2.1 The classical thick subcategory theorem

As the prototypical example, we quickly recall the key steps in the computation of the Balmer spectrum of the homotopy category of finite non-equivariant $p$ -local spectra ${\mathcal{S}}{\mathcal{H}}_{(p)}^{c}$ , following Hopkins and Smith [Reference Hopkins and SmithHS98].

For $n\in \mathbb{N}_{{>}0}$ , let $K(n)$ denote the $n$ th Morava $K$ -theory spectrum at $p$ , with coefficients $K(n)_{\ast }=\mathbb{F}_{p}[v_{n}^{\pm 1}]$ and $|v_{n}|=2(p^{n}-1)$ . Furthermore, $K(0)$ is defined to be the rational Eilenberg–MacLane spectrum $H\mathbb{Q}$ . For $n>0$ we write

$$\begin{eqnarray}P(n)=\{X\in {\mathcal{S}}{\mathcal{H}}_{(p)}^{c}\mid K(n-1)_{\ast }(X)=0\}\end{eqnarray}$$

and $P(\infty )=\bigcap _{n\in \mathbb{N}}P(n)$ for the set of all contractible finite $p$ -local spectra. Every $P(n)$ is a prime ideal, since the $K(n)$ satisfy a Künneth formula. Moreover, note that ${\mathcal{S}}{\mathcal{H}}_{(p)}^{c}$ is generated by the unit of the smash product, the $p$ -local sphere spectrum, so any thick subcategory is automatically a tensor ideal. The thick subcategory theorem [Reference Hopkins and SmithHS98, Theorem 7] for ${\mathcal{S}}{\mathcal{H}}_{(p)}^{c}$ provides a complete computation of the Balmer spectrum $\operatorname{Spc}({\mathcal{S}}{\mathcal{H}}_{(p)}^{c})$ .

Theorem 2.1 (Hopkins–Smith).

If ${\mathcal{C}}\subseteq {\mathcal{S}}{\mathcal{H}}_{(p)}^{c}$ is a proper thick subcategory, then there exists a unique $n\in \overline{\mathbb{N}}_{{>}0}$ such that ${\mathcal{C}}=P(n)$ . Moreover, there are proper inclusions

$$\begin{eqnarray}P(1)\supset P(2)\supset \cdots \supset P(\infty )=\{0\}.\end{eqnarray}$$

The proof can be divided into three steps.

1. (i) Firstly, Ravenel [Reference RavenelRav84, Theorem 2.11] showed that if $n>0$ and $X$ is a finite spectrum for which $K(n)_{\ast }(X)=0$ , then also $K(n-1)_{\ast }(X)=0$ . This translates into the inclusion $P(n+1)\subseteq P(n)$ . Given a finite spectrum $X$ , there hence exists a maximal $n$ such that $X\in P(n)$ . This $n$ is called the type of $X$ . If $X\notin P(1)$ , or in other words, if the rational homology of $X$ is non-trivial, $X$ is said to be of type $0$ .

2. (ii) Secondly, these inclusions are all proper, as was first shown by Mitchell [Reference MitchellMit85], by constructing a finite spectrum $F(n)$ with $K(n)_{\ast }(F(n))\neq 0$ and $K(n-1)_{\ast }(F(n))=0$ .

3. (iii) Thirdly, as an application of the nilpotence theorem, Hopkins and Smith showed that if another finite spectrum $Y$ has type larger than or equal to the type of $X$ , then $Y$ already lies in the tt-ideal $\text{thick}^{\otimes }(X)$ generated by $X$ . As a consequence, any finite spectrum of type $n$ generates $P(n)$ . It follows that the $P(n)$ make up all the proper thick tensor ideals and in particular all the prime ideals in ${\mathcal{S}}{\mathcal{H}}_{(p)}^{c}$ . Moreover, the topology on $\operatorname{Spc}({\mathcal{S}}{\mathcal{H}}_{(p)}^{c})$ is determined by the closure of points, which is given by $\overline{\{P(n)\}}=\{P(n),P(n+1),\ldots ,P(\infty )\}$ .

These steps are also manifest in the computation of the Balmer spectrum of the stable equivariant homotopy category. The goal of the next section is to establish an analogue of (3) for all compact Lie groups $G$ as well as an equivariant generalization of the nilpotence theorem. In contrast, Steps (1) and (2) turn out to be more subtle in the equivariant context, and so far only partial generalizations are known; this will be the subject of the remainder of the paper.

2.2 Stable equivariant homotopy theory

We recall some notions and results from equivariant stable homotopy theory for a compact Lie group $G$ that we will need throughout the paper. Classical references in the Lewis–May approach are [Reference Lewis, May, Steinberger and McClureLMSM86, Reference Greenlees and MayGM95a, Reference MayMay96], a treatment of $G$ -orthogonal spectra is given in [Reference Mandell and MayMM02], and [Reference Mathew, Naumann and NoelMNN17, §§ 5 and 6] contains a discussion of some of the constructions in an $\infty$ -categorical setting.

We denote by ${\mathcal{S}}{\mathcal{H}}_{G,(p)}$ the homotopy category of $p$ -local $G$ -spectra for a compact Lie group  $G$ indexed on a complete universe and write $\operatorname{Sub}(G)$ for the set of closed subgroups of $G$ . For $H\in \operatorname{Sub}(G)$ , the suspension spectrum $G/H_{+}=\unicode[STIX]{x1D6F4}^{\infty }G/H_{+}$ is a $G$ -spectrum, and the $H$ -homotopy groups of $X\in {\mathcal{S}}{\mathcal{H}}_{G,(p)}$ are defined as

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{\ast }^{H}(X)=[G/H_{+},X]_{\ast }^{G},\end{eqnarray}$$

i.e., the graded maps in ${\mathcal{S}}{\mathcal{H}}_{G,(p)}$ from the suspension spectrum of $G/H_{+}$ into $X$ . A map of $G$ -spectra is an equivalence if and only if it induces an isomorphism on $\unicode[STIX]{x1D70B}_{\ast }^{H}$ for all closed subgroups  $H$ of $G$ .

The category ${\mathcal{S}}{\mathcal{H}}_{G,(p)}=({\mathcal{S}}{\mathcal{H}}_{G,(p)},\wedge ,\mathbb{S}_{G})$ is a tensor triangulated category which is compactly generated by the set of dualizable objects $\{G/{H_{+}\}}_{H\in \operatorname{Sub}(G)}$ and with compact unit given by the $G$ -equivariant sphere spectrum $\mathbb{S}_{G}=G/G_{+}$ . For every real $G$ -representation $V$ , the suspension spectrum of the one-point compactification $S^{V}$ is an invertible object in ${\mathcal{S}}{\mathcal{H}}_{G,(p)}$ . As usual, the full subcategory of ${\mathcal{S}}{\mathcal{H}}_{G,(p)}$ spanned by the compact objects is denoted by ${\mathcal{S}}{\mathcal{H}}_{G,(p)}^{c}$ . The objects of ${\mathcal{S}}{\mathcal{H}}_{G,(p)}^{c}$ , which we call the finite $G$ -spectra, can be described in the following way.

Here, to avoid subtleties with $p$ -localization of $G$ -spaces, the finite $G$ -CW complex can always be chosen to be a double suspension, so all fixed points are simply connected and the $p$ -localization is given by the mapping telescope over repeated multiplication by $q$ for all primes $q\neq p$ , using the comultiplication from the suspension coordinates. The lemma is (a $p$ -local variation of) a classical fact about $G$ -spectra and follows from every $G$ -spectrum being a filtered homotopy colimit over $G$ -spectra of the form $S^{-V}\wedge \unicode[STIX]{x1D6F4}^{\infty }(-)$ , see [Reference Lewis, May, Steinberger and McClureLMSM86, Proposition I.4.7].

2.2.2 Restriction and induction

In the case where $\unicode[STIX]{x1D711}$ is the inclusion of a closed subgroup $H$ into $G$ , the pullback functor is called restriction and denoted $\operatorname{Res}_{H}^{G}$ . In that case it has a left adjoint called induction that is denoted $\operatorname{Ind}_{H}^{G}$ . When $A$ is a based $H$ -space, there is a natural equivalence $\operatorname{Ind}_{H}^{G}(\unicode[STIX]{x1D6F4}^{\infty }A)\simeq \unicode[STIX]{x1D6F4}^{\infty }(G_{+}\wedge _{H}A)$ , which follows from the analogous property of $\operatorname{Res}_{H}^{G}$ . Furthermore, for a pair of an $H$ -spectrum $X$ and a $G$ -spectrum $Y$ , Frobenius reciprocity provides an equivalence

$$\begin{eqnarray}\operatorname{Ind}_{H}^{G}(X\wedge \operatorname{Res}_{H}^{G}(Y))\simeq \operatorname{Ind}_{H}^{G}(X)\wedge Y.\end{eqnarray}$$

One way to see this is to note that adjunction gives a natural map from the left to the right, and it is enough to check that this map is an equivalence in the case where $X$ and $Y$ vary through the compact generators. Since these are suspension spectra and $\operatorname{Ind}_{H}^{G},\operatorname{Res}_{H}^{G}$ , and  $\wedge$ commute with $\unicode[STIX]{x1D6F4}^{\infty }$ , this reduces to the space-level analogue, which is easily checked.

2.2.4 Geometric fixed points

For every closed subgroup $H$ of $G$ , there is a geometric fixed point functor

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}^{H}=\unicode[STIX]{x1D6F7}_{G}^{H}:{\mathcal{S}}{\mathcal{H}}_{G,(p)}\rightarrow {\mathcal{S}}{\mathcal{H}}_{(p)},\end{eqnarray}$$

obtained by composing the restriction $\operatorname{Res}_{H}^{G}:{\mathcal{S}}{\mathcal{H}}_{G,(p)}\rightarrow {\mathcal{S}}{\mathcal{H}}_{H,(p)}$ with the (absolute) geometric fixed point functor for $H$ , see [Reference Lewis, May, Steinberger and McClureLMSM86, § II.9], [Reference Mandell and MayMM02, § V.4] or [Reference Mathew, Naumann and NoelMNN17, § 6.2]. This functor is exact, symmetric monoidal, preserves all coproducts, and satisfies $\unicode[STIX]{x1D6F7}^{H}(\unicode[STIX]{x1D6F4}^{\infty }A)\simeq \unicode[STIX]{x1D6F4}^{\infty }(A^{H})$ for all based $G$ -CW complexes $A$ . Since the dualizable objects and the compact objects coincide in ${\mathcal{S}}{\mathcal{H}}_{G,(p)}$ , the geometric fixed point functor $\unicode[STIX]{x1D6F7}^{H}$ preserves compact objects and hence restricts to a functor $\unicode[STIX]{x1D6F7}^{H}:{\mathcal{S}}{\mathcal{H}}_{G,(p)}^{c}\rightarrow {\mathcal{S}}{\mathcal{H}}_{(p)}^{c}$ . Every element $g\in G$ induces conjugation equivalences $\unicode[STIX]{x1D6F7}^{H}\simeq \unicode[STIX]{x1D6F7}^{H^{g}}$ , where $H^{g}=g^{-1}Hg$ . Hence, up to equivalence, the geometric fixed point functors depend only on the conjugacy class of $H$ . Moreover, the collection ${\{\unicode[STIX]{x1D6F7}^{H}\}}_{H\in \operatorname{Sub}(G)}$ is jointly conservative, i.e., a map $f$ in ${\mathcal{S}}{\mathcal{H}}_{G,(p)}$ is an equivalence if and only if $\unicode[STIX]{x1D6F7}^{H}(f)$ is an equivalence in ${\mathcal{S}}{\mathcal{H}}_{(p)}$ for all $H\in \operatorname{Sub}(G)$ (see [Reference SchwedeSch18, Proposition 3.3.10] for a proof).

Finally, geometric fixed points have the following behaviour with respect to pullback along a continuous group homomorphism $\unicode[STIX]{x1D711}:G\rightarrow K$ . For a closed subgroup $H$ of $G$ there are natural equivalences

(1) $$\begin{eqnarray}\unicode[STIX]{x1D6F7}^{H}(\unicode[STIX]{x1D711}^{\ast }(X))\simeq \unicode[STIX]{x1D6F7}^{\unicode[STIX]{x1D711}(H)}(X).\end{eqnarray}$$

These can be derived from their space-level analogues using that both sides are exact, symmetric monoidal, coproduct-preserving functors in $X$ .

3.1 An abstract thick tensor ideal theorem

Let ${\mathcal{C}}=({\mathcal{C}},\otimes ,1)$ be a closed symmetric monoidal triangulated category with arbitrary coproducts and compact unit $1$ , which implies that every dualizable object in ${\mathcal{C}}$ is compact. Following Hovey, Palmieri, and Strickland [Reference Hovey, Palmieri and StricklandHPS97], we say that ${\mathcal{C}}$ is a unital algebraic stable homotopy category if it is compactly generated by a set of dualizable objects.

Given an indexing set $I$ , consider a collection of graded abelian categories $({\mathcal{D}}_{i})_{i\in I}$ with zero objects $0\in {\mathcal{D}}_{i}$ and a collection of stable homological functors $(\unicode[STIX]{x1D705}_{i}:{\mathcal{C}}\rightarrow {\mathcal{D}}_{i})_{i\in I}$ satisfying the following two conditions.

1. (i) Let $i\in I$ and $X,Y\in {\mathcal{C}}$ . Then $\unicode[STIX]{x1D705}_{i}(X\otimes Y)=0$ if and only if $\unicode[STIX]{x1D705}_{i}(X)=0$ or $\unicode[STIX]{x1D705}_{i}(Y)=0$ .

2. (ii) If $R\in \operatorname{Alg}({\mathcal{C}})$ is a monoid in ${\mathcal{C}}$ with $\unicode[STIX]{x1D705}_{i}(R)=0$ for all $i\in I$ , then $R=0$ .

We will refer to such a collection of functors as a coarse support theory for ${\mathcal{C}}$ . The corresponding notion of support is then defined for any $X\in {\mathcal{C}}$ as

$$\begin{eqnarray}\operatorname{supp}(X)=\{i\in I\mid \unicode[STIX]{x1D705}_{i}(X)\neq 0\}.\end{eqnarray}$$

This extends to any collection ${\mathcal{X}}\subseteq {\mathcal{C}}$ of objects in ${\mathcal{C}}$ by setting $\operatorname{supp}({\mathcal{X}})=\bigcup _{X\in {\mathcal{X}}}\operatorname{supp}(X)$ . Since the functors $\unicode[STIX]{x1D705}_{i}$ are stable homological to graded abelian categories ${\mathcal{D}}_{i}$ , then $\operatorname{supp}(X)=\operatorname{supp}(\text{thick}^{\otimes }(X))$ .

The next result is an abstract version of the support-theoretic part of the thick subcategory theorem; the proof is a modification of the argument in [Reference Hovey, Palmieri and StricklandHPS97, Proof of Theorem 5.2.2] and thus essentially due to Hovey, Palmieri, and Strickland.

Proof. For a full subcategory ${\mathcal{S}}\subseteq {\mathcal{C}}$ , let $\operatorname{Loc}^{\otimes }({\mathcal{S}})$ be the smallest localizing ideal of ${\mathcal{C}}$ containing ${\mathcal{S}}$ , i.e., the smallest triangulated subcategory of ${\mathcal{C}}$ containing ${\mathcal{S}}$ which is closed under tensor products with objects of ${\mathcal{C}}$ . The natural inclusion ${\mathcal{D}}\rightarrow {\mathcal{C}}$ then gives rise to a (Verdier) localization sequence

$$\begin{eqnarray}\operatorname{Loc}^{\otimes }({\mathcal{D}})\rightarrow {\mathcal{C}}\rightarrow {\mathcal{C}}/\operatorname{Loc}^{\otimes }({\mathcal{D}}).\end{eqnarray}$$

By [Reference Hovey, Palmieri and StricklandHPS97, Lemma 3.1.6(e) and Theorem 3.3.3], the functor ${\mathcal{C}}\rightarrow {\mathcal{C}}/\operatorname{Loc}^{\otimes }({\mathcal{D}})$ admits a right adjoint. The corresponding composite $L_{{\mathcal{D}}}^{f}:{\mathcal{C}}\rightarrow {\mathcal{C}}/\operatorname{Loc}^{\otimes }({\mathcal{D}})\rightarrow {\mathcal{C}}$ is then a smashing localization functor on ${\mathcal{C}}$ , so there is a canonical isomorphism $L_{{\mathcal{D}}}^{f}Y=Y\otimes L_{{\mathcal{D}}}^{f}1$ for all $Y\in {\mathcal{C}}$ . Suppose $X$ satisfies the assumptions of the proposition, then the conclusion is equivalent to $L_{{\mathcal{D}}}^{f}X=0$ , since $\operatorname{Loc}^{\otimes }({\mathcal{D}})\cap {\mathcal{C}}^{c}={\mathcal{D}}$ by Thomason’s Theorem in the form of [Reference GreenleesGre19, Theorem 7.1(ii)].

Note that we may assume without loss of generality that $X\in \operatorname{Alg}({\mathcal{C}})$ : indeed, since $X$ is a retract of $X\otimes DX\otimes X$ by dualizability, $X$ and the monoid $X\otimes DX$ have the same support. Moreover, the same observation shows that $X\in {\mathcal{D}}$ if and only if $X\otimes DX\in {\mathcal{D}}$ .

We now claim that $\unicode[STIX]{x1D705}_{i}(L_{{\mathcal{D}}}^{f}X)=0$ for all $i\in I$ . To this end, let $i\in I$ and assume that $\unicode[STIX]{x1D705}_{i}(X)=0$ . It follows that $\unicode[STIX]{x1D705}_{i}(L_{{\mathcal{D}}}^{f}(X))=\unicode[STIX]{x1D705}_{i}(X\otimes L_{{\mathcal{D}}}^{f}1)=0$ . If $i\in I$ is instead such that $\unicode[STIX]{x1D705}_{i}(X)\neq 0$ , then $i\in \operatorname{supp}(X)\subseteq \operatorname{supp}({\mathcal{D}})$ , so there exists $Y\in {\mathcal{D}}$ with $\unicode[STIX]{x1D705}_{i}(Y)\neq 0$ . But $L_{{\mathcal{D}}}^{f}Y=0$ , and hence $\unicode[STIX]{x1D705}_{i}((X\otimes L_{{\mathcal{D}}}^{f}1)\otimes Y)=0$ . Therefore $\unicode[STIX]{x1D705}_{i}(X\otimes L_{{\mathcal{D}}}^{f}1)=0$ in this case as well. Consequently, $L_{{\mathcal{D}}}^{f}X$ is a monoid in ${\mathcal{C}}$ with $\unicode[STIX]{x1D705}_{i}(L_{{\mathcal{D}}}^{f}X)=0$ for all $i\in I$ , and thus $L_{{\mathcal{D}}}^{f}X=0$ .◻

We note that the proposition in particular implies that every thick tensor ideal ${\mathcal{D}}$ in ${\mathcal{C}}^{c}$ is determined by its support: it consists of all $X\in {\mathcal{C}}^{c}$ such that $\operatorname{supp}(X)\subseteq \operatorname{supp}({\mathcal{D}})$ .

Example 3.3. Let ${\mathcal{S}}{\mathcal{H}}_{(p)}$ be the $p$ -local stable homotopy category at a prime $p$ . For any $n\in \overline{\mathbb{N}}=\{0,1,\ldots \}\cup \{\infty \}$ , write $K(n)$ for the $p$ -local height $n$ Morava $K$ -theory spectrum, where we set $K(\infty )=H\mathbb{F}_{p}$ . The nilpotence theorem of Devinatz, Hopkins, and Smith [Reference Devinatz, Hopkins and SmithDHS88, Reference Hopkins and SmithHS98] implies that the collection of homological and symmetric monoidal functors

$$\begin{eqnarray}\{K(n)_{\ast }:{\mathcal{S}}{\mathcal{H}}_{(p)}\rightarrow {\operatorname{Mod}_{K(n)_{\ast }}\}}_{n\in \overline{\mathbb{N}}}\end{eqnarray}$$

forms a coarse support theory for ${\mathcal{S}}{\mathcal{H}}_{(p)}$ . In this case, Proposition 3.2 recovers the thick subcategory theorem [Reference Hopkins and SmithHS98, Theorem 7] of Hopkins and Smith, as explained in § 2.1.

3.2 Equivariant support and type functions of finite $G$ -spectra

In order to apply the abstract result of the previous subsection to equivariant stable homotopy theory for a compact Lie group $G$ , we need to construct an appropriate coarse support theory. Fix a prime $p$ and write ${\mathcal{S}}{\mathcal{H}}_{G,(p)}$ for the category of $p$ -local $G$ -spectra. Recall from Example 3.3 that for each $n\in \overline{\mathbb{N}}$ there is a Morava $K$ -theory spectrum $K(n)\in {\mathcal{S}}{\mathcal{H}}_{(p)}$ of height $n$ .

Definition 3.4. Let $I=\operatorname{Sub}(G)\times \overline{\mathbb{N}}$ and define for each $(H,n)\in I$ functors

$$\begin{eqnarray}\unicode[STIX]{x1D705}_{(H,n)}=K(n)_{\ast }(\unicode[STIX]{x1D6F7}^{H}(-)):{\mathcal{S}}{\mathcal{H}}_{G,(p)}\rightarrow \operatorname{Mod}_{K(n)_{\ast }}\!.\end{eqnarray}$$

As in the previous subsection, we shall write $\operatorname{supp}(X)$ for the support of $X\in {\mathcal{S}}{\mathcal{H}}_{G,(p)}$ corresponding to the collection of functors $\{{\unicode[STIX]{x1D705}_{(H,n)}\}}_{(H,n)\in I}$ .

As before, this notion of support extends to any collection ${\mathcal{X}}\subseteq {\mathcal{S}}{\mathcal{H}}_{G,(p)}$ . Since each $\unicode[STIX]{x1D6F7}^{H}$ is exact as well as symmetric monoidal and $K(n)$ has a Künneth isomorphism, we have $\operatorname{supp}(X)=\operatorname{supp}(\text{thick}^{\otimes }(X))$ for all $X\in {\mathcal{S}}{\mathcal{H}}_{G,(p)}$ . Then we have the following result, which was also obtained in unpublished work of Strickland.

Proof. We have to verify conditions (1) and (2) above. For any $(H,n)\in \operatorname{Sub}(G)\times \overline{\mathbb{N}}$ and all $X,Y\in {\mathcal{S}}{\mathcal{H}}_{G,(p)}$ , there is a canonical isomorphism

$$\begin{eqnarray}K(n)_{\ast }(\unicode[STIX]{x1D6F7}^{H}(X))\otimes _{K(n)_{\ast }}K(n)_{\ast }(\unicode[STIX]{x1D6F7}^{H}(Y))\cong K(n)_{\ast }(\unicode[STIX]{x1D6F7}^{H}(X\wedge Y)).\end{eqnarray}$$

Since $K(n)_{\ast }$ is a graded field, this implies that the first condition is satisfied. To see that $\operatorname{supp}$ detects ring objects, suppose that $R\in \operatorname{Alg}({\mathcal{S}}{\mathcal{H}}_{G,(p)})$ with $\operatorname{supp}(R)=\varnothing$ . By the nilpotence theorem in the form of [Reference Hopkins and SmithHS98, Theorem 3 i)], this implies that $\unicode[STIX]{x1D6F7}^{H}(R)=0$ for all $H\in \operatorname{Sub}(G)$ , and hence $R=0$ because ${\{\unicode[STIX]{x1D6F7}^{H}\}}_{H\in \operatorname{Sub}(G)}$ is jointly conservative.◻

For any non-zero finite ( $p$ -local) spectrum $F$ there exists an $n\in \mathbb{N}$ such that $K(n)_{\ast }(F)\neq 0$ , which can be seen using the Atiyah–Hirzebruch spectral sequence.

Definition 3.6. If $X$ is a finite $G$ -spectrum $X$ , we define its type function as the function $\text{type}_{X}:\operatorname{Sub}(G)\rightarrow \overline{\mathbb{N}}$ defined by

$$\begin{eqnarray}\text{type}_{X}(H)=\left\{\begin{array}{@{}ll@{}}\min \{n\in \mathbb{N}\mid K(n)_{\ast }(\unicode[STIX]{x1D6F7}^{H}(X))\neq 0\}\quad & \text{if }\unicode[STIX]{x1D6F7}^{H}(X)\neq 0,\\ \infty \quad & \text{otherwise}.\end{array}\right.\end{eqnarray}$$

It turns out that the central problem for determining the topology on $\operatorname{Spc}({\mathcal{S}}{\mathcal{H}}_{G,(p)}^{c})$ and classifying the tt-ideals is the question of which functions can occur as $\text{type}_{X}$ for some $X\in {\mathcal{S}}{\mathcal{H}}_{G,(p)}^{c}$ .

By a theorem due to Ravenel [Reference RavenelRav84, Theorem 2.11], if $F\in {\mathcal{S}}{\mathcal{H}}_{(p)}^{c}$ has $K(n)_{\ast }(F)=0$ , then $K(n-1)_{\ast }(F)$ as well; so for finite $G$ -spectra $X$ , the function $\text{type}_{X}$ contains the same information as $\operatorname{supp}(X)$ . As before, we may extend $\text{type}_{}$ to subcategories ${\mathcal{X}}\subseteq {\mathcal{S}}{\mathcal{H}}_{G,(p)}^{c}$ by defining

$$\begin{eqnarray}\text{type}_{{\mathcal{X}}}(H)=\min \{\text{type}_{X}(H)\mid X\in {\mathcal{X}}\}.\end{eqnarray}$$

We write $f\geqslant g$ for two functions $f,g:\operatorname{Sub}(G)\rightarrow \overline{\mathbb{N}}$ if $f(H)\geqslant g(H)$ for all $H\in \operatorname{Sub}(G)$ . The next corollary is now an immediate consequence of Propositions 3.2 and 3.5.

When $I$ is given by $\text{thick}^{\otimes }(Y)$ for another finite $G$ -spectrum $Y$ , the corollary implies that $X\in \text{thick}^{\otimes }(Y)$ if and only if $\text{type}_{X}\geqslant \text{type}_{Y}$ . This statement is perhaps slightly surprising, because it means that $X$ lies in $\text{thick}^{\otimes }(Y)$ if and only if $\unicode[STIX]{x1D6F7}^{H}(X)$ lies in $\text{thick}^{\otimes }(\unicode[STIX]{x1D6F7}^{H}(Y))$ for all closed subgroups $H$ , with no condition on how the various geometric fixed points are glued together.

We end this subsection with a result concerning the type function of induced $G$ -spectra, which we shall make repeated use of. Recall from § 2.2.2 that for a closed subgroup $H$ of a compact Lie group $G$ , the restriction functor

$$\begin{eqnarray}\operatorname{Res}_{H}^{G}:{\mathcal{S}}{\mathcal{H}}_{G,(p)}^{c}\rightarrow {\mathcal{S}}{\mathcal{H}}_{H,(p)}^{c}\end{eqnarray}$$

has a left adjoint denoted $\operatorname{Ind}_{H}^{G}$ .

A proof of this result is given in Appendix A.

Remark 3.10. When $G$ is a finite group, there is in fact an equality in part (ii) above, i.e., in that case the type of $\unicode[STIX]{x1D6F7}^{K}(\operatorname{Ind}_{H}^{G}(X))$ equals the minimum of the types of $\unicode[STIX]{x1D6F7}^{K^{\prime }}(\operatorname{Ind}_{H}^{G}(X))$ for all subgroups $K^{\prime }$ of $H$ that are conjugate to $K$ in $G$ , see Remark A.3 in Appendix A.

This is not generally true for compact Lie groups. Note that when $K$ is the trivial group, an equality in part (ii) would mean that the type of $\unicode[STIX]{x1D6F7}^{e}\operatorname{Ind}_{H}^{G}(X)$ is always the same as the type of $\unicode[STIX]{x1D6F7}^{e}(X)$ . Counterexamples to such an equality are given by the following complexes, which play a central role in [Reference Barthel, Hausmann, Naumann, Nikolaus, Noel and StapletonBHN⁺19] and were studied extensively by Arone, Lesh, and Mahowald [Reference AroneAro98, Reference Arone and MahowaldAM99, Reference Arone and LeshAL17]. We refer to [Reference Barthel, Hausmann, Naumann, Nikolaus, Noel and StapletonBHN⁺19, § 4] for more details. Let $n>0$ and let ${\mathcal{P}}_{p^{n}}^{\diamond }$ denote the unreduced suspension of the complex of proper non-trivial partitions of the set $\{1,\ldots ,p^{n}\}$ , equipped with the action of $\unicode[STIX]{x1D6F4}_{p^{n}}$ by permuting the partitions. Further, let $\overline{\unicode[STIX]{x1D70C}}$ denote the reduced standard representation of $\unicode[STIX]{x1D6F4}_{p^{n}}$ , and define $X\in {\mathcal{S}}{\mathcal{H}}_{\unicode[STIX]{x1D6F4}_{p^{n}},(p)}^{c}$ by

$$\begin{eqnarray}X=F(\unicode[STIX]{x1D6F4}^{\infty }{\mathcal{P}}_{p^{n}}^{\diamond },\mathbb{S}_{\unicode[STIX]{x1D6F4}_{p^{n}}})\wedge S^{\overline{\unicode[STIX]{x1D70C}}}.\end{eqnarray}$$

The reduced standard representation (this time over $\mathbb{C}$ ) defines an embedding $\unicode[STIX]{x1D6F4}_{p^{n}}{\hookrightarrow}U(p^{n}-1)$ , so we can consider $\operatorname{Ind}_{\unicode[STIX]{x1D6F4}_{p^{n}}}^{U(p^{n}-1)}X$ . Since ${\mathcal{P}}_{p^{n}}$ is a non-trivial wedge of spheres (see [Reference Orlik and TeraoOT92, 4.109]), the underlying non-equivariant spectrum of $X$ , i.e., $\unicode[STIX]{x1D6F7}^{e}(X)$ , has type $0$ . On the other hand, by work of Arone–Mahowald [Reference Arone and MahowaldAM99] and Arone [Reference AroneAro98], the underlying non-equivariant spectrum of $\operatorname{Ind}_{\unicode[STIX]{x1D6F4}_{p^{n}}}^{U(p^{n}-1)}X$ has type $n$ , see also the proof of [Reference Barthel, Hausmann, Naumann, Nikolaus, Noel and StapletonBHN⁺19, Theorem 2.2].

3.3 The underlying set of $\operatorname{Spc}({\mathcal{S}}{\mathcal{H}}_{G,(p)}^{c})$

In this subsection we describe the set of prime ideals in ${\mathcal{S}}{\mathcal{H}}_{G,(p)}^{c}$ , using the geometric fixed point functors to pull back prime ideals from ${\mathcal{S}}{\mathcal{H}}_{(p)}^{c}$ and applying results of the previous subsections to deduce that every prime ideal is obtained in this way. For any $(H,n)\in \operatorname{Sub}(G)\times \overline{\mathbb{N}}_{{>}0}$ , we write

$$\begin{eqnarray}P_{G}(H,n)=(\unicode[STIX]{x1D6F7}^{H})^{-1}(P(n))=\{X\in {\mathcal{S}}{\mathcal{H}}_{G,(p)}^{c}\mid K(n-1)_{\ast }(\unicode[STIX]{x1D6F7}^{H}(X))=0\}.\end{eqnarray}$$

Since $\unicode[STIX]{x1D6F7}^{H}$ is symmetric monoidal, it follows that each $P_{G}(H,n)$ is again a prime.

We now show that every prime ideal in ${\mathcal{S}}{\mathcal{H}}_{G,(p)}^{c}$ is of the form $P_{G}(H,n)$ for some closed subgroup $H$ and $n\in \overline{\mathbb{N}}_{{>}0}$ , where $n$ is unique and $H$ is unique up to conjugacy. We first prove uniqueness, for which we make use of finite $G$ -spectra of the following form.

Such finite $G$ -spectra always exist, by the following simple construction. Let $Y$ be a non-equivariant finite type $n$ spectrum. We consider the finite $G$ -spectrum $X=G/H_{+}\wedge i_{G}^{\ast }(Y)$ , i.e., the smash product of the based $G$ -space $G/H_{+}$ with the inflation of $Y$ to a $G$ -spectrum.

Proof. For a subgroup $K$ of $G$ we have an equivalence

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}^{K}(X)\simeq (G/H)_{+}^{K}\wedge \unicode[STIX]{x1D6F7}^{K}(i_{G}^{\ast }(Y))\simeq (G/H)_{+}^{K}\wedge Y,\end{eqnarray}$$

which follows from the behaviour of geometric fixed points with respect to smash product, suspension spectra, and inflation, as recalled in § 2.2. Now, if $K$ is conjugate to a subgroup $K^{g}$ of $H$ , then $(G/H)^{K}$ is non-empty, since it contains the coset $g^{-1}H$ . Hence $(G/H)_{+}^{K}$ contains $S^{0}$ as a retract, and it follows that the type of $(G/H)_{+}^{K}\wedge Y$ is equal to the type of $Y$ , namely  $n$ . If, on the other hand, $K$ is not conjugate to a subgroup of $H$ , then the fixed points $(G/H)^{K}$ are empty, and hence $\unicode[STIX]{x1D6F7}^{K}(X)$ is trivial.◻

The existence of such finite $G$ -spectra directly implies the following.

Theorem 3.14. Let $G$ be a compact Lie group. Then the assignment $(H,n)\mapsto P_{G}(H,n)$ defines a bijection

$$\begin{eqnarray}(\operatorname{Sub}(G)/G)\times \overline{\mathbb{N}}_{{>}0}\xrightarrow[{}]{\cong }\operatorname{Spc}({\mathcal{S}}{\mathcal{H}}_{G,(p)}^{c}).\end{eqnarray}$$

Proof. Lemma 3.13 shows that the map is injective; hence it suffices to see that every prime ideal ${\wp}$ is of the form $P_{G}(H,n)$ for some subgroup $H$ and $n\in \mathbb{N}$ . By Corollary 3.7, we have

$$\begin{eqnarray}{\wp}=\{Y\in {\mathcal{S}}{\mathcal{H}}_{G,(p)}^{c}\mid \text{type}_{Y}\geqslant \text{type}_{{\wp}}\}=\mathop{\bigcap }_{H}P_{G}(H,\text{type}_{{\wp}}(H)),\end{eqnarray}$$

where we take $P_{G}(H,\text{type}_{{\wp}}(H))$ to mean the full category ${\mathcal{S}}{\mathcal{H}}_{G,(p)}^{c}$ if $\text{type}_{{\wp}}(H)=0$ . Since ${\wp}$ is prime, it is in particular not the full tt-ideal of all finite $G$ -spectra, so there exists a subgroup $H$ such that $\text{type}_{{\wp}}(H)>0$ . Without loss of generality, we can assume that $H$ is minimal with this property, since every descending chain of compact Lie groups stabilizes after finitely many steps. We claim that ${\wp}$ is equal to $P_{G}(H,\text{type}_{{\wp}}(H))$ . If not, there would exist an element $X\in P_{G}(H,\text{type}_{{\wp}}(H))\setminus {\wp}$ . Furthermore, any finite $G$ -spectrum $Y$ of type $({\leqslant}^{G}\!H,0)$ does not lie in ${\wp}$ , but we claim it is contained in $\bigcap _{H^{\prime }\not \sim H}P_{G}(H^{\prime },\text{type}_{{\wp}}(H^{\prime }))$ . Indeed, if $H^{\prime }$ is conjugate to a proper subgroup of $H$ , then $\text{type}_{{\wp}}(H^{\prime })=0$ by the minimality of $H$ , and hence $P_{G}(H^{\prime },\text{type}_{{\wp}}(H^{\prime }))$ is the ideal of all finite $G$ -spectra. If $H^{\prime }$ is not conjugate to a subgroup of $H$ , then $\text{type}_{Y}(H^{\prime })=\infty$ and hence $Y$ is contained in $P_{G}(H^{\prime },n)$ for all $n$ . However, the smash product $X\wedge Y$ lies in the intersection

$$\begin{eqnarray}P_{G}(H,\text{type}_{{\wp}}(H))\cap \mathop{\bigcap }_{H^{\prime }\not \sim H}P_{G}(H^{\prime },\text{type}_{{\wp}}(H^{\prime }))={\wp},\end{eqnarray}$$

contradicting the fact that ${\wp}$ is prime. Hence, such an $X$ cannot exist, which implies that ${\wp}=P_{G}(H,\text{type}_{{\wp}}(H))$ as desired.◻

3.4 The equivariant nilpotence theorem

The goal of this subsection is to strengthen the observation made in the proof of Proposition 3.5 that $\{{\unicode[STIX]{x1D705}_{(H,n)}\}}_{(H,n)\in \operatorname{Sub}(G)\times \overline{\mathbb{N}}}$ detects ring objects to an equivariant version of the nilpotence theorem. This generalizes [Reference Balmer and SandersBS17, Theorem 4.15] from finite groups to arbitrary compact Lie groups; however, unlike their proof, our argument does not rely on the classification of prime ideals in ${\mathcal{S}}{\mathcal{H}}_{G,(p)}$ , but rather exploits properties of the geometric fixed point functors to reduce to the non-equivariant nilpotence theorem by Devinatz, Hopkins, and Smith.

Lemma 3.15. Let $R$ be a homotopy $G$ -ring spectrum (i.e., an object of $\operatorname{Alg}({\mathcal{S}}{\mathcal{H}}_{G})$ ) and $x\in \unicode[STIX]{x1D70B}_{n}^{G}(R)$ an element. Then $x$ is nilpotent if and only if the mapping telescope

$$\begin{eqnarray}\operatorname{tel}(x)=\underset{n\in \mathbb{N}}{\operatorname{hocolim}}(R\xrightarrow[{}]{x\cdot }S^{-n}\wedge R\xrightarrow[{}]{x\cdot }S^{-2n}\wedge R\xrightarrow[{}]{x\cdot }\cdots \,)\end{eqnarray}$$

is contractible.

Proof. First we note that the homotopy ring $\unicode[STIX]{x1D70B}_{\ast }^{G}(\operatorname{tel}(x))$ is isomorphic to the colimit

$$\begin{eqnarray}\underset{n\in \mathbb{N}}{\operatorname{colim}}(\unicode[STIX]{x1D70B}_{\ast }^{G}(R)\xrightarrow[{}]{x\cdot }\unicode[STIX]{x1D70B}_{\ast }^{G}(R)\xrightarrow[{}]{x\cdot }\cdots \,).\end{eqnarray}$$

Now, if $\operatorname{tel}(x)$ is contractible, then this colimit must be trivial. In particular, the element represented by the class of the element $1\in \unicode[STIX]{x1D70B}_{0}^{G}(R)$ under the inclusion of the first term in the colimit system must be trivial. By standard facts about sequential colimits, this means that $x^{n}\cdot 1=x^{n}$ must be trivial for some $n\in \mathbb{N}$ , and hence $x$ is nilpotent.

If $x$ is nilpotent, say $x^{n}=0$ , then clearly the colimit above is trivial, since any composition of $n$ maps in the colimit system is trivial. So we find that $\unicode[STIX]{x1D70B}_{\ast }^{G}(\operatorname{tel}(x))=0$ . Furthermore, all restrictions $\operatorname{Res}_{H}^{G}(x)$ to closed subgroups $H$ are also nilpotent, since restriction maps are ring homomorphisms. So we conclude that $\unicode[STIX]{x1D70B}_{\ast }^{H}(\operatorname{tel}(\operatorname{Res}_{H}^{G}(x)))$ is trivial for all closed subgroups $H$ . Since we have an equivalence

$$\begin{eqnarray}\operatorname{tel}(\operatorname{Res}_{H}^{G}(x))\simeq \operatorname{Res}_{H}^{G}(\operatorname{tel}(x)),\end{eqnarray}$$

this shows that all the equivariant homotopy groups of $\operatorname{tel}(x)$ vanish, and hence $\operatorname{tel}(x)$ is contractible.◻

Recall that there is a comparison map

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}^{H}:\unicode[STIX]{x1D70B}_{\ast }^{H}(X)\rightarrow \unicode[STIX]{x1D70B}_{\ast }(\unicode[STIX]{x1D6F7}^{H}(X))=\unicode[STIX]{x1D6F7}_{\ast }^{H}(X),\end{eqnarray}$$

which is a ring map if $X$ is a homotopy $G$ -ring spectrum. Given an element $x\in \unicode[STIX]{x1D70B}_{\ast }^{G}(X)$ , we also use the notation $\unicode[STIX]{x1D6F7}^{H}(x)$ for the element $\unicode[STIX]{x1D6F7}^{H}(\operatorname{Res}_{H}^{G}(x))\in \unicode[STIX]{x1D6F7}_{\ast }^{H}(X)$ .

Recall that $F(-,-)$ denotes the function $G$ -spectrum between two $G$ -spectra, see [Reference Lewis, May, Steinberger and McClureLMSM86, § II.3] or [Reference Mandell and MayMM02, § II.3]. The function $G$ -spectra are uniquely determined by being right adjoint to the smash product and make ${\mathcal{S}}{\mathcal{H}}_{G,(p)}$ a closed symmetric monoidal category. In particular, as is generally the case for closed symmetric monoidal categories (see, for example, [Reference KellyKel82, §§ 1.5–1.8]), the function $G$ -spectra $F(-,-)$ define a canonical enrichment of ${\mathcal{S}}{\mathcal{H}}_{G,(p)}$ over itself, and the smash product canonically extends to an enriched bi-functor. As a formal consequence, for any pair of $G$ -spectra $X$ and $Y$ , one can form a homotopy $G$ -ring spectrum $R_{X,Y}=\vee _{n}F(X^{\wedge n},Y^{\wedge n})$ with multiplication given by the enriched smash product

$$\begin{eqnarray}F(X^{\wedge n},Y^{\wedge n})\wedge F(X^{\wedge m},Y^{\wedge m})\rightarrow F(X^{\wedge (n+m)},Y^{\wedge (n+m)}).\end{eqnarray}$$

Its homotopy ring is given by

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{\ast }^{G}(R_{X,Y})\cong \bigoplus _{n}[X^{\wedge n},Y^{\wedge n}]_{\ast }^{G}.\end{eqnarray}$$

Now, if $X$ is a finite $G$ -spectrum, the geometric fixed points $\unicode[STIX]{x1D6F7}^{H}(F(X^{\wedge n},Y^{\wedge n}))$ are naturally equivalent to the non-equivariant function spectrum $F(\unicode[STIX]{x1D6F7}^{H}(X)^{\wedge n},\unicode[STIX]{x1D6F7}^{H}(Y)^{\wedge n})$ . One way to see this is the following. Since $X$ is dualizable, $F(X^{\wedge n},Y^{\wedge n})$ is canonically equivalent to $F(X^{\wedge n},\mathbb{S}_{G})\,\wedge \,Y^{\wedge n}$ (see [Reference Greenlees and MayGM95a, Theorem 4.17]). Now $F(X^{\wedge n},\mathbb{S}_{G})$ is the dual of $X^{\wedge n}$ , whose $H$ -geometric fixed points are equivalent to the dual of $\unicode[STIX]{x1D6F7}^{H}(X^{\wedge n})\simeq \unicode[STIX]{x1D6F7}^{H}(X)^{\wedge n}$ , since $\unicode[STIX]{x1D6F7}^{H}$ is symmetric monoidal. Hence we obtain the above equivalence via

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F7}^{H}(F(X^{\wedge n},Y^{\wedge n})) & \simeq & \displaystyle \unicode[STIX]{x1D6F7}^{H}(F(X^{\wedge n},\mathbb{S}_{G}))\wedge \unicode[STIX]{x1D6F7}^{H}(Y^{\wedge n})\nonumber\\ \displaystyle & \simeq & \displaystyle F(\unicode[STIX]{x1D6F7}^{H}(X)^{\wedge n},\mathbb{S})\wedge \unicode[STIX]{x1D6F7}^{H}(Y)^{\wedge n}\nonumber\\ \displaystyle & \simeq & \displaystyle F(\unicode[STIX]{x1D6F7}^{H}(X)^{\wedge n},\unicode[STIX]{x1D6F7}^{H}(Y)^{\wedge n}),\nonumber\end{eqnarray}$$

where the last step uses dualizability of $\unicode[STIX]{x1D6F7}^{H}(X)^{\wedge n}$ .

The above equivalences are compatible with taking smash products, so it follows that, for finite $X$ , $\unicode[STIX]{x1D6F7}^{H}(R_{X,Y})$ is equivalent as a homotopy ring spectrum to $R_{\unicode[STIX]{x1D6F7}^{H}(X),\unicode[STIX]{x1D6F7}^{H}(Y)}$ . Furthermore, the geometric fixed point map takes a map $g:X^{\wedge n}\rightarrow Y^{\wedge n}$ to $\unicode[STIX]{x1D6F7}^{H}(g):\unicode[STIX]{x1D6F7}^{H}(X)^{\wedge n}\rightarrow \unicode[STIX]{x1D6F7}^{H}(Y)^{\wedge n}$ . Applying Proposition 3.17 to $R_{X,Y}$ hence implies the following.

These results allow us to transport non-equivariant nilpotence theorems to the equivariant world, by testing on all geometric fixed points. In particular, we obtain the following equivariant version of the nilpotence theorem as a consequence of the non-equivariant nilpotence theorem.

As mentioned previously, for finite groups $G$ this theorem was proven using tt-geometric techniques and induction on the order of $G$ by Balmer and Sanders. We close this section with a remark about alternative versions of the equivariant nilpotence theorem.

Remark 3.20. Let $MU_{G}$ be the homotopical $G$ -equivariant complex cobordism spectrum in the sense of tom Dieck [Reference tom DiecktD70]. For any closed subgroup $H\subseteq G$ , the geometric fixed points $\unicode[STIX]{x1D6F7}^{H}MU_{G}$ split into a non-trivial wedge of shifted copies of non-equivariant $MU$ , see [Reference SinhaSin01, Theorem 4.10], so it follows from Corollary 3.18 and the classical nilpotence theorem [Reference Devinatz, Hopkins and SmithDHS88] for $MU$ that $MU_{G}$ detects smash-nilpotence of maps between finite $G$ -spectra.

When $G=C_{2}$ one can also consider the real cobordism spectrum $MU_{\mathbb{R}}$ , where $C_{2}$ acts on the underlying spectrum $MU$ via complex conjugation [Reference LandweberLan68]. This one does not detect smash-nilpotence since its geometric fixed points $\unicode[STIX]{x1D6F7}^{C_{2}}MU_{\mathbb{R}}\simeq MO$ see only $2$ -primary information. For a concrete example of a non smash-nilpotent map between finite $C_{2}$ -spectra which is nilpotent in $MU_{\mathbb{R}}$ , let $C$ be the cofibre of the transfer map $S^{0}\rightarrow (C_{2})_{+}\rightarrow S^{0}$ and consider the multiplication by $2$ map

$$\begin{eqnarray}v=2:C\rightarrow C.\end{eqnarray}$$

We claim that $(MU_{\mathbb{R}})_{\ast }(v)$ is nilpotent, while $v$ is not. To this end, first observe that $\unicode[STIX]{x1D6F7}^{e}(v)$ induces multiplication by $2$ on the mod- $2$ Moore spectrum $\unicode[STIX]{x1D6F7}^{e}C\simeq S^{0}/2$ and hence is smash-nilpotent. Secondly, applying $C_{2}$ -geometric fixed points yields the map

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}^{C_{2}}(v)=2:S^{0}\vee S^{1}\rightarrow S^{0}\vee S^{1},\end{eqnarray}$$

which induces the null map after smashing with $H\mathbb{Z}/2$ and hence also $MO$ . Therefore, the map $MO_{\ast }(\unicode[STIX]{x1D6F7}^{C^{2}}(v))$ is nilpotent. However, multiplication by $2$ is not nilpotent on $\unicode[STIX]{x1D6F7}^{C_{2}}C\simeq S^{0}\vee S^{1}$ and consequently $v$ cannot be smash-nilpotent.

6.1 Change of groups

We begin by proving the following change of groups properties regarding Question 6.1.

The first part can be interpreted as a ‘Going-Up’ theorem for restriction to subgroup functors, see [Reference Balmer and SandersBS17, Proposition 6.9] in the finite group case.