Theorem 0.1. For a finite group G, 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 the full subcategory of finite spectra, this restriction is also full.

Proof. Let G be a finite group. Let X and Y be finite G-CW spectra, and let S be the fixed sphere 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. Let \underline{π}_{∗}^{G}(X) similarly be the graded Mackey functor that is \underline{π}_{n}^{G}(X) in degree n. Note that composition makes \underline{π}_{∗}^{G}(X) a module over \underline{π}_{∗}^{G}(S).

Suppose we have a natural transformation \underline{π}_{∗}^{G}(X) → \underline{π}_{∗}^{G}(Y). For each (conjugacy class of?) 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 : 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 f(α_{H,j,i}) on each wedge summand.

Let K_{n} be the kernel of the W_{n} →^{α} X in the abelian category for equivariant stable homotopy ℱ^{G}. Then the sequence 0 → K_{n} → W_{n} → X is exact. We show that \underline{π}_{∗}^{G}(K_{n} → W_{n} → Y) = 0, so we can apply the faithfulness hypothesis.

Consider \underline{π}_{∗}^{G}(W_{n}). Since S ∧ G/K_{+} is small for all K ≤ G, the \underline{π}_{∗}^{G}(S)-module \underline{π}_{∗}^{G}(W_{n}) breaks up as the coproduct ⨁_{j} ⨁_{H} ⨁_{i} \underline{π}_{∗}^{G}(Σ^{j}(S ∧ G/H_{+}). Consider each \underline{π}_{∗}^{G}(Σ^{j}(S ∧ G/H_{+}))(G/K). At each m, this is the abelian group

[Σ^{m}(S ∧ G/K_{+}), Σ^{j}(S ∧ G/H_{+})]_{G}.

Let Σ^{m}(S∧G/K_{+}] → K_{n} be an element of \underline{π}_{∗}^{G}(K_{n})(G/K). Then Σ^{m}(S∧G/K_{+}) → K_{n} → W_{n} is an element of \underline{π}_{m}^{G}(W_{n})(G/K). This breaks up as a direct sum of maps Σ^{m}(S∧G/K_{+}) → Σ^{j}(S∧G/H_{+}), where the Σ^{j}(S∧G/H_{+}) are the wedge summands of W_{n}. Restricting to one of these summands, we have the following situation:

\xymatrix{Σ^{m}(S ∧ G/K_{+}) \ar[d]^{β} \\ Σ^{j}(S ∧ G/H_{+}) \ar[d]^{α} \ar@{.>}[dr]_{fα} \\ X & Y}

Here β is the restriction of the map Σ^{m}(S ∧ G/K_{+}) → W_{n} to the wedge summand Σ^{j}(S ∧ G/H_{+}), and we can regard β as an element of \underline{π}_{∗}^{G}(Σ^{j}(S ∧ G/H_{+}). We are interested in computing (fα)_{∗}(β) as an element of \underline{π}_{∗}^{G}(Y)(Σ^{m}(S ∧ G/K_{+}). Note that β is also a morphism in the Burnside category, and since f is a natural transformation between the functors \underline{π}_{∗}^{G}(X) and \underline{π}_{∗}^{G}(Y), the following diagram commutes:

\xymatrix{\underline{π}_{∗}^{G}(X)(Σ^{j}(S ∧ G/H_{+})) \ar[r]^{β_{∗}} \ar[d]^{f} & \underline{π}_{∗}^{G}(X)(Σ^{m}(S ∧ G/K_{+})) \ar[d]^{f} \\ \underline{π}_{∗}^{G}(Y)(Σ^{j}(S ∧ G/H_{+})) \ar[r]^{β_{∗}} & \underline{π}_{∗}^{G}(Y)(Σ^{m}(S ∧ G/K_{+}))}

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Reinterpreted in terms of maps, we find that the following diagram commutes:

\xymatrix{Σ^{m}(S ∧ G/K_{+}) \ar[d]^{β} \ar@{.>}[ddr]^{f(α∘β)} \\ Σ^{j}(S ∧ G/H_{+}) \ar[d]^{α} \ar@{.>}[dr]_{fα} \\ X & Y}

Hence we see that (fα)_{∗}(β) is the same as f(α ∘ β). We can thus write the element Σ^{m}(S ∧ G/K_{+}) → K_{n} → W_{n} → Y in \underline{π}_{∗}^{G}(Y) as a sum

∑_{H,j,i} f(α ∘ β) = f( ∑_{H,j,i} α ∘ β).

Since ∑_{H,j,i} α ∘ β is the composition of two maps an exact sequence, it must be zero. Thus ∑_{H,j,i} f(α ∘ β) = 0, as desired.

Therefore we can apply the generating hypothesis, extended to the category ℱ^{G} to conclude that h_{n} : K_{n} → W_{n} → Y is zero. Thus we have:

\xymatrix{K_{n} \ar[r]^{ϕ} \ar[dr]_{0} & W_{n} \ar[d]^{h_{n}} \ar[r] & X \\  & Y}

which gives

\xymatrix{0 \ar[r] & cokerϕ \ar[d] \ar[r] & X \ar@{.>}[dl]^{f_{n}} \\  & Y}

where the map f_{n} is given by the injectivity of Y. Thus we have a sequence of maps {f_{n} : X → Y }.

Since X is finite, we may choose d large enough so that H_{m}(X, \underline{A}) = 0 for all m > d, where \underline{A} is the Burnside ring Mackey functor for the group G. Let C_{d} be the cokernel of W_{d} → X. For m > d, there exists a commutative diagram

\xymatrix{W_{d} \ar[r] \ar[d]_{α} & W_{m} \ar[dl]^{α} \\ X}

by the definition of α. From this, we construct a diagram

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

Hence the map W_{d} → X ⟶^{f_{m}} Y is the same as h_{d}. By the universal property of cokernel, we obtain a map making the following diagram commute:

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

We claim that ℱ^{G}(C_{d}, Y) is finite, that is, that ℱ^{G}(C_{d}, Y)⊗ℚ = 0. First consider H_{∗}^{G}(C_{d}; \underline{A}). Since the functor \underline{H}_{m}^{G}(−; \underline{A}) is exact, for each H ≤ G, we have an exact sequence of abelian groups

\underline{H}_{m}^{G}(W_{d}; \underline{A})(G/H) → \underline{H}_{m}^{G}(X; \underline{A})(G/H) → \underline{H}_{m}^{G}(C_{d}; \underline{A})(G/H) → 0

for all m. When m > d, \underline{H}_{m}^{G}(X; \underline{A}) = 0, and thus \underline{H}_{m}^{G}(C_{d}; \underline{A}) is zero as well. On the other hand, when m ≤ d, we can tensor with the flat module ℚ to get the exact sequence

\underline{H}_{m}^{G}(W_{d}; \underline{A})(G/H) ⊗ ℚ → \underline{H}_{m}^{G}(X; \underline{A})(G/H) ⊗ ℚ → \underline{H}_{m}^{G}(C_{d}; \underline{A})(G/H) ⊗ ℚ → 0

Now, as shown in [1], the rationalization of the fixed sphere spectrum S is H(\underline{A}⊗ℚ). This implies that \underline{H}_{m}^{G}(W_{d}; \underline{A})⊗ℚ = \underline{π}_{m}^{G}(W_{d})⊗ℚ and \underline{H}_{m}^{G}(X; \underline{A})⊗ℚ = \underline{π}_{m}^{G}(X)⊗ℚ. Thus \underline{π}_{m}^{G}(C_{d}) ⊗ ℚ = 0 for m > d. Futhermore, for m ≤ d, the map α_{∗} : [Σ^{m}(S ∧ G/H_{+}), W_{d}] → [Σ^{m}(S ∧ G/H_{+}), X] is rationally surjective by construction, so [Σ^{m}(S ∧ G/H_{+}), C_{d}] ⊗ ℚ is zero for all H ≤ G, and thus \underline{π}_{m}^{G}(C_{d}) ⊗ ℚ = 0 for all m.

Denote by Y_{ℚ} the rationalization of Y; then ℱ^{G}(C_{d}, Y) ⊗ ℚ = ℱ^{G}(C_{d}, Y_{ℚ}). As shown in [1], for any G-spectrum X, the natural map

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

is an isomorphism. Moreover, both sides are cohomology theories on G-spectra, so, since for G-spectra X, Y, [X, Y] = ℱ^{G}(X, Y), we conclude that the columns are exact in the following diagram:

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

By the five lemma, ℱ^{G}(C_{d}, Y_{ℚ}) ≅ ∏ Hom_{ℳ[G]} (\underline{π}_{n}^{G}(W_{d}), \underline{π}_{n}^{G}(Y_{ℚ})). However, since Y_{ℚ} is rational, we can without change replace C_{d} by its rationalization C_{d}_{ℚ}. But \underline{π}_{n}^{G}(C_{d}_{ℚ} = \underline{π}_{n}^{G}(C_{d}) ⊗ ℚ = 0 by the above, and so ℱ^{G}(C_{d}, Y) ⊗ ℚ = 0, as desired.

Therefore, the sequence {f_{m}} has a constant cofinal subsequence, and we can take its value to be the desired map \overline{f} such that \underline{π}_{∗}^{G}(\overline{f}) = f.

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References

[1] J. P. C. Greenlees and J. P. May. Generalized Tate Cohomology. Mem.of the Am. Math. Soc, No. 543. 1995.