THE EQUIVARIANT GENERATING HYPOTHESIS

ANNA MARIE BOHMANN

Abstract. We state the generating hypothesis in the homotopy category of G-spectra for a compact Lie group G, and prove that if G is finite, then the generating hypothesis implies the strong generating hypothesis, just as in the non-equivariant case. We also give an explicit counterexample to the generating hypothesis in the category of rational S^{1}-equivariant spectra.

1. The generating hypothesis in an equivariant context

The Freyd conjecture, also known as the Freyd generating hypothesis, is a longstanding conjecture in stable homotopy theory. Here Ho𝒮 is the homotopy category of spectra.

Conjecture 1.1 (Generating hypothesis). The restriction of the functor π_{∗}(−) : Ho𝒮 → π_{∗}(S)-modules to the subcategory of finite spectra is faithful. That is, if a map f : X → Y between finite spectra X and Y induces the zero map f_{∗}: π_{∗}(X) → π_{∗}(Y), then f is nullhomotopic.

This conjecture was introduced by Freyd in 1965 [3], and remains open today. Recent work has examined analogous statements in algebraic categories that share many properties of the category of spectra, such as the derived category of a ring [6, 8] or the stable module category of a finite group [1, 2]. Here we examine the analogous conjecture for the (topological) category of G-equivariant spectra for a compact Lie group G.

The appropriate generalization of the generating hypothesis to an equivariant context must take into account the homotopy groups of fixed points for each closed subgroup H ≤ G. This is best formulated in the language of Mackey functors.

Conjecture 1.2. Let ℳ[G] be the category of Mackey functors over a compact Lie group G, and HoG𝒮 be the homotopy category of G-spectra. The restriction of the functor \underline{π}_{∗}^{G}(−) : HoG𝒮 → ℳ[G] to the subcategory of finite G-spectra is faithful. That is, if a map f : X → Y between finite G-spectra X and Y induces the zero map \underline{π}_{∗}^{G}(X) → \underline{π}_{∗}^{G}(Y), then f is nullhomotopic.

Note that a map M → N of Mackey functors is zero if and only if the map M(G/H) → N(G/H) is zero for all closed subgroups H ≤ G. Hence this conjecture can also be stated as requiring a map f : X → Y of finite G-spectra to be nullhomotopic if it induces the zero map f_{∗}: π_{∗}^{H}(X) → π_{∗}^{H}(Y) for all H ≤ G.

The above formulation of the Freyd’s conjecture (1.1) is also known as the weak generating hypothesis, as opposed to the strong generating hypothesis. Freyd’s original formulation of the strong generating hypothesis conjectures that the functor π_{∗}(−) is both faithful and full when restricted to the subcategory of finite spectra. In his initial paper on the subject, Freyd also proves Theorem 1.3 ([3, Prop. 9.7]). If the weak generating hypothesis (1.1) holds, then so does the strong generating hypothesis. That is, if the restriction of the functor π_{∗}: Ho𝒮 → π_{∗}(S)−modules to the subcategory of finite spectra is faithful, it is also full on this subcategory.

This type of “faithful implies full” implication does not always hold, as shown by Hovey, Lockridge and Puninski in the case of the derived category of a ring [6]. However, this implication is true in the equivariant case for a finite group G.

Theorem 1.4. Let G be a finite group, and let ℳ[G] be the category of Mackey functors over G. Then, if the functor \underline{π}_{∗}^{G}(−) : G−spectra → ℳ[G] is faithful on restriction to the full subcategory of finite spectra, this restriction is also full.

Our proof does not extend to arbitrary compact Lie groups G. However, we prove that the weak generating hypothesis fails in the simplest case of a non-finite group of equivariance, that is, for the category of rational 𝕋–spectra, where 𝕋 is the compact Lie group S^{1}.

Theorem 1.5. The generalized Freyd conjecture does not hold for the category of rational 𝕋-equivariant spectra.

This result suggests that the generating hypothesis may not hold for other nonfinite groups of equivariance, and also makes immaterial the question of whether the weak generating hypothesis implies the strong generating hypothesis in this category.

In section 2, we prove Theorem 1.4. As a technical tool for this proof, we introduce the abelian category for stable homotopy, which is an abelian category into which the stable category of G-spectra embeds. In section 3, we establish Theorem 1.5 by giving a specific counterexample to the generating hypothesis in the category of rational 𝕋–spectra.

2. The abelian category for stable homotopy and the proof of faithful implies fulls (Or, this section could use a better title)

Our proof of Theorem 1.4 follows Freyd’s proof that the weak generating hypothesis implies the strong generating hypothesis for the category of spectra [3, Prop. 9.7]. In particular, we introduce the abelian category 𝒜(G) associated to the equivariant stable homotopy category. This is a Frobenius abelian category with a full and faithful inclusion ι: G−spectra → 𝒜(G) of the equivariant stable homotopy category with the properties that

• the image of any G-spectrum E in 𝒜(G) is projective and injective, and every projective or injective object in 𝒜(G) is a direct summand of an object in the image of ι,

• 𝒜(G) is self-dual, and

• any homological functor H: G-spectra → 𝒜, where 𝒜 is an abelian category,

extends uniquely to an abelian functor 𝒜(G) → 𝒜. By homological functor, we mean a functor that takes exact triangles of G-spectra to exact sequences in 𝒜. Note that self-duality implies we get a similar extension of any cohomological functor on G-spectra as well.

Any triangulated category has such an abelian category associated to it; this construction was first given in [3]. For a more modern treatment, including several equivalent constructions of this category, see Neeman’s book on triangulated categories [9]. We define 𝒜(G) to be the following quotient of the category of maps in HoG𝒮, which is Freyd’s original description [3, §3] and is also given by Neeman [9, 5.2.1].

Definition 2.1. The objects of 𝒜(G) are maps s → t in HoG𝒮, and morphisms in 𝒜(G) are commutative diagrams

\xymatrix{s \ar[r] \ar[d] & t \ar[d] \\ s′ \ar[r] & t′}

under the additive equivalence relation defined by setting a such a morphism equal to zero if the composite s → t → t′ or s → s′ → t′ is zero. The embedding ι: HoG𝒮 → 𝒜(G) is given by sending a spectrum s to the identity morphsim s → s.

We can think of the functor ι as a universal homological functor from HoG𝒮 to an abelian category. Because ι is full and faithful, we will usually identify a spectrum with its image under ι. In order to make use of the category 𝒜(G) in proving Theorem 1.4, we need to show that if the generating hypothesis holds for G-spectra, it also holds in the category 𝒜(G).

Lemma 2.2. If \underline{π}_{∗}^{G}(−) is faithful on restriction to the category of finite G-spectra, then its extension to 𝒜(G) is also faithful on restriction to objects with a projective resolution by finite G-spectra.

Proof. Objects with a projective resolution by finite G-spectra are the same as objects s → t where s and t are finite G-spectra [9, §5]. The value of a homological functor H on an object s → t in 𝒜(G) is given by the image of H(s) → H(t). If s, t, s′ and t′ are finite G-spectra and we have a morphism,

\xymatrix{s \ar[r] \ar[d] & t \ar[d] \\ s′ \ar[r] & t′}

then the map \underline{π}_{∗}^{G}(s → t) → \underline{π}_{∗}^{G}(s′ → t′) is zero if and only if the map im \underline{π}_{∗}^{G}(s) → im \underline{π}_{∗}^{G}(s′) is zero. But if this is the case, then the map \underline{π}_{∗}^{G}(s) → \underline{π}_{∗}^{G}(s′) → \underline{π}_{∗}^{G}(t′) is zero. Hence, if the generating hypothesis holds for G-spectra, the map s → s′ → t′ is zero, and so our original morphism

\xymatrix{s \ar[r] \ar[d] & t \ar[d] \\ s′ \ar[r] & t′}

is also zero. Thus the generating hypothesis for G-spectra implies the generating hypothesis holds for 𝒜(G). □

We now prove that for a finite group G, the generating hypothesis for G-spectra implies the strong generating hypothesis for G-spectra.

Proof of 1.4. Let G be a finite group. Let X and Y be finite G-CW spectra, and let S be the sphere G-spectrum. Let \underline{π}_{∗}^{G}(S) be the graded Green functor that is \underline{π}_{n}^{G}(S) in degree n. Note that \underline{π}_{0}^{G}(S) is the Burnside ring Green functor. Similarly, let \underline{π}_{∗}^{G}(X) be the graded Mackey functor that is \underline{π}_{n}^{G}(X) in degree n. Composition makes \underline{π}_{∗}^{G}(X) a module over \underline{π}_{∗}^{G}(S).

Suppose we have a natural transformation ϕ : \underline{π}_{∗}^{G}(X) → \underline{π}_{∗}^{G}(Y). To prove that \underline{π}_{∗}^{G}(−) is full, we must find a map f: X → Y such that \underline{π}_{∗}^{G}(f) = ϕ. The structure of the proof is two part. First, we use the embedding ι: HoG𝒮 → 𝒜(G) to construct a sequence of maps \{f_{n}: X → Y \} such that (f_{n})_{∗}: \underline{π}_{j}^{G}(X) → \underline{π}_{j}^{G}(Y) agrees with ϕ for j ≤ n. We then show that {f_{n}} has a constant cofinal subsequence whose value is the desired map f.

For each conjugacy class of closed subgroups H ≤ G and each n, choose a set of generators of \underline{π}_{n}^{G}(X)(G/H) = [Σ^{n}(S ∧ G/H_{+}), X]^{G}. Denote these generators by {α_{H,n,i}: Σ^{n}(S ∧ G/H_{+}) → X}. Let W_{n} = ⋁_{j≤n} ⋁_{H≤G} ⋁_{i} Σ^{j}(S ∧ G/H_{+}), and let α_{n}: W_{n} → X be the map given by α_{H,j,i} on each wedge summand. We then get a map h_{n}: W_{n} → Y that is ϕ(α_{H,j,i}) on each wedge summand.

Let ι_{n} : K_{n} → W_{n} be the kernel of the α_{n} : W_{n} → X in the abelian category for equivariant stable homotopy 𝒜(G). Thus, the sequence 0 → K_{n} → W_{n} → X is exact in 𝒜(G). We first show that \underline{π}_{∗}^{G}(h_{n} ∘ ι_{n}) = 0. Then we apply the assumption that \underline{π}_{∗}^{G}(−) is faithful to conclude that h_{n} ∘ι_{n} is zero. By construction, and because ϕ is a natural transformation of Mackey functors,

\underline{π}_{∗}^{G}(h_{n} ∘ ι_{n}) = \underline{π}_{∗}^{G}(h_{n}) ∘ \underline{π}_{∗}^{G}(ι_{n})
= \underline{π}_{∗}^{G}(ϕ(α_{n}))\underline{π}_{∗}^{G}(ι_{n})
= ϕ(\underline{π}_{∗}^{G}(α_{n}))\underline{π}_{∗}^{G}(ι_{n}).

Since α_{n} ∘ ι_{n} = 0 in 𝒜(G), we see that \underline{π}_{∗}^{G}(h_{n} ∘ ι_{n}) = ϕ(\underline{π}_{∗}^{G}(α_{n} ∘ ι_{n})) = 0. We are assuming the generating hypothesis holds in HoG𝒮, so we can apply Lemma 2.2 to conclude that h_{n} ∘ ι_{n} : K_{n} → W_{n} → Y is zero in 𝒜(G).

Now we construct our maps {f_{n}}. Let ζ_{n}: W_{n} → D_{n} be a cokernel of ι_{n}: K_{n} → W_{n}. Note that as K_{n} is the cokernel of α_{n}: W_{n} → X, the map α_{n}: W_{n} → X factors through an injection ν_{n}: D_{n} → X. Since h_{n} ∘ ι_{n} = 0, the map h_{n} factors through a map γ_{n}: D_{n} → Y. Also let μ_{n}: X → C_{n} be a cokernal of α_{n}: W_{n} → X. The object Y is injective in the category 𝒜(G) [9, 5.1.23], so there is a map f_{n}: X → Y such that f_{n} ∘ ν_{n} = γ_{n}. We summarize these definitions in the following commmutative diagram in 𝒜(G).

\xymatrix{0 \ar[r] & K_{n} \ar[r]^{ι_{n}} \ar[ddr]_{0} & W_{n} \ar[rr]^{α_{n}} \ar[dr]^{ζ_{n}} \ar[dd]_{h_{n}} &  & X \ar[r]^{μ_{n}} \ar[ddll]^{f_{n}} & C_{n} \ar[r] & 0 \\  &  &  & D_{n} \ar[ur]^{ν_{n}} \ar[dl]_{γ_{n}} \\  &  & Y}

We have constructed α_{n} such that \underline{π}_{∗}^{G}(α_{n}) is a surjection \underline{π}_{j}^{G}(W_{n}) → \underline{π}_{j}^{G}(X) for j ≤ n. Since \underline{π}_{∗}^{G}(f_{n})\underline{π}_{∗}^{G}(α_{n}) = \underline{π}_{∗}^{G}(h_{n}) = ϕ\underline{π}_{∗}^{G}(α_{n}), the map \underline{π}_{j}^{G}(f_{n}) = ϕ for j ≤ n. We need to construct a map f = f_{∞} that has this property for all n.

By construction, we have an inclusion i_{n}: W_{n} → W_{n+1} such that α_{n} = α_{n+1} ∘ i_{n} and h_{n} = h_{n+1} ∘ i_{n}. For any m > n, iterating these inclusions of summands gives an inclusion W_{n} → W_{m} which makes the diagram

\xymatrix{W_{n} \ar[r] \ar[d]_{h_{n}} \ar[rr]^{α_{n}} & W_{m} \ar[dl]^{h_{m}} \ar[r]_{α_{m}} & X \ar[dll]^{f_{m}} \\ Y}

commute. Hence the restriction of the map f_{m} ∘ α_{m} : W_{m} → X → Y to the summands contained in W_{n} is the map h_{n}. By the universal property of the cokernel μ_{n}: X → C_{n}, we obtain a factorization of the map f_{m} − f_{n} through C_{n}:

\xymatrix{W_{n} \ar[r]^{α_{n}} \ar[dr]_{0} & X \ar[r]^{μ_{n}} \ar[d]_{f_{n}−f_{m}} & C_{n} \ar@{.>}[dl] \\  & Y}

Since X is finite, we may choose d large enough so that H_{j}(X, \underline{A}) = 0 for all j > d, where \underline{A} is the Burnside ring Mackey functor for the group G. We claim that 𝒜(G)(C_{d}, Y) is finite, that is, that 𝒜(G)(C_{d}, Y) ⊗ ℚ = 0. The desired result the follows, since if 𝒜(G)(C_{d}, Y) is finite, the sequence {f_{m}} has a constant cofinal subsequence, and its common value f satisfies \underline{π}_{∗}^{G}(f) = ϕ.

First consider H_{∗}^{G}(C_{d}; \underline{A}). Since the functor \underline{H}_{j}^{G}(−; \underline{A}): 𝒜(G) → ℳ[G] is exact, for each H ≤ G, we have an exact sequence of abelian groups

\underline{H}_{j}^{G}(W_{d}; \underline{A})(G/H) ⟶^{(α_{d})_{∗}} \underline{H}_{j}^{G}(X; \underline{A})(G/H) → \underline{H}_{j}^{G}(C_{d}; \underline{A})(G/H) → 0 (1)

for all j. When j > d, the Mackey functor \underline{H}_{j}^{G}(X; \underline{A}) = 0, and thus \underline{H}_{j}^{G}(C_{d}; \underline{A}) is zero as well. As May and Greenlees show [5, App. A], the rationalization of the sphere G-spectrum is the Eilenberg–Maclane G-spectrum H(\underline{A} ⊗ ℚ). Thus we can equate the rationalization of the sequence (1) with the sequence

\underline{π}_{j}(W_{d}) ⊗ ℚ ⟶^{(α_{d})_{∗}} \underline{π}_{j}(X) ⊗ ℚ → \underline{π}_{j}(C_{d}) ⊗ ℚ → 0.

When j ≤ d, the map α_{d} is surjective on rational homotopy by construction, so \underline{π}_{j}(C_{d}) ⊗ ℚ = 0 for j ≤ d. Since \underline{H}_{j}^{G}(C_{d}; \underline{A}) ⊗ ℚ = \underline{π}_{j}(C_{d}) ⊗ ℚ = 0 for j > d, the graded Mackey functor \underline{π}_{∗}^{G}(C_{d}) ⊗ ℚ = 0.

Denote by X_{0} the rationalization of a G-spectrum X; then

[X, Y]_{∗}^{G} ⊗ ℚ = [X_{0}, Y_{0}]_{∗}^{G} = [X, Y_{0}]_{∗}^{G}.

By restricting to the category of rational G-spectra, we in principal get an abelian category 𝒜_{0}(G) for rational stable homotopy as in the construction of Definition 2.1. However, the rationalization functor HoG𝒮 → HoG𝒮_{0} induces an extension 𝒜(G) → 𝒜_{0}(G). This functor is an equivalence after tensoring with ℚ, which does not change the category 𝒜_{0}(G) since 𝒜(G) is already rational. Hence we do not distinguish between 𝒜(G) and 𝒜_{0}(G).

As shown in [5, App A], for any G-spectra X and Y, the natural map

ϑ : [X_{0}, Y_{0}]^{G} → ∏_{n} Hom_{ℳ[G]} (\underline{π}_{n}^{G}(X_{0}), \underline{π}_{n}^{G}(Y_{0})) (2)

is an isomorphism. We restate this isomorphism in terms of tensoring with ℚ:

ϑ : [X, Y]^{G} ⊗ ℚ → ∏_{n} Hom_{ℳ[G]} (\underline{π}_{n}^{G}(X) ⊗ ℚ, \underline{π}_{n}^{G}(Y) ⊗ ℚ)

Moreover, for a fixed Y, both sides of (2) are cohomology theories on G-spectra, so they extend to 𝒜(G). Thus the columns are exact in the following diagram:

\xymatrix{[W_{d}, Y]^{G} ⊗ ℚ \ar[r] \ar[d] & ∏_{n} Hom_{ℳ[G]} (\underline{π}_{n}^{G}(W_{d}) ⊗ ℚ, \underline{π}_{n}^{G}(Y) ⊗ ℚ) \ar[d] \\ [X, Y]^{G} ⊗ ℚ \ar[r] \ar[d] & ∏_{n} Hom_{ℳ[G]} (\underline{π}_{n}^{G}(X) ⊗ ℚ, \underline{π}_{n}^{G}(Y) ⊗ ℚ) \ar[d] \\ 𝒜(G)(C_{d}, Y) ⊗ ℚ \ar[r] \ar[d] \ar[r] \ar[d] & ∏_{n} Hom_{ℳ[G]} (\underline{π}_{n}^{G}(C_{d}) ⊗ ℚ, \underline{π}_{n}^{G}(Y) ⊗ ℚ) \ar[d] \\ 0 \ar[r] & 0}

where we are identifying 𝒜(G)(X, Y) with [X, Y] for G-spectra X and Y. By the five lemma, 𝒜(G)(C_{d}, Y) ⊗ ℚ ≅ ∏ Hom_{ℳ[G]} (\underline{π}_{n}^{G}(W_{d}) ⊗ ℚ, \underline{π}_{n}^{G}(Y) ⊗ ℚ). But \underline{π}_{∗}^{G}(C_{d}) ⊗ ℚ = 0, so in fact 𝒜(G)(C_{d}, Y) ⊗ ℚ = 0. Therefore, the sequence \{f_{m}\} has a constant cofinal subsequence, and we can take its value to be the desired map \overline{ϕ}: X → Y such that \underline{π}_{∗}^{G}(\overline{ϕ}) = ϕ. □

Thus, for a finite group G, the weak generating hypothesis implies the strong generating hypothesis, just as in the non-equivariant case.

3. The rational equivariant case

Restricting to the category of rational G-spectra for a group G greatly simplifies the structure of the category. In fact, for a finite group G, Greenlees and May [5] have proved that there is a natural equivalence between rational G-spectra and the category of graded Mackey functors over A(G) ⊗ ℚ. As in the non-equivariant rational case, this equivalence is simply given by the isomorphism (2). This isomorphism completely algebrizes the rational G-stable homotopy category, and so, in particular, the strong generating hypothesis holds.

For infinite compact Lie groups, however, the picture is quite different. We prove that the weak generating hypothesis does not hold in the category of rational 𝕋spectra, where 𝕋 is the circle group. In fact, using Greenlees’s algebraic model for the category of rational 𝕋-spectra, we find an explicit counterexample to the generating hypothesis in this category.

Theorem 1.5. The generalized Freyd conjecture does not hold for the category of rational 𝕋–equivariant spectra.

Before proving this theorem, we prove the following lemma about the equivariant homotopy groups of the suspension spectrum X of a free 𝕋-space \overline{X} with a 𝕋-fixed basepoint.

Lemma 3.1. Let \overline{X} be a free rational 𝕋-space with a 𝕋-fixed basepoint, and let X be its suspension spectrum. Then, for any closed subgroup H ≤ 𝕋,

π_{∗}^{H}(X) = \begin{cases}π_{∗}(ΣX/𝕋) & ifH = 𝕋 \\ π_{∗}(X) & ifH < 𝕋. \end{cases}

Proof. The tom Dieck splitting theorem [7] states that for such an X and for any closed subgroup H ≤ 𝕋,

(3)  π_{∗}^{H}(X) = ∑_{K≤H} π_{∗} (Σ^{∞}(EW_{H}K_{+} ∧_{W_{H}K} Σ^{ad(W_{H}K)}\overline{X}^{K}))

where the sum runs over conjugacy classes of closed subgroups K ≤ H and ad(W_{H}K) is the adjoint representation. Let us first consider π_{∗}^{𝕋}(X). Since \overline{X} is free away from its basepoint, the only summand that contributes to π_{∗}^{𝕋}(X) is the summand corresponding to the trivial group 1 ≤ 𝕋. The Weyl group of 1 ≤ 𝕋 is 𝕋 itself and the adjoint representation of 𝕋 is trivial. Thus

π_{∗}^{𝕋}(X) = π_{∗} (Σ^{∞}(E𝕋_{+} ∧_{𝕋} Σ\overline{X}))
= π_{∗}(ΣX/𝕋).

Now suppose H is a proper closed subgroup of 𝕋. Then H is finite, so ad(W_{H}K) = 0 for all K ≤ H. Since \overline{X} is free away from its basepoint, \overline{X}^{K} is a point unless K is the trivial group. Thus, as above, the summand of (3) corresponding the trivial group is the only summand contributing to π_{∗}^{H}(X). Hence π_{∗}^{H}(X) reduces to

π_{∗}^{H}(X) = π_{∗} (Σ^{∞}(EH_{+} ∧_{H} \overline{X}))
= π_{∗}(X/H).

Since we are working rationally, π_{∗}(X/H) = H_{∗}(X/H). The group H is a proper closed subgroup of 𝕋 and is thus finite; hence H_{∗}(X/H) = H_{∗}(X)/H, but since H is acting as a subgroup of the connected group 𝕋, H must in fact act trivially on H_{∗}(X). Thus π_{∗}(X/H) = H_{∗}(X), and again since we are working rationally, π_{∗}(X/H) = π_{∗}(X). That is, for any proper closed H ≤ 𝕋, the H–homotopy groups π_{∗}^{H}(X) are in fact the non-equivariant homotopy groups π_{∗}(X). □

Proof of 1.5. We restrict our attention to free rational 𝕋-spectra, and use the results of Greenlees in [4]. Throughout the following proof, we assume that all spectra are rational without explicitly indicating this in our notation.

Let X and Y be finite rational 𝕋-spectra. The equivariant Freyd conjecture asserts that a map f : X → Y is zero if and only if it induces the zero map on π_{∗}^{H}(−) for all closed subgroups H ≤ 𝕋. We give a counterexample where X and Y happen to be free rational 𝕋-spectra.

Greenlees proves [4, 3.1.1] that for free rational 𝕋-spectra, there is a natural Adams short exact sequence

0 → Ext_{ℚ[c]}^{∗}(π_{∗}^{𝕋}(ΣX), π_{∗}^{𝕋}(Y)) → [X, Y]_{∗}^{𝕋} → Hom_{ℚ[c]}(π_{∗}^{𝕋}(X), π_{∗}^{𝕋}(Y)) → 0

where ℚ[c] is the graded polynomial algebra given by ℚ adjoin the element c in dimension −2. This element c comes from the Euler class. This short exact sequence is our main tool in finding a counterexample to the equivariant Freyd conjecture.

Let X and Y be the suspension spectra of free, rational 𝕋-spaces \overline{X} and \overline{Y} equipped with 𝕋-fixed basepoints. Assume we have a map f : X → Y inducing the zero map π_{∗}^{H}(X) → π_{∗}^{H}(Y) for all subgroups H ≤ 𝕋. By Lemma 3.1 this is equivalent to assuming that f_{∗} : π_{∗}^{𝕋}(X) → π_{∗}^{𝕋}(Y) and f_{∗} : π_{∗}(X) → π_{∗}(Y) are both zero.

Consider the space 𝕋_{+} ∧ \overline{X} with 𝕋 acting diagonally. There is an isomorphism [Σ^{∞}(𝕋_{+} ∧ \overline{X}), Y]_{∗}^{𝕋} ≅ [X, Y]_{∗} between equivariant homotopy classes of maps from Σ^{∞}𝕋_{+} ∧ \overline{X} to Y and non-equivariant homotopy classes of maps from X to Y, and since X and Y were assumed to be rational, [X, Y]_{∗} is also isomorphic to Hom_{ℚ}(π_{∗}(X), π_{∗}(Y)).

Write ℚ[c] = k and Σ^{∞}(𝕋_{+} ∧ \overline{X}) = Z for conciseness. The projection p : 𝕋_{+} ∧ X → X induces a map of short exact sequences

\xymatrix{0 \ar[r] & Ext_{k}(π_{∗}^{𝕋}(ΣX), π_{∗}^{𝕋}(Y)) \ar[r] \ar[d]_{p_{∗}} & [X, Y]_{∗}^{𝕋} \ar[d]_{p_{∗}} \ar[r] & Hom_{k}(π_{∗}^{𝕋}(X), π_{∗}^{𝕋}(Y)) \ar[r] \ar[d]_{p_{∗}} & 0 \\ 0 \ar[r] & Ext_{k}((π_{∗}^{𝕋}(ΣZ), π_{∗}^{𝕋}(Y)) \ar[r] & [Z, Y]_{∗}^{𝕋} \ar[r] & Hom_{k}(π_{∗}^{𝕋}(Z), π_{∗}^{𝕋}(Y)) \ar[r] & 0}

which can be rewritten as

\xymatrix{0 \ar[r] & Ext_{k}(π_{∗}^{𝕋}(ΣX), π_{∗}^{𝕋}(Y)) \ar[r] \ar[d]_{p_{∗}} & [X, Y]_{∗}^{𝕋} \ar[d]_{p_{∗}} \ar[r] & Hom_{k}(π_{∗}^{𝕋}(X), π_{∗}^{𝕋}(Y)) \ar[r] \ar[d]_{p_{∗}} & 0 \\ 0 \ar[r] & Ext_{k}(π_{∗}^{𝕋}(ΣZ), π_{∗}^{𝕋}(Y)) \ar[r] & Hom_{ℚ}(π_{∗}(X), π_{∗}(Y)) \ar[r] & Hom_{k}(π_{∗}^{𝕋}(Z), π_{∗}^{𝕋}(Y)) \ar[r] & 0}

Let f ∈ [X, Y]_{∗}^{𝕋} be a map satisfying the hypotheses of the generating hypothesis. This assumption implies that p_{∗}(f) is zero in

Hom_{ℚ}(π_{∗}(X), π_{∗}(Y)) = Hom_{ℚ}(π_{∗}^{H}(X), π_{∗}^{H}(Y))

and also that f goes to zero in Hom_{ℚ[c]}(π_{∗}^{𝕋}(X), π_{∗}^{𝕋}(Y)). Hence f is either zero, or lifts to an element of Ext_{ℚ[c]}(π_{∗}^{𝕋}(ΣX), π_{∗}^{𝕋}(Y)). To show that the generating hypothesis fails for the category of rational 𝕋–spectra, it is enough to show that the map

(4) Ext_{ℚ[c]}(π_{∗}^{𝕋}(ΣX), π_{∗}^{𝕋}(Y)) → Ext_{ℚ[c]}(π_{∗}^{𝕋}(ΣZ), π_{∗}^{𝕋}(Y))

has a nontrivial kernel. Any element of this kernel must map to zero in [Σ^{∞}(𝕋_{+} ∧ \overline{X}), Y]_{∗}^{𝕋} by definition, and will map to zero in Hom_{ℚ[c]}(π_{∗}^{𝕋}(X), π_{∗}^{𝕋}(Y)) by exactness, but must be nonzero as an element of [X, Y]_{∗}^{𝕋}.

We now give a concrete example where the map (4) has a nontrivial kernel. The group 𝕋 acts on the complex plane ℂ by multiplication. Consider the diagonal action of 𝕋 on ℂ^{a}, for some positive integer a. Let \overline{S(a)} be the unit sphere in this representation of 𝕋. Consider the rational suspension spectrum of \overline{S(a)}_{+}. We will denote this spectrum by S(a). We claim that for integers a, b > 1, the spectra S(a) and S(b) provide a counterexample to the equivariant generating hypothesis.

To prove this claim, it suffices to calculate the groups

Ext_{ℚ[c]} (π_{∗}^{𝕋}(ΣS(a)), π_{∗}^{𝕋}(S(b))) and Ext_{ℚ[c]} (π_{∗}^{𝕋}(ΣZ∧), π_{∗}^{𝕋}(S(b)))

where now Z = Σ^{∞}(𝕋_{+} ∧ \overline{S(a)}_{+}). First note that Lemmma 3.1 implies

π_{∗}^{𝕋}(S(a)) = π_{∗}(ΣS(a)/𝕋)
= π_{∗}(ΣℂP^{a−1})
= Σ^{2a−1}ℚ[c]/(c^{a})

as a graded module over ℚ[c], and

π_{∗}^{𝕋}(Z) = π_{∗}(ΣS(a))
= π_{∗}(ΣS^{2a−1})
= Σℚ ⊕ Σ^{2a}ℚ,

again as a graded module over ℚ[c]. Hence our Ext calculations reduce to finding the groups

Ext_{ℚ[c]} (Σ^{2a}ℚ[c]/(c^{a}), Σ^{2b−1}ℚ[c]/(c^{b}))

and

Ext_{ℚ[c]} (Σ^{2}ℚ ⊕ Σ^{2a+1}ℚ, Σ^{2b−1}ℚ[c]/(c^{b})) .

To calculate the first group, we use the projective resolution

0 → ℚ[c] ⟶^{c^{a}} Σ^{2a}ℚ[c] → Σ^{2a}ℚ[c]/(c^{a}) → 0

of Σℚ[c]/(c^{a}). Applying Hom_{ℚ[c]} (−, Σ^{2b−1}ℚ[c]/(c^{b})) to this resolution gives an exact sequence

⋯ → Σ^{2b−1−2a}ℚ[c]/(c^{b}) ⟶^{c^{a}} Σ^{2b−1}ℚ[c]/(c^{b})

→ Ext_{ℚ[c]} (Σ^{2a}ℚ[c]/(c^{a}), Σ^{2b−1}ℚ[c]/(c^{b})) → 0

so that

Ext_{ℚ[c]} (Σ^{2a}ℚ[c]/(c^{a}), Σ^{2b−1}ℚ[c]/(c^{b})) = \begin{cases}Σ^{2b−1}ℚ[c]/(c^{a}) & if a ≤ b, \\ Σ^{2b−1}ℚ[c]/(c^{b}) & if a > b. \end{cases}

To calculate Ext_{ℚ[c]} (π_{∗}^{𝕋}(ΣZ), π_{∗}^{𝕋}(S(b))), we use the projective resolution

0 → ℚ[c] ⊕ Σ^{2a−1}ℚ[c] → Σ^{2}ℚ[c] ⊕ Σ^{2a+1}ℚ[c] → Σ^{2}ℚ ⊕ Σ^{2a+1}ℚ → 0

of Σ^{2}ℚ ⊕ Σ^{2a+1}ℚ. Applying Hom (−, Σ^{2b−1}ℚ[c]/(c^{b})) to this sequence gives the exact sequence

⋯ → Σ^{2b−3}ℚ[c]/(c^{b}) ⊕ Σ^{2b−2a−2}ℚ[c]/(c^{b})

→ Σ^{2b−1}ℚ[c]/(c^{b}) ⊕ Σ^{2b−2a}ℚ[c]/(c^{b})
→ Ext_{ℚ[c]}(Σ^{2}ℚ ⊕ Σ^{2a+1}ℚ, Σ^{2b−1}ℚ[c]/(c^{b})) → 0.

Hence

Ext_{ℚ[c]} (Σ^{2}ℚ ⊕ Σ^{2a+1}ℚ, Σ^{2b−1}ℚ[c]/(c^{b})) = Σ^{2b−1}ℚ ⊕ Σ^{2b−2a}ℚ.

For any positive integer n, there are two non-zero maps from Σ^{2b−1}ℚ[c]/(c^{n}) to Σ^{2b−1}ℚ ⊕ Σ^{2b−2a}ℚ, one of even degree and one of odd degree. If n ≥ 2, neither of these maps is injective. Hence, for a, b > 1, the map (4) cannot be injective. The spectra S(a) and S(b) thus provide a counterexample to the equivariant generating hypothesis for the category of rational 𝕋-spectra. □

This result suggests that rationally, the generating hypothesis should fail for a continuous compact Lie group, although we have not shown this. The failure of the generating hypothesis in this case is perhaps surprising in light of the characterization of rings R for which the strong generating hypothesis in the derived category 𝒟(R). Hovey, Lockridge and Puninski [6, Thm 1.3] prove that the generating hypothesis holds in 𝒟(R) if and only if R is von Neumann regular. Since the Burnside ring of a compact Lie group is von Neumann regular, this suggests that the generating hypothesis might hold in equivariant stable category; however, we have shown that this is not true.

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