AUGMENTATION IDEALS OF EQUIVARIANT COHOMOLOGY RINGS.

J.P.C.GREENLEES

Abstract. The purpose of this note is to establish a number of useful results about the augmentation ideal J for the coefficient ring F_{G}^{∗} of a Noetherian complex orientable equivariant cohomology theory. The results show that various naturally occurring substitutes for the ideal have the same radical, and can therefore be used instead of the augmentation ideal in all geometric constructions.

1. Statement of results.

Let G be a finite group, and let F denote a ring G-spectrum, representing an equivariant cohomology theory F_{G}^{∗}(X). When comparing the equivariant theory with the associated non-equivariant theory it is natural to measure the difference by the augmentation ideal

J(G) = ker (F_{G}^{∗} = F_{G}^{∗}(S^{0}) ⟶ F_{G}^{∗}(G_{+}) = F^{∗}) .

When the ambient group is clear from the context it is common to simply write J, and it is one of the purposes of this note to show that it is reasonable to do this.

For example if H is a subgroup of G the restriction map res_{H}^{G} : F_{G}^{∗} = F_{G}^{∗}(S^{0}) ⟶ F_{G}^{∗}(G/H_{+}) = F_{H}^{∗} is a ring homomorphism which allows us to regard any F_{H}^{∗}-module M as a F_{G}^{∗}-module. It is fundamental to inductive proofs to know that completions and local cohomology for the ideals J(G) and J(H) coincide. It is convenient to work entirely with ideals of F_{H}^{∗}; the image of J(G) will not generally be an ideal, so we follow the convention that (res_{H}^{G}J(G)) denotes the ideal it generates. We work throughout with homogeneous ideals; for suitably periodic cohomology theories such as considered in [3] it would be sufficient to work entirely in degree 0.

We assume that the non-equivariant coefficient ring F^{∗} is an integral domain, so that J(G) is a prime ideal; this holds in the applications, but the assumption can be relaxed at the expense of taking radicals. Much more important is that our results all depend on a finiteness condition.

Finiteness Hypothesis 1.1. The ring F_{G}^{∗} is Noetherian, and for any finite complex X the module F_{G}^{∗}(X) is finitely generated over F_{G}^{∗}.

This is equivalent to asking that F_{G}^{∗} is Noetherian, and that for each subgroup H, the restriction map F_{G}^{∗} ⟶ F_{H}^{∗} makes F_{H}^{∗} into a finitely generated module over F_{G}^{∗}.

This is a very natural hypothesis, and satisfied in many cases of interest: for example it is satisfied for equivariant K-theory. It is also satisfied by the Borel theories F_{G}^{∗}(X) = F^{∗}(EG_{+} ∧_{G} X) associated to any non-equivariant multiplicative theory F^{∗}(·) which is Noetherian. This follows by the usual argument with the spectral sequence for the fibration U(n)/G ⟶ BG ⟶ BU(n) associated to a faithful representation G ⟶ U(n). This covers many complex oriented theories of interest, such as those of [3]. Unfortunately equivariant bordism does not satisfy Hypothesis 1.1.

Theorem 1.2. If F_{G}^{∗}(·) satisfies the finiteness hypothesis then

\sqrt{(res_{H}^{G}J(G))} = J(H).

We prove this in Section 5.

The basic way that we get control of the augmentation ideals is to find enough elements of them. We do this using Euler classes. In fact if V is a complex representation we may consider e_{V} : S^{0} ⟶ S^{V}, and the induced map e_{V}^{∗} : F_{G}^{∗}(S^{V}) ⟶ F_{G}^{∗}(S^{0}); this gives an Euler class e(V) ∈ F_{G}^{V}, by pulling back the unit map. It is usually more useful to have an element in integer degrees, so we suppose that F is complex oriented in the sense that it has Thom isomorphisms F_{G}^{∗}(S^{|V |} ∧ X) ≅ F_{G}^{∗}(S^{V} ∧ X) given by multiplication with a suitable element of F_{G}^{|V |}(S^{V}); we then let χ(V) ∈ F_{G}^{|V |} denote the image of e(V) under this Thom isomorphism. The element χ(V) depends on the Thom isomorphism, but only up to a unit multiple; for convenience we assume that the Thom isomorphisms are all given by composites of those for a chosen collection of simple representations V.

Lemma 1.3. (i) If V has a trivial summand then χ(V) = 0. (ii) For two complex representations V and W, we have χ(V ⊕ W) = χ(V)χ(W).

From 1.3 (i) we see that for any representation V we have χ(V) ∈ J(G). It is then natural to consider the ideal

E(G) = (χ(V) | V a simple representation ).

This is a finitely generated ideal containing all Euler classes and E(G) ⊆ J(G).

If G is a p-group E(G) is a good approximation to J(G). In fact it is a good approximation whenever G acts freely on a product of spheres of the form S(V_{1})×⋯×S(V_{n}); it is easy to prove by induction that this applies in particular to all p-groups, but the group A_{4} shows that it does not apply to all soluble groups. If G does act freely on such a product we let E′(G) denote the ideal generated by χ(V_{1}), …, χ(V_{n}); this will usually depend on the particular representations V_{1}, … V_{n}, but we do not display them in the notation. We show that this is also a good approximation to J(G). This is also interesting for compact Lie groups, and we intend to treat this case elsewhere.

Theorem 1.4. If G acts freely on a product of spheres and F_{G}^{∗}(·) satisfies the finiteness hypothesis then

\sqrt{E′(G)} = J(G).

We shall prove this in Section 3.

Note that if G acts freely on a particular product of spheres so does any subgroup, so that if E′(G) is defined, so is E′(H); since res_{H}^{G}e(V) = e(res_{H}^{G}V), we deduce res_{H}^{G}χ(V) is a unit multiple of χ(res_{H}^{G}V), so the special case of Theorem 1.2 in which G acts freely on a product of spheres follows from Theorem 1.4.

On the other hand, if G does not act freely on a product of spheres then E(G) is not a good approximation to J(G). Instead, for each prime p we may choose a Sylow p-subgroup P and form the ideal E′(P) of F_{P}^{∗}, and then the ideal (ind_{P}^{G}E′(P)) of F_{G}^{∗}. We warn that there is no reason to believe this is generated by elements ind_{P}^{G}χ(V) unless the restriction map is surjective. Now form the ideal

IE′(G) = ∑_{p | |G|}(ind_{P}^{G}E′(P)).

Finally, to make transfer arguments work it is worth adjoining certain elements of the augmentation ideal of the Burnside ring A(G) of finite G-sets; explicitly, we let

IA(G) = ([G/P] − |G : P | | p divides |G|).

The unit of the ring spectrum F is given by a map S^{0} ⟶ F. Applying equivariant homotopy, this gives a ring homomorphism A(G) ⟶ F_{G}^{∗}, compatible with induction and restriction, and we may let

J′(G) = IE′(G) + (IA(G)),

noting that this is a finitely generated ideal for any group and any coefficient ring F_{G}^{∗}. We prove that J′(G) is a good approximation to J(G) assuming Finiteness Hypothesis 1.1, and we believe it is a good approximation more generally.

Theorem 1.5. If F_{G}^{∗}(·) satisfies the finiteness hypothesis then

\sqrt{J′(G)} = J(G).

We prove this in Section 4, and this is the main ingredient in the proof of Theorem 1.2. The author thanks Hal Sadofsky for his interest, and many helpful comments, and espe-

cially for finding a mistake in an earlier version of the proof of Theorem 1.4.

2. Finite free complexes.

We defined the augmentation ideal as the kernel of the map induced by G/1_{+} ⟶ S^{0}. More generally, if T_{+} is any finite free G-complex with augmentation π : T_{+} ⟶ S^{0}, we may consider J_{T} (G) = ker(π^{∗} : F_{G}^{∗} ⟶ F_{G}^{∗}T_{+}).

Lemma 2.1. If T_{+} is a finite free complex with N cells then J(G)^{N}F_{G}^{∗}(T_{+}) = 0, and hence

J(G)^{N} ⊆ J_{T} (G).

Proof: Argue by induction on the number of cells of T_{+}. If T_{+} has a single cell then J acts as zero on F_{G}^{∗}(G/1_{+} ∧ S^{n}) by definition, and for the inductive step we consider a cofibre sequence G/1_{+} ∧ S^{n} ⟶ T′_{+} ⟶ T_{+} where T′_{+} has N − 1 cells.

Any 0-cell of T_{+} gives a diagram

G/1_{+} ⟶ S^{0}
↓  ↓=
T_{+} ⟶ S^{0},

which shows immediately that J_{T} (G) ⊆ J(G).

Corollary 2.2. If T_{+} is a finite free complex with N ≥ 1 cells then J(G)^{N} ⊆ J_{T} (G) ⊆ J(G), so that in particular

\sqrt{J_{T} (G)} = J(G).

3. Radicals of Euler class ideals.

Suppose then that G acts freely on the product S(V_{1}) × ⋯ × S(V_{n}). By Lemma 1.3, E′(G) ⊆ J(G), and we complete the proof of Theorem 1.4 in this section by showing that for some finite free complex T, we have J_{T} (G) ⊆ E′(G), so that by Lemma 2.1, E′(G) contains a power of J(G).

Because G acts freely on a product of spheres it is natural to consider the unbased finite free complexes

T (k) = S(kV_{1}) × ⋯ × S(kV_{n}).

The map to a point defines an augmentation T (k)_{+} ⟶ S^{0}, and hence an ideal J_{T (k)}(G). We prove that J_{T (k)}(G) ⊆ E′(G) for sufficiently large k.

First we give a more geometric description of E′(G). By definition χ(V) generates the image of the map induced by e_{V} : S^{0} ⟶ S^{V}, so E′(G) is the image of the map induced by the stable map m : S^{0} ⟶ S^{V_{1}} ∨ … ∨ S^{V_{n}} with Euler classes as components. Define L by the stable cofibre sequence

L ⟶^{l} S^{0} ⟶^{m} S^{V_{1}} ∨ … ∨ S^{V_{n}},

so that E′(G) = im(m^{∗}) = ker(l^{∗}). This is directly analogous to the definition of J_{T} (G) = ker(π^{∗}), but L will usually not be free.

Now define T̃(k) by the cofibre sequence

T (k)_{+} ⟶ S^{0} ⟶ T̃(k).

Note that each projection T (k) ⟶ S(kV_{i}), gives rise to a map p_{i} : T̃(k) ⟶ S^{kV_{i}}, and hence we obtain

p : T̃(k) ⟶ S^{kV_{1}} ∨ … ∨ S^{kV_{n}}

with components p_{i}. We may now form the composite L ⟶ S^{0} ⟶ T̃(k), and J_{T (k)}(G) ⊆ E′(G) if and only if the induced map F_{G}^{∗}(T̃(k)) ⟶ F_{G}^{∗}(L) is zero.

Consider the diagram 3.1.

L ⟶ S^{0} ⟶ T̃(1) ⟶ T̃(2) ⟶ T̃(3) ⟶ ⋯
=↓ =↓ ↓ p
L ⟶ S^{0} ⟶ ⋁_{i} S^{V_{i}}

in which the second row is a cofibre sequence, and the map p : T̃(1) ⟶ ⋁_{i} S^{V_{i}} was described above. Our strategy is to consider the inverse system {F_{G}^{∗}(T̃(k))}_{k}, and to show that for large enough k we have

3.2.

im (F_{G}^{∗}(T̃(k)) ⟶ F_{G}^{∗}(T̃(1))) ⊆ im (F_{G}^{∗}(⋁_{i} S^{V_{i}}) ⟶ F_{G}^{∗}(T̃(1)));

it then follows that the composite

F_{G}^{∗}(T̃(k)) ⟶ F_{G}^{∗}(T̃(1)) ⟶ F_{G}^{∗}(L)

is zero as required.

First we observe there is a natural way to calculate the cohomology of T (k)_{+} and T̃(k) based on Koszul complexes from commutative algebra. Recall that for any ring R and any element we may form the cochain complex K_{x}(R) = (R ⟶^{x} R) non-zero in codegrees 0 and 1. For a sequence x_{1}, …, x_{n} we may consider the unstable Koszul complex K_{x_{1},... ,x_{n}}(R) = K_{x_{1}}(R) ⊗ ⋯ ⊗ K_{x_{n}}(R), and the Koszul cohomology H_{x_{1},... ,x_{n}}^{∗}(R) = H^{∗}(K_{x_{1},... ,x_{n}}(R)).

We shall make use of the observation that the Koszul cohomology is torsion. Indeed, since xH^{∗}(K_{x}(R)) = 0 it follows that for any i the element x_{i}^{2} annihilates the Koszul cohomology.

Lemma 3.3. If σ(k) denotes the sequence χ(V_{1})^{k}, …, χ(V_{n})^{k} then there is a finite spectral sequence

H_{σ(k)}^{∗}(F_{G}^{∗}) ⟹ F_{G}^{∗}(T (k)_{+}).

Proof: The spectral sequence is that of a certain filtration of T (k)_{+}. In fact T (k)_{+} = S(kV_{1})_{+} ∧ … ∧ S(kV_{n})_{+}, and we obtain the filtration as a product of filtrations of the terms. Since we have cofibre sequences S(V)_{+} ⟶ S^{0} ⟶ S^{V}, we may view S(V)_{+} as Σ^{−1}(S^{V} ∪ CS^{0}), and we view this as filtered with 0th filtration Σ^{−1}S^{V} and first filtration S(V)_{+}.

Giving T (k)_{+} the product filtration we have

∗ = T (k)_{+}^{−1} ⊆ T (k)_{+}^{0} ⊆ ⋯ ⊆ T (k)_{+}^{n−1} ⊆ T (k)_{+}^{n} = T (k)_{+},

and use the standard cohomology spectral sequence indexing with E_{1}^{s,t} = F_{G}^{s+t}(T (k)_{+}^{s}/T (k)_{+}^{s−1}). Since T (k)_{+}^{s}/T (k)_{+}^{s−1} is the wedge of all terms Σ^{s−n}S^{kV_{i_{1}}⊕···⊕kV_{i_{n−s}}} with i_{1} < i_{2} < ⋯ < i_{n−s}, we have

E_{1}^{s,∗} = K_{σ(k)}^{s}(F_{G}^{∗}).

Since the inclusions of one sphere in the other induce multiplication by the Euler classes we obtain

E_{2}^{s,∗} = H_{σ(k)}^{s}(F_{G}^{∗})

as claimed.

Associated with the spectral sequence we have a filtration

0 = A^{n} ⊆ A^{n−1} ⊆ ⋯ ⊆ A^{0} ⊆ A^{−1} = F_{G}^{∗}(T (k)_{+})

with A^{s} = ker(F_{G}^{∗}(T (k)_{+}) ⟶ F_{G}^{∗}(T (k)_{+}^{s})). It follows from Lemma 3.3 that for any i multiplication by χ(V_{i})^{2k} reduces the filtration:

χ(V_{i})^{2k}A^{s} ⊆ A^{s+1}

and that a product of n + 1 such elements annihilates F_{G}^{∗}(T (k)_{+}).

Now the map T (k)_{+} ⟶ S^{0} is the quotient by the maximal filtration so we have

3.4.

ΣT (k)_{+}^{n−1} = T̃(k),

and furthermore, the map p : T̃(k) ⟶ ⋁_{i} S^{kV_{i}} can be identified with the suspension of the next subquotient map. Thus we have a precisely similar spectral sequence Ẽ_{∗}^{∗,∗} for F_{G}^{∗}(Σ^{−1}T̃(k)). Its E_{1} term Ẽ_{1}^{s,∗} is simply the truncation of E_{1}^{s,∗} with Ẽ_{1}^{n,∗} = 0. Being explicit at this end we have

⋯ ⟶  E_{1}^{n−2,∗}  ⟶ E_{1}^{n−1,∗} ⟶ E_{1}^{n,∗}
↓=  ↓=  ↓=
⋯ ⟶ F_{G}^{∗}(⋁_{i<j} S^{V_{i}⊕V_{j}}) ⟶ F_{G}^{∗}(⋁_{i} S^{V_{i}}) ⟶ F_{G}^{∗}(S^{0})

It follows that if X is any product of n + 1 terms of the form χ(V_{i})^{2k} we have

XF_{G}^{∗}(T̃(k)) ⊆ im (F_{G}^{∗}(⋁_{i} S^{V_{i}}) ⟶ F_{G}^{∗}(T̃(k))) .

Now the maps T̃(k) ⟶ T̃(k+1) respect the filtration, and the maps on subquotients are given by the relevant smash products of inclusions S^{kV} ⟶ S^{(k+1)V}. These maps therefore induce maps of spectral sequences, and at the E_{1} level these are induced by multiplication by various Euler classes. Indeed at the level of algebra the maps in question arise from

R ⟵^{x} R ⟵^{x} R ⟵^{x} ⋯
x ↓ x^{2} ↓ x^{3} ↓
R ⟵^{1} R ⟵^{1} R ⟵^{1} ⋯

by taking tensor products and deleting the copy of the ring in the largest cohomological degree.

Thus our annihilation result is not by itself sufficient, since multiplication by X would only be achieved by 2k(n + 1) steps of the system, and T̃(k) is only the kth: as observed in 3.2 we need to know that, for large enough k, the image of F_{G}^{∗}(T̃(k)) under the map T̃(1) ⟶ T̃(k) lies in im(F_{G}^{∗}(⋁_{i} S^{V_{i}}) ⟶ F_{G}^{∗}(T̃(1)). To upgrade our information to achieve this we need the finiteness hypothesis.

Lemma 3.5. If R is a Noetherian ring and σ(k) denotes the sequence x_{1}^{k}, …, x_{n}^{k}, then for each fixed j < n the inverse system {H_{σ(k)}^{j}(R)}_{k} is pro-zero.

Proof: We prove this by induction on n. For n = 1 the statement is that the inverse system

⋯ ⟶^{x} ann(x^{3}, R) ⟶^{x} ann(x^{2}, R) ⟶^{x} ann(x, R)

is pro-zero. This follows since the sequence of ideals

ann(x, R) ⊆ ann(x^{2}, R) ⊆ ann(x^{3}, R) ⊆ ⋯

is eventually constant, since R is Noetherian. If ann(x^{k_{0}}, R) = ann(x^{k_{0}+s}, R) for s ≥ 0 then the composite of any k_{0} successive maps in the system is zero.

For the inductive step we enrich the notation and let σ_{n+1}(k) denote the sequence x_{1}^{k}, …, x_{n+1}^{k}. Thus K_{σ_{n+1}(k)}(R) = K_{σ_{n}(k)}(R) ⊗ K_{x_{n+1}^{k}}(R), so that we have a fibre sequence

K_{σ_{n+1}(k)}(R) ⟶ K_{σ_{n}(k)}(R) ⟶^{x_{n+1}^{k}} K_{σ_{n}(k)}(R).

This gives a long exact sequence of pro-groups, which shows by induction that for j < n the inverse system {H^{j}(K_{σ_{n+1}(k)}(R))}_{k} is pro-zero. Since H^{n}(K_{σ_{n}(k)}(R)) = R/(σ_{n}(k)) and {H^{n−1}(K_{σ_{n}(k)}(R))} is pro-zero, it also gives the exact sequence

0 ⟶ {H^{n}(K_{σ_{n+1}(k)}(R))} ⟶ {R/(σ_{n}(k))} ⟶^{x_{n+1}^{k}} {R/(σ_{n}(k))}

of pro-groups. Thus {H^{n}(K_{σ_{n+1}(k)}(R))} is pro-isomorphic to the inverse system {ann(x_{n+1}^{k}, R/(σ_{n}(k)))}. However, since R is Noetherian the sequence

ann(x_{n+1}, R/(σ_{n}(k))) ⊆ ann(x_{n+1}^{2}, R/(σ_{n}(k))) ⊆ ann(x_{n+1}^{3}, R/(σ_{n}(k))) ⊆ ⋯

is eventually constant, at ann(x_{n+1}^{k_{0}}, R/(σ_{n}(k))) say. The map

ann(x_{n+1}^{k+k_{0}}, R/(σ_{n}(k + k_{0}))) ⟶ ann(x_{n+1}^{k}, R/(σ_{n}(k)))

thus factors through the zero map

ann(x_{n+1}^{k+k_{0}}, R/(σ_{n}(k))) ⟶ ann(x_{n+1}^{k}, R/(σ_{n}(k))),

and {H^{n}(K_{σ_{n+1}(k)}(R))} is pro-zero as required.

Remark 3.6. The fact that {H^{n}(K_{σ_{n+1}(k)}(R))} is pro-zero is exactly the statment that x_{1}, x_{2}, …, x_{n+1} is pro-regular in the sense of [2, 1.8], so the proof applies to that more general case.

Applying this with R = F_{G}^{∗}, we see that for each k we can find a number s so that T (k)_{+} ⟶ T (k +s)_{+} induces zero on the E_{2}^{s,t} for s < n. The corresponding result therefore applies also to the map ΣT̃(k) ⟶ ΣT̃(k + s) except in filtration n − 1. Thus there are numbers k(0) ≤ k(1) ≤ k(2) ≤ ⋯ ≤ k(n − 2) so that

im (F_{G}^{∗}(T̃(k(j))) ⟶ F_{G}^{∗}(T̃(1))) ⊆ ker (F_{G}^{∗}(T̃(1)) ⟶ F_{G}^{∗}(ΣT (1)_{+}^{j})) .

We observed in 3.4 that T (k)_{+}^{n−1} = Σ^{−1}T̃(k), so we find

\begin{aligned}im (F_{G}^{∗}(T̃(k(n − 2))) ⟶ F_{G}^{∗}(T̃(1))) & ⊆ & ker (F_{G}^{∗}(T̃(1)) ⟶ F_{G}^{∗}(ΣT (1)_{+}^{n−2})) \\ & = & im (F_{G}^{∗}(⋁_{i} S^{V_{i}}) ⟶ F_{G}^{∗}(T̃(1))) , \end{aligned}

which maps to zero in F_{G}^{∗}(L) by Diagram 3.1 as required.

This completes the proof of Theorem 1.4.

4. Radicals of explicit approximations.

In this section we prove Theorem 1.5. The containment J′(G) ⊆ J(G) is obvious. We show that any prime ideal containing J′(G) must also contain J(G). Let 𝔭 be a prime containing J′(G) and let p be its residue characteristic (ie (p) = 𝔭 ∩ ℤ). The case p = 0 is included, with the convention that the unique Sylow 0-subgroup is the trivial group.

Lemma 4.1. If P is a Sylow p-subgroup of G then 𝔭 = (res_{P}^{G})^{∗}𝔮 for some prime 𝔮 of F_{P}^{∗}.

Proof: Note that the image of the element [G/P] of the Burnside ring does not lie in 𝔭 since otherwise |G : P | = (|G : P | − [G/P]) + [G/P] ∈ 𝔭 which is impossible since |G : P | ∉ (p) = 𝔭 ∩ ℤ. Thus [G/P] acts as a unit on (F_{G}^{∗})_{𝔭}, and since ind_{P}^{G} res_{P}^{G} is multiplication by [G/P], restriction is injective. Since F_{G}^{∗} ⟶ F_{P}^{∗} is finite integrally by hypothesis, the going up theorem implies that Spec((F_{P}^{∗})_{𝔭}) ⟶ Spec((F_{G}^{∗})_{𝔭}) is surjective. The lemma follows since 𝔭 is in the image of Spec((F_{G}^{∗})_{𝔭}) ⟶ Spec(F_{G}^{∗}).

Note that this completes the proof if p = 0 since J(G) = (res_{1}^{G})^{∗}(0), so we may now suppose p > 0.

It is convenient to continue localised at 𝔭, and we need only show that J(G)_{𝔭} is not the unit ideal. Since J(G) = (res_{P}^{G})^{∗}J(P), it suffices to show that J(P)_{𝔮} is not the unit ideal. However, since J′(G) ⊆ 𝔭 = (res_{P}^{G})^{∗}𝔮 we find res_{P}^{G} ind_{P}^{G}E′(P) ⊆ res_{P}^{G}J′(G) ⊆ 𝔮. It therefore suffices to prove the following theorem, for which it is only necessary to localize at the integer prime p.

Theorem 4.2. For any Sylow p-subgroup P, and any p-local cohomology theory F_{G}^{∗}(·) satisfying the finiteness hypothesis

\sqrt{(res_{P}^{G} ind_{P}^{G}E′(P))} = J(P).

The proof has two steps. Firstly, on a large part of E′(P) the map res_{P}^{G} ind_{P}^{G} acts as multiplication by a unit in the p-local Burnside ring, and secondly that the large part of E′(P) is sufficient to determine the primes containing J(P).

Recall that conjugation induces a ring isomorphism c_{g} : F_{P}^{∗} ⟶ F_{P^{g}}^{∗} where P^{g} = g^{−1}P g, and that an element x ∈ F_{P}^{∗} is called stable if

res_{P ∩P^{g}}^{P^{g}}c_{g}x = res_{P ∩P^{g}}^{P}x

for all g ∈ G. Since restriction and conjugation are ring homomorphisms the subset stable(F_{P}^{∗}) of stable elements is a subring of F_{P}^{∗}.

Proposition 4.3. It x is stable then res_{P}^{G} ind_{P}^{G}x = ux for some element u ∈ A(P) which is a p-local unit

Proof: The double coset decomposition of G/P × G/P implies that in general

res_{P}^{G} ind_{P}^{G}x = ∑_{P gP} ind_{P ∩P^{g}}^{P}res_{P ∩P^{g}}^{P^{g}}c_{g}x;

if x is stable, the right hand side becomes ux where

u = ∑_{P gP} ind_{P ∩P^{g}}^{P}res_{P ∩P^{g}}^{P} = ∑_{P gP}[P/P ∩ P^{g}].

It remains only to observe this is a p-local unit. However, A(P)_{(p)} is a local ring with maximal ideal ℘(P, p) [1], so it is enough to observe that u ∉ ℘(P, p). In fact ℘(P, p) = {x ∈ A(P) | ϕ_{P} (x) = 0 mod p}, where ϕ_{P} : A(P) ⟶ ℤ is the ring homomorphism induced by counting the number of P -fixed points in a P -set. Now note that if P ∩ P^{g} ≠ P then ϕ_{P} ([P/P ∩ P^{g}]) = 0 mod p, whilst if P ∩ P^{g} = P then g ∈ N_{G}(P). Thus

ϕ_{P} (u) = |N_{G}(P)/P | ≠ 0 mod p,

as required.

Thus we have

stable(E′(P)) ⊆ res_{P}^{G} ind_{P}^{G}E′(P) ⊆ J(P),

and it suffices to show that J(P) is the unique minimal prime containing stable(E′(P)). It is to obtain the first containment that we have used ind_{P}^{G}E′(P), rather than the smaller ideal generated by the induced Euler classes. Since we are localised at p, it seems one might be able to avoid this, at the expense of complicating the argument.

If P is normal in G the argument is quite easy, since W = G/P acts on F_{P}^{∗}, and thus stability coincides with invariance: in particular stable(I) = I^{W} for any ideal I. Hence stable(F_{P}^{∗}) = (F_{P}^{∗})^{W}, and it is easy to check that

Spec([F_{P}^{∗}]^{W}) = Spec(F_{P}^{∗})/W.

More precisely we shall use the following well known lemma.

Lemma 4.4. Suppose a finite group W acts on a ring R. Given two primes 𝔔, 𝔔′ of R with (𝔔)^{W} ⊆ (𝔔′)^{W} we have w𝔔 ⊆ 𝔔′ for some w ∈ W.

In particular, we have seen that stable(E′(P)) lies in any prime 𝔮 containing res_{P}^{G} ind_{P}^{G}E′(P). But by 1.4 we have \sqrt{E′(P)} = J(P); since the stable elements form a subring, passage to

stable elements commutes with taking radicals, so that \sqrt{stable(E′(P))} = stable(J(P)), and therefore stable(J(P)) ⊆ 𝔮. By 4.4 this means wJ(P) ⊆ 𝔮 for some w; since wJ(P) = J(P) this completes the proof in case P is normal.

It remains to adapt this argument slightly to deal with the case when P is not normal. We let r_{g} : G/H ⟶ G/H^{g} denote the G-map given by right multiplication by g in the sense that r_{g}(xH) = xg(g^{−1}Hg). Thus for example c_{g} = (r_{g^{−1}})^{∗} : F_{P}^{∗} ⟶ F_{P^{g}}^{∗}.

Lemma 4.5. J(P) is invariant in the sense that r_{g}^{∗}J(P^{g}) = J(P).

Proof: The square

G/P ⟶^{r_{g}} G/P^{g}
↑  ↑
G/1 ⟶^{r_{g}} G/1

commutes.

Now form the category Sub_{p}(G) of p-subgroups of G, with inclusions as morphisms. The group G acts on Sub_{p}(G) by conjugation. Now F gives a contravariant functor Q ⟼ F_{Q}^{∗} on Sub_{p}(G), and we think of this as a sheaf on a discrete category. The sheaf theoretic point of view becomes important in the case of compact Lie groups. Let F_{Sub_{p}(G)}^{∗} denote the graded abelian group of global sections of the sheaf: this consists of functions x : P ⟼ x(P) ∈ F_{P}^{∗} on the space of objects compatible with restrictions, and can be expressed as an inverse limit:

F_{Sub_{p}(G)}^{∗} = lim_{←}_{Q} F_{Q}^{∗}.

Note that F_{Sub_{p}(G)}^{∗} still has an action of G. The purpose of this construction is to give a functorial description of the stable elements. Indeed, any stable element s of F_{P}^{∗} determines a section i(s): if Q ⊆ P^{g} we take i(s)(Q) = res_{Q}^{P^{g}}c_{g}s; by stability this is well defined. It is easy to check that i(s) is fixed by G.

Lemma 4.6. The above construction induces a ring isomorphism

stable(F_{P}^{∗}) ⟶^{≅} (F_{Sub_{p}(G)}^{∗})^{G}.

Proof: The inverse of i is evaluation of a section at P.

In view of our finiteness assumptions, we may apply Quillen’s results [4, Appendix B] to conclude that we understand primes. Quillen works with ungraded rings and ideals, but the result for homogeneous primes follows by considering the even graded subrings as ungraded rings and then restricting to homogeneous parts.

Corollary 4.7.

Spec(stable(F_{P}^{∗})) = (lim_{→}_{Q} Spec(F_{Q}^{∗})) /G.

We may now complete the argument by proving the following lemma.

Lemma 4.8. J(P) is the unique minimal prime containing stable(E′(P)).

Proof: We factorise the inclusion

stable(F_{P}^{∗}) = (F_{Sub_{p}(G)}^{∗})^{G} ⟶^{i} F_{Sub_{p}(G)}^{∗} ⟶^{j} F_{P}^{∗},

where j is evaluation at the stalk over P. Obviously i^{∗}j^{∗}I = stable(I) for any ideal I.

Suppose 𝔮 is a prime of F_{P}^{∗} with 𝔮 ⊇ stable(E′(P)) and hence 𝔮 ⊇ stable(J(P)) by 1.4, since passage to stable parts commutes with taking radicals. Obviously stable(𝔮) ⊇ stable(J(P)), and we must show that 𝔮 ⊇ J(P). By Lemma 4.4 it follows that j^{∗}𝔮 ⊇ gj^{∗}J(P) for some g ∈ G, and by Lemma 4.5 gj^{∗}J(P) = j^{∗}J(P) is invariant.

It remains to show that the direct limit does not create any new inclusions. First note that the category Sub_{p}(G) has the Sylow p-subgroups as maximal elements, so that the colimit is a quotient of the disjoint union of the spaces Spec(F_{P′}^{∗}) with P′ a Sylow subgroup. If 𝔔_{1} ∈ Spec(F_{P_{1}}^{∗}) and 𝔔_{2} ∈ Spec(F_{P_{2}}^{∗}) we declare 𝔔_{1} ∼ 𝔔_{2} if there is a subgroup Q ⊆ P_{1}∩P_{2} and a prime 𝔔 ∈ Spec(F_{Q}^{∗}) with 𝔔_{i} = (res_{Q}^{P_{i}})^{∗}𝔔 for i = 1 and 2. The equivalence relation giving the quotient is generated by ∼.

Lemma 4.9. If 𝔔_{1} ∼ 𝔔_{2} as above then 𝔔_{1} ⊇ J(P_{1}) implies 𝔔_{2} ⊇ J(P_{2}).

Proof: With the notation above, if 𝔔_{1} ⊇ J(P_{1}) then 𝔔 ⊇ res_{Q}^{P_{1}}J(P_{1}), and so by 1.4 we have 𝔔 = \sqrt{𝔔} ⊇ \sqrt{(res_{Q}^{P_{1}}J(P_{1}))} = J(Q). Now pull back along res_{Q}^{P_{2}}.

This completes the proof of Theorem 1.5.

5. Radicals of restrictions.

In this section we prove Theorem 1.2. Suppose H is a subgroup of G; we claim that the radical of the ideal generated by res_{H}^{G}J(G) is J(H). We have essentially proved this at each integer prime, so we need only formulate this precisely and deduce the integral statement.

Lemma 5.1. The ideal \sqrt{(res_{Q}^{G}J(G))} is p-locally J(Q) for all p-subgroups Q.

Proof: Choose a Sylow p-subgroup P containing Q, and note that J(P) ⊇ res_{P}^{G}J(G) ⊇ res_{P}^{G} ind_{P}^{G}E′(P), therefore by 4.2, J(P)_{(p)} = \sqrt{(res_{P}^{G}J(G))}_{(p)}.

This is sufficient since E′(P) ⊆ J(P) and (res_{Q}^{P}E′(P)) = E′(Q). In more detail,

E′(Q)_{(p)} = (res_{Q}^{P}E′(P))_{(p)} ⊆ (res_{Q}^{P}J(P))_{(p)} = (res_{Q}^{P}\sqrt{(res_{P}^{G}J(G))})_{(p)} = \sqrt{(res_{Q}^{P}res_{P}^{G}J(G))}_{(p)}.

Thus

\sqrt{E′(Q)}_{(p)} ⊆ \sqrt{res_{Q}^{G}J(G)}_{(p)} ⊆ J(Q)_{(p)},

and by 4.2 we have equality at all stages and

J(Q)_{(p)} = \sqrt{res_{Q}^{G}J(G)}_{(p)},

as required.

It therefore suffices to apply the following proposition with J^{∗} = (res_{H}^{G}J(G)).

Proposition 5.2. If J^{∗} ⊆ J(H) is an ideal of F_{H}^{∗} such that for all primes p and Sylow p-subgroups Q of H, J^{∗} is a pullback from F_{Q}^{∗} and the p-local ideal generated by res_{Q}^{H}J^{∗} has radical J(Q) then \sqrt{J^{∗}} = J(H).

Proof: For each prime p and Sylow subgroup Q of H the ideal J(H) is a pullback: J(H) = (res_{Q}^{H})^{∗}J(Q). By hypothesis, p-locally \sqrt{(res_{Q}^{H}J^{∗})} = J(Q); pulling back we find

\sqrt{((res_{Q}^{H})^{∗}(res_{Q}^{H}J^{∗}))} = (res_{Q}^{H})^{∗}\sqrt{(res_{Q}^{H}J^{∗})} = (res_{Q}^{H})^{∗}J(Q) = J(H).

Since J^{∗} is a pullback (res_{Q}^{H})^{∗}(res_{Q}^{H}J^{∗}) = J^{∗}, so that p-locally \sqrt{J^{∗}} = J(H). The theorem follows, since if the quotient J(H)/\sqrt{J^{∗}} were nontrivial, some p-localization would be nontrivial.

References

1. A.W.M.Dress ‘A characterization of soluble groups.’ Math. Z. 110 (1969) 213-217

2. J.P.C.Greenlees and J.P.May “Derived functors of completion and local homology.” J. Algebra 149 (1992) 438-453

3. J.P.C.Greenlees and Hal Sadofsky “Tate cohomology of theories with one dimensional coefficient ring” Preprint (1995) 14pp.

4. D.G.Quillen “The spectrum of an equivariant cohomology ring I and II.” Annals of Maths 94 (1971) 549-572 and 573-602.

School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH. UK.

E-mail address: j.greenlees@sheffield.ac.uk