2.1 Equivariant geometry

Set $S\in \mathrm {Sch}^G_B$ . If $\phi :G\to K$ is a group homomorphism, we write

$$ \begin{align*} \phi^{-1}:\mathrm{Sch}^K_S\to \mathrm{Sch}^G_{\phi^{-1}S} \end{align*} $$

for the restriction functor, which regards a K-scheme over S as a G-scheme over S via $\phi $ . When no confusion should arise, we write $\mathrm {Sch}^G_{S}$ instead of $\mathrm {Sch}^G_{\phi ^{-1}S}$ .

Set $X\in \mathrm {Sch}^G_B$ . Say that G acts freely on X if the action of $G(T)$ on $X(T)$ is free for any $T\in \mathrm {Sch}_B$ . If G acts freely on X, then the fppf-quotient $(X/G)_{\mathrm {fppf}}$ is representable by an object $X/G\in \mathrm {Sch}_B$ (since all of our schemes are quasi-projective; see [37, Tag 07S7]. Since G is smooth, the fppf-torsor $X\to X/G$ is an étale torsor (as it is a smooth map and so étale locally admits sections).

Definition 2.1. The stabilizer of a point $x\in X$ is the subgroup $\mathrm {Stab}(x)\leq G$ , defined by

$$ \begin{align*} \mathrm{Stab}(x) = \{g\in G \mid g\smash \cdot x = x\text{ and }g \text{ acts as }\mathrm{id}\text{ on }k(x)\}. \end{align*} $$

Then G acts freely on X, provided $\mathrm {Stab}(x)=\{e\}$ for all $x\in X$ . More generally, if $H\leq G$ is a subgroup which acts freely on X, then the quotient $X/H$ inherits an action of the Weyl group $\mathrm {W}(H)=\mathrm {W}_G(H):=\mathrm {N}_G(H)/H$ , and so defines a functor $(-)/H:\mathrm {Sch}^{G,H\textrm {-}\mathrm {free}}_B \to \mathrm {Sch}^{\mathrm {W} H}_{B}.$ This is the composite of the restriction functor $\mathrm {Sch}^{G,H\textrm {-}\mathrm {free}}_B\to \mathrm {Sch}^{\mathrm {N} H,H\textrm {-}\mathrm {free}}_B$ and followed by the quotient $\mathrm {Sch}^{\mathrm {N} H,H\textrm {-}\mathrm {free}}_B \to \mathrm {Sch}^{\mathrm {W} H}_{B}$ .

Let $N\trianglelefteq G$ be a normal subgroup. Suppose that either

1. (i) N acts trivially on S or

2. (ii) N acts freely on S.

In either case, the quotient functor yields a functor

$$ \begin{align*} \mathrm{Sm}^{G,N\textrm{-}\mathrm{free}}_S\to \mathrm{Sm}^{G/N}_{S/N}. \end{align*} $$

Write $q:S\to S/N$ for the quotient map of schemes and $q^{-1}:\mathrm {Sm}^{G/N}_{S/N} \to \mathrm {Sm}^{G/N}_S$ for the functor defined by $q^{-1}(Y) = Y\times _{S/N}S$ .

Proposition 2.2. Suppose that N acts freely on S. Then $(-)/N$ and $\pi ^{-1}q^{-1}$ are inverse equivalences

$$ \begin{align*} (-)/N:\mathrm{Sm}^G_S \rightleftarrows \mathrm{Sm}^{G/N}_{S/N}:\pi^{-1}q^{-1}. \end{align*} $$

Proof Let $f:X\to S$ be in $\mathrm {Sm}^G_S$ . By descent, we have a Cartesian square in $\mathrm {Sm}_S \left (\text {hence in }\mathrm {Sm}^G_S\right )$

It follows that $(-)/N$ is fully faithful. It is also essentially surjective, since if $Y\in \mathrm {Sm}^{G/N}_{S/N}$ , then $Y\cong \left (Y\times _{S/N}S\right )/N$ .

Let $\iota :H\hookrightarrow G$ be a monomorphism of groups. The induction-restriction adjunction

$$ \begin{align*} \iota_!:\mathrm{Sch}^H_S\rightleftarrows \mathrm{Sch}^G_S:\iota^{-1} \end{align*} $$

restricts to an adjunction

$$ \begin{align*} \iota_!: \mathrm{Sm}^H_S\rightleftarrows \mathrm{Sm}^G_S: \iota^{-1}. \end{align*} $$

When $H\leq G$ is a subgroup and $\iota $ is the inclusion, we often write $ \iota _!(X) = G\times _{H} X $ , and this scheme is described concretely as follows. The scheme $G\times X$ becomes an H-scheme under the action $h(g,x) = \left (gh^{-1}, hx\right )$ , and we define

$$ \begin{align*} G\times_{H} X = (G\times X)/H. \end{align*} $$

The scheme $G\times _H X$ has a left G-action through the action of G on itself. We can describe $G\times _{H}X$ in slightly more concrete terms as follows. Choose a complete set of left coset representatives $g_i$ ; then $G\times _{H}X= \coprod _{g_{i}}X_i$ , each $X_i$ is a copy of X, and $g\in G$ acts as $k:X_{i}\to X_{j}$ , where $k\in H$ satisfies $gg_i=g_jk$ .

Set $X\in \mathrm {Sch}^G_B$ . The presheaf of sets $X^G$ is the presheaf of sets on $\mathrm {Sch}_B$ defined by

$$ \begin{align*} X^G(Y) = \{ y\in X(Y)\mid y\text{ is fixed by }G\}. \end{align*} $$

If $X\to B$ is seperated, the presheaf $X^G$ is represented by a closed subscheme of X which is finitely presented over B, which is moreover smooth over B if X is (see [Reference Demazure and Grothendieck12, Proposition XII.9.2, Corollaire XII.9.8] or [Reference Conrad, Gabber and Prasad10, Proposition A.8.10]). Note that if $H\leq G$ is a subgroup, the fixed-point subscheme $X^{H}$ comes equipped with an action of the Weyl group $\mathrm {W}(H)$ .

Now, suppose that $N\unlhd G$ is a normal subgroup which acts trivially on S. Write $\pi :G\to G/N$ for the quotient map. Restricting action along $\pi $ defines a functor $\pi ^{-1}:\mathrm {Sch}_S^{G/N} \to \mathrm {Sch}^{G}_S$ , which is left adjoint to fixed points. We will usually simply write again X instead of $\pi ^{-1}X$ , whenever context makes the meaning clear. Now, restricting attention to smooth S-schemes, we obtain the adjunction

$$ \begin{align*} \pi^{-1}:\mathrm{Sm}_S^{G/N}\rightleftarrows \mathrm{Sm}^{G}_{S}:(-)^N. \end{align*} $$

2.2 Families of subgroups

Families of subgroups provide a convenient way to filter equivariant motivic homotopy theory.

If $\mathcal {F}$ is a family, we write $\mathrm {co}(\mathcal {F}):=\mathcal {F}_{\mathrm {all}} - \mathcal {F}$ for its complement. Note that $\mathrm {co}(\mathcal {F})$ is not a family.Footnote ²

A family $\mathcal {F}$ determines a sieve on $\mathrm {Sm}^G_S$ by letting $\mathrm {Sm}^G_S[\mathcal {F}]\subseteq \mathrm {Sm}^G_S$ be the full subcategory whose objects are smooth G-schemes over S such that all stabilizers are contained in $\mathcal {F}$ . It is useful to make the following more general definition:

Notation 2.7. Set $X\in \mathrm {Sch}^G_S$ . Write

$$ \begin{align*} X^{\mathcal{F}} := \bigcup_{H\in\mathrm{co}(\mathcal{F})} X^{H} \end{align*} $$

and

$$ \begin{align*} X(\mathcal{F}) := X - X^{\mathcal{F}}. \end{align*} $$

The subset $X(\mathcal {F})\subseteq X$ is the set of points whose stabilizers are in $\mathcal {F}$ . Observe that $X(\mathcal {F})\subseteq X$ is an open invariant subscheme and $X^{\mathcal {F}}\subseteq X$ is a closed invariant subscheme, since $X^{\mathcal {F}}$ is a finite union of closed subschemes.

2.3 Motivic G-spaces

Recall that the Nisnevich topology can be defined via a $cd$ -structure.

Let $\mathcal {C}_S\subseteq \mathrm {Sch}^G_S$ be a full subcategory containing $\varnothing $ . We often require $\mathcal {C}_S$ to satisfy one or both of the following properties:

1. (P) If $Y\to X$ is fixed-point reflecting and étale, then $Y\in \mathcal {C}_S$ whenever $X\in \mathcal {C}_S$ .

2. (H) If $X\in \mathcal {C}_S$ , then so is .

Our primary examples of interest are the categories $\mathrm {Sm}_S^G[\mathcal {E}]$ , where $\mathcal {E}$ is a set of subgroups closed under conjugacy. More generally, we could also consider the following property:

• (P ^′ ) If $Y\to X$ is an equivariant étale map, then $Y\in \mathcal {C}_S$ whenever $X\in \mathcal {C}_S$ .

The condition (P) on $\mathcal {C}_S$ guarantees that the fixed-point Nisnevich $cd$ -structure (defined later) is complete, while the condition (P $^\prime $ ) guarantees that the Nisnevich $cd$ -structure is complete. Some categories of interest in this paper – for example, $\mathrm {Sm}_S^G[\mathrm {co}(\mathcal {F})]$ for a family $\mathcal {F}$ – do not satisfy (P $^\prime $ ) but do satisfy the weaker property (P). We will see in Proposition 2.13 that when $\mathcal {C}_S$ satisfies (P $^\prime $ ), then the topology associated to the fixed-point Nisnevich $cd$ -structure coincides with the Nisnevich topology.

Proof Every $t_{\mathrm {fpNis}}$ -cover is a Nisnevich cover. We show that the reverse implication holds. It suffices to show that any Nisnevich square

admits a $t_{\mathrm {fpNis}}$ -refinement.

Write $\mathrm {fpr}(Y)\subseteq Y$ for the set of points where p is fixed-point reflecting. Since $Y\to X$ is unramified and $\mathrm {Stab}(X)\to X$ is universally closed, [Reference Rydh35, Proposition 3.5] applies to show that the set $\mathrm {fpr}(Y)\subseteq Y$ is an invariant open subset. We have that $Y\setminus V\subseteq \mathrm {fpr}(Y)$ . It follows that the outer square

(2.14)

is a fixed-point Nisnevich square (as is the top square). In particular, $\{j,pi\}$ is a $t_{\mathrm {fpNis}}$ -cover which refines $\{j, p\}$ .

Write $\mathcal {P}(\mathcal {C}_S)$ for the $\infty $ -category of presheaves of spaces on $\mathcal {C}_S$ . Note that $\mathcal {C}_S$ does not necessarily contain a terminal object; in particular, a terminal object $\mathcal {P}(\mathcal {C}_S)$ is in general not representable. We write $\mathrm {pt}\in \mathcal {P}(\mathcal {C}_S)$ for this terminal object. Of course, if $S\in \mathcal {C}_S$ , then $\mathrm {pt}$ is representable; it is the presheaf represented by S.

Temporarily, say that F is ‘fixed-point Nisnevich excisive’ if $F(\varnothing )\simeq \mathrm {pt}$ and $F(\mathcal {Q})$ is cartesian for any fixed-point Nisnevich square $\mathcal {Q}$ . There is no real need for this extra terminology, by the following proposition:

The property that a presheaf is Nisnevich excisive is defined by a small set of conditions. It follows from [Reference Lurie28, Section 5.5.4] that the inclusion $\mathcal {S}\mathrm {hv}_{\mathrm {Nis}}(\mathcal {C}_S)\subseteq \mathcal {P}(\mathcal {C}_S)$ is an accessible localization. Write $\mathrm {L}_{\mathrm {Nis}}$ for the resulting localization endofunctor on $\mathcal {P}(\mathcal {C}_S)$ . Moreover, this localization is left-exact in the sense of [Reference Lurie28, Section 6.2.2].

Similarly, the property that a presheaf is -homotopy invariant is defined by a small set of conditions, so that the inclusion is also an accessible localization. Write for the resulting localization endofunctor. It can be described explicitly by the formula , where

(2.18)

(for details, see, for instance, [Reference Antieau and Elmanto2, Theorem 4.25]). A map $f:F_1\to F_2$ in $\mathcal {S}\mathrm {hv}_{\mathrm {Nis}}(\mathcal {C}_S)$ is a Nisnevich equivalence provided that $\mathrm {L}_{\mathrm {Nis}}(f)$ is an equivalence, and an -equivalence provided that is an equivalence.

When $\mathcal {E}$ is a set of subgroups, closed under conjugacy, we use the notation

$$ \begin{align*} \mathcal{S}\mathrm{pc}^{G,\mathcal{E}}(S)&:= \mathcal{S}\mathrm{pc}\left(\mathrm{Sm}^{G}_S[\mathcal{E}] \right) \\ \mathcal{S}\mathrm{pc}_{\bullet}^{G,\mathcal{E}}(S)&:=\mathcal{S}\mathrm{pc}_{\bullet}\left(\mathrm{Sm}^{G}_S[\mathcal{E}] \right). \end{align*} $$

The inclusion $\mathcal {S}\mathrm {pc}(\mathcal {C}_S)\subseteq \mathcal {P}(\mathcal {C}_S)$ is an accessible localization, and we write

$$ \begin{align*} \mathrm{L}_{\mathrm{mot}}:\mathcal{P}(\mathcal{C}_S)\to \mathcal{P}(\mathcal{C}_S) \end{align*} $$

for the corresponding localization endofunctor. The motivic localization functor may be computed by the formula [Reference Morel and Voevodsky32, Lemma 3.2.6]

(2.21)

We will make use of the notion of the tensor product of presentable $\infty $ -categories as defined and studied in [Reference Lurie29, Section 4.8.1] (see especially [Reference Lurie29, Proposition 4.8.1.17]). The category $\mathcal {S}\mathrm {pc}(\mathcal {C}_S)$ is Cartesian monoidal (i.e., it is symmetric monoidal with respect to the Cartesian product). The symmetric monoidal product on $\mathcal {S}\mathrm {pc}(\mathcal {C}_S)$ extends to one on

$$ \begin{align*} \mathcal{S}\mathrm{pc}_\bullet(\mathcal{C}_S)\simeq\mathcal{S}\mathrm{pc}(\mathcal{C}_S)\otimes\mathcal{S}_\bullet, \end{align*} $$

where $\mathcal {S}_\bullet :=\mathcal {S}_{\mathrm {pt}/}$ denotes the $\infty $ -category of pointed spaces (see [Reference Gepner, Groth and Nikolaus15, Lemma 3.6] and [Reference Lurie29, Proposition 4.8.2.11]). We write $\wedge $ for the symmetric monoidal product on $\mathcal {S}\mathrm {pc}_\bullet (\mathcal {C}_S)$ . Sometimes we will need to use the symmetric monoidal product on $\mathcal {P}_{\bullet }(\mathcal {C}_S):=\mathcal {P}(\mathcal {C}_S)_{\mathrm {pt}/}$ , which we denote $\wedge ^{\mathcal {P}}$ to avoid confusion. The symmetric monoidal localization functor $\mathrm {L}_{\mathrm {mot}}:\mathcal {P}(\mathcal {C}_S) \to \mathcal {S}\mathrm {pc}(\mathcal {C}_S)$ induces a unique symmetric monoidal localization functor $\mathrm {L}_{\mathrm {mot}}:\mathcal {P}_{\bullet }(\mathcal {C}_S)\to \mathcal {S}\mathrm {pc}_{\bullet }(\mathcal {C}_S)$ . In particular, given $X,Y\in \mathcal {P}_{\bullet }(\mathcal {C}_S)$ , then

$$ \begin{align*} \mathrm{L}_{\mathrm{mot}}(X)\wedge \mathrm{L}_{\mathrm{mot}}(Y)\simeq \mathrm{L}_{\mathrm{mot}}\left(X\wedge^{\mathcal{P}} Y\right). \end{align*} $$

Given a functor $u:\mathcal {C} \to \mathcal {D}$ , the functor $u^*:\mathcal {P}(\mathcal {D}) \to \mathcal {P}(\mathcal {C})$ , defined by precomposition with u, preserves small limits and colimits, as these are computed objectwise in presheaf $\infty $ -categories [Reference Lurie28, Corollary 5.1.2.3], which are presentable $\infty $ -categories [Reference Lurie28, Theorem 5.5.1.1]. It follows from [Reference Lurie28, Corollary 5.5.2.9] that $u^*$ admits both a left adjoint $u_!$ and a right adjoint $u_*$ ,

A functor $u:\mathcal {C}\to \mathcal {D}$ between small $\infty $ -categories equipped with Grothendieck topologies is called topologically co-continuous Footnote ⁴ if for every $Y\in \mathcal {C}$ and every covering sieve R on $u(Y)$ , the sieve $u^*R\times _{u^*u_!Y}Y\hookrightarrow Y$ , consisting of arrows $Z\to Y$ such that $uZ\to uY$ factors through R, is a covering sieve on Y.

For the remainder of this section,

(2.24) $$ \begin{align} u:\mathcal{C}_S\subseteq\mathcal{D}_S \end{align} $$

is the inclusion between full subcategories of $\mathrm {Sch}_S^G$ , both containing $\varnothing $ and both satisfying properties (P) and (H). Main examples of interest to keep in mind are the inclusions $\mathrm {Sm}^G_S[\mathcal {E}]\subseteq \mathrm {Sm}^G_S[\mathcal {E}^{\prime }]$ .

The adjoint pairs $(u_!,u^*)$ and $(u^*,u_*)$ extend to adjoint pairs on -invariant Nisnevich sheaves. Overloading notation, we continue to write $u_!,u^*,u_*$ for the induced functors on categories of -invariant Nisnevich sheaves.

Proposition 2.28. Let $u:\mathcal {C}_S\subseteq \mathcal {D}_S$ be as in formula (2.24).

1. 1. The restriction functor $u^*:\mathcal {P}(\mathcal {D}_S)\to \mathcal {P}(\mathcal {C}_S)$ preserves -invariant Nisnevich sheaves.

2. 2. The induced functor $u^*:\mathcal {S}\mathrm {pc}(\mathcal {D}_S)\to \mathcal {S}\mathrm {pc}(\mathcal {C}_S)$ is symmetric monoidal and has a left adjoint $u_!$ and a right adjoint $u_*$ .

3. 3. Similarly, $u^*:\mathcal {S}\mathrm {pc}_\bullet (\mathcal {D}_S)\to \mathcal {S}\mathrm {pc}_\bullet (\mathcal {C}_S)$ is symmetric monoidal and has a left and a right adjoint, which we again denote respectively by $u_!$ and $u_*$ .

Moreover, these functors fit into commutative diagrams

Proposition 2.30. Let $u:\mathcal {C}_S\subseteq \mathcal {D}_S$ be as before. The functors

$$ \begin{align*} u_!, u_*:\mathcal{S}\mathrm{pc}(\mathcal{C}_S) & \to \mathcal{S}\mathrm{pc}(\mathcal{D}_S), \\ u_!, u_*:\mathcal{S}\mathrm{pc}_\bullet(\mathcal{C}_S) & \to \mathcal{S}\mathrm{pc}_\bullet(\mathcal{D}_S) \end{align*} $$

are all full and faithful. In particular, if $\mathcal {E}$ is a family of subgroups closed under conjugacy and $u:\mathrm {Sm}^G_S[\mathcal {E}]\subseteq \mathrm {Sm}^G_S$ is the inclusion, then $u_!, u_*:\mathcal {S}\mathrm {pc}^{G,\mathcal {E}}(S) \to \mathcal {S}\mathrm {pc}^G(S)$ and $u_!, u_*:\mathcal {S}\mathrm {pc}_\bullet ^{G,\mathcal {E}}(S) \to \mathcal {S}\mathrm {pc}_\bullet ^G(S)$ are full and faithful.

Let $\iota :H\hookrightarrow G$ be a group monomorphism. Let $\mathcal {C}_S^H\subseteq \mathrm {Sch}^H_S$ and $\mathcal {C}^G_S\subseteq \mathrm {Sch}^G_S$ be full subcategories satisfying (P) and (H), as before. Suppose further that the induction-restriction adjunction $\iota _!:\mathrm {Sch}^{H}_S\rightleftarrows \mathrm {Sch}^{G}_S:\iota ^*$ restricts to the adjunction

$$ \begin{align*} \iota_!:\mathcal{C}^{H}_S\rightleftarrows \mathcal{C}^{G}_S:\iota^*. \end{align*} $$

Let $f:T\to S$ be a map in $\mathrm {Sch}_B^G$ . Let $\mathcal {C}_T^G\subseteq \mathrm {Sch}^G_T$ and $\mathcal {C}^G_S\subseteq \mathrm {Sch}^G_S$ be full subcategories satisfying (P) and (H), as before. Suppose that the base change $f^{-1}:\mathrm {Sch}^G_S\to \mathrm {Sch}^G_T$ restricts to a functor $f^{-1}:C_S^G\to C^G_T$ . We write $f_*:\mathcal {P}\left (\mathcal {C}_T^G\right )\to \mathcal {P}\left (\mathcal {C}_S^G\right )$ for precomposition with $f^{-1}$ – that is, $f_*=(f^{-1})^*$ in the notation from before. The functor $f_*$ preserves Nisnevich excisive presheaves and homotopy-invariant presheaves, and thus restricts to a functor $f_*: \mathcal {S}\mathrm {pc}\left (\mathcal {C}_T^G\right )\to \mathcal {S}\mathrm {pc}\left (\mathcal {C}_S^G\right )$ . This functor preserves limits and thus has a left adjoint, which we write as $f^*:\mathcal {S}\mathrm {pc}\left (\mathcal {C}_S^G\right )\to \mathcal {S}\mathrm {pc}\left (\mathcal {C}_T^G\right )$ . Suppose further that the adjunction $u:\mathrm {Sch}^{G}_T\rightleftarrows \mathrm {Sch}^{G}_S:f^{-1}$ restricts to an adjunction

$$ \begin{align*} u:\mathcal{C}^{G}_T\rightleftarrows \mathcal{C}^{G}_S:f^{-1}. \end{align*} $$

Then $f^*:\mathcal {P}\left (\mathcal {C}_S^G\right )\to \mathcal {P}\left (\mathcal {C}_T^G\right )$ is precomposition with the forgetful functor u and it preserves Nisnevich excisive presheaves and homotopy-invariant presheaves. Thus $f^*$ restricts to a functor $f^*:\mathcal {S}\mathrm {pc}\left (\mathcal {C}_S^G\right )\to \mathcal {S}\mathrm {pc}\left (\mathcal {C}_T^G\right )$ , which is left adjoint to $f_*$ . Also in this case, $f^*$ preserves limits and so has a left adjoint $f_\#:\mathcal {S}\mathrm {pc}\left (\mathcal {C}_T^G\right )\to \mathcal {S}\mathrm {pc}\left (\mathcal {C}^G_S\right )$ .

The functors $f^*$ , $f_*$ preserve final objects and so induce an adjunction $f^*:\mathcal {S}\mathrm {pc}_\bullet \left (\mathcal {C}_S^G\right )\rightleftarrows \mathcal {S}\mathrm {pc}_\bullet \left (\mathcal {C}_T^G\right ):f_*$ . In the case when the forgetful functor restricts to $u:\mathcal {C}^{G}_T\to \mathcal {C}^{G}_S$ , then there is an adjunction $f_\#:\mathcal {S}\mathrm {pc}\left (\mathcal {C}_T^G\right )\rightleftarrows \mathcal {S}\mathrm {pc}\left (\mathcal {C}^G_S\right ):f^*$ .

Remark 2.32. The adjunctions of Lemma 2.31 admit the following alternate description in terms of change-of-base functors. There is an equivalence of categories

(2.33) $$ \begin{align} \mathrm{Sch}^G_{G\times_HS} \xrightarrow{\sim}\mathrm{Sch}^H_S \end{align} $$

induced by taking the fiber over $\{e\}\times S\subseteq G\times _HS$ . Write $\mathcal {C}_{G\times _HS}\subseteq \mathrm {Sch}^G_{G\times _HS}$ for the full subcategory corresponding to $\mathcal {C}^H_S$ under formula (2.33). The restriction functor $\iota ^*$ corresponds to pullback along $f:G\times _HS\to S$ . Moreover, the equivalence (2.33) induces an equivalence of motivic spaces

$$ \begin{align*} \mathcal{S}\mathrm{pc}\left(\mathcal{C}_{G\times_HS}\right)\simeq \mathcal{S}\mathrm{pc}\left(\mathcal{C}_S^H\right), \end{align*} $$

and under this equivalence the functors $i_!, i^*, i_*$ are respectively identified with the functors $f_\#, f^*, f_*$ .

2.4 Motivic G-spectra

We recall the construction of categories of motivic G-spectra from [Reference Hoyois24] and some variants.

Let $\mathcal {C}^{\otimes }$ be a presentably symmetric monoidal $\infty $ -category and X a set of objects in $\mathcal {C}$ . If $I = \{x_1,\dotsc , x_n\}$ is a finite subset of X, write $\bigotimes I = x_1\otimes \dotsb \otimes x_n$ . Write

$$ \begin{align*} \mathcal{C}\left[X^{-1}\right] := \operatorname*{{\mathrm{colim}}}_{\substack{I\subseteq X \\ I\text{ finite }}}\mathcal{C}\left[\left(\bigotimes I\right)^{-1}\right], \end{align*} $$

where $\mathcal {C}\left [x^{-1}\right ]$ denotes the symmetric monoidal inversion of an object $x\in \mathcal {C}$ in a presentable symmetric monoidal $\infty $ -category $\mathcal {C}$ [Reference Robalo34, §2.1].

Alternatively, one may consider the stabilization in $\mathrm {Mod}_{\mathcal {C}}$ , the $\infty $ -category of $\mathcal {C}$ -modules in $\mathcal {P}{\mathrm {r}}^{\mathrm {L},\otimes }$ . Recall that if $\mathcal {M}\in \mathrm {Mod}_{\mathcal {C}}$ and $x\in \mathcal {C}$ , then $\mathrm {Stab}_x(\mathcal {M})$ is the colimit, in $\mathrm {Mod}_{\mathcal {C}}$ , of the sequence

$$ \begin{align*} \mathcal{M}\xrightarrow {-\otimes x} \mathcal{M} \xrightarrow{-\otimes x} \mathcal{M} \to\dotsb. \end{align*} $$

More generally, for a set of objects X in $\mathcal {C}$ , define

$$ \begin{align*} \mathrm{Stab}_X(\mathcal{M}) := \operatorname*{{\mathrm{colim}}}_{\substack{I\subseteq X \\ I\text{ finite }}}\mathrm{Stab}_{\otimes I}(\mathcal{M}). \end{align*} $$

An object $x\in \mathcal {C}$ is n-symmetric if the cyclic permutation on $x^{\otimes n}$ is homotopic to the identity. If each $x\in X$ is n-symmetric for some $n\geq 2$ , then the canonical map of $\mathcal {C}\left [X^{-1}\right ]$ -modules is an equivalence

$$ \begin{align*} \mathcal{M}\otimes_{\mathcal{C}}\mathcal{C}\left[X^{-1}\right]\xrightarrow{\sim}\mathrm{Stab}_{X}(\mathcal{M}) \end{align*} $$

(see [Reference Hoyois24, Section 6.1] and [Reference Robalo34, Corollary 2.22]).

Let $\mathcal {E}$ be a finite-rank locally free G-module on S. Write $T^{\mathcal {E}}\in \mathcal {S}\mathrm {pc}_\bullet (S)$ for the associated motivic sphere, defined as the Thom space

where

is the zero section. We will also write $\Sigma ^{\mathcal {E}}$ for the associated endofunctor

$$ \begin{align*} \Sigma^{\mathcal{E}}\simeq T^{\mathcal{E}}\wedge -. \end{align*} $$

We will also be interested in stabilizing the categories $\mathcal {S}\mathrm {pc}^{G,\mathcal {F}}_\bullet (S)$ , for a family $\mathcal {F}$ . Observe that $\mathcal {S}\mathrm {pc}^{G,\mathcal {F}}_\bullet (S)$ is an $\mathcal {S}\mathrm {pc}^{G}_\bullet (S)$ -module and $u_!:\mathcal {S}\mathrm {pc}_\bullet ^{G,\mathcal {F}}(S)\to \mathcal {S}\mathrm {pc}_\bullet ^G(S)$ is a map of $\mathcal {S}\mathrm {pc}^{G}_\bullet (S)$ -modules, since if $X\in \mathrm {Sm}^G_S[\mathcal {F}]$ and $Y\in \mathrm {Sm}^G_S$ , then $X\times Y\in \mathrm {Sm}^G_S[\mathcal {F}]$ . In particular, even though spheres $T^{\mathcal {E}}\in \mathcal {S}\mathrm {pc}^{G}_\bullet (S)$ are generally not objects of $\mathcal {S}\mathrm {pc}^{G,\mathcal {F}}_\bullet (S)$ , they still determine endofunctors $\Sigma ^{\mathcal {E}}:\mathcal {S}\mathrm {pc}^{G,\mathcal {F}}_\bullet (S)\to \mathcal {S}\mathrm {pc}^{G,\mathcal {F}}_\bullet (S)$ .

Write $\mathrm {Sph}_B^G:=\left \{T^{\mathcal {E}}\mid \mathcal {E}\in \mathrm {Rep}_B^G \right \}$ , where $\mathrm {Rep}_B^G$ is the set of finite-rank G-vector bundles over B.

Write

$$ \begin{align*} \Sigma^{\infty}_{\mathcal{T}}:\mathcal{S}\mathrm{pc}^{G,\mathcal{F}}_{\bullet}(S) \to \mathcal{S}\mathrm{pt}^{G,\mathcal{F}}_{\mathcal{T}}(S) \end{align*} $$

for the stabilization functor. In the case when $\mathcal {T} = \mathrm {Sph}_B^G$ , we simply write $\Sigma ^{\infty }$ . When no confusion should arise, given $X\in \mathcal {S}\mathrm {pc}^{G,\mathcal {F}}_{\bullet }(S)$ we will write again X for its image in $\mathcal {S}\mathrm {pt}^{G,\mathcal {F}}_{\mathcal {T}}(S)$ instead of $\Sigma ^{\infty }_{\mathcal {T}}X$ . Given $T^{\mathcal {V}}\in \mathcal {T}$ , $p:S\to B$ , $E\in \mathcal {S}\mathrm {pt}^{G,\mathcal {F}}_{\mathcal {T}}(S)$ , and , we typically write $\Sigma ^{k\mathcal {V}}E$ rather than $\Sigma ^{kp^*\mathcal {V}}E$ when no confusion should arise.

Proof That $\mathcal {S}\mathrm {pt}^{G,\mathcal {F}}_{\mathcal {T}}(S)$ is stable is a consequence of the fact that $\mathcal {T}$ is stabilizing and that

. It is symmetric monoidal by construction. The arguments for the remaining points of (1)–(3) are similar to [Reference Hoyois24, Section 6]. Since $\mathcal {S}\mathrm {pt}^{G,\mathcal {F}}_{\mathcal {T}}(S)$ is stable, in order to establish (4) it suffices to show that $E\simeq 0$ whenever $p^*E\simeq 0$ for all $p:X\to S$ in $\mathrm {Sm}^G_S[\mathcal {F}]$ with X affine. If $T^{\mathcal {V}}\in \mathcal {T}$ , then since $\mathcal {V}\in \mathrm {Rep}_B^G$ , we have

. Now, if $p^*E\simeq 0$ , we have

Given E such that $p^*E\simeq 0$ for all $p:X\to S$ in $\mathrm {Sm}^G_S[\mathcal {F}]$ with X affine, let $\mathcal {C}_E\subseteq \mathcal {S}\mathrm {pt}^{G,\mathcal {F}}_{\mathcal {T}}(S)$ be the full subcategory of whose objects are W such that $\underline {\smash {\mathrm {Map}}}_{\mathcal {S}\mathrm {pt}^{G,\mathcal {F}}_{\mathcal {T}}(S)}(W, E)\simeq 0$ – that is, the E-acyclic objects. Then $\mathcal {C}_E$ is closed under colimits and it contains $\Sigma ^{-k\mathcal {V}}\Sigma ^{\infty }_{\mathcal {T}}X_+$ for any $k\geq 0$ and $T^{\mathcal {V}}\in \mathcal {T}$ . Thus by (3), $\mathcal {C}_E = \mathcal {S}\mathrm {pt}^{G,\mathcal {F}}_{\mathcal {T}}(S)$ , which implies that $E\simeq 0$ as required.

Remark 2.37. Over an affine base, every representation is the quotient of a finite sum of copies of the regular representation $\rho _G$ . This implies that for any S,

$$ \begin{align*} \mathcal{S}\mathrm{pt}^G(S) \simeq \mathcal{S}\mathrm{pt}^G_{T^{\rho_G}}(S). \end{align*} $$

Let $N\trianglelefteq G$ be a normal subgroup and $\pi :G\to G/N$ the quotient homomorphism. This induces a function $\pi ^*: \mathrm {Rep}_B^{G/N} \to \mathrm {Rep}_B^{G}$ , and we write

$$ \begin{align*} N\textrm{-}\mathrm{triv}= \left\{T^{\mathcal{E}} \mid \mathcal{E}\in \pi^*\left( \mathrm{Rep}_B^{G/N}\right) \right\}\subseteq \mathrm{Sph}_B^G \end{align*} $$

for the associated set of ‘N-trivial G-spheres’. This stabilizing set of spheres plays an important role in later sections.

Lemma 2.38. Let $\mathcal {F}$ be a family. The adjunction $u_!\colon\ \mathcal {S}\mathrm {pc}_{\bullet }^{G,\mathcal {F}}(S)\rightleftarrows \mathcal {S}\mathrm {pc}_{\bullet }^G(S):u^*$ of $\mathcal {S}\mathrm {pc}^{G}_{\bullet }(S)$ -modules induces an adjoint pair

$$ \begin{align*} u_!:\mathcal{S}\mathrm{pt}^{G,\mathcal{F}}_{\mathcal{T}}(S)\rightleftarrows \mathcal{S}\mathrm{pt}_{\mathcal{T}}^G(S):u^*. \end{align*} $$

Moreover, $u^*$ is symmetric monoidal and

$$ \begin{align*} u_!:\mathcal{S}\mathrm{pt}^{G,\mathcal{F}}_{\mathcal{T}}(S)\hookrightarrow \mathcal{S}\mathrm{pt}^G_{\mathcal{T}}(S) \end{align*} $$

is full and faithful with essential image the localizing tensor ideal generated by $\Sigma ^{-nV}X_+$ , where $T^V\in p^*\mathcal {T}$ and $X\in \mathrm {Sm}^{G}_S[\mathcal {F}]$ .

Let $\mathcal {C}^{\otimes }$ be a presentably symmetric monoidal $\infty $ -category and $E\in \mathcal {C}$ an idempotent object. Recall that this means there is a map such that and are equivalences [Reference Lurie29, Definition 4.8.2.1]. Tensoring with E is a localization functor $\mathrm {L}=E\otimes -:\mathcal {M}\to \mathcal {M}$ on any $\mathcal {M}\in \mathrm {Mod}_{\mathcal {C}}$ [Reference Lurie29, Proposition 4.8.2.4].

Lemma 2.39. With notation as before,

$$ \begin{align*} \mathrm{L}\mathcal{M} \simeq \mathrm{L}\mathcal{C}\otimes_{\mathcal{C}} \mathcal{M}. \end{align*} $$

We record the following result which we will use a few times. A similar statement can be found in [Reference Bachmann5, Lemma 26].

Lemma 2.40. Let $\mathcal {C}^{\otimes }$ be a presentably symmetric monoidal $\infty $ -category and X a set of objects. Suppose that $E\in \mathcal {C}$ is an idempotent object. Write $\mathrm {L}=E\otimes -$ for the associated symmetric monoidal localization endofunctor. Then there is an equivalence

$$ \begin{align*} \mathrm{L}\left(\mathcal{C}\left[X^{-1}\right]\right)\simeq (\mathrm{L} \mathcal{C})\left[X^{-1}\right] \end{align*} $$

in $\mathrm {CAlg}\left (\mathcal {P}{\mathrm {r}}^{\mathrm {L},\otimes }\right )$ .

Next we record the motivic version of the Wirthmüller isomorphism, which is a special case of the ambidexterity equivalence proved in [Reference Hoyois24]. If $H\leq G$ is a subgroup and $\iota $ is the inclusion, we sometimes write

$$ \begin{align*} G_+\ltimes_{H} X:=\iota_!X. \end{align*} $$

Proposition 2.41 Wirthmüller isomorphism [Reference Hoyois24]

Let $\iota :H\hookrightarrow G$ be a group monomorphism. Let $\mathcal {F}$ , $\mathcal {F}^{\prime }$ be families of subgroups of, respectively, H and G such that the induction-restriction adjunction restricts to $\iota _!:\mathrm {Sm}^H_S[\mathcal {F}]\rightleftarrows \mathrm {Sm}^G_S[\mathcal {F}^{\prime }]:\iota ^{-1}$ . Then there is an induced adjunction

$$ \begin{align*} \iota_!:\mathcal{S}\mathrm{pt}^{H,\mathcal{F}}(S)\rightleftarrows \mathcal{S}\mathrm{pt}^{G,\mathcal{F}^{\prime}}(S):\iota^*, \end{align*} $$

such that $\iota _!(\Sigma ^\infty X)\simeq \Sigma ^\infty (\iota _!X)$ and $\iota ^*(\Sigma ^\infty X)\simeq \Sigma ^\infty \left (\iota ^{-1}X\right )$ . Moreover, $\iota ^*$ admits a right adjoint $\iota _*$ and there is an equivalence

$$ \begin{align*} \iota_!\xrightarrow{\sim} i_*. \end{align*} $$

2.5 Functoriality

We will make use of the functoriality of the $\infty $ -categories of equivariant motivic spaces and spectra with respect to the group G, the family of subgroups $\mathcal {F}$ , and the base scheme S. To establish this, it will be convenient to introduce the following notation:

The inclusion $i:\phi ^{-1}\mathcal {F}\subset \mathcal {F}^{\prime }$ is unique if it exists. When no confusion should arise, we write $(\phi , f)$ instead of $(\phi , i, f)$ for a morphism in .

We identify with the full subcategory of whose objects are triples of the form $(G,\mathcal {F}_{\mathrm {all}}, S)$ . The inclusion is left adjoint to the forgetful functor.

In this subsection we extend constructions in the previous sections to functors

whose respective values on $(G,\mathcal {F},S)$ are the symmetric monoidal $\infty $ -categories $\mathcal {S}\mathrm {pc}^{G,\mathcal {F}}(S)$ , $\mathcal {S}\mathrm {pc}^{G,\mathcal {F}}_{\bullet }(S)$ , and $\mathcal {S}\mathrm {pt}^{G,\mathcal {F}}(S)$ , and on morphisms, $(\phi , i, f)^*\simeq i_!\phi ^*f^*$ .

Corollary 2.44. The assignment $(G,\mathcal {F},S)\mapsto \mathrm {Sm}_S^G[\mathcal {F}]$ , $(\phi ,i,f)\mapsto i\phi ^{-1}f^{-1}$ , extends to a functor

the category of commutative algebra objects of $\mathrm {Cat}$ with respect to the Cartesian symmetric monoidal structure (equivalently, the category of symmetric monoidal categories).

Composing with the symmetric monoidal presheaves functor $\mathrm {Cat}\to \mathrm {Cat}_\infty \to \mathcal {P}{\mathrm {r}}^{\mathrm {L}}$ , we obtain a functor

which sends the object $(G,\mathcal {F}, S)$ to $\mathcal {P}\left (\mathrm {Sm}^G_S[\mathcal {F}]\right )$ and the morphism $(\phi , i, f)$ to $i_!\phi ^*f^*$ .

To obtain the necessary functoriality of equivariant motivic spaces and spectra, we follow the techniques of [Reference Bachmann and Hoyois7, Section 6.1]. Recall from there that we have a commutative diagram of $\infty $ -categories

in which all squares are Cartesian. Here $\mathrm {Pos}$ denotes the $\infty $ -category of (not necessarily small) posets, the lower right-hand horizontal arrow sends an $\infty $ -category to the poset of subsets of the set of equivalence classes of objects, and $\mathcal {E}\to \mathrm {Pos}$ is the universal co-Cartesian fibration, restricted to posets. The $\infty $ -categories $\mathcal {O}\mathrm {Cat}_\infty $ and $\mathcal {M}\mathrm {Cat}_\infty $ are, respectively, the $\infty $ -categories of $\infty $ -categories equipped with a collection of equivalence classes of objects respectively a collection of equivalence classes of arrows.

Consider the subfunctor of the composition

whose value on the object $(G,\mathcal {F},S)$ is the full subcategory

$$ \begin{align*} W_{\left(G,\mathcal{F},S\right)}\subset\operatorname{\mathrm{Fun}}\left(\Delta^1,\mathcal{P}\left(\mathrm{Sm}_S^G[\mathcal{F}]\right)\right) \end{align*} $$

consisting of the motivic equivalences. Since motivic equivalences are stable under the smash product and are preserved by the functors $f^*$ , $\phi ^*$ , and $i_!$ , the assignments

$$ \begin{align*} (G,\mathcal{F},S)\mapsto\left(\mathcal{P}\left(\mathrm{Sm}_S^G[\mathcal{F}]\right),W_{\left(G,\mathcal{F},S\right)}\right),\left(\mathcal{P}_{\bullet}\left(\mathrm{Sm}_S^G[\mathcal{F}]\right), W_{\left(G,\mathcal{F},S\right)}\right) \end{align*} $$

induce functors

The images of $\left (\mathcal {P}\left (\mathrm {Sm}_S^G[\mathcal {F}]\right ), W_{\left (G,\mathcal {F},S\right )}\right )$ and $\left (\mathcal {P}_{\bullet }\left (\mathrm {Sm}_S^G[\mathcal {F}]\right ), W_{\left (G,\mathcal {F},S\right )}\right )$ under the partial left adjoint to

$$ \begin{align*} \mathrm{CAlg}(\mathrm{Cat}_\infty)\to \mathrm{CAlg}(\mathcal{M}\mathrm{Cat}_\infty) \end{align*} $$

are, respectively, $\mathcal {S}\mathrm {pc}^{G,\mathcal {F}}(S)$ and $\mathcal {S}\mathrm {pc}^{G,\mathcal {F}}_{\bullet }(S)$ .

Write $\mathrm {Sph}^G_S$ for the set of spheres $\left \{T^{\mathcal {E}}\right \}$ , where $\mathcal {E}$ is an equivariant vector bundle over S. The assignment $(G,\mathcal {F},S)\mapsto \mathrm {Sph}^G_S$ is a presheaf of sets on

, which we write as $\mathrm {Sph}$ . Let

be a subpresheaf of $\mathrm {Sph}$ , which is closed under the smash product and takes values in stabilizing sets of spheres – that is, there is some $T^{\mathcal {E}}\in \mathcal {T}_{\left (G,\mathcal {F},S\right )}$ such that $T^{\mathcal {E}}\simeq T\wedge T^{\mathcal {E}'}$ . We obtain a functor

which on objects is the assignment $(G,\mathcal {F},S)\mapsto \left (\mathcal {S}\mathrm {pc}_{\bullet }^{G,\mathcal {F}}(S),\mathcal {T}_{\left (G,\mathcal {F},S\right )}\right )$ . By the following lemma, we obtain $\mathcal {S}\mathrm {pt}^{G,\mathcal {F}}_{\mathcal {T}_{\left (G,\mathcal {F},S\right )}}(S)$ as the image of $\left (\mathcal {S}\mathrm {pc}_{\bullet }^{G,\mathcal {F}}(S),\mathcal {T}_{\left (G,\mathcal {F},S\right )}\right )$ under the partial left adjoint of $\mathrm {CAlg}(\mathrm {Cat}_{\infty })\to \mathrm {CAlg}(\mathcal {O}\mathrm {Cat}_{\infty })$ , $\mathcal {C}\mapsto (\mathcal {C},\pi _0\mathrm {Pic}(\mathcal {C}))$ .

Lemma 2.46. Let $\left (\mathcal {C}^\otimes ,U\right )$ be an object of $\mathrm {CAlg}(\mathcal {O}\mathrm {Cat}_\infty )$ such that $\mathcal {C}$ is presentable symmetric monoidal and U is small. Then the partial adjoint of

$$ \begin{align*} \mathrm{CAlg}(\mathrm{Cat}_{\infty})\to \mathrm{CAlg}(\mathcal{O}\mathrm{Cat}_{\infty}), \quad\mathcal{C}\mapsto (\mathcal{C},\pi_0\mathrm{Pic}(\mathcal{C})), \end{align*} $$

is defined at $\left (\mathcal {C}^\otimes ,U\right )$ .

3.1 Universal motivic $\mathcal {F}$ -spaces

Let $\mathcal {F}$ be a family of subgroups. In classical equivariant homotopy theory, there is a G-space $\mathrm {E}_{\bullet } \mathcal {F}$ characterized by the property that a G-space X admits a unique map $X\to \mathrm {E}_{\bullet } \mathcal {F}$ if all of the stabilizers of X are in $\mathcal {F}$ , and no maps from X to $\mathrm {E}_{\bullet } \mathcal {F}$ otherwise. The G-space $\mathrm {E}_{\bullet } \mathcal {F}$ formally exists as a presheaf on $\mathrm {Sm}^G_B$ and hence as a motivic G-space over B, but it does not have the correct universality property.

Example 3.1. Let $G\neq \{e\}$ , $B=\operatorname {{\mathrm {Spec}}}(k)$ a field, and let $L/k$ be a finite Galois extension such that $G\subseteq \mathrm {Gal}(L/k)$ , and consider $\operatorname {{\mathrm {Spec}}}(L)$ as a smooth G-scheme over k via the Galois action. Then $\operatorname {{\mathrm {Spec}}}(L)^{H} = \varnothing $ for all $e\neq H\subseteq G$ – that is, it has a free G-action. However, we claim that

$$ \begin{align*} \operatorname{\mathrm{Map}}_{\mathcal{S}\mathrm{pc}^G(k)}(\operatorname{{\mathrm{Spec}}}(L), \mathrm{E}_{\bullet} G) = \varnothing. \end{align*} $$

Indeed,

$$ \begin{align*} (\mathrm{E}_{\bullet} G)_{0}(\operatorname{{\mathrm{Spec}}}(L))= \operatorname{\mathrm{Hom}}_{\mathrm{Sm}_k^G}\left(\operatorname{{\mathrm{Spec}}}(L), \coprod_{G}\operatorname{{\mathrm{Spec}}}(k)\right) = \varnothing, \end{align*} $$

and so the claim follows, since $ (\mathrm {E}_{\bullet } G)_{0}(\operatorname {{\mathrm {Spec}}}(L))$ surjects onto $\pi _{0}(\operatorname {\mathrm {Map}}_{\mathcal {S}\mathrm {pc}^G(k)}(\operatorname {{\mathrm {Spec}}}(L), \mathrm {E}_{\bullet } G))$ (see, e.g., [Reference Morel and Voevodsky32, Corollary 2.3.22, Remark 3.2.5]).

Definition 3.2. Let $\mathcal {F}$ be a family of subgroups in G. The universal motivic $\mathcal {F}$ -space over S is the object $\mathbf {E}\mathcal {F}_S\in \mathcal {P}\left (\mathrm {Sm}^G_S\right )$ whose value on $X\in \mathrm {Sm}^G_S$ is

$$ \begin{align*} \mathbf{E}\mathcal{F}_S(X) = \begin{cases} \mathrm{pt}, & X\in \mathrm{Sm}^G_S[\mathcal{F}], \\ \varnothing, & \text{else}. \end{cases} \end{align*} $$

When the base S is understood, we simply write $\mathbf {E} \mathcal {F}$ .

Proof We need to check that $\mathbf {E}\mathcal {F}$ is Nisnevich excisive and

-homotopy invariant. From the definition we have that $\mathbf {E}\mathcal {F}(\varnothing )=\mathrm {pt}$ . Given a Nisnevich square (2.11), the possible values of

are the squares

which are all Cartesian squares. It follows that $\mathbf {E}\mathcal {F}$ is Nisnevich excisive. Also, $\mathbf {E}\mathcal {F}$ is

-homotopy invariant, since

.

Totaro [Reference Totaro39] and Morel and Voevodsky [Reference Morel and Voevodsky32, Section 4.2] constructed a geometric model for the classifying space of an algebraic group. This is generalized by Hoyois in [Reference Hoyois23, Section 2] to construct a geometric model for certain equivariant classifying spaces. Similar considerations lead to geometric models for the spaces $\mathbf {E} \mathcal {F}$ .

Example 3.5. Write $\rho =\rho _G$ . Let

be the open invariant subscheme

The inclusions

induce maps $U_{n}\to U_{n+1}$ .

Set $f:S\to B$ in $\mathrm {Sch}_B^G$ . Then is a system of approximations to $\mathbf {E}\mathcal {F}_S$ . Conditions (1)–(3) are clear. To check the last condition, we may assume that B and S are affine (since S is equivariant locally affine). Let $X\in \mathrm {Sm}_S^G[\mathcal {F}]$ be affine. Then for n sufficiently large, there is an equivariant closed immersion of B-schemes . For any $H\not \in \mathcal {F}$ , we have , which means that factors to give a map $X\to U_n$ over B. This defines the desired section $X\to f^*U_n$ .

If $(U_i)_{i\in I}$ is a system of approximations to $\mathbf {E}\mathcal {F}_S$ , define

$$ \begin{align*} U_{\infty} := \operatorname*{{\mathrm{colim}}}_IU_i \in \mathcal{S}\mathrm{pc}^G(S). \end{align*} $$

Proposition 3.7. Let $\mathcal {F}$ be a family of subgroups in G and $(U_i)_{i\in I}$ be a system of approximations to $\mathbf {E}\mathcal {F}_S$ . Then there is an equivalence

$$ \begin{align*} U_{\infty} \xrightarrow{\simeq} \mathbf{E}\mathcal{F}_S \end{align*} $$

in $\mathcal {S}\mathrm {pc}^{G}(S)$ .

Proposition 3.8. Let $\mathcal {F}$ be a family of subgroups and let $f:S^{\prime }\to S$ be a morphism in $\mathrm {Sch}_B^G$ . Then

$$ \begin{align*} f^*(\mathbf{E}\mathcal{F}_{S})\simeq \mathbf{E}\mathcal{F}_{S^{\prime}}. \end{align*} $$

Write $i:\mathrm {Sm}^{G}_S[\mathcal {F}] \subseteq \mathrm {Sm}^G_S$ for the inclusion of categories.

Proposition 3.9. There is a canonical equivalence of endofunctors

$$ \begin{align*} i_!i^*\simeq \mathbf{E}\mathcal{F}\times -:\mathcal{S}\mathrm{pc}^G(S)\to \mathcal{S}\mathrm{pc}^G(S) \end{align*} $$

and

$$ \begin{align*} i_!i^*\simeq \mathbf{E}\mathcal{F}_+\wedge -:\mathcal{S}\mathrm{pc}_{\bullet}^G(S) \to \mathcal{S}\mathrm{pc}_{\bullet}^G(S). \end{align*} $$

Proof We treat the unbased case; the based case then follows. Set $X\in \mathcal {S}\mathrm {pc}^G(S)$ . We have $(i^*X)(W) \simeq X(W)$ for $W\in \mathrm {Sm}^G_S[\mathcal {F}]$ . In particular, the projection $\mathbf {E}\mathcal {F}\to \mathrm {pt}$ induces the equivalences $i^*(\mathbf {E}\mathcal {F}\times X) \simeq i^*(X)$ and thus

$$ \begin{align*} i_!i^*(\mathbf{E}\mathcal{F}\times -) \xrightarrow{\sim} i_!i^*(-). \end{align*} $$

Now $\mathbf {E}\mathcal {F}$ is equivalent to $\operatorname *{{\mathrm {colim}}}_i U_i$ , where $U_i\in \mathrm {Sm}^{G}_S[\mathcal {F}] $ . If $W\in \mathrm {Sm}^G_S$ , then each $U_i\times W$ is in $\mathrm {Sm}^{G}_S[\mathcal {F}] $ . It follows that if X is any object of $\mathcal {S}\mathrm {pc}^G(S)$ , then, writing X as a colimit of objects of $\mathrm {Sm}_S^G$ , we see that $\mathbf {E}\mathcal {F}\times X \simeq i_!(E)$ for some $E\in \mathcal {S}\mathrm {pc}^{G,\mathcal {F}}(S)$ . In particular, since $\eta :\mathrm {id}\simeq i^*i_! $ is an equivalence by Proposition 2.30, we have $i_!\eta :i_!(E)\simeq i_!i^*i_!(E)$ . From the triangle identity for the unit and counit, we have that $\epsilon i_!:i_!i^*(i_!(E)) \simeq i_!(E)$ is also an equivalence. It follows that we have equivalences

$$ \begin{align*} \mathbf{E}\mathcal{F}\times - \simeq i_!i^*(\mathbf{E}\mathcal{F}\times -) \simeq i_!i^*.\\[-36pt] \end{align*} $$

Recall that $i_!:\mathcal {S}\mathrm {pt}^{G,\mathcal {F}}_{\mathcal {T}}(S)\to \mathcal {S}\mathrm {pt}^G_{\mathcal {T}}(S)$ is full and faithful with essential image the localizing tensor ideal generated by $T^{-\mathcal {E}}\otimes X_+$ , where $X\in \mathrm {Sm}_S^{G}[\mathcal {F}] $ and $T^{\mathcal {E}}\in \mathcal {T}$ (see Lemma 2.38).

Proposition 3.11. There is an equivalence of colocalization endofunctors

$$ \begin{align*} \mathbf{E}\mathcal{F}_+\otimes -\simeq i_!i^*:\mathcal{S}\mathrm{pt}^G_{\mathcal{T}}(S)\to \mathcal{S}\mathrm{pt}^G_{\mathcal{T}}(S). \end{align*} $$

In particular, there are natural equivalences

and

$$ \begin{align*} i_*i^*\simeq F_{\mathcal{S}\mathrm{pt}^G_{\mathcal{T}}(S)}\left(\mathbf{E}\mathcal{F}_+,\mathrm{id}\right). \end{align*} $$

Proof The first statements follow from Proposition 3.9. The last statement follows from the natural equivalences

$$ \begin{align*} \operatorname{\mathrm{Map}}_{\mathcal{S}\mathrm{pt}^G_{\mathcal{T}}(S)}(X, i_*i^*Y)\simeq \operatorname{\mathrm{Map}}_{\mathcal{S}\mathrm{pt}^G_{\mathcal{T}}(S)}(i_!i^*X, Y)\simeq \operatorname{\mathrm{Map}}_{\mathcal{S}\mathrm{pt}^G_{\mathcal{T}}(S)}\left(\mathbf{E}\mathcal{F}_+\otimes X, Y\right). \end{align*} $$

3.2 Filtrations by adjacent families

We recall the definition of adjacent families.

Of course, if $N=G$ then N-adjacent families are exactly adjacent families. Since G is finite, one can always find a filtration

$$ \begin{align*} \varnothing = \mathcal{F}_{-1} \subseteq \mathcal{F}_{0}\subseteq \mathcal{F}_{1}\subseteq \dotsb \subseteq \mathcal{F}_{n}=\mathcal{F}_{\mathrm{all}}, \end{align*} $$

such that each pair $\mathcal {F}_i\subseteq \mathcal {F}_{i+1}$ is N-adjacent. For example, one can be produced as follows. Define the sequence of families

$$ \begin{align*} \{e\}=\mathrm{Fil}_0^G\subset \mathrm{Fil}_1^G\subset \dotsb \subset \mathrm{Fil}_i^G\subset \dotsb \subset \mathrm{Fil}_{N}^G=\mathcal{F}_{{\mathrm{all}}} \end{align*} $$

by setting $\mathrm {Fil}_0 = \{e\}$ and inductively defining $\mathrm {Fil}_i^G$ by

$$ \begin{align*} \mathrm{Fil}_{i}^G := \left\{ H\leq G \mid \text{each proper subgroup } K<H \text{ is in }\mathrm{Fil}_{i-1}^G\right\}. \end{align*} $$

(Since G is finite, this sequence terminates at a finite stage.) Each $\mathrm {Fil}_i^G$ is a family. More generally, define

$$ \begin{align*} \mathrm{Fil}_{i}^{N \trianglelefteq G}:= \left\{ K\leq G \mid K\cap N\in \mathrm{Fil}_{i}^{N} \right \}. \end{align*} $$

The families just defined are not adjacent, but $\mathrm {Fil}_{i}^{N \trianglelefteq G}\setminus \mathrm {Fil}_{i-1}^{N \trianglelefteq G}$ is a finite union of conjugacy classes. Let $\{(H_i)\}$ be the set of these conjugacy classes. Then the families

$$ \begin{align*} \mathrm{Fil}_{i-1}^{N \trianglelefteq G}\subseteq \mathrm{Fil}_{i-1}^{N \trianglelefteq G}\cup \{(H_1)\}\subseteq \mathrm{Fil}_{i-1}^{N \trianglelefteq G}\cup\{(H_1), (H_2)\} \subseteq \dotsb \subseteq \mathrm{Fil}_{i}^{N \trianglelefteq G} \end{align*} $$

are all N-adjacent.

In any case, a filtration $\varnothing \subseteq \mathcal {F}_{0}\subseteq \mathcal {F}_{1}\subseteq \dotsb \subseteq \mathcal {F}_{\mathrm {all}}$ gives rise to a filtration

(3.13) $$ \begin{align} \ast \to \mathbf{E}\mathcal{F}_{0+}\wedge X\to \mathbf{E}\mathcal{F}_{1+}\wedge X\to \dotsb \to \mathbf{E}\mathcal{F}_{n+}\wedge X\simeq X \end{align} $$

of an object $X\in \mathcal {C}$ whenever $\mathcal {C}$ is an $\mathcal {S}\mathrm {pc}^G_{\bullet }(S)$ -module, for example, $\mathcal {S}\mathrm {pt}^G(S)$ .

To use this filtration, we need to analyze the filtration quotients, which we do in Proposition 3.27 and Proposition 4.12.

3.3 Universal spaces for pairs

Definition 3.14. Let $\mathcal {F}\subseteq \mathcal {F}^{\prime }$ be a subfamily. Define the based motivic G-space $\mathbf {E}(\mathcal {F}^{\prime }, \mathcal {F})$ so that it sits in the cofiber sequence

$$ \begin{align*} \mathbf{E}\mathcal{F}_{+}\to \mathbf{E}\mathcal{F}^{\prime}_{+} \to \mathbf{E}(\mathcal{F}^{\prime}, \mathcal{F}). \end{align*} $$

If $\mathcal {F}^{\prime }=\mathcal {F}_{\mathrm {all}}$ , define $\widetilde {\mathbf {E}}\mathcal {F} := \mathbf {E}(\mathcal {F}_{\mathrm {all}},\mathcal {F})$ .

Note that $\mathbf {E}(\mathcal {F}^{\prime },\varnothing )\simeq \mathbf {E}(\mathcal {F}^{\prime })_+$ . At the other extreme, since $\mathbf {E}\mathcal {F}_{\mathrm {all}} \simeq \mathrm {pt}$ , the space $\widetilde {\mathbf {E}}\mathcal {F}$ sits in the cofiber sequence

(3.15) $$ \begin{align} \mathbf{E}\mathcal{F}_{+} \to S^{0} \to \widetilde{\mathbf{E} }\mathcal{F} \end{align} $$

in $\mathcal {S}\mathrm {pc}_{\bullet }^G(S)$ .

Proof This follows from the commutative diagram

induced by inclusions of families, in which the rows are each cofiber sequence and the left and middle vertical arrows are equivalences.

Corollary 3.17. Set $X\in \mathcal {S}\mathrm {pc}^G_{\bullet }(S)$ . The map $X\to \widetilde {\mathbf {E}}\mathcal {F}\wedge X$ induced by $S^0\to \widetilde {\mathbf {E}}\mathcal {F}$ induces an equivalence

$$ \begin{align*} \operatorname{\mathrm{Map}}_{\mathcal{S}\mathrm{pc}_{\bullet}^G(S)}\left(\widetilde{\mathbf{E}}\mathcal{F}\wedge X, \widetilde{\mathbf{E}}\mathcal{F}\wedge Y\right) \simeq \operatorname{\mathrm{Map}}_{\mathcal{S}\mathrm{pc}_{\bullet}^G(S)}\left( X,\widetilde{\mathbf{E}}\mathcal{F}\wedge Y\right). \end{align*} $$

Lemma 3.18. Let $\mathcal {F}\subseteq \mathcal {F}^{\prime }$ be an inclusion of families and $\mathcal {E}$ a family such that $\mathcal {E}\cap \mathcal {F} = \mathcal {E}\cap \mathcal {F}^{\prime }$ . Then the map $S^{0}\to \widetilde {\mathbf {E} } \mathcal {E}$ induces an equivalence of based motivic G-spaces

$$ \begin{align*} \mathbf{E} (\mathcal{F}^{\prime},\mathcal{F})\xrightarrow{\sim}\mathbf{E} (\mathcal{F}^{\prime},\mathcal{F})\wedge \widetilde{\mathbf{E} } \mathcal{E}. \end{align*} $$

Proof Smashing $\mathbf {E}\mathcal {E}_+$ with the defining cofiber sequence for $\mathbf {E}(\mathcal {F}^{\prime }, \mathcal {F})$ yields the cofiber sequence

$$ \begin{align*} \mathbf{E} \mathcal{F}_+\wedge\mathbf{E} \mathcal{E}_+\xrightarrow{f} \mathbf{E} \mathcal{F}^{\prime}_{+}\wedge \mathbf{E} \mathcal{E}_{+} \to \mathbf{E} (\mathcal{F}^{\prime},\mathcal{F})\wedge\mathbf{E} \mathcal{E}_+. \end{align*} $$

Smashing $\mathbf {E}(\mathcal {F}^{\prime },\mathcal {F})$ with the defining cofiber sequence for $\widetilde {\mathbf {E}}\mathcal {E}$ yields the cofiber sequence

$$ \begin{align*} \mathbf{E} (\mathcal{F}^{\prime},\mathcal{F})\wedge\mathbf{E} \mathcal{E}_+ \to \mathbf{E} (\mathcal{F}^{\prime},\mathcal{F}) \xrightarrow{i} \mathbf{E} (\mathcal{F}^{\prime},\mathcal{F})\wedge \widetilde{\mathbf{E} } \mathcal{E}. \end{align*} $$

By the hypothesis and Proposition 3.21, f is an equivalence, and so $\mathbf {E} (\mathcal {F}^{\prime },\mathcal {F})\wedge \mathbf {E} \mathcal {E}_+$ is contractible. This implies that i is an equivalence.

To continue the analysis, we pass to the $\infty $ -categorical stabilization – that is, $S^1$ -spectra – of the various categories of motivic G-spaces. We write

$$ \begin{align*} \mathcal{S}\mathrm{pt}_{S^1}^{G,\mathcal{E}}(S):=\mathrm{Stab}\left(\mathcal{S}\mathrm{pc}^{G,\mathcal{E}}_{\bullet}(S)\right). \end{align*} $$

Recall that $\mathcal {S}\mathrm {pt}_{S^1}^{G,\mathcal {E}}(S)$ can be identified with the category of -invariant Nisnevich sheaves of spectra. We write $\underline {\smash {\mathrm {Map}}}_{\mathcal {C}}(X,Y)$ for the spectrum of maps in a stable $\infty $ -category $\mathcal {C}$ .

Let $u:\mathcal {C}_S\subseteq \mathcal {D}_S$ be as in formula (2.24). Then we have induced functors

Since $u^*: \mathcal {P}(\mathcal {D}_S)\to \mathcal {P}(\mathcal {C}_S)$ is a symmetric monoidal left adjoint, it follows that $\mathcal {S}\mathrm {pt}_{S^1}(\mathcal {D}_S)\to \mathcal {S}\mathrm {pt}_{S^1}(\mathcal {C}_S)$ is as well. Since $u_!,u_*: \mathcal {P}(\mathcal {C}_S)\to \mathcal {P}(\mathcal {D}_S)$ are full and faithful by Proposition 2.30, it follows that $u_!,u_*:\mathcal {S}\mathrm {pt}_{S^1}(\mathcal {C}_S)\to \mathcal {S}\mathrm {pt}_{S^1}(\mathcal {D}_S)$ are as well.

We introduce a minor technical condition on pairs $\mathcal {F}\subseteq \mathcal {F}^{\prime }$ over S, which we sometimes require:

We will say that $\mathcal {F}$ satisfies this condition if the pair $\mathcal {F}\subseteq \mathcal {F}_{\mathrm {all}}$ satisfies the condition.

Proof The first two items are immediately equivalent.

To see that (3) implies (2), we have $ u^*(\mathbf {E}(\mathcal {F}^{\prime },\mathcal {F})\otimes X)\simeq u^*(\mathbf {E}(\mathcal {F}^{\prime },\mathcal {F}))\otimes u^*(X)$ and it is straightforward that $u^*(\mathbf {E}(\mathcal {F}^{\prime },\mathcal {F}))\simeq S^{0}$ in $\mathcal {S}\mathrm {pt}_{S^1}^{G,\mathcal {E}}(S)$ .

Now we show that (2) implies (3). First, we assume that all elements of $\mathcal {E}$ act trivially on S. Filter the inclusion $\mathcal {F}\subseteq \mathcal {F}^{\prime }$ by adjacent families and consider the resulting sequence

$$ \begin{align*} \mathbf{E}\mathcal{F}_+=\mathbf{E}\mathcal{F}_{0+}\to\mathbf{E}\mathcal{F}_{1+}\to \mathbf{E}\mathcal{F}_{2+}\to \dotsb \to \mathbf{E}\mathcal{F}_{n-1+}\to \mathbf{E}\mathcal{F}_{n+}=\mathbf{E}\mathcal{F}^{\prime}_+. \end{align*} $$

An inductive argument shows that it suffices to establish that

$$ \begin{align*} \mathbf{E}(\mathcal{F}_{r},\mathcal{F}_{r-1})\otimes X_1 \to \mathbf{E}(\mathcal{F}_{r},\mathcal{F}_{r-1})\otimes X_2 \end{align*} $$

is an equivalence for $1\leq i\leq n$ . Thus, we may assume that $\mathcal {F}^{\prime },\mathcal {F}$ are adjacent and say that $\mathcal {F}^{\prime }\setminus \mathcal {F} = \{(H)\}$ and H acts trivially on S.

By Proposition 3.9 and Proposition 3.16, in order to show that the map displayed here is an equivalence, it suffices to show that

$$ \begin{align*} \underline{\smash{\mathrm{Map}}}_{\mathcal{S}\mathrm{pt}^G_{S^1}(S) }\left(W_+,\widetilde{\mathbf{E}}\mathcal{F}\otimes f\right) \end{align*} $$

is an equivalence for any $p:W\to S$ in $\mathrm {Sm}^G_S[\mathcal {F}^{\prime }]$ , with W affine. From the gluing sequence [Reference Hoyois24, Proposition 5.2], we have the cofiber sequence

$$ \begin{align*} W(\mathcal{F})_+\to W_+ \to p_\#i_*\left(W^{\mathcal{F}}_+\right) \end{align*} $$

in $\mathcal {S}\mathrm {pt}^G_{S^1}(S) $ , where $i:W^{\mathcal {F}}\subseteq W$ is the inclusion (see Notation 2.7). By Proposition 3.16 we have

$$ \begin{align*} \underline{\smash{\mathrm{Map}}}_{\mathcal{S}\mathrm{pt}^G_{S^1}(S) }\left(-,\widetilde{\mathbf{E}}\mathcal{F}\otimes f\right)\simeq \underline{\smash{\mathrm{Map}}}_{\mathcal{S}\mathrm{pt}^G_{S^1}(S) }\left(\widetilde{\mathbf{E}}\mathcal{F}\otimes -, \widetilde{\mathbf{E}}\mathcal{F}\otimes f\right). \end{align*} $$

We have $\widetilde {\mathbf {E}}\mathcal {F}\otimes W(\mathcal {F})_+\simeq \mathrm {pt}$ , using Corollary 3.10 and Proposition 3.16, and so we conclude that

$$ \begin{align*} \underline{\smash{\mathrm{Map}}}_{\mathcal{S}\mathrm{pt}^G_{S^1}(S) }\left(p_\#i_*\left(W^{\mathcal{F}}_+\right),\widetilde{\mathbf{E}}\mathcal{F}\otimes f\right)\simeq \underline{\smash{\mathrm{Map}}}_{\mathcal{S}\mathrm{pt}^G_{S^1}(S) }\left(W_+,\widetilde{\mathbf{E}}\mathcal{F}\otimes f\right). \end{align*} $$

The canonical map $G\times _{\mathrm {N}(H)}W^{H}\to W^{\mathcal {F}}$ is an isomorphism. Indeed, it suffices to check that $W^{H}\cap W^{gHg^{-1}} = \varnothing $ if $H\neq gHg^{-1}$ . Now, if $w\in W^{H}\cap W^{gHg^{-1}}$ , then $\mathrm {Stab}(w)$ contains both of these subgroups. But since $\mathrm {Stab}(w)\in \mathcal {F}^{\prime }$ and $(H)$ is a maximal element of this poset, we have that $H=\mathrm {Stab}(w)=gHg^{-1}$ . In particular, $W^{\mathcal {F}}$ is smooth over S, since $W^{H}\to S$ is smooth. It follows from purity [Reference Hoyois24, Proposition 5.7] that $p_\#i_*W^{\mathcal {F}} \simeq \mathrm {Th}(\mathcal {N}_{i})$ . We show that if $V\to W^{\mathcal {F}}$ is an equivariant vector bundle, then $\underline {\smash {\mathrm {Map}}}_{\mathcal {S}\mathrm {pt}^G_{S^1}(S)}\left (\mathrm {Th}(V),\widetilde {\mathbf {E}}\mathcal {F}\otimes f\right )$ is an equivalence. For this, we may assume that H is normal and in particular that $W^{\mathcal {F}}=W^H$ . Indeed, we have $\mathrm {Th}(V)\simeq G_+\ltimes _{\mathrm {N} H}\mathrm {Th}\left (V\rvert _{W^H}\right )$ , so via the induction-restriction adjunction we can replace G by $\mathrm {N} H$ and $\mathcal {F}, \mathcal {F}^{\prime }$ by $\mathcal {F}\rvert _{\mathrm {N} H}$ , $\mathcal {F}^{\prime }\rvert _{\mathrm {N} H}$ , if necessary. Since G is linearly reductive, we can write $V\cong V^{\prime }\oplus V^H$ . Since $(V^{\prime })^H=0$ , all stabilizers of $V^{\prime }\setminus \{0\}$ are in $\mathcal {F}$ and so $\widetilde {\mathbf {E}}\mathcal {F}\otimes V^{\prime }\setminus \{0\}_+\simeq \ast $ , which implies that

$$ \begin{align*} \widetilde{\mathbf{E}}\mathcal{F}\otimes \mathrm{Th}(V^{\prime})\simeq \widetilde{\mathbf{E}}\mathcal{F}\otimes W^{\mathcal{F}}_+. \end{align*} $$

Now it follows that

$$ \begin{align*} \underline{\smash{\mathrm{Map}}}_{\mathcal{S}\mathrm{pt}^G_{S^1}(S)}\left(\mathrm{Th}(V),\widetilde{\mathbf{E}}\mathcal{F}\otimes f\right)& \simeq \underline{\smash{\mathrm{Map}}}_{\mathcal{S}\mathrm{pt}^G_{S^1}(S)}\left(\mathrm{Th}(V^{\prime})\otimes \mathrm{Th}\left(V^H\right),\widetilde{\mathbf{E}}\mathcal{F}\otimes f\right) \\ &\simeq \underline{\smash{\mathrm{Map}}}_{\mathcal{S}\mathrm{pt}^{G}_{S^1}(S)}\left(W^{\mathcal{F}}_+\otimes \mathrm{Th}\left(V^H\right), \widetilde{\mathbf{E}}\mathcal{F}\otimes f\right)\\ &\simeq \underline{\smash{\mathrm{Map}}}_{\mathcal{S}\mathrm{pt}^{G,\mathcal{E}}_{S^1}(S)}\left(W^{\mathcal{F}}_+\otimes \mathrm{Th}\left(V^H\right),u^*\left(\widetilde{\mathbf{E}}\mathcal{F}\otimes f\right)\right)\\ &\simeq \underline{\smash{\mathrm{Map}}}_{\mathcal{S}\mathrm{pt}^{G,\mathcal{E}}_{S^1}(S)}\left(W^{\mathcal{F}}_+\otimes \mathrm{Th}\left(V^H\right),u^*( f)\right), \end{align*} $$

which is an equivalence, as needed.

Next we consider the case when $\mathcal {F}=\mathcal {F}^{\prime }\cap \mathcal {F}[N]$ , where N is a normal subgroup of G which acts trivially on S. Again we consider $p:W\to S$ in $\mathrm {Sm}^G_S[\mathcal {F}^{\prime }]$ and the cofiber sequence

$$ \begin{align*} W(\mathcal{F}[N])_+\to W_+ \to p_\#i_*\left(W^{N}_+\right) \end{align*} $$

in $\mathcal {S}\mathrm {pt}^G_{S^1}(S) $ , where now $i:W^{N}\subseteq W$ is the inclusion. Since $W(\mathcal {F}[N])_+\in \mathrm {Sm}^G_S[\mathcal {F}]$ we have that $\widetilde {\mathbf {E}}\mathcal {F}\otimes W(\mathcal {F}[N])_+\simeq \mathrm {pt}$ , and we conclude that

$$ \begin{align*} \underline{\smash{\mathrm{Map}}}_{\mathcal{S}\mathrm{pt}^G_{S^1}(S) }\left(p_\#i_*\left(W^{N}_+\right),\widetilde{\mathbf{E}}\mathcal{F}\otimes f\right)\simeq \underline{\smash{\mathrm{Map}}}_{\mathcal{S}\mathrm{pt}^G_{S^1}(S) }\left(W_+,\widetilde{\mathbf{E}}\mathcal{F}\otimes f\right). \end{align*} $$

Since N acts trivially on S, $W^N\to S$ is again smooth, and so we have $p_\#i_*W^{N} \simeq \mathrm {Th}(\mathcal {N}_{i})$ . A similar argument as in the previous paragraph shows that

$$ \begin{align*} \underline{\smash{\mathrm{Map}}}_{\mathcal{S}\mathrm{pt}^G_{S^1}(S)}\left(\mathrm{Th}(\mathrm{N}_{i}),\widetilde{\mathbf{E}}\mathcal{F}\otimes f\right) \end{align*} $$

is an equivalence.

Now we consider the general case. Let N be the normal subgroup as in Condition 3.19 and consider the inclusions $\mathcal {F}\subseteq \mathcal {F}^{\prime \prime }\subseteq \mathcal {F}^{\prime }$ , where we write $\mathcal {F}^{\prime \prime } = \mathcal {F}^{\prime }\cap \mathcal {F}[N]$ . From the cofiber sequence

$$ \begin{align*} \mathbf{E}(\mathcal{F}^{\prime\prime},\mathcal{F}) \to \mathbf{E}(\mathcal{F}^{\prime},\mathcal{F}) \to \mathbf{E}(\mathcal{F}^{\prime},\mathcal{F}^{\prime\prime}), \end{align*} $$

we see that it suffices to show that $\mathbf {E}(\mathcal {F}^{\prime \prime },\mathcal {F})\otimes f$ and $\mathbf {E}(\mathcal {F}^{\prime },\mathcal {F}^{\prime \prime })\otimes f$ are both equivalences. The case $\mathcal {F}^{\prime \prime }\subseteq \mathcal {F}'$ is covered by the previous paragraph, and since all elements of $\mathcal {F}^{\prime \prime }\setminus \mathcal {F}$ act trivially on S, this case is covered by the first.

Corollary 3.22. Let $\mathcal {F}\subseteq \mathcal {F}^{\prime }$ be a subfamily which satisfies Condition 3.19. Write $\mathcal {E}=\mathcal {F}^{\prime }\setminus \mathcal {F}$ , and $u:\mathrm {Sm}^G_S[\mathcal {E}]\subseteq \mathrm {Sm}^G_S$ the inclusion. Then the map

$$ \begin{align*} \mathbf{E}(\mathcal{F}^{\prime},\mathcal{F})\otimes u_!u^*X \to \mathbf{E}(\mathcal{F}^{\prime},\mathcal{F})\otimes X \end{align*} $$

is an equivalence for any $X\in \mathcal {S}\mathrm {pt}_{S^1}^G(S)$ .

3.4 Localization at a cofamily

Proposition 3.23. Let $\mathcal {F}$ be a family of subgroups which satisfies Condition 3.19 and write $j:\mathrm {Sm}^G_S[\mathrm {co}(\mathcal {F})]\subseteq \mathrm {Sm}^G_S$ for the inclusion. Then the natural equivalence $j^*\left (\widetilde {\mathbf {E}}\mathcal {F}\otimes -\right )\simeq j^*$ induces an equivalence of localization endofunctors

$$ \begin{align*} \widetilde{\mathbf{E}}\mathcal{F}\otimes - \simeq j_*j^*: \mathcal{S}\mathrm{pt}_{S^1}^G(S)\to \mathcal{S}\mathrm{pt}_{S^1}^G(S). \end{align*} $$

Proof We have natural equivalences

$$ \begin{align*} \underline{\smash{\mathrm{Map}}}(W, j_*j^* X) & \simeq \underline{\smash{\mathrm{Map}}}(j^*W, j^*X) \\ & \simeq \underline{\smash{\mathrm{Map}}}\left(j_!j^*W, \widetilde{\mathbf{E}}\mathcal{F}\otimes X\right) \\ & \simeq \underline{\smash{\mathrm{Map}}}\left(\widetilde{\mathbf{E}}\mathcal{F}\otimes j_!j^*W, \widetilde{\mathbf{E}}\mathcal{F}\otimes X\right) \\ & \simeq \underline{\smash{\mathrm{Map}}}\left(\widetilde{\mathbf{E}}\mathcal{F}\otimes W, \widetilde{\mathbf{E}}\mathcal{F}\otimes X\right) \\ & \simeq \underline{\smash{\mathrm{Map}}}\left(W, \widetilde{\mathbf{E}}\mathcal{F}\otimes X\right), \end{align*} $$

where the second follows by adjunction and the equivalence $j^*X\simeq j^*\left (\widetilde {\mathbf {E}}\mathcal {F}\otimes X\right )$ , the fourth from Corollary 3.22, and the third and fifth from Corollary 3.17.

Write $\mathrm {L}_{\mathrm {co}(\mathcal {F})} \mathcal {S}\mathrm {pt}^G_{\mathcal {T}}(S)$ for the essential image of $\widetilde {\mathbf {E}}\mathcal {F}\wedge -:\mathcal {S}\mathrm {pt}^G_{\mathcal {T}}(S)\to \mathcal {S}\mathrm {pt}^G_{\mathcal {T}}(S)$ . Since $\mathbf {E}\mathcal {F}_+\wedge -$ is a colocalization endofunctor, $\widetilde {\mathbf {E}}\mathcal {F}\wedge -$ is a localization endofunctor. In particular, $\widetilde {\mathbf {E}}\mathcal {F}$ is an idempotent object of $\mathcal {S}\mathrm {pt}^G_{\mathcal {T}}(S)$ [Reference Lurie29, Proposition 4.8.2.4], and so $\mathcal {S}\mathrm {pt}^G_{\mathcal {T}}(S) \to \mathrm {L}_{\mathrm {co}(\mathcal {F})}\mathcal {S}\mathrm {pt}^G_{\mathcal {T}}(S)$ is a symmetric monoidal localization.

We will abuse notation and terminology slightly by saying that a stabilizing set of spheres $\left \{T^{\mathcal {E}}\in \mathcal {S}\mathrm {pc}^G_{\bullet }(S)\right \}$ is in $\mathcal {S}\mathrm {pc}^{G,\mathrm {co}(\mathcal {F})}_{\bullet }(S)$ if it is in the essential image of $j_!$ . Suppose that $\mathcal {T}\subseteq \mathrm {Sph}^G_B$ is a set of spheres such that $p^*\mathcal {T}$ is in $\mathcal {S}\mathrm {pc}^{G,\mathrm {co}(\mathcal {F})}_{\bullet }(S)$ . In this case, we can stabilize $\mathcal {S}\mathrm {pc}^{G,\mathrm {co}(\mathcal {F})}_{\bullet }(S)$ with respect to $\mathcal {T}$ and we define

$$ \begin{align*} \mathcal{S}\mathrm{pt}^{G,\mathrm{co}(\mathcal{F})}_{\mathcal{T}}(S):= \mathcal{S}\mathrm{pc}_{\bullet}^{G,\mathrm{co}(\mathcal{F})}(S)\left[(p^*\mathcal{T})^{-1}\right]. \end{align*} $$

Equivalently, $\mathcal {S}\mathrm {pt}^{G,\mathrm {co}(\mathcal {F})}_{\mathcal {T}}(S)\simeq \mathcal {S}\mathrm {pt}_{S^1}^{G,\mathrm {co}(\mathcal {F})}(S)\left [(p^*\mathcal {T})^{-1}\right ]$ .

Proof Note that $S\in \mathrm {Sm}^{G}_S[\mathrm {co}(\mathcal {F})]$ , so that $j_!$ preserves terminal objects. That it is symmetric monoidal then follows from the fact that j preserves products. Since $j_!$ is a fully faithful left adjoint, the unit map $\mathrm {id}\to j^*j_!$ is an equivalence. Precomposing $j_!$ with the equivalence $\widetilde {\mathbf {E}}\mathcal {F}\land (-)\simeq j_*j^*$ of Proposition 3.23, we obtain equivalences $\widetilde {\mathbf {E}}\mathcal {F}\land j_!(-)\simeq j_*j^*j_!\simeq j_*$ and consequently a commutative diagram

In particular, the functor $\overline {j}_*:\mathcal {S}\mathrm {pt}^{G,\mathrm {co}(\mathcal {F})}_{S^1}(S)\to \mathrm {L}_{\mathrm {co}(\mathcal {F})}\mathcal {S}\mathrm {pt}^G_{S^1}(S)$ is a composite of symmetric monoidal functors, hence symmetric monoidal itself. It is fully faithful, since the composite functor $\mathcal {S}\mathrm {pt}^{G,\mathrm {co}(\mathcal {F})}_{S^1}(S)\to \mathrm {L}_{\mathrm {co}(\mathcal {F})}\mathcal {S}\mathrm {pt}^G_{S^1}(S)\subseteq \mathcal {S}\mathrm {pt}^G_{S^1}(S)$ (also denoted $j_*$ ) is fully faithful, by Proposition 2.30. To see that $j_*$ is also essential surjective, suppose $X\in \mathcal {S}\mathrm {pt}_{S^1}^G(S)$ and consider $j^*X\in \mathcal {S}\mathrm {pt}_{S^1}^{G,\mathrm {co}(\mathcal {F})}(S)$ . By the previous equivalence $\widetilde {\mathbf {E}}\mathcal {F}\land j_!(-)\simeq j_*$ and Proposition 3.23, we see that

$$ \begin{align*} \widetilde{\mathbf{E}}\mathcal{F}\land j_!j^*X\simeq\widetilde{\mathbf{E}}\mathcal{F}\land X, \end{align*} $$

since both are equivalent to $j_*j^*X$ . Thus $\overline {j}_*:\mathcal {S}\mathrm {pt}^{G}_{S^1}(S)\to \mathcal {S}\mathrm {pt}^{G,\mathrm {co}(\mathcal {F})}_{S^1}(S)$ is an equivalence of symmetric monoidal $\infty $ -categories.

Proof Since $j_!$ is a symmetric monoidal left adjoint, it induces a symmetric monoidal functor on $p^*\mathcal {T}$ -stabilizations.

By Proposition 3.24, the induced map

$$ \begin{align*} \mathcal{S}\mathrm{pt}_{\mathcal{T}}^{G,\mathrm{co}(\mathcal{F})}(S)\simeq\mathcal{S}\mathrm{pt}_{S^1}^{G,\mathrm{co}(\mathcal{F})}(S)\left[p^*\mathcal{T}^{-1}\right]\to \left(\mathrm{L}_{\mathrm{co}(\mathcal{F})}\mathcal{S}\mathrm{pt}_{S^1}^G(S)\right)\left[p^*\mathcal{T}^{-1}\right] \end{align*} $$

is an equivalence in $\mathrm {CAlg}\left (\mathcal {P}{\mathrm {r}}^{\mathrm {L},\otimes }\right )$ . The result now follows from the equivalence from Lemma 2.40:

$$ \begin{align*} \left(\mathrm{L}_{\mathrm{co}(\mathcal{F})}\mathcal{S}\mathrm{pt}^G_{\mathcal{S}^1}(S)\right)\left[p^*\mathcal{T}^{-1}\right]\simeq \mathrm{L}_{\mathrm{co}(\mathcal{F})}\left(\mathcal{S}\mathrm{pt}^G_{\mathcal{S}^1}(S)\right)\left[p^*\mathcal{T}^{-1}\right]. \\[-32pt] \end{align*} $$

3.5 Adjacent pairs

Lemma 3.26. Suppose that $\mathcal {F}\subseteq \mathcal {F}^{\prime }$ is N-adjacent at H and that H acts trivially on S. Set $X\in \mathrm {Sm}^G_S\left [\mathcal {F}^{\prime }\setminus \mathcal {F}\right ]$ . Then the canonical map

$$ \begin{align*} f:G\times_{N_GH}X^H\to X \end{align*} $$

is an isomorphism.

Let $H\leq N$ be a subgroup. Then $\mathrm {W}_NH$ is a normal subgroup of $\mathrm {W}_GH$ , and we will often write $\mathcal {W} H = \mathrm {W}_G H/\mathrm {W}_N H$ for the quotient of Weyl groups. We will often write $\mathbf {E}_{\mathrm {W}_NH}(\mathrm {W}_GH) $ instead of $\mathbf {E}\mathcal {F}(\mathrm {W}_NH)$ for the universal $\mathrm {W}_NH$ -free motivic $\mathrm {W}_GH$ -motivic space, in order to emphasize the ambient group, where $\mathcal {F}(\mathrm {W}_NH)$ is the family of subgroups $\{K\leq \mathrm {W}_GH \mid K\cap \mathrm {W}_NH = \{e\}\}$ .

Proof It suffices to prove these equivalences in the case $S=B$ ; the general case follows by applying $f^*$ , where $f:S\to B$ is the structure map. For the first item, by Proposition 3.21 it suffices to show that the projection

$$ \begin{align*} p:\left(G\times_{\mathrm{N}_GH} \mathbf{E}_{\mathrm{W}_NH}(\mathrm{W}_GH)\right)_+ \to S^0 \end{align*} $$

induces equivalences $ \operatorname {\mathrm {Map}}_{\mathcal {S}\mathrm {pc}^G(B)}(X,p)$ , for any $X\in \mathrm {Sm}^G_B\left [\mathcal {F}^{\prime }\setminus \mathcal {F}\right ]$ . But for such an X we have, by Lemma 3.26, that $X\cong G\times _{\mathrm {N}_GH}X^H$ , so we have

$$ \begin{align*} \underline{\smash{\mathrm{Map}}}_{\mathcal{S}\mathrm{pt}_{S^1}^G(B)}\left(X_+,\right. & \left.\left(G\times_{\mathrm{N}_GH} \mathbf{E}_{\mathrm{W}_NH}(\mathrm{W}_GH)\right)_+\right) \\ & \simeq \underline{\smash{\mathrm{Map}}}_{\mathcal{S}\mathrm{pt}_{S^1}^{\mathrm{N}_GH}(B)}\left(X^{H}_+, \left(G\times_{\mathrm{N}_GH} \mathbf{E}_{\mathrm{W}_NH}(\mathrm{W}_GH)\right)_+\right) \\ & \simeq \underline{\smash{\mathrm{Map}}}_{\mathcal{S}\mathrm{pt}_{S^1}^{\mathrm{W}_GH}(B)}\left(X^{H}_+, \left(G\times_{\mathrm{N}_GH} \mathbf{E}_{\mathrm{W}_NH}(\mathrm{W}_GH)\right)^H_+\right)\\ & \simeq \underline{\smash{\mathrm{Map}}}_{\mathcal{S}\mathrm{pt}_{S^1}^{\mathrm{W}_GH}(B)}\left(X^{H}_+, \left(S^0\right)^H\right)\\ & \simeq \underline{\smash{\mathrm{Map}}}_{\mathcal{S}\mathrm{pt}_{S^1}^{\mathrm{N}_GH}(B)}\left(X^{H}_+, S^0\right) \\ & \simeq \underline{\smash{\mathrm{Map}}}_{\mathcal{S}\mathrm{pt}_{S^1}^{G}(B)}\left(\left(G\times_{\mathrm{N}_GH}X^{H}\right)_+, S^0\right). \end{align*} $$

For the second item, we have that $\mathcal {F}_{\mathrm {N}_GH}[H]\cap \mathcal {F}^{\prime }\rvert _{\mathrm {N}_GH} = \mathcal {F}_{\mathrm {N}_GH}[H]\cap \mathcal {F}\rvert _{\mathrm {N}_GH}$ , and so this follows from Lemma 3.18.

4.1 Fixed-point motivic spaces

We begin by extending the adjunction

$$ \begin{align*} \pi^{-1}:\mathrm{Sm}_S^{G/N}\rightleftarrows \mathrm{Sm}^{G}_{S}:(-)^N \end{align*} $$

to motivic G-spaces. Note that $\pi ^{-1}$ is full and faithful and induces an equivalence

(4.1) $$ \begin{align} \mathrm{Sm}_S^{G/N}\simeq \mathrm{Sm}^{G}_S[\mathrm{co}(\mathcal{F}[N])]\subseteq \mathrm{Sm}^G_S \end{align} $$

with the subcategory of smooth G-schemes over S on which N acts trivially.

The functor $\pi _*$ in the following propositions is the N-fixed-point functor on motivic G-spaces. In keeping with standard notation, we sometimes write

$$ \begin{align*} (-)^N := \pi_*. \end{align*} $$

Proposition 4.2. The functor $\pi ^{-1}:\mathrm {Sm}^{G/N}_S\to \mathrm {Sm}^G_S$ induces adjoint pairs of functors

$$ \begin{align*} \pi^{*}:\mathcal{S}\mathrm{pc}^{G/N}(S) & \rightleftarrows \mathcal{S}\mathrm{pc}^G(S) : \pi_* \\ \pi^{*}:\mathcal{S}\mathrm{pc}^{G/N}_{\bullet}(S) & \rightleftarrows \mathcal{S}\mathrm{pc}^G_{\bullet}(S) : \pi_* \end{align*} $$

such that the following diagrams commute:

Moreover, $\pi ^*$ and $\pi _*$ are symmetric monoidal, and $\pi _*$ preserves colimits.

4.2 Fixed-point spectra

We view $G/N$ -equivariant vector bundles on S as G-equivariant vector bundles via the quotient homomorphism $\pi :G\to G/N$ . Given a stabilizing subset $\mathcal {T}\subseteq \mathrm {Sph}^{{G/N}}_{B}$ , we have the stabilizing subset

$$ \begin{align*} \pi^*\mathcal{T} = \left\{T^{\pi^*\mathcal{E}} \mid T^{\mathcal{E}}\in \mathcal{T}\right\} \subseteq \mathrm{Sph}^{G}_B, \end{align*} $$

which for simplicity we usually write again as $\mathcal {T}$ . We call a G-sphere of the form $\pi ^*\left (T^{\mathcal {E}}\right )$ an N-trivial G-sphere. Recall also that we write $N\textrm {-}\mathrm {triv} = \pi ^*\left (\mathrm {Sph}^{G/N}_B\right )$ .

Proposition 4.3. Let $\mathcal {T}\subseteq \mathrm {Sph}^{{G/N}}_{B}$ be a stabilizing set of spheres. There is an adjoint pair of symmetric monoidal functors

$$ \begin{align*} \pi^*:\mathcal{S}\mathrm{pt}_{\mathcal{T}}^{{G/N}}(S) \rightleftarrows \mathcal{S}\mathrm{pt}_{\mathcal{T}}^G(S):\pi_* \end{align*} $$

such that $\pi _*\left (\Sigma ^{\infty }_{\mathcal {T}}Y\right )\simeq \Sigma ^{\infty }_{\mathcal {T}}\left (Y^N\right )$ for $Y\in \mathcal {S}\mathrm {pc}_\bullet ^G(S)$ .