Algebraic models for 1-dimensional categories of rational $G$ -spectra

J.P.C.Greenlees Mathematics Institute, Zeeman Building, Coventry CV4, 7AL, UK [email protected]

Abstract.

In this paper we give algebraic models for rational $G$ -spectra for a compact Lie group $G$ when the geometric isotropy is restricted to lie in a 1-dimensional block of conjugacy classes. This includes all blocks of all groups of dimension 1, semifree spectra, and 1-dimensional blocks for many other groups $G$ .

The author is grateful for comments, discussions and related collaborations with S.Balchin, D.Barnes, T.Barthel, M.Kedziorek, L.Pol, J.Williamson. The work is partially supported by EPSRC Grant EP/W036320/1. The author would also like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Equivariant homotopy theory in context, where later parts of work on this paper was undertaken. This work was supported by EPSRC grant EP/Z000580/1.

1. Introduction

If we pick a set $\mathcal{V}$ of conjugacy classes of (closed) subgroups of a compact Lie group $G$ we can consider the category $\mbox{$G$-{\bf spectra}}|\mathcal{V}$ of $G$ -spectra whose geometric isotropy is contained in $\mathcal{V}$ , and it is conjectured [9] that there is a small and calculable abelian model $\mathcal{A}(G|\mathcal{V})$ so that the category DG- $\mathcal{A}(G|\mathcal{V})$ of differential graded objects in $\mathcal{A}(G|\mathcal{V})$ is a model for $\mbox{$G$-{\bf spectra}}|\mathcal{V}$ in the sense that there is a Quillen equivalence

\mbox{$G$-{\bf spectra}}|\mathcal{V}\simeq DG-\mathcal{A}(G|\mathcal{V}).

The purpose of the present paper is to verify the conjecture in a range of particularly simple cases.

The space $\mathfrak{X}_{G}=\mathrm{Sub}(G)/G$ of conjugacy classes of closed subgroups of $G$ has two topologies of interest. First of all, it has the h-topology, which is the quotient topology of the Hausdorff metric topology on $\mathrm{Sub}(G)$ . It also has the Zariski topology, whose closed sets are the h-closed sets which are also closed under passage to cotoral subgroups¹¹1 $L$ is cotoral in $K$ if $L$ is normal in $K$ with quotient being a torus. For conjugacy classes $(L)$ is cotoral in $(K)$ if the relation holds for some representative subgroups.. Since $\mathcal{V}$ is a subset of $\mathfrak{X}_{G}$ it also acquires two topologies, though it will not necessarily be a spectral space.

The very simplest case is when $\mathcal{V}$ is h-discrete and there are no non-trivial cotoral relations (we say $\mathcal{V}$ is of Thomason height 0). In this case the conjecture is immediate from results of [21], and the model is described by

\mathcal{A}(G|\mathcal{V})\simeq\prod_{H\in\mathcal{V}}\mathcal{A}(G|H)

and

\mathcal{A}(G|H)\simeq\mbox{tors-}H^{*}(BW_{G}^{e}(H))[W_{G}^{d}(H)]\mbox{-mod},

where $W_{G}(H)=N_{G}(H)/H$ has identity component $W_{G}^{e}(H)$ and component group $W_{G}^{d}(H)$ .

It is the purpose of the present paper to prove the conjecture in a range of cases when $\mathcal{V}$ is one step more complicated, so that it has a subspace $\mathcal{K}$ of Thomason height 0 and one additional point $K^{*}$ . The symbol $K^{*}$ will be used for this point throughout this paper. Thus we either have a cotoral inclusion $K\leq_{ct}K^{*}$ for some $K\in\mathcal{K}$ or else $K^{*}$ is an h-limit of points of $\mathcal{K}$ . We will also restrict the group theory by assuming $\mathcal{K}$ consists of finite groups, and $K^{*}$ has finite index in its normalizer (most examples we make explicit also have $K^{*}=G$ ). This is restrictive, but it permits a simple exposition and covers the cases of immediate interest. Section 10 describes some of the complications that arise as more cases are covered.

The method is closely based on that used for the circle group and for the two blocks of $O(2)$ , but it is more uniform and covers many other cases. The cases covered include all blocks of 1-dimensional groups and semi-free $G$ -spectra when $G$ is a torus. It also covers the block of full subgroups of a toral group (the identity component $G_{e}=\mathbb{T}$ is a torus) where the $G_{d}$ -module $H_{1}(T;\mathbb{Q})$ is simple. Already these cases display a range of different behaviour.

It is helpful to bear some examples in mind.

Example 1.1.

(1)

The space $\mathcal{V}$ need not be irreducible. This means that $K^{*}$ is not necessarily a generic point.

For example if $\mathcal{V}$ in the h-topology is the one point compactification of a countable set and the cotoral relation is trivial. This is the case for the dihedral subgroups of $O(2)$ .
(2)

The space need not be compact in the h-topology. For example we may add infinitely many points to a compact example without adding the limit point. It is easy to construct such examples with $G$ the 2-torus (see (4)).
(3)

A more extreme form of this is when every neighbourhood of $K^{*}$ contains an infinite set of which it is not the limit. For example $G=K^{*}=T^{2}$ is the 2-torus, and the other points are the finite subgroups of $C_{1}\times T,C_{2}\times T,C_{3}\times T,\ldots$ .
(4)

The space $\mathcal{V}$ may be discrete in the h-topology. For example we may take $G=K^{*}=T^{2}$ to be the 2-torus and $\mathcal{K}$ to consist of the proper subgroups of $T\times 1\subseteq T\times T=G$ .

It is natural to want to generalize these results. The arguments presented here also prove the conjecture in many other 1-dimensional cases, and the natural level of generality is not yet clear; some cases may best be treated as subcategories of higher dimensional examples. In any case, our focus here will be on expository simplicity rather than greatest generality. Section 10 discusses ways to relax the restrictions.

1.A. Associated work in preparation

This paper is the second in a series of 5 constructing an algebraic category $\mathcal{A}(SU(3))$ and showing it gives an algebraic model for rational $SU(3)$ -spectra. This series gives a concrete illustrations of general results in small and accessible examples.

The first paper [13] describes the group theoretic data that feeds into the construction of an abelian category $\mathcal{A}(G)$ for all toral groups $G$ and makes them explicit for toral subgroups of rank 2 connected groups. The present paper (which does not logically depend on [13]) constructs algebraic models for all relevant 1-dimensional blocks. The paper [14] gives an algebraic model for the maximal torus normalizer in $U(2)$ .

The paper [15] assembles this information and that from [4] to give an abelian category $\mathcal{A}(U(2))$ in 7 blocks and shows it is an algebraic model for rational $U(2)$ -spectra. Finally, the paper [16] constructs $\mathcal{A}(SU(3))$ in 18 blocks and shows it is equivalent to the category of rational $SU(3)$ -spectra. The most complicated parts of the model for $G=U(2)$ and $G=SU(3)$ are the toral blocks, which are based on the work in the present paper.

This series is part of a more general programme. Future installments will consider blocks with Noetherian Balmer spectra [19] and those with no cotoral inclusions [17]. An account of the general nature of the models is in preparation [18], and the author hopes that this will be the basis of the proof that the category of rational $G$ -spectra has an algebraic model in general.

1.B. Contents

The paper is divided into two parts. Part 1 is essentially algebraic and Part 2 shows that appropriate categories of rational $G$ -spectra are equivalent to the algebraic models.

The main part of the paper gives an algebraic model for a 1-dimensional block with a single dominant subgroup. To show that this is useful, we start in Section 2 by showing that for a 1-dimensional group, the space of subgroups can be decomposed into such blocks, and furthermore the auxiliary data can be describe in these terms. Thereafter we have in mind a single 1-dimensional block. In Section 3 we describe the data required to build the model, and in Section 4 we describe how this data can be obtained for compact Lie groups. In Section 5 we explain how to build the model from the data, and we show how to work with the category, inparticular proving it is of injective dimension 1.

We begin Part 2 in Section 6 by describing the general strategy for showing a 1-dimensional block has an algebraic model.

It is rather well known that for a connected compact Lie group $G$ , the category of cofree $G$ -spectra is a category of complete modules over $H^{*}(BG)$ , and that this can easily be adapted to disconnected groups. In Section 7 we show that the category of modules over $DEG_{+}$ is the corresponding category of modules over $H^{*}(BG)$ without any completeness condition, and that the model for cofree $G$ -spectra is obtained by completion. This gives the stalkwise model, and it is quite easy to assemble this algebraically over the height 0 subgroups. However for the splicing to the height 1 subgroup we need to have a common framework. The new complication here is having to deal with the varying component structure, and the method for doing that is described in Section 8. This completes the account of the model over height 0 subgroups, and finally in Section 9 we assemble the information over height 0 and height 1 groups to establish that 1-dimensional blocks are algebraic. In Section 10 we review the technical difficulties that need to be tackled to give further generalization.

Part I Algebra

2. Full subgroups

The purpose of this section is to show that for 1-dimensional groups $G$ the category of $G$ -spectra breaks into pieces each one of which is covered by our general analysis. This comes in two steps. Firstly, we establish a decomposition indexed by conjugacy classes of subgroups $V$ of the component group $W$ . Secondly, we show that it suffices to consider the case when $V=W$ , in the sense that the model in the general case can often be easily deduced from the full case.

For the purpose of this section, we suppose that $G$ has identity component $G_{e}=\mathbb{T}$ a torus and lives in an extension

1\longrightarrow\mathbb{T}\longrightarrow G\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}W\longrightarrow 1.

2.A. Partition by subgroups of the component group

For any compact Lie group $G$ , one may show that the space $\mathrm{Sub}(G)/G$ can be partitioned into blocks dominated by a single group. When $G$ is a toral group there is a partition that takes a particularly simple form. Since any 1-dimensional group $G$ has identity component a circle, this case is covered automatically.

Lemma 2.1.

[13, 2.1] For a toral group $G$ as above, the space $\mathfrak{X}_{G}=\mathrm{Sub}(G)/G$ of conjugacy classes of subgroups of the toral group $G$ is partitioned into pieces, $\mathcal{V}^{G}_{\overline{H}}$ one for each conjugacy class of subgroups $\overline{H}$ of $W$ .

If $\overline{H}\subseteq W$ , the set

\mathcal{V}_{\overline{H}}^{G}=\{(K)\;|\;\pi(K)=\overline{H}\}

is clopen in the Hausdorff metric topology and closed under passage to cotoral subgroups. Furthermore, $\mathcal{V}^{G}_{\overline{H}}$ is dominated by $\pi^{-1}(\overline{H})$ in the sense that it consists of all subgroups cotoral in $\pi^{-1}(\overline{H})$ .

Accordingly, the Balmer spectrum with its Zariski topology is a coproduct

\mathfrak{X}_{G}=\coprod_{(\overline{H})}\mathcal{V}_{\overline{H}}^{G}.

Remark 2.2.

(a) The partition is crude, in the sense that the sets $\mathcal{V}^{G}_{\overline{H}}$ can often be decomposed further.

(b) Subgroups mapping onto $W$ are called full, and the component of full subgroups is $\mathcal{V}^{G}_{W}$ . Most of the rest of the paper will focus on full subgroups because the component of $\mathcal{V}^{G}_{V}$ can be studied in the group $\tilde{V}:=\pi^{-1}V$ as explained in Subsection 2.B below.

(c) Since subgroups of $G$ with image $V$ are by definition subgroups of $\tilde{V}$ , the map $\mathcal{V}^{\tilde{V}}_{V}\longrightarrow\mathcal{V}^{G}_{V}$ is surjective, and the only effect is fusion of $\tilde{V}$ -conjugacy classes to form $G$ -conjugacy classes. Fusion can nonetheless have significant effects (for example the map $\mathfrak{X}_{\mathbb{T}}\longrightarrow\mathfrak{X}_{G}$ factors through $\mathfrak{X}_{\mathbb{T}}\longrightarrow\mathfrak{X}_{\mathbb{T}}/W$ ).

2.B. Reduction to full subgroups

Continuing with a toral group $G$ , in analysing $\mathcal{V}^{G}_{V}$ , we may reduce to the case that $V$ is normal in $W$ .

To see this, let $N=N_{W}(V)$ . The $W$ -conjugacy class $(V)_{W}$ may split into several $N$ -conjugacy classes $(V)_{W}=\coprod_{i}(V_{i})_{N}$ . There is a corresponding splitting of the idempotent $e_{V}$ : $\mathrm{res}^{W}_{N}(e_{V})=\sum_{i}e_{V_{i}}$ . These idempotents may be inflated to $G$ , and we see

\tilde{N}\mbox{-{\bf spectra}}|\mathcal{V}^{G}_{V}\simeq e_{V}\tilde{N}\mbox{-{\bf spectra}}\simeq\prod_{i}e_{V_{i}}\tilde{N}\mbox{-{\bf spectra}}\simeq\prod_{i}\tilde{N}\mbox{-{\bf spectra}}|\mathcal{V}^{\tilde{N}}_{V_{i}}.

Since the $V_{i}$ are conjugate in $G$ , the factors are equivalent using conjugation by elements of $G$ . For subgroups $H$ with $\pi H=V$ , the relevant map of underlying spaces is

\coprod_{i}\mathcal{V}^{\tilde{N}}_{V_{i}}\longrightarrow\mathcal{V}^{G}_{V}.

Lemma 2.3.

Suppose $V\subseteq W$ and let $N=N_{W}(V)$ . Restriction induces a full and faithful functor

\mbox{$G$-{\bf spectra}}|\mathcal{V}^{G}_{V}\longrightarrow\tilde{N}\mbox{-{\bf spectra}}|\mathcal{V}^{\tilde{N}}_{V}.

The essential image consists of $\tilde{N}$ -spectra which are constant on the factors $\tilde{N}\mbox{-{\bf spectra}}|\mathcal{V}^{\tilde{N}}_{V_{i}}$ in the sense that they correspond under conjugation by $G$ . Composing with restriction to one factor, we obtain an equivalence

\mbox{$G$-{\bf spectra}}|\mathcal{V}^{G}_{V}\simeq\tilde{N}\mbox{-{\bf spectra}}|\mathcal{V}^{\tilde{N}}_{V}.

Remark 2.4.

This shows directly that the two categories should have equivalent models. In fact we observe that $N_{G}(V)=N_{\tilde{N}}(V)$ , so that the sheaf of rings and component structures also agree. Thus, corresponding to the equivalence of categories of $G$ -spectra we have an equivalence

\mathcal{A}(G|\mathcal{V}^{G}_{V})=\mathcal{A}(\tilde{N}|\mathcal{V}^{\tilde{N}}_{V}).

Proof : We must show the map is full, faithful and essentially surjective.

The map

[X,Y]^{G}\longrightarrow[G_{+}\wedge_{N}eX,Y]^{G}=[eX,Y]^{N}

is induced by $G_{+}\wedge_{\tilde{N}}S^{0}\wedge X\simeq G_{+}\wedge_{\tilde{N}}eX\longrightarrow X$ of $G$ -spectra. Now $G/\tilde{N}=W/N$ , and the $V$ fixed points consist of cosets $wN$ so that $V^{w}\subseteq N$ . The fact that restriction is an isomorphism follows since the map $W_{+}\wedge_{V}eS^{0}\longrightarrow S^{0}$ is an equivalence in $V$ -fixed points. ∎

Next, we may further reduce to working with $\mathcal{V}^{\tilde{V}}_{V}$ if we take into account the action of the Weyl group $N_{W}(V)/V$ .

Lemma 2.5.

If $N=N_{W}(V)$ , the geometric isotropy of $X$ consists of subgroups of $\tilde{V}$ then restriction induces an isomorphism

[X,Y]^{\tilde{N}}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\left([X,Y]^{\tilde{V}}\right)^{N/V}.

Proof : We have an equivalence $EN/V_{+}\wedge X\simeq X$ , and the spectral sequence of the skeletal filtration of $EN/V_{+}$ gives the isomorphism. ∎

Remark 2.6.

This shows that to understand $\tilde{N}$ -spectra we need only understand $\tilde{V}$ spectra together with an action of the finite group $N/V$ . However one does need to bear in mind that $N/V$ acts on the space of subgroups as well as all other elements of the construction.

Combining this with Lemma 2.3, as in [10, 6.10], this shows we have a reduction to the case of full subgroups, and we may take

\mathcal{A}(G|\mathcal{V}^{G}_{V})\simeq\mathcal{A}(\tilde{N}|\mathcal{V}^{\tilde{N}}_{V})\simeq``\mathcal{A}(\tilde{V}|\mathcal{V}^{\tilde{V}}_{V})[N/V]^{\prime\prime}.

Note that in this statement $N/V$ may act non-trivially on $\mathcal{V}^{\tilde{V}}_{V}$ , so the right hand category is not simply $\mathcal{A}(\tilde{V}|\mathcal{V}^{\tilde{V}}_{V})$ with an action of $N/V$ , and considerable elucidation along the lines of [10] is necessary. With these caveats, we may reduce to the case of full subgroups.

This strategy is especially effective if $G$ is 1-dimensional since the action of $N/V$ on the identity component does not permute subgroups. In this case we find

\mathcal{V}^{\tilde{N}}_{V}=\mathcal{V}^{\tilde{V}}_{V},

and in fact the Weyl groups are of the same dimension so that the structure sheaves of rings will agree. The component groups will differ since the group $N/V$ certainly acts for $\tilde{N}$ -spectra. Indeed, if $\pi(K)=V$ there is a map

N_{\tilde{N}}(K)\longrightarrow N_{N}(V)=N,

since $K\cap T$ is characteristic, and hence there is a map

W_{\tilde{N}}(K)\longrightarrow N/V,

but this need not be an isomorphism for all $K$ .

3. Ingredients for the abelian models

The rest of our analysis supposes that we are considering $G$ -spectra with geometric isotropy in a 1-dimensional space $\mathcal{V}$ of a special form. In fact we suppose given a countable set $\mathcal{K}$ of conjugacy classes subgroups. As a subspace of $\mathfrak{X}_{G}$ it is discrete and we assume $\mathcal{V}=\mathcal{K}\amalg\{K^{*}\}$ .

For example if $G$ is a 1-dimensional group we may choose a subgroup $V$ of the component group and suppose $\mathcal{K}$ consists of the finite subgroups $H$ of $G$ with $\pi(H)=V$ and $K^{*}=\pi^{-1}(V)$ .

We discuss the form of the space $\mathcal{V}$ in Subsection 3.A, and the auxiliary data in Subsection 3.B.

3.A. Decomposing $\mathcal{V}$

We start with a countable set $\mathcal{K}$ of conjugacy classes of finite subgroups, and $\mathcal{V}=\mathcal{K}\amalg\{K^{*}\}$ . It is convenient to partition $\mathcal{K}$ .

Lemma 3.1.

There is a partition $\mathcal{K}=\mathcal{K}_{0}\amalg\mathcal{K}_{1}\amalg\mathcal{K}\mathcal{R}$ into Zariski clopen sets where

•

$\mathcal{K}_{1}$ consists of subgroups cotoral in $K^{*}$ .
•

$\mathcal{K}_{0}$ consists of subgroups not cotoral in $K^{*}$ but $\mathcal{K}_{0}$ has $K^{*}$ as a limit point and
•

the remainder $\mathcal{K}\mathcal{R}$ has no limit points in $\mathcal{V}$ .

Proof : If $K^{*}$ is not a limit point when we remove $\mathcal{K}_{1}$ we take $\mathcal{K}\mathcal{R}=\mathcal{K}\setminus\mathcal{K}_{1}$ and $\mathcal{K}_{0}=\emptyset$ . Otherwise, we choose an h-neighbourhood $U$ of $K^{*}$ , and take $\mathcal{K}_{0}^{\prime}:=(\mathcal{K}\setminus\mathcal{K}_{1})\cap U$ and $\mathcal{K}\mathcal{R}:=(\mathcal{K}\setminus\mathcal{K}_{1})\setminus U$ . ∎

Remark 3.2.

It is disappointing that we cannot generally give a canonical decomposition, and that the pieces cannot be simpler.

(i) Example 1.1 (3) shows that it may happen that every choice of $\mathcal{K}_{0}$ contains an infinite set $\mathcal{K}\mathcal{R}^{\prime}$ without $K^{*}$ as a limit point, so $\mathcal{K}\mathcal{R}^{\prime}$ could be moved into $\mathcal{K}\mathcal{R}$ .

(ii) Even if $K^{*}$ is a limit point of every infinite subset of $\mathcal{K}_{0}$ , the partition is not canonical, since any finite set of points can be moved between $\mathcal{K}_{0}$ and $\mathcal{K}\mathcal{R}$ .

Remark 3.3.

(i) If $\mathcal{K}\mathcal{R}=\emptyset$ and $K^{*}$ is a limit point of every infinite subset of $\mathcal{K}_{0}$ , we say $\mathcal{V}$ is almost irreducible.

(ii) The topology is determined by the closed subsets of $\mathcal{V}$ not containing $K^{*}$ . This contains the finite subsets of $\mathcal{K}$ , and if it contains no other subsets, $\mathcal{K}\cup\{K^{*}\}$ is the one point compactification of $\mathcal{K}$ .

We discuss some examples and then show that in many cases it suffices to deal with the special cases $\mathcal{K}=\mathcal{K}_{0}$ (Type 0) and $\mathcal{K}=\mathcal{K}_{1}$ (Type 1).

The models differ in character according to the sizes of $\mathcal{K}_{0}$ and $\mathcal{K}_{1}$ : even in this very special context, a wide variety of behaviours is possible.

Example 3.4.

Here is a small selection of almost irreducible examples with $\mathcal{K}_{1}$ infinite.

(1)
If $G=T^{2}$ all subgroups are cotoral in $G$ , so in all cases $\mathcal{K}=\mathcal{K}_{1}$ . Nonetheless these cases can vary in character.
1. (a)
  
  If $\mathcal{K}$ consists of all subgroups $T^{2}[n]$ then $G=K^{*}$ is the 1-point compactification.
2. (b)
  
  If $\mathcal{K}=\mathcal{K}_{1}=\{T[n]\times T\;|\;n\geq 1\}$ then the h-topology is discrete.
3. (c)
  
  If $\mathcal{K}=\mathcal{K}_{1}=\{T^{2}[n]\;|\;n\geq 1\}\cup\{C_{m}\times 1\;|\;m\geq 1\}$ then there is an infinite subset of $\mathcal{K}_{1}$ without limit points.
4. (d)
  
  If $\mathcal{K}$ consists of all finite subgroups, then it has infinitely many limit points.
(2)

We may have both $\mathcal{K}_{0}$ and $\mathcal{K}_{1}$ infinite. For example $G=T\times O(2)$ and then take $K^{*}=G$ with $\mathcal{K}_{1}=\{C_{m}\times O(2)\;|\;m\geq 1\}$ , and $\mathcal{K}_{0}=\{T\times D_{2n}\;|\;n\geq 1\}$ .

A property holds almost everywhere in $\mathcal{K}$ if it holds except for a closed subset of $\mathcal{V}$ lying in $\mathcal{K}_{0}\amalg\mathcal{K}\mathcal{R}$ (in the almost irreducible case, this means for all but a finite number of points of $\mathcal{K}_{0}$ ).

Decompositions of $\mathcal{V}$ lead to decompositions of the model. We illustrate with two special cases.

Example 3.5.

(i) We may always write $\mathcal{V}=\mathcal{V}^{\prime}\amalg\mathcal{K}\mathcal{R}$ and then

\mathcal{A}(G|\mathcal{V})=\mathcal{A}(G|\mathcal{V}^{\prime})\times\mathcal{A}(G|\mathcal{K}\mathcal{R}),

where $\mathcal{A}(G|\mathcal{K}\mathcal{R})$ is described in the introduction. The existence of idempotents in the Burnside ring $A(G)$ shows there is a corresponding decomposition of spectra, so for most purposes we may assume $\mathcal{K}\mathcal{R}=\emptyset$ .

(ii) If $\mathcal{V}$ is almost irreducible then $\mathcal{V}=\mathcal{V}_{0}\vee\mathcal{V}_{1}$ (with the wedge point being $K^{*}$ ) where $\mathcal{V}_{0}=\mathcal{K}_{0}^{\#}$ (one point compactification) and $\mathcal{V}_{1}=\mathcal{K}_{1}\cup\{K^{*}\}$ . There is then a pullback square

and similarly for spectra.

An important example example consists of subroups of $G=N_{U(2)}(T^{2})$ of dimension $\geq 1$ with $K^{*}=G$ , where $\mathcal{K}_{0}$ consists of 1-dimensional subgroups containing the central circle, and $\mathcal{K}_{1}$ consists of 1-dimensional subgroups containing all elements $\mathrm{diag}(\lambda,\lambda^{-1})$ .

3.B. Auxiliary data

The model for $G$ -spectra with geometric isotropy in $\mathcal{V}$ is a category whose objects are equivariant sheaves of modules over a sheaf of rings over $\mathcal{V}$ . In the 1-dimensional case, rather than making explicit all of the adjectives to make this precise, we will give the data required directly. We require the additional data of a ‘sheaf of rings’ $\mathcal{R}$ and a ‘component structure’ $\mathcal{W}$ which specifies the equivariance together with a ‘coordinate structure’ $\mathcal{S}$ linking the topology and algebra.

Definition 3.6.

(a) A sheaf of rings $\mathcal{R}$ on $\mathcal{V}$ consists of commutative polynomial rings $\mathcal{R}(F)$ for $F\in\mathcal{K}$ and $\mathcal{R}(K^{*})$ , and a ring homomorphisms $\mathcal{R}(K^{*})\longrightarrow\mathcal{R}(F)$ for almost all $F$ .

(b) A component structure on $\mathcal{V}$ consists of finite groups $\mathcal{W}_{F}$ for $F\in\mathcal{K}$ and $\mathcal{W}_{K^{*}}$ , together with a homomorphism $\mathcal{W}_{F}\longrightarrow\mathcal{W}_{K^{*}}$ for almost all $F$ .

(c) The component structure $\mathcal{W}$ has an action on $\mathcal{R}$ if $\mathcal{W}_{F}$ acts on $\mathcal{R}(F)$ and $\mathcal{W}_{G}$ acts on $\mathcal{R}(K^{*})$ so that $\mathcal{R}(G)\longrightarrow\mathcal{R}(F)$ is $\mathcal{W}_{F}$ -equivariant.

We also need to relate the topology on $\mathcal{V}$ to the algebra of the sheaves. For this we need functions whose vanishing determines the topology.

Definition 3.7.

An coordinate structure on a sheaf of rings om $\mathcal{V}$ is a collection of multiplicatively closed sets $\mathcal{S}_{K^{*}/F}$ in $\mathcal{R}(F)$ for $F\in\mathcal{K}_{1}$ . For $F\in\mathcal{K}_{0}$ , we take $\mathcal{S}_{K^{*}/F}=\{0,1\}$ , and for $F\in\mathcal{K}\mathcal{R}$ we take $\mathcal{S}_{K^{*}/F}=\{0\}$ .

If there is a compatible component structure we say $\mathcal{S}_{K^{*}/F}$ is compatible if the elements of $\mathcal{S}_{K^{*}/F}$ are invariant under the action.

We will show in Section 4 below that there is a sheaf of rings and a compatible component structure arising in the Lie group context.

4. Equivariant sheaf data in the geometric context

In this section we explain describe the necessary auxiliary data $(\mathcal{R},\mathcal{W},\mathcal{S})$ from Subsection 3.B in the motivating example of a compact Lie group. It is notable that we need to use the fact we are working over the rationals to give a splitting principle to obtain the structure in full generality.

4.A. Sheaves and component structures

The simplest way to ensure a sheaf of rings with compatible component structure is to have a homomorphism $N_{G}(K)\longrightarrow N_{G}(K^{*})$ extending an inclusion $K\subseteq K^{*}$ . In this case it is straightforward since there is an induced map

W_{G}(K)=N_{G}(K)/K\longrightarrow N_{G}(K^{*})/K^{*}=W_{G}(K^{*}).

The map on identity components supplies a map $\mathcal{O}_{K}\longleftarrow\mathcal{O}_{K^{*}}$ , and it is equivariant for the map $W_{K}=W_{G}^{d}(K)\longrightarrow W_{G}^{e}(K^{*})=W_{K^{*}}$ .

We may ensure this is the case for $K\in\mathcal{K}_{0}$ since the normalizer construction is upper semi-continuous [1, 9.8] so that $N_{G}(K)\subseteq N_{G}(K^{*})$ for almost all $K\in\mathcal{K}$ . It is often reasonable for cotoral inclusions too: for example we might suppose $G=N_{G}(K^{*})$ for all $H\in\mathcal{V}_{1}$ . Often we may be able to reduce to this case. For example if $\mathcal{V}_{1}$ is a singleton $(K^{*})$ we have the familiar question of how to allow for the restriction from $G$ to $N_{G}(K^{*})$ .

However we want to cover the general case (for example if $1\in\mathcal{K}$ ). For this we proceed as follows. After conjugation, we may suppose $K\subseteq K^{*}$ with $K^{*}/K$ is a torus. We may then choose a maximal torus of $W_{G}(K)$ containing $K^{*}/K$ and write $\tilde{T}$ for its inverse image in $G$ . We see (since the cotoral relation is transitive [6]) that both $K^{*}$ and $K$ are cotoral $\tilde{T}$ . The image $\tilde{T}/K^{*}$ of $\tilde{T}$ in $W_{G}(K^{*})$ is a torus, so we can choose the maximal torus of $W_{G}(K^{*})$ to contain it. In fact $\tilde{T}/K^{*}$ is itself a maximal torus, since otherwise its inverse image $\tilde{T}_{1}$ would have image $\tilde{T}_{1}/K$ in $W_{G}(K)$ properly containing the maximal torus $TW_{G}(K)$ . The situation is depicted below.

Now if $g$ is an element of $G$ whose image in $W_{G}(K)$ normalizes $TW_{G}(K)$ , $g$ normalizes $\tilde{T}$ , and hence its image in $W_{G}(K^{*})$ normalizes $TW_{G}(K^{*})$ .

Writing $\sim$ for Borel’s rational cohomology isomorphism $N_{G}(T)\sim G$ , we have a map $\alpha$

W_{G}(K^{*})\sim N_{W_{G}(K^{*})}(TW_{G}(K^{*}))\stackrel{{\scriptstyle\alpha}}{{\longleftarrow}}N_{W_{G}(K)}(TW_{G}(K))\sim W_{G}(K).

\alpha^{*}:H^{*}(BW_{G}(K^{*}))\longrightarrow H^{*}(BW_{G}(K))

as required.

4.B. Localization

In the geometric context we take

\mathcal{E}_{K}=\{e(W)\;|\;W\in\mathrm{Rep}(N_{G}(K)),W^{K}=0\}.

\mathcal{I}_{K}=\{1,e_{K}\},

and on $\mathcal{K}\mathcal{R}$ we take the trivial multiplicative set.

5. The abelian models

In Subsection 5.A we describe the standard model $\mathcal{A}(\mathcal{V},\mathcal{R},\mathcal{W},\mathcal{S})$ formed from the space $\mathcal{V}=\mathcal{K}\cup\{K^{*}\}$ and its auxiliary data $(\mathcal{R},\mathcal{W},\mathcal{S})$ . In Subsections 5.B and 5.C we describe a stalkwise construction and the relation to the standard model. In Subsection 5.D we name two cases of somewhat different characters, based on the dimension of the stalks of the sheaf $\mathcal{R}$ . In Subsection 5.E we construct injective resolutions in the standard model, showing it is of injective dimension 1. Finally, in Subsection 5.F we make this explicit in some familiar examples (some readers may wish to flick forward to the examples as they read).

5.A. The standard model

Given $\mathcal{V}$ , a sheaf of rings, with Euler classes and a compatible component structure we may describe the standard model $\mathcal{A}(\mathcal{V},\mathcal{R},\mathcal{S},\mathcal{W})$ .

We take $\mathcal{O}_{\mathcal{K}}=\prod_{F\in\mathcal{K}}\mathcal{R}(F)$ , and we form the multiplicatively closed set

\mathcal{S}:=\{(s_{F})\;|\;s_{F}\in\mathcal{S}_{K^{*}/F}\mbox{ and }s_{F}=1\mbox{ almost everywhere }\}\subseteq\mathcal{O}_{\mathcal{K}}.

Now we consider the diagram of rings

\mathcal{O}^{\lrcorner}:=\left(\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 9.06668pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\\&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 33.06668pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathcal{R}(K^{*})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 49.6403pt\raise-5.5pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 49.6403pt\raise-28.95583pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-9.06668pt\raise-40.59358pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathcal{O}_{\mathcal{K}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 9.06668pt\raise-40.59358pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 35.46251pt\raise-40.59358pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 35.46251pt\raise-40.59358pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathcal{S}^{-1}\mathcal{O}_{\mathcal{K}}}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}\right)

The standard model $\mathcal{A}=\mathcal{A}(\mathfrak{X},\mathcal{R},\mathcal{S},\mathcal{W})$ consists of $\mathcal{O}^{\lrcorner}$ -modules

where (1) the $\mathcal{R}(K^{*})$ -module $V$ is torsion, (2) (quasicoherence) the horizontal induces an isomorphism $\mathcal{S}^{-1}N\cong P$ and (3) (extendedness) the vertical induces an isomorphism $\mathcal{S}^{-1}\mathcal{O}_{\mathcal{K}}\otimes V\cong P$ .

We refer to $N$ as the nub and $V$ as the vertex, and informally we write $N\longrightarrow\mathcal{S}^{-1}\mathcal{O}_{\mathcal{K}}\otimes V$ for the above object.

There are compatible actions of the finite groups. Thus the $F$ th component of each element of $\mathcal{S}$ is $\mathcal{W}_{F}$ invariant, there is an action of $\mathcal{W}_{F}$ on $e_{F}N$ , and of $\mathcal{W}_{K^{*}}$ on $V$ , and the map $V\longrightarrow\mathcal{S}^{-1}N\longrightarrow\mathcal{S}^{-1}e_{F}N$ is $\mathcal{W}_{F}$ -equivariant.

5.B. Separating subgroups

As described in [3] one expects two different models where the subgroups are considered separately (the ‘separated model’ where there is still a weak condition on the vertical map, and the ‘complete model’, where the $\mathcal{O}_{\mathcal{K}}$ -module is complete in a suitable sense but there is no condition on the vertical map). We begin by describing functors for separating and recombining the subgroups.

The pre-separated model $\mathcal{A}_{ps}=\mathcal{A}_{ps}(\mathfrak{X},\mathcal{R},\mathcal{S},\mathcal{W})$ consists of a torsion $\mathcal{R}(K^{*})$ -module $V$ with an action of $\mathcal{W}_{K^{*}}$ and $\mathcal{R}(F)$ -modules $N(F)$ with an action of $\mathcal{W}_{F}$ for each $F$ together with a spreading map

\sigma:V\longrightarrow\mathcal{S}^{-1}\prod_{F}N(F).

We require that the map $V\longrightarrow\mathcal{S}^{-1}\prod_{F}N(F)\longrightarrow\mathcal{S}^{-1}N(F)$ is $W_{F}$ -equivariant.

5.C. Adjunction

There is an adjunction

relating the standard and preseparated models. The constructions we describe are all consistent with the actions of the component structure, so actions will not be mentioned explicitly.

If $X=(N\longrightarrow\mathcal{S}^{-1}\mathcal{O}_{\mathcal{K}}\otimes V)$ then $eX$ has the same vertex $V$ and the separated stalk at $F$ is $N(F)=e_{F}N$ where $e_{F}$ is the idempotent supported at $F$ . The structure map is the composite

V\longrightarrow\mathcal{S}^{-1}\mathcal{O}_{\mathcal{K}}\otimes V=\mathcal{S}^{-1}N\longrightarrow\mathcal{S}^{-1}\prod_{F}N(F).

If $Y=(\{N(F)\}_{F},V,\sigma)$ then $pY$ has the same vertex $V$ and the nub $N$ is defined by the pullback square

Since $\Gamma_{\mathcal{S}}M\simeq\bigoplus_{F}e_{F}\Gamma_{\mathcal{S}}M$ , the unit gives an isomorphism $X\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}peX$ .

On the other hand the counit need not be an isomorphism. For example, if $Y$ has the property that $N(F)=0$ for all $F\in\mathcal{K}$ , we find $pY$ has nub $N=\mathcal{S}^{-1}\mathcal{O}_{\mathcal{K}}\otimes V$ , and $epY$ has stalk $\mathcal{S}^{-1}\mathcal{R}(F)\otimes V$ at $F\in\mathcal{K}$ , which need not be zero.

5.D. Two structure sheaves

Finally we consider two structure sheaves with somewhat different behaviours. These correspond to the two cases $\mathcal{K}=\mathcal{K}_{1}$ and $\mathcal{K}=\mathcal{K}_{0}$ . The present account focuses on a particularly simple choice of sheaf $\mathcal{R}$ that covers examples of immediate interest.

Definition 5.1.

The Type 0 structure sheaf has $\mathcal{R}(K^{*})=\mathbb{Q}$ and $\mathcal{R}(F)=\mathbb{Q}$ for all $F\in\mathcal{K}$ and $\mathcal{S}$ is the multiplicatively closed set of functions $\mathcal{K}\longrightarrow\{0,1\}$ with finite support and $\mathcal{K}$ is infinite.

The Type 1 structure sheaf has $\mathcal{R}(K^{*})=\mathbb{Q}$ and $R(F)=\mathbb{Q}[c]$ for all $F$ and $\mathcal{S}$ is the multiplicatively closed set of functions $e:\mathcal{K}\longrightarrow\mathbb{Z}_{\geq 0}$ with finite support.

The crudest difference between the types is about whether the standard and separated models are essentially different.

Lemma 5.2.

For the Type 0 structure sheaf, the $p-e$ adjunction is an equivalence of categories.

Proof : The statement about Type 0 structure sheaves follows since $\mathcal{S}^{-1}R(F)=0$ , so that applying $e_{F}$ to the defining pullback for $p$ we recover $(epN)(F)=N(F)$ . ∎

To explain the situation with Type 1 structure sheaf we need to recall the models of the generating basic cells

\sigma_{G}=(\mathcal{O}_{\mathcal{K}}\longrightarrow\mathcal{S}^{-1}\mathcal{O}_{\mathcal{K}}\otimes\mathbb{Q})\mbox{ and }\sigma_{F}=f_{F}(\mathbb{Q})=(\mathbb{Q}_{F}\longrightarrow 0).

Lemma 5.3.

The Type 1 structure sheaf the counit of the adjunction is not an equivalence, but it is a cellular equivalence in the sense that it induces an isomorphism of $\mathrm{Hom}(\sigma_{G},\cdot)$ and $\mathrm{Hom}(\sigma_{F},\cdot)$

Proof : With Type 1 structure sheaves, the counit is not an isomorphism in the given example of a skyscraper sheaf at $G$ .

For the cells $\sigma_{F}$ the cellular equivalence is clear since $\mathrm{Hom}(\sigma_{F},X)=\Gamma_{c}e_{F}N$ , which does not change under the functor $p$ . For the cell $\sigma_{G}$ we note that $\mathrm{Hom}(\sigma_{G},X)$ is calculated as a pullback square:

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0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 0.0pt\raise-30.3222pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 78.52367pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathrm{Hom}(\mathbb{Q},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 105.96814pt\raise-5.5pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 105.96814pt\raise-29.18445pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern 0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 194.53487pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 201.56264pt\raise-3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 201.56264pt\raise-70.2822pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-33.5792pt\raise-40.8222pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathrm{Hom}_{\mathcal{O}_{\mathcal{K}}}(\mathcal{O}_{\mathcal{K}},N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 33.5792pt\raise-40.8222pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 57.5792pt\raise-40.8222pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 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0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 7.56248pt\raise-81.91995pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 178.35709pt\raise-81.91995pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 102.96814pt\raise-81.91995pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 178.35709pt\raise-81.91995pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathcal{S}^{-1}\mathcal{O}_{\mathcal{K}}\otimes V}$}}}}}}}\ignorespaces}}}}\ignorespaces.

In other words the comparison map is the map relating to the inner and outer pullbacks in the diagram

Since the square is a pullback by definition, the map is an isomorphism. ∎

5.E. Homological algebra of the standard model

It is an easy exercise to understand this abelian category, but it is useful to work it through for reference in higher dimensional contexts. By Maschke’s Theorem the actions of the component groups $\mathcal{W}_{K}$ do not affect the homological dimension.

Lemma 5.4.

If the structure sheaf is of Type 0 or Type 1, the standard abelian category $\mathcal{A}$ is of injective dimension 1.

Remark 5.5.

(a) One point of writing this down is to highlight the slightly different formal structure of the proof in Type 0 and Type 1.

(b) In Remark 5.8 below that the argument is easily adapted to give an estimate of the injective dimension when $\mathcal{R}$ is a more general diagram of polynomial rings, but we take the opportunity to be a bit more precise in the present case.

Proof : We may write down enough injectives. Indeed, for any graded $\mathbb{Q}$ -vector space $W$ with an action of $\mathcal{W}_{K^{*}}$ , the object

e(W)=\left(\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 24.59445pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\\&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 64.77223pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 73.1889pt\raise-3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 73.1889pt\raise-27.05444pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-24.59445pt\raise-38.6922pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathcal{S}^{-1}\mathcal{O}_{\mathcal{K}}\otimes W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 24.59445pt\raise-38.6922pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 48.59445pt\raise-38.6922pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 48.59445pt\raise-38.6922pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathcal{S}^{-1}\mathcal{O}_{\mathcal{K}}\otimes W}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}\right)

has the property $\mathrm{Hom}(X,e(W))=\mathrm{Hom}(V,W)$ . It is therefore injective.

Similarly if $T_{F}$ is a torsion $R(F)$ -module, the object

f_{F}(T_{F})=\left(\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 8.80577pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\\&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 32.80577pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 38.30577pt\raise-3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 38.30577pt\raise-28.1511pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-8.80577pt\raise-37.59554pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{T_{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 8.80577pt\raise-37.59554pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 32.80577pt\raise-37.59554pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 32.80577pt\raise-37.59554pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}\right)

lies in the standard model and has the property

\mathrm{Hom}(X,f_{F}(T_{F}))=\mathrm{Hom}(V(F),T_{F}).

It is therefore injective if $T_{F}$ is an injective $\mathbb{Q}[c]$ -module.

Now suppose $X=(N\longrightarrow\mathcal{S}^{-1}\mathcal{O}_{\mathcal{K}}\otimes V)$ . For an arbitrary object $X$ we may take the map $X\longrightarrow e(V)$ corresponding to the identity on $V$ ; its kernel is at the nub, where it is a module $T$ with $\mathcal{S}^{-1}T=0$ . In either Type 0 or Type 1, this means that $T=\bigoplus_{F}e_{F}T$ (direct sum!). In Type 0, $e_{F}T=I_{F}$ is already injective, and in Type 1 we choose a resolution $0\longrightarrow e_{F}T\longrightarrow I_{F}\longrightarrow J_{F}\longrightarrow 0$ . In any case we may construct a monomorphism

\mu:X\longrightarrow e(V)\oplus\prod_{F}f_{F}(I_{F}).

It is at this point that the two cases differ.

In the Type 0 case, we consider the product $\prod_{F}f_{F}(I_{F})$ : the value at $K^{*}$ is $\mathcal{S}^{-1}\prod_{F}I_{F}$ , which is non-zero if infinitely many terms $I_{F}$ are nonzero. The cokernel of $\mu$ is thus $e(W)$ for some vector space $W$ , and we have found an injective resolution of length 1. We observe that not all objects are injective since $\mathrm{Ext}^{1}(e(\mathbb{Q}),\underline{\mathbb{Q}})=\mathcal{S}^{-1}\prod_{F}\mathbb{Q}\neq 0$ .

In the Type 1 case, we may again proceed with the resolution, but as in Type 0, $\prod_{F}f_{F}(I_{F})$ may be non-zero at $K^{*}$ . We may instead embed $T$ in the injective $\mathcal{E}$ -torsion module $\bigoplus_{F}f_{F}(I_{F})$ . This sum of injectives is also injective, since $\mathrm{Ext}(A,\bigoplus_{F}I_{F})=0$ for all $\mathcal{O}_{\mathcal{K}}$ -modules $A$ which occur as a nub (as in [8, 5.3.1]). This is clear for torsion modules $A$ , and it is clear for $A=\mathcal{O}_{\mathcal{K}}$ , and follows in general from this.

Accordingly, we have found an injective resolution of length 1. It is easy to see that the are non-split extensions (for example the short exact sequence of torsion $R(F)$ -modules $0\longrightarrow\mathbb{Q}\longrightarrow\mathbb{Q}[c]^{\vee}\longrightarrow\Sigma^{-2}\mathbb{Q}[c]^{\vee}\longrightarrow 0$ gives non-split extensions on applying $f_{F}(\cdot)$ for any $F$ ). ∎

Remark 5.6.

We could have used the same argument for Type 0 as for Type 1, and we would still have reached the conclusion that the category is of injective dimension 1. The point is that to use that argument we need to think in terms of modules over the ring $\mathcal{O}_{\mathcal{K}}$ (rather than sheaves over $\mathcal{K}^{*}$ ), and the ring is not of injective dimension 0. The given argument relying on the fact that stalks are fields seemed more transparent.

It is useful to record the following criterion for injectivity for later reference.

Lemma 5.7.

In the Type 0 case, any object $X=(N\longrightarrow\mathcal{S}^{-1}\mathcal{O}_{\mathcal{K}}\otimes V)$ for which only finitely many of the modules $T_{H}=e_{H}(\Gamma_{\mathcal{S}}N)$ are non-zero is injective.

In the Type 1 case, a module $X=(N\longrightarrow\mathcal{S}^{-1}\mathcal{O}_{\mathcal{K}}\otimes V)$ with the property that $e_{F}N$ is divisible for all $F\in\mathcal{K}$ is injective.

Proof : In the Type 0 case, only finitely many of the terms $f_{F}(T_{F})$ are non-zero, so the product of them has zero vertex.

In the Type 1 case, the map $X\longrightarrow e(V)\oplus\bigoplus_{F}f_{F}(I_{F})$ constructed in the previous lemma is an isomorphism. The point is that the kernel $T$ is injective and so we can take $I_{F}=T_{F}$ and $J_{F}=0$ . ∎

Remark 5.8.

Essentially the same argument will show that the algebraic model is of finite injective dimension if the rings $\mathcal{R}(K^{*})$ and $\mathcal{R}(K)$ are polynomial rings. Indeed, the full subcategory of objects $e(W)$ is equivalent to $\mathcal{R}(K^{*})$ modules. For an arbitrary object $X$ we consider the map $X\longrightarrow e(W)$ where $W$ is the value of $X$ at $K^{*}$ and then obtain two exact sequences

0\longrightarrow K\longrightarrow X\longrightarrow\overline{X}\longrightarrow 0\mbox{ and }0\longrightarrow\overline{X}\longrightarrow e(W)\longrightarrow C\longrightarrow 0

where $K=f(K^{\prime})$ and $C=f(C^{\prime})$ for torsion $\mathcal{O}_{\mathcal{K}}$ -modules $K^{\prime},C^{\prime}$ .

If $\mathcal{R}(K^{*})$ is a polynomial ring on $a$ variables and $\mathcal{R}(K)$ has at most $b$ variables for $K\in\mathcal{K}$ then $\mathrm{injdim}(f(C^{\prime})),\mathrm{injdim}(f(K^{\prime}))\leq b$ and hence $\mathrm{injdim}(\overline{X})\leq\max(a,b+1)$ , and therefore the same bound applies to $\mathrm{injdim}(X)$ .

5.F. Examples

First, there are three examples where the answer is well known [8, 7].

Example 5.9.

(i) The very simplest case comes from the circle group $G=T$ . This has a single block, and we take $\mathcal{K}=\mathcal{C}$ to be the set of finite cyclic subgroups (in bijection to the positive integers by order of subgroup).

Then $\mathfrak{X}_{T}=\mathfrak{X}(\mathcal{C})$ , we take the Type 1 structure sheaf $\mathcal{R}(F)=\mathbb{Q}[c]$ for all $F$ , and the component structure is trivial ( $W_{F}=W_{G}=1$ ). The set $\mathcal{S}$ consists of Euler classes of representations $V$ with $V^{T}=0$ , so that $e(V)(F)=c^{\dim(V^{F})}$ .

The model $\mathcal{A}$ is the standard model $\mathcal{A}(T)$ for rational $T$ -spectra.

(ii) Next we may take $G=O(2)$ and look at the toral block, again taking $\mathcal{K}=\mathcal{C}$ . Now the compactification again consists of closed subgroups of the circle $SO(2)$ . (The compactifying point $K^{*}$ in this example is $SO(2)$ ). The rings are Type 1 as in the previous example, but we take each of the groups $\mathcal{W}_{F}$ and $\mathcal{W}_{G}$ to be of order 2 and the maps $\mathcal{W}_{F}\longrightarrow\mathcal{W}_{G}$ to be isomorphisms. The action of $\mathcal{W}_{F}$ on $\mathbb{Q}[c]$ takes $c$ to $-c$ .

The model $\mathcal{A}$ is the model $\mathcal{A}(O(2)|toral)=\mathcal{A}(SO(2)][W]$ for the toral block of rational $O(2)$ -spectra.

(iii) The model for the toral block of $G=Pin(2)$ is precisely the same as in Part (ii).

For the other cases we will consider only the full subgroups.

Example 5.10.

(i) For $G=T\times C_{2}$ . This has two blocks. The first consists of subgroups of $T$ , and the second (considered here) consists of full subgroups. We take $\mathcal{K}$ to consist of the full subgroups of $T\times C_{2}$ and $K^{*}=G$ is the compactifying point. The subgroups in $\mathcal{K}$ are either $C_{n}\times C_{2}$ or else the cyclic groups $C_{2n}^{\prime}$ generated by $(e^{2\pi i/2n},-1)$ .

This example is another Type 1 example, essentially like $\mathcal{A}(T)$ except that $\mathcal{C}$ has been replaced by a different countable set.

(ii) For $G=O(2)$ the space of subgroups again divides into two blocks. The first consists of subgroups of $SO(2)$ , and the second (considered here) consists of full subgroups. We take $\mathcal{K}=\mathcal{D}$ to consist of the conjugacy classes of dihedral subgroups and $K^{*}=G$ . This time the ring is Type 0, with $\mathcal{R}(F)=\mathbb{Q}$ for all $F$ ; the groups $\mathcal{W}_{F}$ are all of order 2, and $\mathcal{W}_{G}$ is trivial.

The multiplicatively closed set $\mathcal{S}$ consists of the characteristic functions of the cofinite sets of $\mathcal{D}$ .

The model $\mathcal{A}$ is the model $\mathcal{A}(O(2)|\mathrm{full})$ for the dihedral component of rational $O(2)$ -spectra.

(iii) For $G=Pin(2)$ the model is essentially the same as that for $O(2)$ . The toral block is identical to that of $O(2)$ and the block of full subgroups is like that for except that $\mathcal{K}=\mathcal{Q}$ consists of the quaternion subgroups.

Put together we have the following algebraic models for 1-dimensional groups.

\mathcal{A}(T\times W)\simeq\mathcal{A}(T)[W]\times\mathcal{A}(T)

\mathcal{A}(O(2))\simeq\mathcal{A}(SO(2))[W]\times W\mbox{-Sh}/(\mathcal{D}^{\#})

\mathcal{A}(Pin(2))\simeq\mathcal{A}(Spin(2))[W]\times W\mbox{-Sh}/(\mathcal{Q}^{\#})

There is an equivalence $\mathcal{A}(O(2))\simeq\mathcal{A}(Pin(2))$ , but note this is given by different bijections on the two components (in the non-toral part it is natural to choose the bijection $Q_{4a}\leftrightarrow D_{2a}$ quotienting out the central subgroup of order 2, but in the toral part we must choose a bijection between cyclic subgroups of $SO(2)$ and those of $Spin(2)$ , so we cannot use the quotient map).

Example 5.11.

(i) We may consider the block corresponding to full subgroups of the normalizer in the maximal torus in $SU(3)$ . This consists of $\mathcal{W}$ -sheaves over $\mathcal{K}^{\#}$ where $\mathcal{K}$ is the discrete space of conjugacy classes of finite subgroups and $\mathcal{W}_{F}$ is a group of order 3 for all $F$ and $\mathcal{W}_{K^{*}}=1$ . In this case the full subgroups are $T^{2}[n]\rtimes\Sigma_{3}$ , and therefore in bijection with the positive integers (see [13, Section 13] for more details).

(ii) We may consider the block corresponding to full subgroups of $T^{2}\rtimes C_{3}$ (a subgroup of $T^{2}\rtimes\Sigma_{3}$ ). Again $\mathcal{W}$ -sheaves over $\mathcal{K}_{3}^{\#}$ where $\mathcal{K}_{3}$ is the discrete space of conjugacy classes of finite subgroups and agan $\mathcal{W}_{F}$ is of order 3 for all $F$ and $\mathcal{W}_{K^{*}}=1$ . The set $\mathcal{K}_{3}$ is described in [13, Section 12].

Part II Topology

6. The abelian models are Quillen models: general strategy

We will show that the abelian categories provide models in all the cases we study. The structure of the argument is the same as that for tori in [24]: we show that the sphere spectrum is the pullback of rings which are isotropically concentrated and formal in a strong sense.

6.A. Outline

The core of the proof is the fact that the sphere spectrum $S^{0}$ is a pullback of an isotropic cube, using the general inductive argument of [2, 8.1], adapted to the non-Noetherian setting.

This allows us to outline the proof: we give a symbolic description and then explain the notation and discuss the ingredients in the argument. Details will be given in the rest of the paper.

\mbox{$G$-{\bf spectra}}|\mathcal{V}\stackrel{{\scriptstyle 0}}{{\simeq}}S^{0}\mbox{-mod}-\mbox{$G$-{\bf spectra}}|\mathcal{V}\stackrel{{\scriptstyle 1}}{{\simeq}}\mathrm{Cell}((S^{0})^{\lrcorner}\mbox{-mod}-\mbox{$G$-{\bf spectra}})\\ \stackrel{{\scriptstyle 2}}{{\simeq}}\mathrm{Cell}((\widetilde{(S^{0})}^{\lrcorner})^{G}\mbox{-mod}-\mathcal{W}-\mathrm{spectra})\stackrel{{\scriptstyle 3}}{{\simeq}}\mathrm{Cell}(C^{G}_{*}(\widetilde{(S^{0})}^{\lrcorner})\mbox{-mod}-\mathbb{Q}[\mathcal{W}]\mbox{-mod})\\ \stackrel{{\scriptstyle 4}}{{\simeq}}\mathrm{Cell}(\pi^{G}_{*}(\widetilde{(S^{0})}^{\lrcorner})\mbox{-mod}-\mathbb{Q}[\mathcal{W}]\mbox{-mod})\stackrel{{\scriptstyle 5}}{{\simeq}}DG-\mathcal{A}(G|\mathcal{V})

Equivalence 0 simply uses the fact that $G$ -spectra are modules over the sphere spectrum. Equivalence 1 uses the fact [23, 4.1] that the category of modules over a homotopy pullback ring is equivalent to the cellularization of the category of generalized diagrams over the individual modules, together with the fact that the localized sphere is the pullback.

From Equivalence 2 onwards, we introduce variants on the terms of the initial cube so as to keep track of the finite Weyl groups. The cospan $(\widetilde{S^{0}})^{\lrcorner}$ of $G$ -spectra replaces terms of $(S^{0})^{\lrcorner}$ by coinductions which vary by subgroup. The new objects have the property that their $G$ -fixed point spectra are products of spectra with homology $H^{*}(BW_{G}^{e}(K))$ for relevant subgroups $K$ and the category of modules take values in the corresponding product of categories with $W_{G}^{d}(K)$ -action. The cellularization will pick out the appropriate abelian category.

Equivalence 2 uses the results of [22]. To explain, for each subgroup $K$ we consider the normalizer $N=N_{G}(K)$ , the Weyl group $W=N_{G}(K)/K$ with identity component $W^{e}$ and discrete quotient $W^{d}$ . Finally, we write $N^{f}$ for the inverse image of $W^{e}$ in $N$ . With this notation, there are equivalences

DE\langle K\rangle\mbox{-mod}-G-spectra\simeq DE\langle K\rangle\mbox{-mod}-N-spectra\simeq\\ DE\langle K\rangle^{K}\mbox{-mod}-W-spectra\simeq DE\langle K\rangle^{N^{f}}\mbox{-mod}-W^{d}-spectra

where the first is the forgetful map and the second is passage to $K$ fixed ponts (an equivalence because $DE\langle K\rangle$ lies over $K$ ) and the third is passage to $N^{f}$ -fixed points under $W^{e}$ (an equivalence by the Eilenberg-Moore theorem because $W^{e}$ is connected). It requires some care to assemble these equivalences when $\mathcal{K}$ is infinite, and we will explain in Section 8.

Equivalence 3 follows from Shipley’s Theorem [26], and is easily adapted to the type of diagram we have. Equivalence 4 is a formality statment. Finally, Equivalence 5 folows from the Cellular Skeleton Theorem, which will identify the cellularization of the algebraic category of modules with the derived category of an abelian category.

We will first expla The abelian models are Q models gen star in the argument for individual subgroups and then discuss how to assemble these for the whole category.

7. Modules over completions and completions of modules

In this section we consider the stalks over a single subgroup. We deal with two particular matters we address. The first is reflected in the title: the splicing data comes about because we have models for arbitrary modules over the completed rings, not just complete modules. The primitive example to bear in mind is that there many more modules over $\mathbb{Z}_{p}^{\wedge}$ than there are complete modules (for example $\mathbb{Z}_{p}^{\wedge}[1/p]$ ) . Roughly speaking the complete modules are the ingredients but modules over the completed ring are used in the splicing. The second matter is that we clarify the necessary generators when the group is not connected in a way that will be important when we allow infinitely many subgroups.

7.A. Trivial coisotropy (connected)

Starting with the simplest case we consider a connected compact Lie group $G$ and focus on trivial isotropy or coisotropy. First of all, there is an equivalence

\mbox{free-$G$-spectra}\simeq\mbox{cofree-$G$-spectra}

using the functors $(\cdot)\wedge EG_{+}$ and $F(EG_{+},\cdot)$ . Similarly in algebra

\mbox{torsion-$H^{*}(BG)$-modules}\simeq\mbox{complete-$H^{*}(BG)$-modules}

using (derived) torsion and derived completion functors $\Gamma_{I}(\cdot)$ and $\Lambda_{I}(\cdot)$ . (A model for torsion $H^{*}(BG)$ -modules is given by DG modules in the abelian category of torsion modules. In the complete case, the category of $I$ -adically complete modules is not abelian, so one needs to use the abelian category of $L_{0}^{I}$ -complete modules [25].)

We have equivalences

We will focus on the right hand end (cofree spectra and complete modules).

One method of proving the vertical equivalences is to observe that $S^{0}\longrightarrow DEG_{+}$ is a non-equivariant equivalence, and hence

\mbox{cofree-$G$-spectra}=L_{EG_{+}}(S^{0}\mbox{-mod-$G$-spectra})\simeq L_{EG_{+}}(DEG_{+}\mbox{-mod-$G$-spectra}).

We may then prove $DEG_{+}$ is formal.

Proposition 7.1.

We have equivalences

DEG_{+}\mbox{-mod-$G$-spectra}\simeq DBG_{+}\mbox{-mod-spectra}\simeq C^{*}(BG)\mbox{-mod}\simeq\mbox{$H^{*}(BG)$-mod}

Proof : Writing $R=DEG_{+}$ , the first equivalence is in [22], using the fixed point functor $R\mbox{-mod-$G$-spectra}\longrightarrow R^{G}\mbox{-mod-spectra}$ and its left adjoint. This is an equivalence by the Cellularization Principle [20] provided $R$ -modules are generated by $R$ .

This is true when $G$ is connected (there are various proofs, but one giving this generality is in [11]). ∎

Theorem 7.2.

If $G$ is connected, we have a commutative square

Proof : In summary, we have the equivalences

∎

7.B. Trivial coisotropy (disconnected)

We explain how to cover the case of disconnected groups.

Theorem 7.3.

For an arbitrary group $G$ we have a commutative square

Proof : From the Eilenberg-Moore spectral sequence $DEG_{+}$ -modules are generated by $G^{d}/F_{+}$ -modules where $F$ is a finite subgroup of $G^{d}$ . The map of $G_{d}$ -spectra $S^{0}\longrightarrow eS^{0}$ is a non-equivariant equivalence where $e\in A(G_{d})$ is the idempotent supported at 1. Thus $S^{0}\wedge DEG_{+}\longrightarrow eS^{0}\wedge DEG_{+}$ is an equivalence. The result follows from the fact that free $G^{d}$ -spectra are generated by $G^{d}_{+}$ . ∎

8. Treating infinitely many subgroups at once

In the previous section, we argued that $DEG_{+}$ is strongly isotropically formal in the sense that we could take fixed points for a single subgroup and obtain a formal ring spectrum. It is rather easy to adapt this to $DE\langle H\rangle$ for any single subgroup $H$ , by the method of [12]. We recall this argument below, but the main point of the present section is to explain how to deal with infinite products of such spectra.

In the previous section we explained how to deal with a single subgroup, which we took to be the trivial group for convenience. The argument involved a subgroup $K$ and some associated subgroups. Principally this means its normalizer $N=N_{G}(K)$ , and its Weyl group $W=N/K$ , but we also need to mention the identity component $W^{e}$ of $W$ , and its discrete quotient $W^{d}=W/W^{e}$ , and finally $N^{f}$ , which is the inverse image of $W^{e}$ in $N$ , so that $N/N^{f}\cong W/W^{e}=W^{d}$ .

The argument is that (for suitable ring spectra $R$ ) we have equivalences

R\mbox{-mod}-G-\mathrm{spectra}\stackrel{{\scriptstyle(1)}}{{\simeq}}R\mbox{-mod}-N-\mathrm{spectra}\stackrel{{\scriptstyle(2)}}{{\simeq}}R^{K}\mbox{-mod}-W-\mathrm{spectra}\stackrel{{\scriptstyle(3)}}{{\simeq}}R^{N^{f}}\mbox{-mod}-W^{d}-\mathrm{spectra}

Equivalence (1) is the forgetful map from $G$ -spectra to $N$ -spectra, and relies on fusion of $N$ -conjugacy classes being favourable.

Equivalence (2) is passage to $K$ -fixed points, and relies on $R$ having geometric isotropy consisting of subgroups containing $K$ . Equivalence (3) is passage to categorical fixed points, and uses the Eilenberg-Moore theorem for the connected group $W^{e}$ .

We now wish to treat many subgroups $K$ at once, but the intermediate categories and functors in the above argument involve $N=N_{G}(K)$ and therefore depend on $K$ . We explain here that we may instead factorize the composite so that the dependency on $K$ occurs in the ring and whilst the categories and functors are independent of $K$ .

The subtlety is that both $N$ and $N^{f}$ play a role. In our examples $R\simeq F_{N}(G_{+},R)$ but $R\not\simeq F_{N^{f}}(G_{+},R)$ (consider $K=D_{2n}$ insider $G=O(2)$ (or even inside $G=D_{4n}$ )). Thus we need to use $N$ . On the other hand, taking $N$ -fixed points loses the action of the $W^{d}$ . The solution is to reinsert the action of $W^{d}$ as an endomorphism of the functor $F_{N^{f}}(G_{+},\cdot)$ .

Lemma 8.1.

The diagram

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commutes where the top horizontal is $F_{N^{f}}(G_{+},\bullet)$ . The composite is a right Quillen functor.

Proof : The commutativity is just the formula

F_{N^{f}}(G_{+},X)^{G}\simeq X^{N^{f}}.

All functors in the diagram are right adjoints. ∎

The maps in the diagram are equivalences under various circumstances.

Lemma 8.2.

(i) The restriction functor

R\mbox{-mod-$G$-spectra}\langle\mathcal{V}\rangle\longrightarrow R\mbox{-mod-$N$-spectra}\langle\mathcal{V}\rangle

is an equivalence if, for all $K\in\mathcal{V}\cap\mathrm{supp}(R)$ a containment ${}^{g}K\subseteq N$ implies $g\in N$ , so that $(G/N)^{K}$ is a singleton.

(ii) The map

F_{N}(G_{+},R)\mbox{-mod}-G-\mathrm{spectra}\longrightarrow F_{N^{f}}(G_{+},R)\mbox{-mod}-G-\mathrm{spectra}[W^{d}]

is an equivalence if $R$ is $N^{f}$ -free.

(iii) The map

\Psi^{K}:R\mbox{-mod}-N-\mathrm{spectra}\langle\mathcal{V}\rangle\longrightarrow R^{K}\mbox{-mod}-W-\mathrm{spectra}\langle\mathcal{V}\rangle

is an equivalence if every element of $\mathrm{supp}(R)\cap\mathcal{V}$ contains $K$ .

(iv) The map

\Psi^{W^{e}}:R^{K}\mbox{-mod}-W-\mathrm{spectra}\langle\mathcal{V}\rangle\longrightarrow R^{N^{f}}\mbox{-mod}-W^{d}-\mathrm{spectra}\langle\mathcal{V}\rangle

is an equivalence if $R^{N^{f}}$ is $W^{e}$ -free and $W^{d}_{+}\wedge R$ generates $R^{K}\mbox{-mod}-W-\mathrm{spectra}$ .

Proof : Part (i) follows since $G/N_{+}\wedge M\simeq G_{+}\wedge_{N}M\longrightarrow M$ is an equivalence for all $R$ -modules $M$ .

Part (ii) is the fact that free $W^{d}$ -spectra are generated by $W^{d}_{+}$ and hence by Morita theory equivalent to spectra with a $W^{d}$ -action.

Part (iii) is [22, Theorem 7.1].

Part (iv) follows from [21]. ∎

9. The abelian models are Quillen models in dimension 1

In this section we prove a general theorem. Amongst the small cases covered are the following familiar examples

•

$G=T$ , the circle group
•

$G=O(2)$ , toral subgroups $\mathcal{V}^{O(2)}_{1}$
•

$G=O(2)$ , full subgroups $\mathcal{V}^{O(2)}_{W}$
•

$G=Pin(2)$ , toral subgroups $\mathcal{V}^{Pin(2)}_{1}$
•

$G=Pin(2)$ , full subgroups $\mathcal{V}^{Pin(2)}_{W}$
•

$G=T\times C_{2}$ , toral subgroups $\mathcal{V}^{T\times C_{2}}_{1}$
•

$G=T\times C_{2}$ , full subgroups $\mathcal{V}^{T\times C_{2}}_{W}$
•

$G$ of rank 2 and $W=C_{3}$ or $\Sigma_{3}$ and $\Lambda_{0}$ non-trivial.

We saw in Lemma 5.4 that $\mathcal{A}(G|\mathcal{V})$ is of finite injective dimension in these cases and hence $DG-\mathcal{A}(G|\mathcal{V})$ admits the injective model structure with homology isomorphisms as weak equivalences.

Theorem 9.1.

Suppose $G$ is a compact Lie group and $\mathcal{V}$ is obtained from a set $\mathcal{K}$ of conjugacy classes of finite subgroups of $G$ by adjoining the conjugacy class of a subgroup $K^{*}$ with finite Weyl group. Then there is a Quillen equivalence

\mbox{$G$-{\bf spectra}}|\mathcal{V}\simeq DG-\mathcal{A}(G|\mathcal{V}).

where

\mathcal{A}(G|\mathcal{V})\simeq\mathcal{A}(\mathcal{K},\mathcal{R},\mathcal{S},\mathcal{W}),

where the data is as in Section 4

Since we know how to deal with 0-dimensional summands, we suppose $\mathcal{K}\mathcal{R}=\emptyset$ . The first step is to express the category as a pullback, which we do by giving a pullback square of ring spectra. We will then calculate the homotopy of the ring spectra, and their strong isotropic formality. Finally we will prove the algebraic category has the model as a cellular skeleton.

9.A. The pullback square

Broadly speaking there are two ways of presenting the argument. (1) Giving a pullback square of rings in $G$ -spectra and then localizing to give a pullback square in $G$ -spectra with geometric isotropy $\mathcal{V}$ . (2) Constructing the local objects directly and showing they form a pullback square. We will adopt the first option, since we quickly get a pullback square and stay with familiar objects as long as possible. When the filtration is more complicated, one can imagine that (2) may be more attractive.

For our present case, the pullback square is extremely familiar: we begin with the homotopy Tate square

where $\mathcal{F}$ is the family of all finite subgroups. This is a pullback square in $G$ -spectra, and hence also in the category of spectra over $\mathcal{V}$ . Furthermore, the objects are all commutative rings.

In the present case it is very easy to find more economical representatives of the homotopy types in the category of spectra over $\mathcal{V}$ .

Lemma 9.2.

In the category of $G$ -spectra over $\mathcal{V}$ we have equivalences

(i) $\widetilde{E}{\mathcal{F}}\simeq E\langle K^{*}\rangle$ and

(ii) $E\mathcal{F}_{+}\simeq\bigvee_{K\in\mathcal{K}}E\langle K\rangle=E\langle\mathcal{K}\rangle$

Proof : By hypothesis, $\mathcal{V}=\mathcal{K}\amalg\{K^{*}\}$ for a collection $\mathcal{K}$ of finite subgroups, not usually a family.

Since, $K^{*}$ is the only infinite subgroup in $\mathcal{V}$ , Part (i) is immediate.

By [5, Theorem 1.2], there is an equivalence $E\mathcal{F}_{+}\simeq\bigvee_{F\in\mathcal{F}}E\langle\mathcal{F}\rangle$ . The inclusion of $E\langle\mathcal{K}\rangle$ is a $\mathcal{V}$ -equivalence. ∎

Remark 9.3.

Using the splitting cited in the lemma, there is an idempotent selfmap $e_{\mathcal{K}}$ of $E\mathcal{F}_{+}$ , with $e_{\mathcal{K}}E\mathcal{F}_{+}\simeq E\langle\mathcal{K}\rangle$ . We then have $e_{\mathcal{K}}DE\mathcal{F}_{+}\simeq\prod_{K\in\mathcal{K}}DE\langle K\rangle\simeq DE\langle\mathcal{K}\rangle$ .

The spectra $DE\mathcal{F}_{+}$ and $\prod_{K\in\mathcal{K}}DE\langle K\rangle$ are usually not equivalent in spectra over $\mathcal{V}$ , but we show their difference is entirely concentrated at $K^{*}$ .

Corollary 9.4.

The square

is a homotopy pullback in $G$ -spectra over $\mathcal{V}$ .

Proof : This follows since the horizontal fibres of the original pullback are $E\mathcal{F}_{+}$ and in the second, the fibre of the lower horizontal is $E\langle\mathcal{K}\rangle$ . ∎

9.B. Coefficient rings

It remains to study the categories of modules over each of the terms. For this we adapt the work Sections 7 and 8. For individual subgroups we have already seen modules over $\mathcal{R}(K)=DE\langle K\rangle$ in $G$ -spectra are equivalent to modules over $\mathcal{R}(K)^{N^{f}_{G}(K)}\simeq C^{*}(BW_{G}^{e}(K))$ , which is a non-equivariant spectrum with an action of $W_{G}^{d}(K)$ .

For each $K\in\mathcal{K}$ we consider the $G$ -spectrum $\tilde{\mathcal{R}}(K)=F_{N^{f}_{G}(K)}(G_{+},\mathcal{R}(K))$ and then let $R=\prod_{K\in\mathcal{K}}\tilde{\mathcal{R}}(K)$ . We have

R^{G}=\left(\prod_{K}F_{N_{G}^{f}(K)}(G_{+},\mathcal{R}(K))\right)^{G}\simeq\prod_{K}\mathcal{R}(K))^{N_{G}^{f}(K)}

and

\mathcal{R}(K)^{N_{G}^{f}(K)}=(\mathcal{R}(K)^{K})^{W_{G}^{e}(K)}=(DEN/K_{+})^{W_{G}^{e}(K)}=D\tilde{B}W_{G}^{e}(K).

Stripping away specifics, we will argue that $\mathcal{R}(K)$ -module $G$ -spectra are equivalent to modules over a commutative ring $\pi_{*}^{N^{f}}(\mathcal{R}(K))$ for a subgroup $N^{f}$ of $G$ in a category of representations of a finite group $W^{d}$ . There are steps in equivariant homotopy, leading from $\mathcal{R}(K)$ -modules in $G$ -spectra to $\mathcal{R}(K)^{N^{f}}$ -modules in spectra with an action of $W^{d}$ . Shipley’s Theorem shows this is equivalent to $\mathbb{Q}[W^{d}]$ modules over a DGA with homology $\pi_{*}(R^{N^{f}})$ , and this is shown to be formal.

From an expository point of view, the first thing to identify is the target ring $\pi_{*}(\mathcal{R}(K)^{N^{f}})$ , even though it doesn’t play a role in the argument until the end.

Lemma 9.5.

We have

\pi^{G}_{*}(\tilde{\mathcal{R}}(1))=\pi^{G}_{*}(DE\langle\mathcal{K}\rangle)=\prod_{K}H^{*}(BW_{G}^{e}(K))=\mathcal{O}_{\mathcal{K}}

\pi^{G}_{*}(\tilde{\mathcal{R}}(K^{*}))=\pi^{G}_{*}(\mathcal{R}(K^{*}))=\pi^{G}_{*}(E\langle K^{*}\rangle)=H^{*}(BW_{G}^{e}(K^{*}))=\mathcal{O}_{K^{*}}=\mathbb{Q}

Proof : The first part is straightforward from the isomorphisms

\pi^{G}_{*}(DE\langle K\rangle)=[S^{0},DE\langle K\rangle]^{G}_{*}=[E\langle K\rangle,S^{0}]^{G}=[E\langle K\rangle,S^{0}]^{N}=\\ [EN/K_{+},S^{0}]^{N/K}=[BN/K_{+},S^{0}]=H^{*}(BN/K)

The second part is simple since $K^{*}$ has finite Weyl group (otherwise, we would need to take careful account of the fact the calculation is in the $\mathcal{V}$ -local category). ∎

Lemma 9.6.

There is a multiplicatively closed set $\mathcal{S}$ so that

\pi^{G}_{*}(E\langle K^{*}\rangle\wedge DE\langle\mathcal{K}\rangle)=\mathcal{S}^{-1}\prod_{K\in\mathcal{K}}H^{*}(BW_{G}^{e}(K))=\mathcal{S}^{-1}\mathcal{O}_{\mathcal{K}}

To describe the localization we split the product up into three factors

\mathcal{O}_{\mathcal{K}}=\mathcal{O}_{\mathcal{K}_{1}}\times\mathcal{O}_{\mathcal{K}_{0}}\times\mathcal{O}_{\mathcal{K}\mathcal{R}}

•

On $\mathcal{K}_{1}$ the multiplicatively closed set is $\mathcal{E}=\{e(W)\;|\;W\in\mathrm{Rep}(N_{G}(K^{*})),W^{K^{*}}=0\}$
•

On $\mathcal{K}_{0}$ , the multiplicatively closed set is $\mathcal{I}=\{i(U)\;|\;U\subseteq\mathcal{K}_{0}\mbox{ closed in }\mathcal{V}\}$ .
•

On $\mathcal{K}\mathcal{R}$ , the multiplicatively closed set is $\{0\}$ .

Proof : The splitting and the $\mathcal{K}\mathcal{R}$ part are clear.

We start with the $\mathcal{K}_{0}$ part. Since supports are closed under cotoral specialization, any idempotent nonzero on $K^{*}$ will be nonzero on $\mathcal{K}_{1}$ . Next we note that for any $\mathcal{V}$ -closed subset $U$ of $\mathcal{K}_{0}$ there is an idempotent $e_{U}$ with support $U\cup\{K^{*}\}\cup\mathcal{K}_{1}$ because the latter is open and closed in $\mathcal{V}$ . We may then observe

E\langle K^{*}\rangle\wedge E\langle\mathcal{K}_{0}\rangle=\mathop{\mathop{\mathrm{lim}}\limits_{\rightarrow}}\nolimits_{U}e_{U}E\langle\mathcal{K}_{0}\rangle,

because $e_{U}E\langle K\rangle\simeq*$ for $K\not\in V$ .

For the $\mathcal{K}_{1}$ part we will show directly that the algebraic localization

DE\langle\mathcal{K}\rangle\longrightarrow\mathcal{E}^{-1}DE\langle\mathcal{K}\rangle

is the isotropic localization. For this we need to show that $\mathcal{E}^{-1}DE\langle\mathcal{K}\rangle$ is $K$ -equivariantly contractible for all $K\in\mathcal{K}$ and that $\Gamma_{\mathcal{E}}DE\langle\mathcal{K}\rangle$ has non-equivariantly contractible geometric $K^{*}$ -fixed points.

For the first statement, it suffices to say that for each $K\in\mathcal{K}$ there is an element $e\in\mathcal{E}$ so that $e$ is $K$ -equivariantly null. For this we choose $e(V)$ for a representation of the torus $K^{*}/K$ . For the second statement, we may work $K^{*}$ -equivariantly, and we note that inverting $\mathcal{E}$ commutes with restriction of subgroups. Thus we have a $K^{*}$ -equivariant equivalence $\mathcal{E}^{-1}DE\langle\mathcal{K}\rangle\simeq S^{\infty V(K^{*})}\wedge DE\langle\mathcal{K}\rangle$ and $\Gamma_{\mathcal{E}}DE\langle\mathcal{K}\rangle\simeq E\mathcal{P}_{+}\wedge DE\langle\mathcal{K}\rangle$ , which has not $K^{*}$ -fixed points as required.

∎

Remark 9.7.

In the $\mathcal{K}_{1}$ part one might hope to make an argument using representation theory to give a model of $\widetilde{E}{\mathcal{F}}\wedge DE\langle\mathcal{K}\rangle$ of the form $S^{\infty V(K^{*})}$ . This will often work, for example when $K^{*}=G$ . However if $G=SO(3)$ and $K^{*}=SO(2)$ , all representations of $SO(3)$ have fixed points under elements of order 2, so no such model exists.

Corollary 9.8.

(i) For any $DE\langle\mathcal{K}\rangle$ -module $N$ , the map

N\longrightarrow E\langle K^{*}\rangle\wedge N

induces inversion of $\mathcal{S}$ , so that

\pi^{G}_{*}(E\langle K^{*}\rangle\wedge N)=\mathcal{S}^{-1}\pi^{G}_{*}(N).

(ii) The map $E\langle K^{*}\rangle\wedge X\longrightarrow DE\langle\mathcal{K}\rangle\wedge E\langle K^{*}\rangle\wedge X$ induces extension of scalars along $\mathbb{Q}=\mathcal{R}(K^{*})\longrightarrow\mathcal{S}^{-1}\mathcal{O}_{\mathcal{K}}$ so that

\pi^{G}_{*}(E\langle K^{*}\rangle\wedge DE\langle\mathcal{K}\rangle\wedge X)=\mathcal{S}^{-1}\mathcal{O}_{\mathcal{K}}\otimes\pi_{*}(\Phi^{K^{*}}X)

Proof : (i) We note that $\pi^{G}_{*}=\pi^{N}_{*}$ for these spectra, and then the models of $E\langle K^{*}\rangle$ given in the lemma give the result.

(ii) The map $S^{0}\longrightarrow DE\langle\mathcal{K}\rangle$ induces a map $\pi_{*}(\Phi^{K^{*}}X)\longrightarrow\pi^{G}_{*}(E\langle K^{*}\rangle\wedge DE\langle\mathcal{K}\rangle\wedge X)$ which extends to a natural transformation. Since $\mathcal{S}^{-1}\mathcal{O}_{\mathcal{K}}$ is flat it is a natural transformation of homology theories compatible with the action of $\mathcal{W}_{K}$ . In checking it is an isomorphism we may ignore the action of $\mathcal{W}_{K}$ . The result therefore follows from the case when $X$ is an $N^{f}$ -equivariant sphere. ∎

9.C. Uniformization and formality

There are two convenient facts about module categories that we now need to exploit. Firstly, for a finite group $W$ , the category of $A[W]$ -modules is equivalent to the category of $A$ -modules with a $W$ -action: $A[W]\mbox{-mod}\simeq(A\mbox{-mod})[W]$ (requires $W$ to be finite). Secondly, the category of modules over a product $A_{1}\times A_{2}$ is equivalent to the product of the module categories: $(A_{1}\times A_{2})\mbox{-mod}\simeq(A_{1}\mbox{-mod})\times(A_{2}\mbox{-mod})$ (requires that this is a finite product).

By tom Dieck’s finiteness theorem [27], $W_{G}^{d}(K)$ takes only finitely many values, so there is a partition of $\mathcal{K}=\mathcal{K}_{1}\amalg\cdots\amalg\mathcal{K}_{N}$ into finitely many pieces, each of which has a single value $W_{i}$ of $W_{G}^{d}(K)$ . For brevity we write

(\prod_{K}R_{K})\mbox{-mod}[\mathcal{W}]:=(\prod_{K\in\mathcal{K}_{1}}R_{K}\mbox{-mod})[W_{1}]\times\cdots\times(\prod_{K\in\mathcal{K}_{N}}R_{K}\mbox{-mod})[W_{N}]

Lemma 9.9.

With $R=DE\langle\mathcal{K}\rangle$ , there is an equivalence

R\mbox{-mod}-G-\mathrm{spectra}\simeq\left(\prod_{K\in\mathcal{K}}D\tilde{B}W_{G}^{e}(K)_{+}\right)\mbox{-mod}-\mathrm{spectra}[\mathcal{W}].

Proof : Using Lemma 8.1, we have a right Quillen functor

R\mbox{-mod}-G-\mathrm{spectra}\longrightarrow(R^{G}\mbox{-mod}-\mathrm{spectra})[\mathcal{W}]

which is an equivalence on each individual factor.

Breaking the product up into finitely many parts we may assume that there is a single finite group $W$ associated to each factor $R_{K}$ , so that each factor is generated by $\tilde{\mathcal{R}}(K)=F_{N_{G}^{f}(K)}(G_{+},\mathcal{R}(K))$ corresponding to $D\tilde{B}W_{G}^{e}(K)_{+}\wedge W_{+}$ . The product on the right is generated by $(\prod_{K}D\tilde{B}W_{G}^{e}(K)_{+})\wedge W_{+}$ .

Equivalences of cells are detected on the factors, so that the unit and counit are equivalences on generators. Hence by the Cellularization Principle [20], it is an equivalence as required. ∎

Corollary 9.10.

(i) The category of $DE\langle\mathcal{K}\rangle$ -module $\mbox{$G$-{\bf spectra}}\langle\mathcal{V}\rangle$ is equivalent to the category of DG $\mathcal{W}$ -equivariant objects in $\mathcal{O}_{\mathcal{K}}$ -modules.

(ii) The category of $DE\langle K^{*}\rangle$ -module $\mbox{$G$-{\bf spectra}}\langle\mathcal{V}\rangle$ is equivalent to the category of DG $\mathcal{W}_{K^{*}}$ -equivariant $\mathcal{O}_{K^{*}}$ -modules.

Proof : Case (ii) is straightforward.

In Case (i) we have a Quillen pair showing that $DE\langle\mathcal{K}\rangle$ -module $\mbox{$G$-{\bf spectra}}\langle\mathcal{V}\rangle$ are equivalent to $D\tilde{E}\langle\mathcal{K}\rangle^{G}$ -module $\mathcal{W}$ -spectra by Lemma 9.9. By Shipley’s Theorem this is equivalent to DG modules in $\mathcal{W}$ -spectra over a DGA with homology $\pi_{*}D\tilde{E}\langle\mathcal{K}\rangle^{G}=\prod_{K\in\mathcal{K}}H^{*}(BW_{G}^{e}(K))=\mathcal{O}_{\mathcal{K}}$ by Lemma 9.5.

To see that the DGA is formal, we argue as follows. For a single factor we have a CDGA $\mathcal{O}_{K}^{\prime}$ with homology $\mathcal{O}_{K}$ . Thus $H_{*}(\mathcal{O}_{K^{*}}^{\prime})=Z(\mathcal{O}_{K^{*}}^{\prime})/B(\mathcal{O}_{K^{*}}^{\prime})$ is isomorphic to $\mathcal{O}_{K^{*}}=\mathcal{R}(K^{*})$ , which is the symmetic algebra on a finite dimensional vector space $V$ with an action of $\mathcal{W}_{K}^{*}$ . By Maschke’s Theorem we may find a submodule $V^{\prime}$ of $Z(\mathcal{O}_{K}^{\prime})$ mapping to $V$ , and thus we may choose an equivariant homology isomorphism $\mathcal{V}=\mathrm{Symm}(V)\longrightarrow\mathcal{O}_{K}^{\prime}$ . This covers the generic point $K^{*}$ . For $\mathcal{K}$ , we then use the equivalence $\mathcal{O}_{\mathcal{K}}^{\prime}\longrightarrow\prod_{K}e_{K}\mathcal{O}_{\mathcal{K}}^{\prime}$ followed by the above maps on each factor. ∎

We need to show that the objectwise equivalences may be assembled to an equivalence of cospans.

Lemma 9.11.

The diagram of $G$ -spectra is intrinsically formal.

Proof : The proof is essentially the same as for the circle. Let us suppose we have a cospan $\mathcal{O}^{\prime}_{K^{*}}\longrightarrow\mathcal{T}^{\prime}\longleftarrow\mathcal{O}^{\prime}$ with homology $\mathcal{O}_{K^{*}}\longrightarrow\mathcal{S}^{-1}\mathcal{O}_{\mathcal{K}}\longleftarrow\mathcal{O}_{\mathcal{K}}$ .

Starting at $\mathcal{O}_{K^{*}}^{\prime}$ , we know $H_{*}(\mathcal{O}_{K^{*}}^{\prime})=Z(\mathcal{O}_{K^{*}}^{\prime})/B(\mathcal{O}_{K^{*}}^{\prime})$ is isomorphic to $\mathcal{O}_{K^{*}}=\mathcal{R}(K^{*})$ , which is the symmetic algebra on a finite dimensional vector space $V$ with an action of $\mathcal{W}_{K}^{*}$ . By Maschke’s Theorem we may find a submodule $V^{\prime}$ of $Z(\mathcal{O}_{K^{*}}^{\prime})$ mapping to $V$ , and thus we may choose an equivariant homology ismorphism $\mathcal{V}=\mathrm{Symm}(V)\longrightarrow\mathcal{O}_{K^{*}}^{\prime}$ .

This gives an equivalence

and we may suppose $\mathcal{O}_{K^{*}}^{\prime}$ is standard. Now proceed as in the diagram below.

The original diagram $\mathcal{O}_{K^{*}}\longrightarrow\mathcal{T}\longleftarrow\mathcal{O}$ has homology $\mathcal{R}(K^{*})\longrightarrow\mathcal{S}^{-1}\mathcal{O}_{\mathcal{K}}\longleftarrow\mathcal{O}_{\mathcal{K}}$ . In the following diagram, all horizontals are homology isomorphisms. Most maps are self explanatory, but we note that inverting additional classes already inverted in homology induces a weak equivalence. We therefore choose $\hat{\mathcal{S}}$ by adding representative cycles to $\mathcal{S}$ so the the relevant DGA receives a map.

∎

9.D. The cellular skeleton theorem

We have shown that the category of $G$ -spectra over $\mathcal{V}$ has a purely algebraic model, which is the cellularization of a category of modules. The cellularization means that the weak equivalences are not obvious. We complete the picture by showing that this algebraic cellularization is equivalent to DG objects in the abelian category $\mathcal{A}(G|\mathcal{V})$ with the weak equivalences being homology isomorphisms.

More precisely, $\mathcal{A}(G|\mathcal{V})$ is a coreflective subcategory of the category of modules with cellular equivalences of DG objects being homology isomorphisms, and the inclusion of abelian categories induces a Quillen equivalence.

Lemma 9.12.

The cellularization of the category of $\pi^{G}_{*}((S^{0})^{\lrcorner})$ -modules has $\mathcal{A}(G|\mathcal{V})$ as a cellular skeleton.

Proof : First we note that the images of any $G$ -spectrum lie in $\mathcal{A}(G|\mathcal{V})$ . This is because the homotopy of the horizontal map is inverting the multiplicatively closed set $\mathcal{S}$ by Corollary 9.8 (i), and the vertical map is extension of scalars along a flat map by Corollary 9.8 (ii). In particular, the cells are in $\mathcal{A}(G|\mathcal{V})$ .

To see that any object of $\mathcal{A}(G|\mathcal{V})$ is cellular, we use the injective resolutions from Subsection 5.E. The point is that any object of $\mathcal{A}(G|\mathcal{V})$ admits an embedding into an injective which is a sum of injectives $e(H_{*}(BW_{G}^{e}(K^{*})))$ and $f_{K}(H_{*}(BW_{G}^{e}(K))$ , whose cokernel is a sum of injectives of the second type. Each of these injectives are cellular because they are the images of $G$ -spectra. Indeed $e(H_{*}(BW_{G}^{e}(K^{*}))$ is the image of $\widetilde{E}{\mathcal{F}}$ , and $f_{K}(H_{*}(BW_{G}^{e}(K))$ is the image of $DE\langle K\rangle\wedge F_{N}(G_{+},EW_{G}(K)_{+})$ .

Finally, we argue that any object in the module category is cellularly equivalent to a an object of $\mathcal{A}(G|\mathcal{V})$ . Suppose then that $X=(V\longrightarrow P\longleftarrow N)$ is an object of the module category. The functor $e$ is a right adjoint to evaluation so there is a map $X\longrightarrow e(V)$ with fibre $X^{\prime}=(0\longrightarrow P^{\prime}\longleftarrow N^{\prime})$ . Since $e(V)$ is a wedge of copies of $e(\mathbb{Q})$ it is cellular, so it suffices to show $X^{\prime}$ is cellularly equivalent to an object of $\mathcal{A}(G|\mathcal{V})$ . Inverting $\mathcal{S}$ gives a map $(0\longrightarrow P^{\prime}\longleftarrow N^{\prime})\longrightarrow(0\longrightarrow P^{\prime}\longleftarrow\mathcal{S}^{-1}N^{\prime})$ . The fibre $X^{\prime\prime}$ is $(0\longrightarrow 0\longleftarrow\Gamma_{\mathcal{S}}N^{\prime})$ with $\Gamma_{\mathcal{S}}N^{\prime}$ being $\mathcal{S}$ -torsion and hence cellular. It remains to observe $(0\longrightarrow P^{\prime}\longleftarrow\mathcal{S}^{-1}N^{\prime})$ is cellularly trivial, since then we have $X^{\prime\prime}\simeq X^{\prime}$ and a cofibre sequence $X^{\prime}\longrightarrow X\longrightarrow e(V).$

To show cellular triviality of $(0\longrightarrow P^{\prime}\longleftarrow\mathcal{S}^{-1}N^{\prime})$ we check maps out of cells. There are no maps from $\sigma_{K}$ for $K\in\mathcal{K}$ because $\mathcal{S}^{-1}N^{\prime}$ is torsion free. Finally maps from $\sigma_{K^{*}}$ are calculated from the pullback square

which in our case is

∎

10. Towards the general 1-dimensional case

We consider the restrictions currently imposed, and how they might be avoided.

drop the restriction that $K^{*}$ is of finite index in its normalizer:: The first issue is that $E\langle K^{*}\rangle$ needs to be replaced by a different ring spectrum. If $K^{*}$ is normal in $G$ this means $S^{\infty V(K^{*})}\wedge DEG/K^{*}_{+}$ . Probably in general it should be the $K^{*}$ -localization of the $K^{*}$ -completion of the sphere. Adapting the proof of the pullback square should be a straightforward. Under appropriate group theoretic conditions there is a more explicit model of the ring $G$ -spectrum, and this will enable calculations of homotopy groups to link to algebra.

The effect on the algebraic model is that $\mathcal{O}_{K^{*}}$ will be a more general polynomial ring, so the models will be of higher homological dimension. One also needs to discuss the relationships between normalizers of $K^{*}$ and subgroups in $\mathcal{K}$ . If $N_{G}(K^{*})$ contains all subgroups $N_{G}(F)$ it is straightforward, but this should be weakened, so that it covers cases like $K^{*}=SO(2)$ in $G=SO(3)$ .
drop the restriction that the subgroups $F$ in $\mathcal{K}$ are finite:: Our proof of the formality of $DE\langle\mathcal{K}\rangle$ involves a certain uniformity of behaviour. It should be sufficient that all elements of $\mathcal{K}$ have a common identity component. Since we may always use idempotents, it should suffice that $\mathcal{K}$ can be broken into finitely many parts each of which consists of subgroups with a common identity component.
drop the restriction that $\mathcal{V}$ has a single compactifying point:: The general case that $\mathcal{V}$ is a 1-dimensional spectral space introduces significant complication. For example, even with only finitely many height 1 subgroups the model needs to incorporate the combinatorics of specialization and how it interacts with the containments of normalizers.

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Algebraic models for 1-dimensional categories of rational GG-spectra