The spectrum of the Chern subring

David J. Green and Ian J. Leary

13 November 1996

Abstract. For certain subrings of the mod-p-cohomology of a compact Lie group, we give a description of the spectrum, analogous to Quillen’s description of the spectrum of the whole cohomology ring. Subrings to which our theorem applies include the Chern subring. Corollaries include a characterization of those groups for which the Chern subring is F-isomorphic to the cohomology ring.

1. Introduction. Let G be a compact Lie group (e.g., a finite group) and let H^{∗}(G) = H^{∗}(BG; F_{p}) be its mod-p cohomology ring. This ring is a finitely generated graded-commutative F_{p}-algebra. In [16], D. Quillen studied this ring from the viewpoint of commutative algebra. His results may be stated in terms of the prime ideal spectrum of H^{∗}(G), but the cleanest statement concerns the variety, X_{G}(k), of algebra homomorphisms from H^{∗}(G) to an algebraically closed field k of characteristic p. The Chern subring, Ch(G) ⊆ H^{∗}(G), is the subring generated by Chern classes of unitary representations of G. We give a description of X′_{G}(k), the variety of algebra homomorphisms from Ch(G) to k, analogous to Quillen’s description of X_{G}(k). As corollaries of this result, we classify the minimal prime ideals of Ch(G), and characterize those groups G for which the natural map from X_{G}(k) to X′_{G}(k) is a homeomorphism.

In the case when G = E is an elementary abelian p-group, i.e., a direct product of copies of the cyclic group of order p, X_{E}(k) is naturally isomorphic to E ⊗ k, where E is viewed as a vector space over F_{p} and the tensor product is taken over F_{p}. For general G, Quillen describes X_{G}(k) as the colimit of the functor (−) ⊗ k over a category A = A(G) with objects the elementary abelian subgroups of G, and morphisms those group homomorphisms that are induced by conjugation in G. Our description of X′_{G}(k) is as the colimit of the same functor over a category A′. This category has the same objects as Quillen’s category, but a morphism in A′ is a group homomorphism that merely preserves conjugacy in G. In other words, a group homomorphism f: E_{1} → E_{2} satisfies:

\begin{aligned}f ∈ A & ⇔ ∃g ∀e, f(e) = g^{−1}eg, \\ f ∈ A′ & ⇔ ∀e ∃g, f(e) = g^{−1}eg. \end{aligned}

This theorem is a corollary of a more general colimit theorem, which says, roughly, that the variety for any subring of H^{∗}(G) that is both ‘large’ and ‘natural’ may be expressed as such a colimit. Other corollaries of this theorem, in the case when G is finite, include a description of the variety for the subring of Ch(G) generated by Chern classes of representations realizable over any subfield of C, and a slight variation on the usual proof of Quillen’s theorem in which transfers of Chern classes are used instead of the Evens norm map.

Acknowledgements. This work was started while the first named author held a Max Kade visiting fellowship at the University of Chicago, and the second named author held a Leibniz fellowship at the Max Planck Institut für Mathematik, Bonn. The first named author gratefully acknowledges the support of the Deutsche Forschungsgemeinschaft and the Max Kade Foundation. The second named author gratefully acknowledges the support of the Leibniz Fellowship Programme.

The authors thank Dave Benson, Jon Carlson, Hans-Werner Henn, David Kirby and Jeremy Rickard for their helpful comments on this work.

2. Representations and the Chern subring. First we recall some facts concerning Chern classes [3,21]. Let U(n) be the group of n × n unitary matrices, and T (n) its subgroup of diagonal matrices. Then H^{∗}(T (n)) is a free polynomial algebra F_{p}[x_{1}, …, x_{n}] on n generators of degree two. H^{∗}(U(n)) is isomorphic to F_{p}[c_{1}, …, c_{n}], where c_{i} has degree 2i. The map from H^{∗}(U(n)) to H^{∗}(T (n)) is injective and sends c_{i} to the ith elementary symmetric function in the x_{i}.

Let G be a compact Lie group. G has faithful finite-dimensional complex representations, and any finite-dimensional representation is equivalent to a unitary representation. If ρ: G → U(n) is a unitary representation of G, write ρ: BG → BU(n) for the induced map of classifying spaces, whose homotopy class depends only on the equivalence class of ρ. The ith Chern class of ρ is defined by c_{i}(ρ) = ρ^{∗}(c_{i}) ∈ H^{2i}(G). Define c_{·}(ρ), the total Chern class of ρ, to be 1 + c_{1}(ρ) + ⋯ + c_{n}(ρ). Chern classes enjoy the following properties (‘Whitney sum formula’ and ‘naturality’), for any θ: G → U(m) and any f: H → G:

c_{·}(ρ ⊕ θ) = c_{·}(ρ)c_{·}(θ) c_{·}(ρ ∘ f) = f^{∗}c_{·}(ρ).

There is a unique way to define Chern classes for virtual representations so that they continue to enjoy the above properties. Let c′_{·} = 1 + c′_{1} + c′_{2} + ⋯ be the unique power series in F_{p}[[c_{1}, …, c_{n}]] satisfying c′_{·}c_{·} = 1, and define c_{i}(−ρ) = ρ^{∗}(c′_{i}). In general infinitely many of the c_{i}(−ρ) will be non-zero, but note that they are all expressible in terms of the c_{i}(ρ). Definition. The Chern subring Ch(G) of H^{∗}(G) is the subring generated by the c_{i}(ρ) for all i and all virtual representations ρ.

By the above remarks, the c_{i}(ρ) as ρ ranges over the irreducible representations of G suffice to generate Ch(G). In the case when G is finite, it follows that Ch(G) is finitely generated, since G has only finitely many inequivalent irreducible representations. For general G it is also true that Ch(G) is finitely generated. This is a special case of the following proposition.

Proposition 2.1. Let G be a compact Lie group, and ρ: G → U(n) a faithful unitary representation of G. If R is a subring of H^{∗}(G) containing each c_{i}(ρ), then R is finitely generated.

Proof. Venkov showed that H^{∗}(G) is finitely generated by showing that H^{∗}(G) is finite over (i.e., is a finitely generated module for) H^{∗}(U(n)) [22,16,5]. Now R is an H^{∗}(U(n))- submodule of H^{∗}(G), so is finitely generated since H^{∗}(U(n)) is Noetherian.

Remark. For G finite, the finite-generation of H^{∗}(G) is due independently to Evens and to Venkov by completely different proofs [9,22]. There is another proof (closer to Evens’ than to Venkov’s) in [8].

Definition. A virtual representation ρ of G is said to be p-regular if the virtual dimension of ρ is strictly positive and for every elementary abelian subgroup E ≅ (Z/p)^{n} of G, the restriction to E of ρ is a direct sum of copies of the regular representation of E.

Proposition 2.2. For each prime p, every compact Lie group G has a p-regular representation.

Proof. G has a faithful representation in U(n) for some n, and every elementary abelian subgroup of U(n) is conjugate to a subgroup of T (n), the torus consisting of diagonal matrices. Thus it suffices to show that U(n) has a virtual representation whose restriction to (Z/p)^{n} ⊆ T (n) is the regular representation.

Recall that the representation ring R(T (n)) of T (n) is isomorphic to the Laurent polynomial ring Z[τ_{1}, τ_{1}^{−1}, …, τ_{n}, τ_{n}^{−1}], where τ_{i} is the 1-dimensional representation

τ_{i}: diag(ξ_{1}, …, ξ_{n}) ↦ ξ_{i}.

R(U(n)) maps injectively to R(T (n)) with image the subring Z[σ_{1}, σ_{2}, …, σ_{n}, σ_{n}^{−1}], where σ_{i} is the ith elementary symmetric function in the τ_{j}.

The polynomial

P = ∏_{i=1}^{n}(1 + τ_{i} + ⋯ + τ_{i}^{p−1})

is a symmetric polynomial in the τ_{j}, and so is expressible in terms of σ_{1}, …, σ_{n}. The corresponding (p^{n}-dimensional) representation of U(n) restricts to (Z/p)^{n} ⊆ T (n) as the regular representation.

Remark. For G finite, the regular representation of G is of course p-regular.

Using Quillen’s result that we state as Theorem 5.1, it may be shown that H^{∗}(G) is finite over the subring generated by the Chern classes of any p-regular representation. For genuine (as opposed to virtual) representations, this can be deduced from Venkov’s result: If ρ: G → U(n) is a p-regular representation of G, the kernel of ρ contains no elements of order p, and is therefore a finite group of order coprime to p. It follows that H^{∗}(ρ(G)) ≅ H^{∗}(G) (consider the spectral sequence for the extension ker(ρ) → G → ρ(G)), and hence H^{∗}(G) is finite over the image of ρ^{∗}. When p = 2, the representation constructed in the proof of Proposition 2.2 is a genuine representation. David Kirby has shown us an argument to prove that U(n) has a p-regular genuine representation if and only if either p = 2, or n = 1, or (p, n) = (3, 2).

3. Varieties for cohomology. Let k be an algebraically closed field of characteristic p, and let R be a finitely generated commutative F_{p}-algebra. Define V_{R}(k), the variety for R, to be the set of ring homomorphisms from R to k, with the Zariski topology, i.e., the smallest topology in which the set

F_{I} = {ϕ: R → k | ker(ϕ) ⊇ I}

is closed for each ideal I of R. A ring homomorphism f: R → S gives rise to a continuous map f^{∗}: V_{S}(k) → V_{R}(k). If S is finite over f(R) (i.e., S is a finitely generated f(R)- module) then f^{∗} is a closed mapping and has finite fibres, by the ‘going up’ or ‘lying over’ theorem [4,6]. If S is finite over f(R) and ker(f) is nilpotent, then f^{∗} is surjective.

There is a continuous map from V_{R}(k) to Spec(R), the prime ideal spectrum of R, that sends the map ϕ: R → k to the ideal ker(ϕ). If the transcendence degree of k over F_{p} is sufficiently large (as large as a generating set for R will suffice), then this map is surjective. Thus information about V_{R}(k) gives rise to information about Spec(R).

We shall also require the following:

Proposition 3.1. (a) Let S be a subring of R containing R^{p}, the subring of pth powers of elements of R. Then the natural map from V_{R}(k) to V_{S}(k) is a homeomorphism.

(b) Let a finite group G act on R, with fixed point subring S = R^{G}. Then the natural map V_{R}(k) → V_{S}(k) induces a homeomorphism V_{R}(k)/G → V_{S}(k).

Proof. Each of these claims may be proved by showing that R is finite over S, and deducing that the map given is continuous, closed, and a bijection. See for example [4,6,16].

For p = 2, the ring H^{∗}(G) is commutative. For p an odd prime, elements of H^{∗}(G) of odd degree are nilpotent, so although H^{∗}(G) is not commutative, the quotient of H^{∗}(G) by its radical, h^{∗}(G) = H^{∗}(G)/\sqrt{0}, is commutative. Any homomorphism from H^{∗}(G) to k factors through h^{∗}(G). Define X_{G}(k) to be the variety V_{R}(k) for R = h^{∗}(G). By the above remark, points of X_{G}(k) may be viewed as homomorphisms from H^{∗}(G) to k. Let S be the subring of elements of H^{∗}(G) of even degree. A homomorphism ϕ: S → k extends uniquely to a homomorphism from H^{∗}(G) to k (if x is in odd degree, then either p is odd, and x^{2} = 0, or p = char(k) = 2, so in either case ϕ(x) is the unique square root of ϕ(x^{2})). It follows that the natural map X_{G}(k) → V_{S}(k) is a homeomorphism, since it is a closed, continuous bijection. Hence X_{G}(k) could equally be defined in terms of S.

Note that a group homomorphism f: H → G induces a map f_{∗}: X_{H}(k) → X_{G}(k). We write ι_{H}^{G} for f_{∗} in the case when f is the inclusion of a subgroup H in G. A theorem of Evens and Venkov [9,22] states that in this case H^{∗}(H) is finite over H^{∗}(G). (To deduce this from the result quoted in the proof of Proposition 2.1, note that a faithful representation of G restricts to a faithful representation of H.) It follows that ι_{H}^{G} is closed and has finite fibres.

Define X′_{G}(k) to be V_{Ch(G)}(k). By Proposition 2.1 the natural map from X_{G}(k) to X′_{G}(k) is surjective, closed, and has finite fibres.

Proposition 3.2. Let ρ be a representation of G, and let R(n) be the subring of H^{∗}(G) generated by the Chern classes of nρ = ρ^{⊕n} = ρ⊕⋯⊕ρ. Then the natural map V_{R(1)}(k) → V_{R(n)}(k) is a homeomorphism.

Proof. If p does not divide n, then

c_{i}(nρ) = nc_{i}(ρ) + P (i, n),

for some expression P (i, n) in the c_{j}(ρ) for j < i. So in this case R(n) = R(1). On the other hand, if n = pm then

c_{·}(nρ) = c_{·}(mρ)^{p} = 1 + c_{1}(mρ)^{p} + c_{2}(mρ)^{p} + ⋯,

so in this case R(n) = R(m)^{p}, the subring of pth powers of elements of R(m), and the map V_{R}(k) → V_{R^{p}}(k) is a homeomorphism.

The methods that we shall use to study the Chern subring apply equally to the Stiefel-Whitney subring, defined analogously, in the case when p = 2. (For information concerning Stiefel-Whitney classes see [21]). As an alternative, the following proposition may be applied.

Proposition 3.3. Let p = 2 and let S be the subring of H^{∗}(G) generated by Stiefel-Whitney classes of real representations of G. Then

S^{2} ⊆ Ch(G) ⊆ S,

and the natural map from V_{S}(k) to X′_{G}(k) is a homeomorphism.

Proof. If θ is an n-dimensional real representation of G, then θ^{C}, the complexification of θ, is an n-dimensional complex representation of G with c_{i}(θ^{C}) = w_{i}^{2}(θ). Conversely, if ψ is an n-dimensional complex representation of G and ψ_{R} is the same representation viewed as a 2n-dimensional real representation, then w_{i}(ψ_{R}) = 0 for i odd and w_{2i}(ψ_{R}) = c_{i}(ψ). This proves the claimed inclusions. The claimed homeomorphism follows from Proposition 3.1(a).

4. Examples. In this section we discuss the case of an elementary abelian p-group, and also give an example to show that the map X_{G}(k) → X′_{G}(k) is not always a homeomorphism. This example was the starting point for the work of this paper.

Let E be an elementary abelian p-group of rank n, E ≅ (Z/p)^{n}. Then E may be viewed as a vector space over F_{p}. Write E^{∗} for Hom(E, F_{p}). There is a natural isomorphism E^{∗} ≅ H^{1}(E). For p = 2, H^{∗}(E) is isomorphic to the symmetric algebra on H^{1}(E), or equivalently, the ring of polynomial functions on E viewed as a vector space:

H^{∗}(E) ≅ S(E^{∗}) ≅ F_{p}[E].

For p > 2, the Bockstein β: H^{1}(E) → H^{2}(E) is injective, and H^{∗}(E) is isomorphic to the tensor product of the exterior algebra on H^{1}(E) tensored with the symmetric algebra on B = β(H^{1}(E)):

H^{∗}(E) ≅ Λ(E^{∗}) ⊗ S(B) ≅ Λ(E^{∗}) ⊗ F_{p}[E].

In any case, h^{∗}(E) is naturally isomorphic to F_{p}[E], generated in degree one for p = 2 and in degree two for odd p. It follows that X_{E}(k) is naturally isomorphic to E ⊗ k, where E is viewed as a vector space over F_{p} and the tensor product is over F_{p}, so that E ⊗ k ≅ k^{n}.

Irreducible representations of E are 1-dimensional, and the map ρ ↦ c_{1}(ρ) is a natural bijection between the set of irreducible representations of E and B = β(H^{1}(E)). (When p = 2, β = Sq^{1}, and so β(x) = x^{2}.) The Chern subring Ch(E) of H^{∗}(E) is the subalgebra of H^{∗}(E) generated by B. For p > 2 this subring maps onto h^{∗}(E), and for p = 2 it maps onto h^{∗}(E)^{2}, the subring of squares of elements of h^{∗}(E). In any case, the map from X_{E}(k) to X′_{E}(k) is a homeomorphism.

Proposition 4.1. Let ρ be a direct sum of copies of the regular representation of E, and let R be the subring of H^{∗}(E) generated by the Chern classes of ρ. Then the natural map from X_{E}(k) to V_{R}(k) factors through a homeomorphism

k^{n}/GL_{n}(F_{p}) ≅ X_{E}(k)/GL(E) → V_{R}(k).

Proof. By Proposition 3.2 it suffices to consider the case when ρ is the regular representation. Identify Ch(E) with F_{p}[E], generated in degree two. The total Chern class of ρ is

c_{·}(ρ) = ∏_{x∈E^{∗}}(1 + x).

This is invariant under the full automorphism group, GL(E), of E. By a theorem of Dickson, the only i > 0 for which c_{i}(ρ) is non-zero are i = p^{n} − p^{j}, where 0 ≤ j < n = dim_{F_{p}}(E). Moreover, these c_{i}(ρ) freely generate a polynomial subring of F_{p}[E], and this is the complete ring of GL(E)-invariants in F_{p}[E] [6,23]. The claim follows by part (b) of Proposition 3.1. Remark. Let A be a non-identity element of GL_{n}(F_{p}), and let v be an element of k^{n} fixed by A. Then v is in the kernel of I − A, a non-zero matrix with entries in F_{p}, and so v lies in a proper subspace of k^{n} defined over F_{p} (i.e., a subspace of the form V ⊗ k for some proper F_{p}-subspace V of F_{p}^{n}). It follows that GL_{n}(F_{p}) acts freely on the complement of all such subspaces. For an elementary abelian group E, let

X_{E}^{+}(k) = X_{E}(k) \ ⋃_{F <E} ι_{F}^{E}(X_{F} (k)).

By the above argument, GL(E) acts freely on X_{E}^{+}(k). Example. Let q = p^{n} for some n ≥ 2. Let G be the affine transformation group of the line over F_{q}. Then G is expressible as an extension with kernel E = (F_{q}, +), an elementary abelian p-group of rank n, and quotient Q = GL_{1}(F_{q}), cyclic of order q−1. The conjugation action of Q on E is transitive on non-identity elements of E. One example of such a group is the alternating group A_{4} (p = 2, n = 2). An easy transfer argument shows that H^{∗}(G) maps isomorphically to the ring of invariants, H^{∗}(E)^{Q}, and it follows from Proposition 3.1 that X_{G}(k) is homeomorphic to X_{E}(k)/Q = k^{n}/Q, where Q = GL_{1}(F_{q}) ≤ GL_{n}(F_{p}) ≤ GL_{n}(k).

It is easy to check that G has exactly q distinct irreducible representations. All but one of these are 1-dimensional and restrict to E as the trivial representation. The other one is (q −1)-dimensional and restricts to E as the regular representation minus the trivial representation. Hence by Proposition 4.1, X′_{G}(k) is homeomorphic to X_{E}(k)/GL(E) = k^{n}/GL_{n}(F_{p}). Thus the map from X_{G}(k) to X′_{G}(k) is not a homeomorphism.

5. Quillen’s colimit theorem. In [16], Quillen showed that for general G, X_{G}(k) is determined by the elementary abelian subgroups of G. Roughly speaking, he showed that X_{G}(k) is equal to the union of the images of the X_{E}(k), where E ranges over the elementary abelian subgroups of G, and that as little identification takes place between the points of the X_{E}(k) as is consistent with the fact that inner automorphisms of G act trivially on H^{∗}(G). More precisely, let f: E_{1} → E_{2} be a homomorphism between elementary abelian subgroups of G that is induced by an inner automorphism of G. Then the following diagram commutes.

H^{∗}(G) ⟵^{Id} H^{∗}(G)
↓^{Res}  ↓^{Res}
H^{∗}(E_{1}) ⟵^{f^{∗}} H^{∗}(E_{2})

Consequently the following diagram commutes.

X_{G}(k) ⟶^{Id} X_{G}(k)
↑^{ι}  ↑^{ι}
X_{E_{1}}(k) ⟶^{f_{∗}} X_{E_{2}}(k)

This fact motivates the following definition. Definition. The Quillen category A for a compact Lie group G and a prime p is the category whose objects are the elementary abelian p-subgroups of G, with morphisms from E_{1} to E_{2} being those group homomorphisms that are induced by conjugation in G. Any such group homomorphism is of course injective.

In general G will have infinitely many elementary abelian p-subgroups. These subgroups form finitely many conjugacy classes though ([16], lemma 6.3). Thus although the Quillen category for G is infinite in general, it contains only finitely many isomorphism types of object (or is ‘skeletally finite’).

The morphisms f: E_{1} → E_{2} in the Quillen category are precisely the maps for which the diagram above commutes. It follows that the natural map

∐_{{E≤G \atop E el. ab.}} X_{E}(k) ⟶ X_{G}(k)

factors through a map α: colim_{A}X_{E}(k) → X_{G}(k).

Theorem 5.1. (Quillen [16]) The map α: colim_{A}X_{E}(k) → X_{G}(k) is a homeomorphism.

The map α is continuous and is closed because A is skeletally finite. Thus the main content of the theorem is that α is a bijection. We shall use only half of this theorem, the statement that α is surjective, in our main theorem. The surjectivity of α is equivalent to the statement ‘an element of H^{∗}(G) is nilpotent if and only if its image in each H^{∗}(E) is nilpotent’.

6. A new colimit theorem. Motivated by the Quillen category, we define: Definition. A category of elementary abelian subgroups of G is a category whose objects are (all of) the elementary abelian p-subgroups of G, and whose morphisms from E_{1} to E_{2} are injective group homomorphisms.

The Quillen category, A(G), is of course a category of elementary abelian subgroups of G. Another example is the category C_{reg}(G), with the morphism set C_{reg}(E_{1}, E_{2}) equal to the set of all 1-1 group homomorphisms from E_{1} to E_{2}. Any category of elementary abelian subgroups of G is a subcategory of C_{reg}(G).

Any subring R of H^{∗}(G) gives rise to a category C(R) of elementary abelian subgroups of G, where f: E_{1} → E_{2} is a morphism in C(R) if and only if

R  ⟵^{Id}  R
↓^{Res}  ↓^{Res}
h^{∗}(E_{1}) ⟵^{f^{∗}} h^{∗}(E_{2})

commutes. Note that we use h^{∗}(E_{i}), the cohomology ring modulo its radical, rather than H^{∗}(E_{i}). Note that if f is an isomorphism of groups and is a morphism in C(R), f^{−1} is also in C(R). Each C(R) contains the Quillen category, and hence is skeletally finite. By the argument given in Section 5, the map α: colim_{A}X_{E}(k) → X_{G}(k) induces a map γ = γ(R): colim_{C(R)}X_{E}(k) → V_{R}(k). Definition. Say that a subring of H^{∗}(G) is large if it contains the Chern classes of some p-regular representation of G. Say that a subring of H^{∗}(G) is natural if it is generated by homogeneous elements and is closed under the action of the Steenrod algebra.

The new colimit theorem of the title of this section is:

Theorem 6.1. Let G be a compact Lie group, and let R be a subring of H^{∗}(G) that is both large and natural. Then the map

γ: colim_{C(R)}X_{E}(k) → V_{R}(k)

is a homeomorphism.

It is possible that this theorem could be proved using more general colimit theorems due to S. P. Lam, to D. Rector, and to H.-W. Henn, J. Lannes and L. Schwartz [14,18,12]. These theorems say, roughly speaking, that the variety for any Noetherian algebra over the Steenrod algebra should be expressible as a similar sort of colimit. Even with these theorems, Quillen’s description of X_{G}(k) would still be needed to identify the categories that arise with categories of elementary abelian subgroups of G. The proof given below is more elementary, in that it relies on no work that is more recent than that of Quillen.

The proof of the theorem uses the following lemma.

Lemma 6.2. Let S be the subring of H^{∗}(G) generated by the Chern classes of some p-regular representation of G. Then C(S) is equal to the category C_{reg} defined above, and the map γ(S) is a homeomorphism.

Proof. Let F be a maximal elementary abelian subgroup of G. Note that the natural map from X_{F} (k) (mapping the category with one object and one morphism to C_{reg}) induces a homeomorphism

X_{F} (k)/GL(F) ≅ colim_{C_{reg}}X_{E}(k).

By Proposition 4.1, the image of ι_{F}^{G}: X_{F} (k) → V_{S}(k) is homeomorphic to X_{F} (k)/GL(F). If E is any elementary abelian subgroup of G and f: E ↪ F is any injective group homomorphism, then Res_{E}^{G}(ρ) and f^{∗}Res_{F}^{G}(ρ) are equal to a sum of (the same number of) copies of the regular representation of E. Hence C(S) is equal to C_{reg}, and Im(γ) = Im(ι_{F}^{G}). It follows that γ is a homeomorphism onto its image. Finally, by Theorem 5.1, this image is the whole of V_{S}(k). Proof of the theorem. Since R is large, it contains a subring S satisfying the conditions of Lemma 6.2. Let E, F be two elementary abelian subgroups of G, suppose that the rank of E is less than or equal to that of F, and suppose that ϕ ∈ X_{E}(k) and ψ ∈ X_{F} (k) define the same point of V_{R}(k). A fortiori ϕ and ψ define the same point of V_{S}(k), and so by Lemma 6.2 there is an injective group homomorphism f: E ↪ F such that ψ = ϕ ∘ f^{∗} = f_{∗}(ϕ). It suffices to show that such an f is in C(R).

For any such f: E ↪ F, let S be the set of subgroups of E such that f restricted to E is a morphism in C(R):

S = {E′ ≤ E | (f|_{E′}: E′ → F) ∈ C(R)},

and define a subset X(f) of X_{E}(k) by

X(f) = {ϕ ∈ X_{E}(k) | f_{∗}(ϕ) ∘ Res_{E′}^{G}|_{R} = ψ ∘ Res_{F}^{G}|_{R}}.

From the definitions,

X(f) ⊇ ⋃_{E′∈S} ι_{E′}^{E}X_{E′}(k),

and it suffices to show that equality holds. Note that a subgroup E′ ≤ E is in S if and only if ιX_{E′}(k) is a subset of X(f). Hence it suffices to show that X(f) is equal to some union of sets of the form ιX_{E′}(k). Now let I(f) be the ideal of H^{∗}(E) generated by all elements of the form Res_{E}^{G}(r) − f^{∗}Res_{F}^{G}(r), where r ∈ R. The subvariety of X_{E}(k) defined by I(f) is the set X(f) defined above. Since R is natural (in the sense defined above the statement), the ideal I(f) is homogeneous and closed under the action of the Steenrod algebra. But by a theorem of Serre [19,16], the variety corresponding to any such ideal of H^{∗}(E) has the required form.

Minimal prime ideals of a commutative F_{p}-algebra R correspond to irreducible components of V_{R}(k). Hence one obtains:

Corollary 6.3. Let R be a large, natural subring of H^{∗}(G). The minimal prime ideals of R are in bijective correspondence with the isomorphism types of maximal objects in C(R).

An object of a category is called maximal if every map from it is an isomorphism. An isomorphism class of maximal objects in the Quillen category is a conjugacy class of maximal elementary abelian subgroups of G.

Corollary 6.4. Let R and S be large natural subrings of H^{∗}(G), and suppose that R is a subring of S. The natural map V_{S}(k) → V_{R}(k) is a homeomorphism if and only if the categories C(R) and C(S) are equal.

Proof. A direct proof could be given at this stage, but it is easier to apply Proposition 9.1, which implies that no subcategory of C_{reg} that strictly contains C(S) gives rise to the same variety.

7. Applications to rings of Chern classes. We start by defining some categories of elementary abelian subgroups of G. Definition. Define categories of elementary abelian subgroups of G, A′, A′_{R}, A′_{P}, and for each d dividing p − 1, A_{d}, by stipulating that an injective homomorphism f: E → F is in:

A′ if ∀e, f(e) is conjugate (in G) to e; A′_{R} if ∀e, f(e) is conjugate to e or to e^{−1}; A′_{P} if ∀e, the subgroups ⟨e⟩ and ⟨f(e)⟩ are conjugate; A′_{d} if ∀e, f(e) is conjugate to ξ(e) for some ξ in the order d subgroup of Aut(⟨e⟩).

Note that for p = 2, A′ = A′_{R} = A′_{P}, and for odd p, A′_{R} = A′_{2}, and A′_{P} = A′_{p−1}. Note also that the difference between A′ and the Quillen category A is the difference between ‘∀e ∃gf(e) = g^{−1}eg’ and ‘∃g ∀ef(e) = g^{−1}eg’. The reason for introducing these categories is the following proposition.

Proposition 7.1. Let K be a subfield of C and let |K(ζ_{p}): K| = d, where ζ_{p} is a primitive pth root of 1. Let G be a compact Lie group, and in cases (c) and (d) suppose that G is finite. Let R be the subring of H^{∗}(G) generated by Chern classes of:

(a) All representations of G;

(b) Representations of G realisable over the reals;

(c) Permutation representations of G;

(d) Representations of G realisable over K.

In each case, the variety V_{R}(k) is homeomorphic to colim_{C(R)}X_{E}(k). The category C(R) is:

(a) A′, (b) A′_{R}, (c) A′_{P}, (d) A′_{d}.

Proof. In each case, the morphisms in the category given are precisely those group homomorphisms for which χ(e) = χ(f(e)) for all characters χ coming from representations of the given type. (See [20] Chapter 12 for case (d), and for case (c) note that if e, e′ are elements of G of order p, then the permutation action of e on G/⟨e′⟩ has a fixed point if and only if ⟨e⟩ is conjugate to ⟨e′⟩.) The proposition therefore follows from the lemma below.

Lemma 7.2. Let A be an additive subgroup of the representation ring of G, generated by genuine representations, and containing a p-regular representation. Let R be the subring of H^{∗}(G) generated by the Chern classes of all elements of A. Then R is large and natural, and hence by Theorem 6.1

γ: colim_{C(R)}X_{E}(k) → V_{R}(k)

is a homeomorphism. Furthermore, f: E ↪ F is a morphism in C(R) if and only if for all e ∈ E, and all characters χ of elements of A, χ(e) = χ(f(e)).

Proof. First, suppose that A is generated by a single representation ρ. The image of ρ^{∗}: H^{∗}(U(n)) → H^{∗}(G) is natural since H^{∗}(U(n)) is graded and acted upon by the Steenrod algebra. The general case follows from the Cartan formula. By hypothesis, R is large. The claimed homeomorphism follows from Theorem 6.1, and it only remains to describe C(R).

A representation is determined up to equivalence by its character. Hence for any f as in the statement and ρ a generator of A, f^{∗}Res_{F}^{G}c_{·}(ρ) = Res_{E}^{G}c_{·}(ρ), and so any such f is in C(R). For the converse, note that since Ch(E) is a unique factorization domain, a representation of E is determined up to equivalence by its dimension and its total Chern class. Thus if f: E ↪ F is a homomorphism for which there exists e and χ with χ(f(e)) ≠ χ(e), there exists i and ρ, a generator of A, such that Res_{E}^{G}(c_{i}(ρ)) − f^{∗}Res_{F}^{G}(c_{i}(ρ)) ≠ 0. Hence f is not in C(R).

Quillen’s description of X_{G}(k) (Theorem 5.1, and Theorem 8.1), Corollary 6.4 and Proposition 7.1 together yield:

Corollary 7.3. The natural map X_{G}(k) → X′_{G}(k) is a homeomorphism if and only if the categories A(G) and A′(G) are equal.

Example. (A p-group G for which the map X_{G}(k) → X′_{G}(k) is not a homeomorphism.) Let E be the additive group F_{p}^{n} for some n > 2, and let U ≤ GL(E) = GL_{n}(F_{p}) be the Sylow p-subgroup of GL(E) consisting of upper triangular matrices. Let Q be the subgroup of U consisting of all matrices (a_{i,j}) that are constant along diagonals, i.e., a_{i,j} = a_{i+1,j+1} whenever 1 ≤ i < m and 1 ≤ j < m. Finally, let G be the split extension with kernel E and quotient Q. The group E is a maximal elementary abelian subgroup of G. Easy matrix calculations show that the orbits of the action of Q on elements of E are equal to the orbits of the action of U, and that any element of GL(E) that preserves the U-orbits in E is fact an element of U. It follows that the image of X_{E}(k) in X′_{G}(k) is X_{E}(k)/U, whereas the image of X_{E}(k) in X_{G}(k) is of course X_{E}(k)/Q. Thus G is a p-group such that the fibres of the map X_{G}(k) → X′_{G}(k) above general points of one irreducible component have order |U: Q| = p^{(n−1)(n−2)/2}.

Example. (A group G for which X′_{G}(k) has fewer irreducible components than X_{G}(k).) Let G be GL_{3}(F_{p}). There are two distinct Jordan forms for elements of order p in G (resp. one if p = 2), and hence G has two conjugacy classes (resp. one conjugacy class if p = 2) of elements of order p. All maximal elementary abelian subgroups of G have rank two. The subgroups

E_{1} = ( \begin{matrix}1 & ∗ & ∗ \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) and E_{2} = ( \begin{matrix}1 & 0 & ∗ \\ 0 & 1 & ∗ \\ 0 & 0 & 1 \end{matrix} )

are maximal elementary abelian subgroups, and are not conjugate, although every nonidentity element of E_{1} is conjugate to every non-identity element of E_{2}. It follows that the images of X_{E_{1}}(k) and X_{E_{2}}(k) in X_{G}(k) are distinct irreducible components of X_{G}(k), whereas their images in X′_{G}(k) are equal.

8. Transfers of Chern classes. Throughout this section, G shall be a finite group. Following Moselle [15], we consider the ‘Mackey closure’ of Ch(G), or in other words the smallest natural subring of H^{∗}(G) that contains Ch(G) and is closed under transfers. More formally, we make the following: Definition. Let G be a finite group. Define \overline{Ch}(G) recursively as the subring of H^{∗}(G) generated by Ch(G) and the image of \overline{Ch}(H) under the transfer Cor_{H}^{G} for each proper subgroup H < G.

We shall prove:

Theorem 8.1. Let G be a finite group, with Quillen category A, and let R = \overline{Ch}(G). The map α induces a homeomorphism α: colim_{A}X_{E}(k) → V_{R}(k).

We do not use the injectivity of Quillen’s map in proving Theorem 8.1, so two immediate corollaries of Theorem 8.1 are:

Corollary 8.2. (Quillen) For a finite group G, the map α: colim_{A}X_{E}(k) → X_{G}(k) is injective.

Corollary 8.3. For a finite group G, the inclusion of \overline{Ch}(G) in H^{∗}(G) induces a homeomorphism of varieties.

Proof of the theorem. The transfer map Cor_{H}^{G} commutes with the action of the Steenrod algebra, by either a topological argument [1] or an algebraic one [10]. It follows that R = \overline{Ch}(G) is a large natural subring of H^{∗}(G). By Theorem 6.1, it suffices to show that C(R) = A.

First we show that if E, F are elementary abelian subgroups of G such that E is not conjugate to a subgroup of F, then there is no map in C(R) from E to F. Since any map is the composite of an isomorphism followed by the inclusion of a subgroup, it suffices to consider the case when E and F have the same rank. (Note that if f is any map in C(R) that is an isomorphism of elementary abelian groups, then the inverse of f is also in C(R), so f is an isomorphism in C(R).)

Let N = N_{G}(F) be the normalizer of F in G, let θ = CF − 1 be the (|F | − 1)- dimensional representation of F given by the difference of the regular representation and the trivial representation, and let ρ = Ind_{F}^{N}(θ) be the induced representation of N. Equivalently, ρ is the regular representation of N minus the permutation representation on the cosets N/F of F. Note that ρ is a genuine representation of N of dimension |N| − |N: F |.

Now let F′ be any elementary abelian subgroup of N. The regular representation of N restricts to F′ as a sum of |N: F′| copies of the regular representation of F′. The number of orbits of F′ on the cosets N/F, or equivalently the number of trivial F′-summands of the permutation module CN/F, is equal to |N: F′F |. It follows that Res_{F′}^{N}(ρ), the restriction to F′ of ρ, contains the trivial F′-module as a direct summand if and only if F′F ≠ F′, i.e., if and only if F is not a subgroup of F′.

Let n be the dimension of ρ, so that n = |N|(1 − 1/|F |), and note that since the restriction to F of ρ does not contain the trivial representation, Res_{F}^{N}(c_{n}(ρ)) is non-zero. Let x_{F} = Cor_{N}^{G}(c_{n}(ρ)). For E an elementary abelian subgroup of G of the same rank as F, the Mackey formula affords a calculation of Res_{E}^{G}(x_{F}):

Res_{E}^{G}(x_{F}) = ∑_{EgN} Cor_{E∩gNg^{−1}}^{E}c_{g}^{∗}Res_{g^{−1}Eg∩N}^{N}(c_{n}(ρ)),

where for any subgroup K of G, c_{g} is the homomorphism k ↦ g^{−1}kg, and the sum is over a set of double coset representatives for E\G/N. The restriction map from E to any subgroup is surjective in cohomology, and Cor_{E′}^{E}Res_{E′}^{E} is equal to multiplication by |E: E′|. Hence the transfer Cor_{E′}^{E} is zero for any proper subgroup E′ of E. Thus the only non-zero contributions to the above sum come from terms in which g^{−1}Eg ≤ N. On the other hand, we know that the restriction of c_{n}(ρ) to an elementary abelian subgroup of N is non-zero if and only if that subgroup contains F. Since we are assuming that E has the same rank as F, it follows that the only non-zero terms will come from g such that F = g^{−1}Eg. If F = g^{−1}Eg = h^{−1}Eh, then g^{−1}hF h^{−1}g = F, so g^{−1}h ∈ N, and so EgN = EhN. It follows that, for E and F of the same rank, Res_{E}^{G}(x_{F}) = c_{g}^{∗}(c_{n}(ρ)) for any g such that g^{−1}Eg = F, and is zero if there is no such g, i.e., if E and F are not conjugate in G. Since Res_{F}^{N}c_{n}(ρ) is non-zero in h^{∗}(F), it follows that when E and F have the same rank, there are morphisms from E to F in C(R) only if E is conjugate to F.

It remains to show that the automorphisms of F in the category C(R) are precisely the maps induced by conjugation in G. Let C = C_{G}(F) be the centralizer of F in G, and suppose that |C: F | = p^{m}r, for some r coprime to p. For any representation λ of F, Res_{F}^{C}Ind_{F}^{C}(λ) = p^{m}rλ. It follows that the image of Ch(C) in Ch(F) contains the subring of p^{m}th powers. In F_{p}[F] = Ch(F), there is a homogeneous element y_{1} such that the F_{p}GL(F)-submodule generated by y_{1} is free (see [2], pp. 45–46). The element y_{2} = y_{1}^{p^{m}} also has this property, and y_{2} = Res_{F}^{C}(y′_{2}) for some y′_{2} ∈ Ch(C). Now let y′ = y′_{2}Res_{C}^{N}(c_{n}(ρ)), where ρ and n are as in the previous paragraph. Then y′ is an element of Ch(C) whose restriction to an elementary abelian subgroup of C is non-zero only if that subgroup contains F. Moreover, the restriction to F of the representation ρ is invariant under GL(F), and so y = Res_{F}^{C}(y′) generates a free F_{p}GL(F)-submodule of Ch(F). Now define z_{F} ∈ H^{∗}(G) by z_{F} = Cor_{C}^{G}(y′). The Mackey formula gives

Res_{F}^{G}(z_{F}) = ∑_{F gC} Cor_{F ∩gCg^{−1}}^{F}c_{g}^{∗}Res_{g^{−1}F g∩C}^{C}(y′).

Now Res_{F′}^{C}(y′) = 0 unless F′ contains F, and so only those terms for which g^{−1}F g = F can be non-zero. Thus

Res_{F}^{G}(z_{F}) = ∑_{g∈N/C} c_{g}^{∗}(y),

where the sum is over cosets of C = C_{G}(F) in N = N_{G}(F). Since y generates a free F_{p}GL(F)-summand of Ch(F), an automorphism f of F satisfies f^{∗}Res_{F}^{G}(z_{F}) = Res_{F}^{G}(z_{F}) if and only if f is equal to conjugation by some element of N. Remark. The first part of this proof is very similar to Quillen’s second proof of this statement, using the Evens norm map [17,11,5]. The second part is less similar however. Our argument is complicated by the weaker technique that we are using to construct elements, but is simplified by our use of Theorem 6.1 which means that we do not need to construct as many elements as are needed in [17].

Corollary 8.3 seems to be fairly well-known, although we have been unable to find it stated in the literature. Our first proof of Corollary 8.3 was essentially independent of the rest of this paper, but used a comparatively recent theorem of Carlson ([7], or theorem 10.2.1 of [11]): For G a p-group, with centre Z, the radical of ker(Res_{Z}^{G}) is equal to the radical of the ideal generated by the images of Cor_{H}^{G}, where H ranges over all proper subgroups of G.

To prove Corollary 8.3 directly, note that if G_{p} is a Sylow p-subgroup of G, the transfer from H^{∗}(G_{p}) to H^{∗}(G) is surjective. Thus it suffices to consider the case when G is a pgroup. Let G be a p-group, with centre Z. Using representations induced from Z up to G it may be shown that the image of Ch(G) in H^{∗}(Z) contains the subring of p^{m}th powers for sufficiently large m (as in the proof of Theorem 8.1). Thus if y ∈ H^{∗}(G), there exists m and x_{1} ∈ Ch(G) such that y_{1} = y^{p^{m}} − x_{1} is in the kernel of Res_{Z}^{G}. By Carlson’s theorem, there exists n, subgroups H(1), …, H(l) of G and x′_{i} ∈ H^{∗}(H(i)) such that

y_{2} = y_{1}^{p^{n}} = ∑_{i} Cor_{H(i)}^{G}(x′_{i}).

By induction on the order of G, there exists N such that, for each i, (x′_{i})^{p^{N}} ∈ \overline{Ch}(H(i)). Noting that for x of degree 2i,

Cor_{H}^{G}(x^{p}) = Cor_{H}^{G}(P^{i}x) = P^{i}Cor_{H}^{G}(x) = Cor_{H}^{G}(x)^{p},

it follows that

y_{2}^{p^{N}} = ∑_{i} Cor_{H(i)}^{G}((x′_{i}))^{p^{N}} = ∑_{i} Cor_{H(i)}^{G}((x′_{i})^{p^{N}}) ∈ \overline{Ch}(G).

9. A closure operation.

Definition. Let C be a category of elementary abelian subgroups of a group G. Define \overline{C}, the closure of C, to be the smallest subcategory of C_{reg} such that:

1. C is contained in \overline{C};

2. if f : E_{1} → E_{2} is in \overline{C}, and F_{i} ≤ E_{i} with f(F_{1}) ≤ F_{2}, then f : F_{1} → F_{2} is in \overline{C};

3. if f : E_{1} → E_{2} is in \overline{C} and is an isomorphism of groups, then f^{−1} : E_{2} → E_{1} is in \overline{C}.

Say that C is closed if C = \overline{C}. Note that the categories A, A′, and C(R) for any R ≤ H^{∗}(G) are closed.

Proposition 9.1. For any C containing the Quillen category A, the category \overline{C} is the unique largest subcategory of C_{reg} such that the natural map

colim_{C}X_{E}(k) → colim\overline{C}X_{E}(k)

is a homeomorphism.

Proof. Let D be the subcategory of C_{reg} whose morphisms f : E → F are those group homomorphisms that make the diagram

X_{E}(k)  ⟶^{f}  X_{F} (k)
↓^{ι}  ↓^{ι}
colim_{C}X_{E}(k) ⟶^{Id} colim_{C}X_{E}(k)

commute. Then D has the property claimed, and it suffices to show that \overline{C} = D. Note also that \overline{C} is contained in D and that D is closed. Let f: E_{1} → E_{2} be a morphism in D. Since f: E_{1} → E_{2} is in D (resp. in \overline{C}) if and only if f: E_{1} → f(E_{1}) is, it may be assumed that f is a group isomorphism. Let ϕ be an element of X_{E_{1}}^{+}(k), i.e., an element of X_{E_{1}}(k) not contained in X_{F} (k) for any proper subgroup F of E_{1}. Since GL(E_{1}) acts freely on X_{E_{1}}^{+}(k), it follows that f: E_{1} → E_{2} is uniquely determined by ψ = f_{∗}(ϕ).

By definition of D, ψ and ϕ have the same image in colim_{C}X_{E}(k). Since C is skeletally finite (because it contains A), there are chains (F_{0}, …, F_{m}) of objects of C and (f_{1}, …, f_{m}) of morphisms in C, where

f_{i}: F_{i−1+ϵ(i)} → F_{i−ϵ(i)}

for some ϵ(i) ∈ {0, 1}, and (ψ_{0}, …, ψ_{m}), ψ_{i} ∈ X_{F_{i}}(k), with

F_{0} = E_{1},  F_{m} = E_{2},  ψ_{0} = ϕ,  ψ_{m} = ψ,  f_{i}_{∗}(ψ_{i−1+ϵ(i)}) = ψ_{i−ϵ(i)}.

Let F′_{i} be the unique subgroup of F_{i} such that ψ_{i} ∈ X_{F′_{i}}^{+}(k). Then F′_{i} has the same rank as E_{1} and f_{i} restricts to an isomorphism f′_{i} from F′_{i−1+ϵ(i)} to F′_{i−ϵ(i)}. Letting δ(i) = 1−2ϵ(i),

f_{i}^{′δ(i)} is a morphism in \overline{C} from F′_{i−1} to F′_{i}, and the composite

f′ = f_{m}^{′δ(m)} ∘ ⋯ ∘ f_{1}^{′δ(1)}

is a morphism in \overline{C} from E_{1} to E_{2} such that f′(ϕ) = ψ. Hence f′ = f, and f is a morphism in \overline{C} as claimed.

For any category C of elementary abelian subgroups of a group G, one may define a subring R(C) of H^{∗}(G) as the inverse image of lim_{C} H^{∗}(E). This subring is large and is natural because lim_{C} H^{∗}(E) is.

Proposition 9.2. For C any category of elementary abelian subgroups of G containing the Quillen category A, C(R(C)) = \overline{C}.

Proof. Clearly, C(R(C)) contains \overline{C}, and is closed. Hence it suffices to show that the induced map of varieties is a homeomorphism. Quillen showed that the map from H^{∗}(G) to lim_{A} H^{∗}(E) contains the subring of p^{n}th powers for some n (in fact this is equivalent to the injectivity of the map colim_{A}X_{E}(k) → X_{G}(k)) [16]. Let S = lim_{C} H^{∗}(E), and note that if x is any element of S, the p^{n}th power of x is in the image of R = R(C). It follows that the map V_{S}(k) → V_{R}(k) is a homeomorphism, and it suffices to show that the natural map

colim_{C}X_{E}(k) → V_{S}(k)

is a homeomorphism. But this is a special case of lemma 8.11 of [16].

The proposition shows that there is a sort of ‘Galois correspondence’ between large natural subrings of H^{∗}(G) and categories of elementary abelian subgroups of G.

10. Some other categories. For each n ≥ 0, define a category A^{(n)} of elementary abelian subgroups of a group G by declaring that the morphism f: E ↪ F is in A^{(n)} if and only if for all e_{1}, …, e_{n} ∈ E, there exists g ∈ G such that f(e_{i}) = g^{−1}eg.

Note that A^{(0)} is the category C_{reg} of Section 6, and A^{(1)} is the category A′. For each n, A^{(n)} ⊇ A^{(n+1)}, and when n is greater than or equal to the p-rank of G, A^{(n)} is equal to Quillen’s category A. This suggests that A^{(∞)} should be defined to be A. Each of the categories A^{(n)} is closed in the sense of section 9, and the subrings R^{(n)} = R(A^{(n)}) form a natural filtration of H^{∗}(G) = R^{(∞)}.

The categories A^{(n)}(G) are related to the generalized characters of G due to Hopkins, Kuhn and Ravenel [13] in the same way that the category A′ is related to ordinary characters. It seems possible that there should be a description of the variety for the subring of elements of H^{∗}(G) coming from E^{0}(BG), where E is a generalized cohomology theory to which Hopkins-Kuhn-Ravenel’s work applies, in the same way that Chern classes are elements of H^{∗}(G) coming from K^{0}(BG). We shall not make a precise conjecture, but shall give examples to show that the categories A^{(n)} can be distinct from each other.

Proposition 10.1. For each n ≥ 0 and each prime p, there is a p-group G for which A^{(n)}(G) ≠ A^{(n+1)}(G).

Proof. For n = 0, the cyclic group of order p (for p odd), or the elementary abelian group of order four (for p = 2), will suffice. Hence we may assume that n > 0. Let C be a cyclic group of order p, let E be a faithful F_{p}C-module of F_{p}-dimension n + 1, and let c ∈ GL(E) represent the action on E of a generator for C. Now let Z be a vector space over F_{p} with basis {z_{M}} indexed by the maximal F_{p}-subspaces of E, so that Z has dimension (p^{n+1} − 1)/(p − 1).

For each maximal subspace M of E, pick a linear map ψ_{M}: E → Z, with kernel M and image generated by z_{M}. For each M, define b_{M} ∈ GL(E ⊕ Z) by the equation

b_{M}(e, z) = (c(e), z + ψ_{M}(e)).

Let A be the subgroup of GL(E ⊕ Z) generated by the b_{M}, and let G be the semidirect product (E ⊕ Z): A.

The subgroup Z is left invariant by A, so is central in G. Let ϕ be the homomorphism sending A ≤ GL(E ⊕ Z) to GL((E ⊕ Z)/Z) ≅ GL(E), and let B = ker(ϕ) ≤ A. Note that elements of B act trivially on Z and on (E ⊕ Z)/Z, and so B may be identified with a subgroup of the elementary abelian p-group Hom(E, Z).

We claim that the automorphism c of E is a morphism in A^{(n)}(G), but is not a morphism in A^{(n+1)}(G). If M is any rank-n subgroup of E, then the element b_{M} ∈ G acts on M in the same way as c. On the other hand, if c were a morphism in A^{(n+1)}(G), there would have to be an element d of G, acting on E ⊕ Z as d(e, z) = (c(e), z). But then, for any M, d′ = d^{−1}b_{M} would be an element of (E ⊕Z) : B acting as d′(e, z) = (e, z +ψ_{M}(e)). To complete the proof, it suffices to show that there can be no such element d′.

Let R be the image of F_{p}C in the ring End(E). Since F_{p}C is a commutative local ring, it follows that R is too. In particular, the non-units in R form an ideal. Fix M, a maximal subgroup of E. The group B ≤ A ≤ GL(E) is generated as a normal subgroup by the elements b_{M}^{p}, and b_{M}^{−1}b_{N}, where N ranges over all other maximal subgroups of E. The action of these generators on E ⊕ Z is given by:

b_{M}^{p}(e, z) = (c^{p}(e), z + ∑_{i=0}^{p−1} ψ_{M}(c^{i}(e))) = (e, ψ_{M}(r̄e) + z),

where r̄ is the image of c̄ = ∑_{i=0}^{p−1} c^{i} in R = End(E), and

b_{M}^{−1}b_{N}(e, z) = (e, z + ψ_{N}(e) − ψ_{M}(e)).

We therefore have to show that the element d′ described above does not lie in the subgroup of GL(E ⊕ Z) generated by

(e, z) ↦ (e, z + ψ_{M}(r̄e))  and  (e, z) ↦ (e, z + ψ_{N}(c^{i}e) − ψ_{M}(c^{i}e)),

for all N ≠ M and 0 ≤ i ≤ p − 1.

If d′: (e, z) ↦ (e, z + ψ_{M}(e)) is in the subgroup B, there are μ, λ_{N} ∈ R ≤ End(E) such that

Imλ_{N} ⊆ ker(ψ_{N}) = N for all N ≠ M, and

Im (1 − μr̄ + ∑_{N≠M} λ_{N} ) ⊆ ker(ψ_{M}) = M.

From the first family of equations, it follows that each λ_{N} is a non-unit in R, and of course r̄ is a non-unit. But then 1 − μr̄ + ∑ λ_{N} must be a unit in R, contradicting the second equation. Remark. We exhibit p-group examples above because p-groups control the behaviour of mod-p cohomology, and hence p-group examples tend to be harder to find. Similar examples can be constructed for other cyclic groups C. One interesting example is the case when C = GL_{1}(F_{q}) for q = p^{n+1} > 2 and E is the additive group of F_{q}. In this case F_{p}C is not a local ring, but its image in End(E) is isomorphic to F_{q}, and r̄ = 0, so the argument used above still applies.

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David J. Green Inst. f. Experimentelle Math., Ellernstraße 29, D-45326 Essen Germany Ian J. Leary Faculty of Math. Studies, Univ. of Southampton, Southampton, SO17 1BJ United Kingdom