒬 ⊂ 𝒫(G, p, N − r) in \overline{\{𝒫(G, p, N − r)\}} with 𝒬 ∉ Z_{p,N}. By Theorem 4.9 (and Rem. 8.9), we have 𝒬 = 𝒫(H, p, ℓ) for some H ≤ G and some 1 ≤ ℓ ≤ ∞. Let s = log_{p} |H|, in which case 𝒬 ∉ Z_{p,N} implies that N − 1 ≥ ℓ + s. By Proposition 6.2 and Corollary 8.3, we have 𝒫(1, p, N − 1) ⊂ 𝒫(1, p, ℓ + s) ⊂ 𝒫(H, p, ℓ) ⊂ 𝒫(G, p, N − r), which contradicts (A) for n = N − 1 ≥ r.

(B)⇒(C): In Spc(SH(G)_{(p)}^{c}), the closed subset Z_{p,N} has finite (hence quasicompact) complement. (The picture (8.18) below may be a useful guide.) It follows from general tt-geometry that any closed subset with quasi-compact complement is the support of some object: see [Bal05, Prop. 2.14].

(C)⇒(A): Let n ≥ r and let x ∈ SH(G)_{(p)}^{c} be such that supp(x) = Z_{p,n+1}, that is, use (C) for N := n + 1 > r. Then 𝒫(G, p, n − r + 1) ∈ Z_{p,n+1} = supp(x) means x ∉ 𝒫(G, p, n − r + 1), whereas 𝒫(1, p, n) ∉ Z_{p,n+1} = supp(x) means x ∈ 𝒫(1, p, n). Together, this gives 𝒫(1, p, n) ⊈ 𝒫(G, p, n − r + 1) as desired. □

8.16. Remark. One can almost make the same statement as above with SH(G)^{c} instead of the p-local version SH(G)_{(p)}^{c}. However, there is some “fringe effect” when N is small, typically if 𝒫(G, p, 1) is allowed in Z_{p,N}; in that case, Z_{p,N} is not closed but for the rather disjoint issue that 𝒫(G, p, 1) = 𝒫(G, q, 1) for q ≠ p and thus contains in its closure many other primes 𝒫(H, q, n) for q ≠ p.

8.17. Remark. As in Remark 7.3, it can be useful to draw a picture of Spc(SH(G)_{(p)}^{c}). Let us assume that G is a p-group for simplicity, say |G| = p^{r}. So subgroups H ≤ G can be filtered by the integer s = log_{p}(|H|), which runs between 0 and r. We read s horizontally and the chromatic degree m of 𝒫(H, p, m) vertically:

[figure: m; n −; n − 1 −; ⋮; n − r −; ⋮; 3 −; 2 −; 1 −; s; |; |; |; |; |; r; ···; s; ···; 0; (H=G); (|H|=p^{s}); (H=1); •; •; •; ⋮; •; •; •; •; •; •; ⋮; •; •; •; ⋮; •; •; •; •; •; •; ⋮; •; •; •; ⋮; •; •; •; •; •; •; ⋮; •; •; •; ⋮; •; •; •; •; •; •; ⋮; •; •; •; ⋮; •; •; •; •; •; •; ⋮; Spc(SH(G)_{(p)}^{c}) =]

We warn the reader that this picture is slightly misleading since of course G can have non-conjugate subgroups of a fixed order p^{s}. So, the intermediate columns (for 0 < s < r) actually have several layers. This picture is nonetheless useful to build some intuition. (The picky reader can restrict to G = C_{p^{r}} if necessary.) The inclusions depicted above are the only ones if Conjecture 8.7 holds true.

Truncating below p-chromatic level n, i.e., localizing via SH(G) ↠ 𝒮ℋ(G)_{p,n} = triv(f_{p,n}) ⊗ SH(G), the spectrum would be as above but truncated to only include those m ≤ n. In particular, it is a finite space (but not a discrete one).

Also, the subset Z_{p,N} of Proposition 8.15 looks as follows (the •, not the ∘):

(8.18)  [figure: m; N −; N − 1 −; ⋮; N − r −; ⋮; 3 −; 2 −; 1 −; s; |; |; |; |; |; r; ···; s; ···; 0; (H=G); (|H|=p^{s}); (H=1); ∘; ∘; ∘; ⋮; ∘; ∘; ∘; ∘; •; •; ⋮; ∘; ∘; ∘; ⋮; ∘; ∘; ∘; •; •; •; ⋮; ∘; ∘; ∘; ⋮; ∘; ∘; •; •; •; •; ⋮; ∘; ∘; ∘; ⋮; ∘; •; •; •; •; •; ⋮; ∘; ∘; ∘; ∘; •; •; •; •; •; •; ⋮; ⋮; Z_{p,N}; 𝒫(G, p, N − r)]

Let us conclude this section with a summary of what we know about the topology:

8.19. Corollary. Let G be a finite group. Every closed subset of Spc(SH(G)^{c}) is a finite union of irreducible closed subsets, and every irreducible closed set is of the form \overline{\{𝒫\}} = \{ 𝒬 ∈ Spc(SH(G)^{c}) | 𝒬 ⊆ 𝒫 \} for a unique prime 𝒫 ∈ Spc(SH(G)^{c}).

Let us consider two arbitrary primes in SH(G)^{c} (see Theorems 4.9 and 4.14)

𝒬 = 𝒫(K, q, n)  and  𝒫 = 𝒫(H, p, m)

for subgroups H, K ≤ G, primes p, q and chromatic integers 1 ≤ m, n ≤ ∞. We fix 𝒫 (i.e., we fix H, p and m) and we discuss when the inclusion 𝒬 ⊆ 𝒫 holds, i.e., when 𝒬 ∈ \overline{\{𝒫\}}, as function of K, q and n.

(a) If K is not G-conjugate to a q-subnormal subgroup of H then 𝒬 ∉ \overline{\{𝒫\}}.

(b) If m > 1 and p ≠ q then 𝒬 ∉ \overline{\{𝒫\}}. (If m = 1 then 𝒫 = 𝒫(H, q, 1) anyway.) These statements were independent of n. In the remaining cases, the inclusion 𝒬 ⊆ 𝒫 does always happen, for all n in a connected interval. More precisely:

(c) Suppose that K is G-conjugate to a q-subnormal subgroup of H and suppose that either m = 1 or p = q. Then 𝒫(K, q, ∞) ∈ \overline{\{𝒫\}} and there is a welldefined chromatic integer n_{K} := min \{ 1 ≤ ℓ ≤ ∞ | 𝒫(K, q, ℓ) ∈ \overline{\{𝒫\}} \} such that

m ≤ n_{K} ≤ m + log_{q}(|H|/|K|)

(for m = ∞, this simply means n_{K} = m = ∞) with the property that

𝒬 ∈ \overline{\{𝒫\}} if and only if n_{K} ≤ n ≤ ∞.

If n_{K} equals m and is finite, then K ∼_{G} H. Finally, if Conjecture 8.7 holds for the q-group H/O^{q}(H) – for instance if G has square-free order – then

n_{K} = m + log_{q}(|H|/|K|).

Proof. Parts (a) and (b) follow from Corollary 6.4, Remark 6.5 and Proposition 6.9. Part (c) follows from Theorem 8.2, Proposition 8.1 and Theorem 8.11. □

8.20. Remark. In (c), if m = 1, one can replace p by q without changing the problem since 𝒫 is 𝒫(H, q, 1) as well. So one can read p instead of q everywhere in (c).

9. Tate re-interpretation of the log_{p}-Conjecture

In this section we provide equivalent formulations of the log_{p}-Conjecture 8.7, in terms of blue shift phenomena for Tate cohomology. Let G be a p-group of order p^{r} and let us denote by ℱ_{≤p^{s}} the family of subgroups { H ≤ G | |H| ≤ p^{s} } of order at most p^{s}; see Ex. 5.17. We denote the associated idempotent triangle by e_{≤p^{s}} → 𝟙 → f_{≤p^{s}} → · and the associated Tate functor by

t_{≤p^{s}} = [f_{≤p^{s}}, Σe_{≤p^{s}} ⊗ −] ≅ f_{≤p^{s}} ⊗ [e_{≤p^{s}}, −] : SH(G) → SH(G)

as in Section 5. Finally, let us agree that 𝒞_{p,0} := SH_{(p)}^{c} denotes the whole category of compact p-local spectra, and recall that x is said to be of type-n if x ∈ 𝒞_{p,n} \𝒞_{p,n+1}.

9.1. Theorem. Let G be a group of order p^{r}. The following are equivalent: (A) Conjecture 8.7 holds : 𝒫(1, p, n) ⊈ 𝒫(G, p, n − r + 1) whenever r ≤ n < ∞. (B) For all integers s, t and n such that 0 ≤ s < t ≤ r and n ≥ r − s, and for

every subgroup H ≤ G with |H| = p^{t}, we have

Φ^{H}(t_{≤p^{s}}(triv(f_{p,n}))) ⊗ 𝒞_{p,n−(t−s)} = 0.

In a slogan: the geometric fixed points of the Tate functor Φ^{H}(t_{≤p^{s}}(−)) lowers chromatic degree by the (p-exponent) “distance” t − s from the subgroup H to the family ℱ_{≤p^{s}}.

Proof. (A)⇒(B): Let N := s+n+1 > r. Then by Proposition 8.15, there exists an object x ∈ SH(G)_{(p)}^{c} whose support is Z_{p,N} = { 𝒫(K, p, ℓ) ∈ Spc(SH(G)_{(p)}^{c}) | K ≤ G, N − log_{p} |K| ≤ ℓ ≤ ∞ }. Localizing further to 𝒮ℋ(G)_{p,n} = f_{p,n} ⊗ SH(G) (see Prop. 5.24), we obtain an object x′ := x ⊗ f_{p,n} ∈ 𝒮ℋ(G)_{p,n}^{c} whose support is

supp(x′) = { 𝒫(K, p, ℓ) | N − log_{p} |K| ≤ ℓ ≤ n }.

Note that any point 𝒫(K, p, ℓ) in supp(x′) must satisfy log_{p} |K| > s (otherwise n + 1 = N − s ≤ ℓ ≤ n). It follows that for every y ∈ 𝒮ℋ(G)_{p,n}^{c} in thick_{⊗}{G/K_{+} | K ∈ ℱ_{≤p^{s}}} we have y ⊗x′ = 0 and [y, x′] = 0 for they have disjoint support in 𝒮ℋ(G)_{p,n}^{c}. Hence the same vanishing is true for the corresponding left idempotent e_{≤p^{s}} in place of y. Repatriated into SH(G)_{(p)} this reads e_{≤p^{s}}⊗x⊗f_{p,n} = 0 and [e_{≤p^{s}}, x⊗f_{p,n}] = 0. From either of them we deduce

(9.2)  t_{≤p^{s}}(f_{p,n} ⊗ x) = t_{≤p^{s}}(f_{p,n}) ⊗ x = 0.

Now for every subgroup H ≤ G with |H| = p^{t}, we have by the construction of x that 𝒫(H, p, n−(t−s)) ∉ Z_{p,N} = supp(x) whereas 𝒫(H, p, n−(t−s)+1) ∈ Z_{p,N} = supp(x). Unfolding the definitions, this means that Φ^{H}(x) belongs to 𝒞_{p,n−(t−s)} but not to 𝒞_{p,n−(t−s)+1}. In other words, Φ^{H}(x) has type exactly n − (t − s) (in SH_{(p)}^{c}) and therefore 𝒞_{p,n−(t−s)} = thick(Φ^{H}(x)) in SH_{(p)}^{c}. So, to prove (B), it suffices to prove the vanishing of

Φ^{H}(t_{≤p^{s}}(f_{p,n})) ⊗ Φ^{H}(x).

But this follows by applying Φ^{H} to (9.2).

(B)⇒(A): Again via Proposition 8.15, it suffices to prove that for every N > r, the irreducible closed subset \overline{\{𝒫(G, p, N − r)\}} in Spc(SH(G)_{(p)}^{c}) is exactly

Z_{p,N} = { 𝒫(K, p, ℓ) | K ≤ G, 1 ≤ ℓ ≤ ∞ such that ℓ + log_{p} |K| ≥ N }.

We already have one inclusion, \overline{\{𝒫(G, p, N − r)\}} ⊇ Z_{p,N}, by Corollary 8.3 and we now discuss the reverse inclusion. Since the complement of \overline{\{𝒫(G, p, N − r)\}} in Spc(SH(G)_{(p)}^{c}) is compact (in fact, even the complement of Z_{p,N} is finite), there exists an object z ∈ SH(G)_{(p)}^{c} whose support is exactly

supp(z) = \overline{\{𝒫(G, p, N − r)\}}.

Suppose ab absurdo that there exists a “bad point” 𝒫_{0} = 𝒫(K_{0}, p, n) belonging to \overline{\{𝒫(G, p, N − r)\}} but with 𝒫_{0} ∉ Z_{p,N}; the latter reads n + log_{p} |K_{0}| < N. We can assume that n is minimal among such “bad points”. This means that we can assume the following:

(9.3)  for all ℓ < n, if ℓ + log_{p} |H| < N then 𝒫(H, p, ℓ) ∉ \overline{\{𝒫(G, p, N − r)\}}.

Let s := N −n−1. Since n < N, we have s ≥ 0 and we can consider the family ℱ_{≤p^{s}}. We claim that the following Tate object vanishes in SH(G)_{(p)}:

t_{≤p^{s}}(f_{p,n} ⊗ z).

To see this, it suffices to prove that Φ^{H}(t_{≤p^{s}}(f_{p,n}) ⊗ z) vanishes in SH_{(p)} for each H ≤ G. When H ∈ ℱ_{≤p^{s}} this is automatic from Example 5.18 and Remark 5.15. Let us then take H ≤ G with |H| = p^{t} and t > s. We are in the situation of hypothesis (B) for 0 ≤ s < t ≤ r. (Note we indeed have n ≥ r − s because we started with N > r.) Hence the vanishing of

Φ^{H}(t_{≤p^{s}}(f_{p,n}) ⊗ z) = Φ^{H}(t_{≤p^{s}}(f_{p,n})) ⊗ Φ^{H}(z)

would follow from Φ^{H}(z) ∈ 𝒞_{p,n−(t−s)}. Now s = N − n − 1 by definition, so n − (t − s) = N − t − 1 and we are left to prove Φ^{H}(z) ∈ 𝒞_{p,N−t−1}. If N − t − 1 = 0 then there is nothing to prove. Otherwise, set ℓ := N − t − 1 < N − s − 1 = n and consider the prime 𝒫(H, p, ℓ). We have ℓ + log_{p} |H| = N − t − 1 + t = N − 1 < N and we can apply (9.3), which tells us that 𝒫(H, p, ℓ) ∉ \overline{\{𝒫(G, p, N − r)\}} = supp(z). The latter reads z ∈ 𝒫(H, p, ℓ) hence Φ^{H}(z) ∈ 𝒞_{p,ℓ} = 𝒞_{p,N−t−1} as desired.

At this stage, we have proved the announced vanishing of the Tate object: t_{≤p^{s}}(f_{p,n} ⊗ z) = 0 in SH(G)_{(p)}. Tensoring by the dual of z, we get by rigidity

0 = t_{≤p^{s}}(f_{p,n} ⊗ z) ⊗ z^{∨} = [f_{≤p^{s}} ⊗ z, Σe_{≤p^{s}} ⊗ z ⊗ f_{p,n}]
= [f_{≤p^{s}} ⊗ z ⊗ f_{p,n}, Σe_{≤p^{s}} ⊗ z ⊗ f_{p,n}]

where the last equality uses Remark 5.5 again. In the localization 𝒮ℋ(G)_{p,n} = f_{p,n} ⊗ SH(G), the last equality implies Hom_{𝒮ℋ (G)_{p,n}}(f_{≤p^{s}} ⊗ z, Σe_{≤p^{s}} ⊗ z) = 0. Consequently, the triangle Δ_{≤p^{s}} ⊗ z splits in 𝒮ℋ(G)_{p,n}, where we thus have:

z ≃ z_{1} ⊕ z_{2}  with  z_{1} = e_{≤p^{s}} ⊗ z and z_{2} = f_{≤p^{s}} ⊗ z .

Since z ∈ 𝒮ℋ(G)_{p,n}^{c} is compact, so are z_{1} and z_{2} and their support must then be contained in Y_{1} := { 𝒫(K, p, n) ∈ Spc(𝒮ℋ(G)_{p,n}^{c}) | |K| ≤ p^{s} } and Y_{2} := { 𝒫(K, p, n) ∈ Spc(𝒮ℋ(G)_{p,n}^{c}) | |K| > p^{s} } respectively. Moreover, the “bad point” 𝒫(K_{0}, p, n) belongs to Y_{1}, since log_{p} |K_{0}| < N −n = s+1, and 𝒫(G, p, N −r) belongs to Y_{2}. Even more, this point 𝒫(G, p, N − r) is the generic point of supp(z), which is irreducible (in Spc(SH(G)_{(p)}), hence in the open Spc(𝒮ℋ(G)_{p,n}^{c}) as well). The non-trivial decomposition supp(z) = Y_{1} ⊔Y_{2} is therefore absurd. Hence there was no “bad point” and we have the desired equality \overline{\{𝒫(G, p, N − r)\}} = Z_{p,N}. □

9.4. Corollary. If G is a group of order p^{r} which satisfies the log_{p}-Conjecture 8.7, then the geometric G-fixed points of the Tate construction lowers chromatic degree by r, namely for every n ≥ r, we have Φ^{G}(t_{G}(triv(f_{p,n}))) ⊗ 𝒞_{p,n−r} = 0.

Proof. This is the extreme case s = 0 and t = r in part (B) of Theorem 9.1. □

If one is willing to study the log_{p}-Conjecture as a whole then global geometric fixed points are also enough.

9.5. Corollary. Let p be a prime. The following are equivalent:

(A) The log_{p}-Conjecture 8.7 holds for all p-groups.

(B) For every group G of order p^{r}, every 0 ≤ s < r, and every n ≥ r − s, we have Φ^{G}(t_{≤p^{s}}(triv(f_{p,n}))) ⊗ 𝒞_{p,n−(r−s)} = 0.

Proof. (A)⇒(B) is the special case t = r of Theorem 9.1 (B). The converse (B)⇒(A) also follows from Theorem 9.1 by induction on |G|. Indeed, the “missing” cases Φ^{H}(t_{≤p^{s}}(triv(f_{p,n})))⊗𝒞_{p,n−(t−s)} = 0 with |H| = p^{t} and t < r hold by the induction hypothesis applied to H, using that res_{H}^{G} preserves t_{≤p^{s}} (Ex. 5.18). □

10. The classification of tt-ideals in SH(G)^{c}

As recalled in Remark 5.1, tt-ideals of SH(G)^{c} are in bijection with the Thomason subsets of Spc(SH(G)^{c}), i.e., arbitrary unions of closed subsets each having quasicompact complement.

10.1. Proposition. Let Z = \overline{\{𝒫_{1}\}} ∪ ⋯ ∪ \overline{\{𝒫_{k}\}} be a closed subset of Spc(SH(G)^{c}) (see Prop. 6.1). We can assume that this is irredundant, that is, 𝒫_{i} ⊈ 𝒫_{j} for all 1 ≤ i ≠ j ≤ k. Let 𝒫_{i} = 𝒫(H_{i}, p_{i}, m_{i}) for all i = 1, …, k. Then the complement of Z is quasi-compact if and only if all chromatic integers m_{1}, …, m_{k} are finite.

Proof. The overall architecture of the proof resembles that of Proposition 6.1. The non-equivariant case (G = 1) can be found in [Bal10a, Cor. 9.5 (d)].

For a general G, let us show that if all m_{1}, …, m_{k} are finite then Z has quasicompact complement. As { supp(x) | x ∈ SH(G)^{c} } is a closed basis for the topology, we need to prove that whenever Z = ∩_{i∈I} supp(x_{i}) for a family {x_{i}}_{i∈I} of objects in SH(G)^{c} there exists a finite subset J ⊆ I such that Z = ∩_{i∈J} supp(x_{i}). Since Z ⊆ ∩_{i∈J} supp(x_{i}) for any J ⊆ I, we just need to make J big enough so that we get equality. We shall do this by collecting together a finite union of finite subsets of I, one for each subgroup K ≤ G (up to conjugacy). Indeed, since Spc(SH(G)^{c}) = ∪_{K≤G} Im(φ^{K,G}), it suffices to show that we can find a finite subset J_{K} ⊆ I so that the desired equality holds when intersected with Im(φ^{K,G}):

(10.2)  Z ∩ Im(φ^{K,G}) =^{?} ∩_{i∈J_{K}} supp(x_{i}) ∩ Im(φ^{K,G}).

Since φ^{K,G} = Spc(Φ^{K,G}) is an inclusion-preserving bijection Spc(SH^{c}) →^{∼} Im(φ^{K,G}), with inverse Spc(triv) suitably restricted (see Cor. 4.6), we can transport the question of (10.2) into Spc(SH^{c}). Let us study both sides separately.

For H, K ≤ G, p a prime and 1 ≤ m < ∞ finite, it is a direct consequence of Corollary 8.19 that the intersection of \overline{\{𝒫(H, p, m)\}} = \{ 𝒬 | 𝒬 ⊆ 𝒫(H, p, m) \} with Im(φ^{K,G}) = { 𝒫(K, q, n) | q, n } is either empty or equals the closure of a single point, namely 𝒫(K, q, n_{K}) in the notation of Corollary 8.19 (c); note that this n_{K} is finite since m is. Also note that 𝒫(K, q, n_{K}) = φ^{K,G}(𝒞_{q,n_{K}}). Hence

(10.3)  (φ^{K,G})^{−1}(\overline{\{𝒫(H, p, m)\}} ∩ Im(φ^{K,G})) = \overline{\{𝒞_{q,n_{K}} \}} or ∅.

Furthermore, for every object x ∈ SH(G)^{c}, we have by a general tt-fact that

(10.4) (φ^{K,G})^{−1}( supp(x) ∩ Im(φ^{K,G})) = (φ^{K,G})^{−1}(supp(x)) = supp(Φ^{K,G}(x)).

We can now use (10.3) and (10.4) to reduce the question of (10.2) to the known non-equivariant case, thus producing the desired finite J_{K} ⊆ I. Hence the result.

Conversely, suppose that one of the m_{i} = ∞. Write Z = Z′ ∪ \overline{\{𝒫(H, p, ∞)\}}, regrouping the other irreducibles under Z′. Since 𝒫(H, p, ∞) = ∩_{1≤n<∞}𝒫(H, p, n), we have Z = ∩_{1≤n<∞}(Z′ ∪ \overline{\{𝒫(H, p, n)\}}). If the complement of Z is quasi-compact then there is a finite n such that Z = Z′ ∪ \overline{\{𝒫(H, p, n)\}}. But then 𝒫(H, p, n) ∈ Z = Z′ ∪\overline{\{𝒫(H, p, ∞)\}} and since 𝒫(H, p, n) ⊈ 𝒫(H, p, ∞), it follows that 𝒫(H, p, n) ∈ Z′. However this forces 𝒫(H, p, ∞) ∈ \overline{\{𝒫(H, p, n)\}} ⊆ Z′, or in other words 𝒫(H, p, ∞) is contained in one of the other irreducibles of Z, contradicting the assumption that the 𝒫_{1}, …, 𝒫_{k} were irredundant. □

10.5. Corollary. The Thomason subsets of Spc(SH(G)^{c}) are the arbitrary unions of \overline{\{𝒫(H, p, m)\}} = \{ 𝒬 | 𝒬 ⊆ 𝒫(H, p, m) \} for arbitrary H ≤ G, p prime and finite chromatic integer 1 ≤ m < ∞.

Proof. This is direct from Proposition 6.1 and Proposition 10.1. □

10.6. Corollary (Classification of tt-ideals in SH(G)^{c}). Let G be a finite group and let ℕ_{∞} := {0, 1, 2, …} ∪ {∞} with the obvious relation < (setting ∞ ≮ ∞). Consider all functions f : { (H, p) | H ≤ G and p prime } → ℕ_{∞} and let us say that f is admissible if it satisfies the following property:

(A) If 𝒫(K, q, n) ⊆ 𝒫(H, p, m) and m > f(H, p) then n > f(K, q). If the log_{p}-Conjecture 8.7 is satisfied by every p-group subquotient of G (for all primes p), then this property can equivalently be expressed without any knowledge of inclusions between equivariant primes as follows:

(A’) For every prime p and for every p-subnormal K ≤ H, we have f(K, p) ≤ f(H, p) + log_{p}[H : K]. Moreover, if f(H, p) = 0 for any pair (H, p) then f(H, q) = 0 for all primes q.

Independently of the log_{p}-Conjecture, there is a one-to-one correspondence between such admissible functions and Thomason subsets of Spc(SH(G)^{c}) mapping f to Y_{f} := { 𝒫(H, p, m) | m > f(H, p) }. Consequently, there is a one-to-one correspondence between admissible functions and tt-ideals in SH(G)^{c} mapping f to

𝒥_{f} := { x ∈ SH(G)^{c} | Φ^{H}(x) ∈ 𝒞_{p,f(H,p)} , ∀ H ≤ G and p such that f(H, p) > 0 }.

Proof. If Y ⊆ Spc(SH(G)^{c}) is a Thomason subset, then Y = ∪_{H≤G, p prime}X(H, p) where X(H, p) := { 𝒫(H, p, n) | 𝒫(H, p, n) ∈ Y }. By Corollaries 8.19 and 10.5, we see that each non-empty X(H, p) is of the form { 𝒫(H, p, n) | n_{H,p} ≤ n ≤ ∞ } for some finite n_{H,p} ≥ 1. The link with our function f is simply that we set f(H, p) = ∞ when X(H, p) = ∅ and f(H, p) = n_{H,p} − 1 when X(H, p) is non-empty. This is made in such a way that X(H, p) = { 𝒫(H, p, n) | n > f(H, p) } in both cases. The definitions of “admissible” are translations of Corollary 8.19.

The classification of tt-ideals then follows from general tt-geometry (see Remark 5.1) since one easily checks that 𝒥_{f} = 𝒦_{Y_{f}} for 𝒦 = SH(G)^{c}. □

References

[Bal05] Paul Balmer. The spectrum of prime ideals in tensor triangulated categories. J. Reine Angew. Math., 588:149–168, 2005.

[Bal10a] Paul Balmer. Spectra, spectra, spectra – tensor triangular spectra versus Zariski spectra of endomorphism rings. Algebr. Geom. Topol., 10(3):1521–1563, 2010.

[Bal10b] Paul Balmer. Tensor triangular geometry. In International Congress of Mathematicians, Hyderabad (2010), Vol. II, pages 85–112. Hindustan Book Agency, 2010.

[Bal11] Paul Balmer. Separability and triangulated categories. Adv. Math., 226(5):4352–4372, 2011.

[Bal13] Paul Balmer. Separable extensions in tensor-triangular geometry and generalized Quillen stratification. 2013. Preprint, 17 pages, available online at arXiv:1309.1808. To appear in Ann. Sci. Éc. Norm. Supér. (4).

[Bal14] Paul Balmer. Splitting tower and degree of tt-rings. Algebra Number Theory, 8(3):767– 779, 2014.

[BCR97] David J. Benson, Jon F. Carlson, and Jeremy Rickard. Thick subcategories of the stable module category. Fund. Math., 153(1):59–80, 1997.

[BDS15] Paul Balmer, Ivo Dell’Ambrogio, and Beren Sanders. Restriction to finite-index subgroups as étale extensions in topology, KK-theory and geometry. Algebr. Geom. Topol., 15(5):3025–3047, 2015.

[BDS16] Paul Balmer, Ivo Dell’Ambrogio, and Beren Sanders. Grothendieck-Neeman duality and the Wirthmüller isomorphism. 2016. Preprint, 36 pages, available online at arXiv:1501.01999. To appear in Compos. Math.

[BF11] Paul Balmer and Giordano Favi. Generalized tensor idempotents and the telescope conjecture. Proc. Lond. Math. Soc. (3), 102(6):1161–1185, 2011.

[DHS88] Ethan S. Devinatz, Michael J. Hopkins, and Jeffrey H. Smith. Nilpotence and stable homotopy theory. I. Ann. of Math. (2), 128(2):207–241, 1988.

[Dre69] Andreas Dress. A characterisation of solvable groups. Math. Z., 110:213–217, 1969.

[FP07] Eric M. Friedlander and Julia Pevtsova. Π-supports for modules for finite group schemes. Duke Math. J., 139(2):317–368, 2007.

[GM95] J. P. C. Greenlees and J. P. May. Generalized Tate cohomology. Mem. Amer. Math. Soc., 113(543):viii+178, 1995.

[Gre01] J. P. C. Greenlees. Tate cohomology in axiomatic stable homotopy theory. In Cohomological methods in homotopy theory (Bellaterra, 1998), volume 196 of Progr. Math., pages 149–176. Birkhäuser, Basel, 2001.

[HHR16] M. A. Hill, M. J. Hopkins, and D. C. Ravenel. On the nonexistence of elements of Kervaire invariant one. Ann. of Math. (2), 184(1):1–262, 2016.

[HPS97] Mark Hovey, John H. Palmieri, and Neil P. Strickland. Axiomatic stable homotopy theory. Mem. Amer. Math. Soc., 128(610), 1997.

[HS96] Mark Hovey and Hal Sadofsky. Tate cohomology lowers chromatic Bousfield classes. Proc. Amer. Math. Soc., 124(11):3579–3585, 1996.

[HS98] Michael J. Hopkins and Jeffrey H. Smith. Nilpotence and stable homotopy theory. II. Ann. of Math. (2), 148(1):1–49, 1998.

[Joa15] R. Joachimi. Thick ideals in equivariant and motivic stable homotopy categories. 2015. Preprint, 115 pages, available online at arXiv:1503.08456.

[Kuh04] Nicholas J. Kuhn. Tate cohomology and periodic localization of polynomial functors. Invent. Math., 157(2):345–370, 2004.

[LMS86] L. G. Lewis, Jr., J. P. May, and M. Steinberger. Equivariant stable homotopy theory, volume 1213 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1986. With contributions by J. E. McClure.

[MM02] M. A. Mandell and J. P. May. Equivariant orthogonal spectra and S-modules. Mem. Amer. Math. Soc., 159(755):x+108, 2002.

[Nee92] Amnon Neeman. The connection between the K-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel. Ann. Sci. École Norm. Sup. (4), 25(5):547–566, 1992.

[Str12] Neil P. Strickland. Thick ideals of finite G-spectra. Unpublished notes, 2012.

[Tho97] R. W. Thomason. The classification of triangulated subcategories. Compositio Math., 105(1):1–27, 1997.

Paul Balmer, Mathematics Department, UCLA, Los Angeles, CA 90095-1555, USA

E-mail address: balmer@math.ucla.edu
URL: http://www.math.ucla.edu/∼balmer

Beren Sanders, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark

E-mail address: sanders@math.ku.dk
URL: http://beren.sites.ku.dk