CHERN APPROXIMATIONS FOR GENERALISED GROUP COHOMOLOGY

N. P. STRICKLAND

Let G be a finite group, and let E^{∗} be a generalised cohomology theory, subject to certain technical conditions (“admissibility” in the sense of [4]). Our aim in this paper is to define and study a certain ring C(E, G) that is in a precise sense the best possible approximation to E^{0}BG that can be built using only knowledge of the complex representation theory of G. There is a natural map C(E, G) ⟶ E^{0}BG, whose image is the subring \overline{C}(E, G) ≤ E^{0}BG generated over E^{0} by all Chern classes of all complex representations. There is ample precedent for considering this subring in the parallel case of ordinary cohomology; see for example [13, 14, 3]. However, although the generators of \overline{C}(E, G) come from representation theory, the same cannot be said for the relations; one purpose of our construction is to remedy this. We also also develop a kind of generalised character theory which gives good information about ℚ ⊗ C(E, G). In the few cases that we have been able to analyse completely, either ℚ ⊗ C(E, G) ≠ ℚ ⊗ E^{0}BG for easy character-theoretic reasons, or we have C(E, G) = E^{0}BG.

Rather than working directly with rings, we will study the formal schemes X(G) = spf(E^{0}BG) and X_{Ch}(G) = spf(C(E, G)); note that the map C(E, G) ⟶ E^{0}BG corresponds to a map X(G) ⟶ X_{Ch}(G). See [4, 9, 8] for foundational material on formal schemes. Suitably interpreted, our main definition is that X_{Ch}(G) is the scheme of homomorphisms from the Λ-semiring R^{+}(G) of complex representations of G to the Λ-semiring scheme of divisors on the formal group 𝔾 associated to E.

We start by fixing some conventions in Section 1. We then recall the basic theory of Λ-semirings (Section 2), set up the parallel theory of Λ-semiring schemes, and define the Λ-semiring scheme of divisors (Section 3). We then recall the definition of Adams operations and study their basic properties (Section 4). Using this we give a precise definition of X_{Ch}(G) and an implicit presentation of C(E, G) by generators and relations (Section 5). In Section 6, we work out the case of the symmetric group Σ_{3} at the prime 3, and show that X(Σ_{3}) = X_{Ch}(Σ_{3}). In Section 7 we show that X(G) = X_{Ch}(G) when G is Abelian, and in Section 8 we show that the same is true when E is the p-adic completion of complex K-theory and G is a p-group. We then use Adams operations to reduce certain questions to the Sylow subgroup of G (Section 9) and to prove that X_{Ch}(G) is finite over X = spf(E^{0}) (Section 10). In Section 11, we recall the Hopkins-Kuhn-Ravenel generalised character theory, which relates ℚ ⊗ E^{0}BG to the set Ω(G) of conjugacy classes of homomorphisms ℤ_{p}^{n} ⟶ G. We give a parallel (but less precise) relationship between ℚ ⊗ C(E, G) and the set Ω_{Ch}(G) of Λ-semiring homomorphisms R^{+}(G) ⟶ ℕ[(ℚ_{p}/ℤ_{p})^{n}]. These descriptions are related by a map κ: Ω(G) ⟶ Ω_{Ch}(G). In Section 12, we compare Ω(G) and Ω_{Ch}(G) with two other sets that are sometimes easier to understand. We next return to examples: in Section 14 we show that X(Σ_{4}) = X_{Ch}(Σ_{4}) at the prime 2, and in Section 15 we study Ω(G) and Ω_{Ch}(G) when G is an extraspecial group at an odd prime. We then show that a certain approach which appears more precise actually captures no more information (Section 16). We conclude in Section 17 by proving a result in representation theory that was used in Section 9.

1. Notation and conventions

Fix a prime p. Throughout this paper, E will denote a p-local generalised cohomology theory with an associative and unital product. We write E^{k} for E^{k}(point), so E^{∗} is a ℤ-graded ring and E^{0} is an ungraded ring. We assume that E has the following properties:

1. E^{0} is a commutative complete local Noetherian ring, with maximal ideal 𝔪 say.

2. E^{k} = 0 whenever k is odd.

3. E^{−2} contains a unit (so E^{k}X ≃ E^{k−2}X for all X).

Date: August 4, 2000.

4. Either p > 2 and E is commutative, or p = 2 and E is quasi-commutative, which means that there is a natural derivation Q: E^{k}X ⟶ E^{k+1}X and an element v ∈ E^{−2} such that 2v = 0 and ab−(−1)^{|a||b|}ba = vQ(a)Q(b) for all a, b ∈ E^{∗}X.

There is one more condition, which needs some background explanation. Note that the quasi-commutativity condition means that whenever E^{1}X = 0, the ring E^{0}X is commutative (in the usual ungraded sense.) In particular, E^{∗} = E^{∗}(point) is commutative. A collapsing Atiyah-Hirzebruch spectral sequence argument shows that

E^{∗}ℂP^{∞} ≃ E^{∗} ˆ⊗H^{∗}ℂP^{∞} = E^{∗} ˆ⊗ℤ[[y]] = E^{∗}[[y]],

where y ∈ Ẽ^{2}ℂP^{∞}; it follows that E is complex-orientable. We can multiply y by a unit in E^{−2} to get an element x ∈ Ẽ^{0}ℂP^{∞} such that E^{∗}ℂP^{∞} = E^{∗}[[x]]. We fix such an element x once and for all (although we will state our results in a form independent of this choice as far as possible). This gives rise in the usual way to a formal group law F over E^{0}.

5. In addition to the above properties, we will assume that the reduction of F modulo the maximal ideal of E^{0} has height n < ∞.

In the language of [4], our conditions say that E is a K(n)-local admissible theory. The only difference is that previously we insisted that E should be commutative rather than quasi-commutative; the reader can easily check that everything in [4] goes through in the quasi-commutative case.

We will describe our results in the language of formal schemes. Most of the formal schemes that we consider have the form spf(R), where R is a complete local Noetherian E^{0}-algebra. For these the foundational setting discussed in [9] is satisfactory: one can regard the category of formal schemes as the opposite of the category of complete semilocal Noetherian rings and continuous homomorphisms. We also make some use of formal schemes such as spf(∏_{k∈ℤ} E^{0}[[c_{1}, c_{2}, …]]); a set of foundations covering these is developed in [8]. The older category of formal schemes embeds as a full subcategory of the newer one.

Definition 1.1. We let X be the formal scheme spf(E^{0}), and write 𝔾 for the formal group spf(E^{0}ℂP^{∞}) over X. Note that our element x ∈ Ẽ^{0}ℂP^{∞} can be regarded as a coordinate on 𝔾, with the property that

x(a + b) = F (x(a), x(b)) = x(a) +_{F} x(b).

Remark 1.2. Many of our constructions work with an arbitary formal group 𝔾 over a formal scheme X; it is not usually necessary to assume that 𝔾 comes from a cohomology theory, although that is the case of most interest for us.

We will let G denote a finite group. We write e = e(G) for the exponent of G, in other words the least common multiple of the orders of the elements. We factor this in the form e = p^{v}e′ = p^{v(G)}e′(G), where e′ ≠ 0 (mod p). We also choose a Sylow p-subgroup P ≤ G.

2. Λ-(semi)rings

We will use the following definition:

Definition 2.1. A semiring is a set R equipped with the following structure.

• A commutative and associative addition law with neutral element (written as 0); we do not assume that there are additive inverses.

• A commutative and associative multiplication law with neutral element 1, which distributes over addition.

A Λ-semiring is a semiring R equipped with Maps λ^{k}: R ⟶ R for k ≥ 0 satisfying λ^{0}(x) = 1 and λ^{1}(x) = x and λ^{k}(x + y) = ∑_{k=i+j} λ^{i}(x)λ^{j}(y).

A Λ-ring is a Λ-semiring which has additive inverses.

The initial Λ-semiring is ℕ and the initial Λ-ring is ℤ; in both cases we have

λ^{k}(n) = ( \begin{matrix}n \\ k \end{matrix}) = n(n − 1) … (n − k + 1)/k!.

Definition 2.2. An ℕ-augmented Λ-semiring is a Λ-semiring R equipped with a homomorphism dim: R ⟶ ℕ of Λ-semirings. A ℤ-augmented Λ-ring is a Λ-ring R equipped with a homomorphism dim: R ⟶ ℤ of Λ-rings.

Example 2.3. Let R^{+}(G) be the semiring of isomorphism classes of complex representations of G. It is well-known that this is a Λ-semiring with operations λ^{k} given by exterior powers. There is an augmentation dim: R^{+}(G) ⟶ ℕ sending each representation to its dimension.

Example 2.4. Let A be an Abelian group, and let ℕ[A] be the group semiring of A, in other words the set of expressions ∑_{a} n_{a}[a] with n_{a} ∈ ℕ and n_{a} = 0 for all but finitely many a. Equivalently, we have ℕ[A] = ∐_{n} A^{n}/Σ_{n}. This has a canonical structure as a Λ-semiring, with

λ^{k}([a_{1}] + … + [a_{n}]) = ∑_{I}[a_{i_{1}} + … + a_{i_{k}}],

where the sum on the right runs over all lists I = (i_{1}, …, i_{k}) such that 1 ≤ i_{1} < … < i_{k} ≤ n. There is an augmentation dim: ℕ[A] ⟶ ℕ defined by

dim([a_{1}] + … + [a_{n}]) = n.

If A is finite and A^{∗} = Hom(A, S^{1}) then ℕ[A] = R^{+}(A^{∗}) as Λ-semirings.

Example 2.5. For any space X, let Vect^{+}(X) denote the semiring of isomorphism classes of complex vector bundles over X. This is a Λ-semiring with operations as for R^{+}(G). We will always allow vector bundles to have different dimensions over different components of the base, so we do not have a natural map dim: Vect^{+}(X) ⟶ ℤ. We write Vect_{d}^{+}(X) for the set of isomorphism classes of bundles all of whose fibres have dimension d, and we put Pic(X) = Vect_{1}(X) ≃ H^{2}(X). This is an Abelian group, and there is an evident map ℕ[Pic(X)] ⟶ Vect^{+}(X). In the case X = BG, there is a well-known homomorphism R^{+}(G) ⟶ Vect^{+}(BG) sending a representation V to the bundle V ×_{G} EG.

Remark 2.6. In the important examples of Λ-(semi)rings, some extra identities hold that relate the elements λ^{i}λ^{j}(x) and λ^{i}(xy) to the elements λ^{k}(x) and λ^{l}(y). For many purposes it would be preferable to take these identities as part of the definition of a Λ-(semi)ring. However, it turns out that this would make no difference for us and the identities are complicated (particularly in the semiring case) so we omit them. In Section 16 we will discuss an approach which is apparently even more precise, and show that it actually gives no more information than our approach using Λ-semirings without extra identities.

Remark 2.7. Let R^{+} be a Λ-semiring, and let R be its Grothendieck completion, or in other words the group completion of R^{+} considered as a monoid under addition. It is well-known that this can be made into a Λ-ring in a canonical way, and that any homomorphism from R^{+} to a Λ-ring factors uniquely through R. Moreover, if R^{+} is augmented over ℕ then R is augmented over ℤ.

Example 2.8. The Grothendieck completion of R^{+}(G) is of course the ring R(G) of virtual representations of G, and the completion of ℕ[A] is the group ring ℤ[A]. We write Vect(X) for the Grothendieck completion of Vect^{+}(X). It is well-known that the complex K-theory K^{0}(X) is a ℤ-augmented Λ-semiring and that there is a natural map Vect(X) ⟶ K^{0}(X) which is an isomorphism whenever X is compact Hausdorff.

Remark 2.9. We will occasionally use the notation ℤ[A]^{+} = ℕ[A] and ℤ[A]_{d}^{+} = A^{d}/Σ_{d} ⊂ ℕ[A].

3. Λ-(semi)ring schemes

The theory of Λ-semirings is an instance of universal algebra: it is defined in terms of operations ω: R^{k} ⟶ R with k = 0, 1 or 2, and identities between operations derived from these. It is thus formal to define the notion of a Λ-semiring object in any category C with finite products: such a thing is an object R ∈ C equipped with maps

0, 1: 1 ⟶ R
+, ×: R^{2} ⟶ R
λ^{k}: R ⟶ R  for all k ∈ ℕ

making the evident diagrams commute. (Here the object 1 ∈ C is the terminal object.) Similar remarks apply to Λ-rings.

Next, suppose that C has arbitrary coproducts such that the natural map

∐_{i,j} X_{i} × Y_{j} ⟶ ∐_{i} X_{i} × ∐_{j} X_{j}

is always an isomorphism. We then have a product-preserving functor S ↦ \underline{S} := ∐_{s∈S} 1 from sets to C, so \underline{ℕ} is a Λ-semiring object in C. Similarly, \underline{ℤ} is a Λ-ring object in C.

Example 3.1. Take C = \overline{h}T , the homotopy category of unbased CW-complexes. We have a functor Vect^{+}(−) from \overline{h}T^{op} to the category of Λ-semirings, which is represented by the space ∐_{d} BU(d). It follows by Yoneda’s lemma that ∐_{d} BU(d) is a Λ-semiring in \overline{h}T . Similarly, the functor K^{0}(−) from \overline{h}T^{op} to the category of Λ-rings is represented by the Λ-ring space ℤ × BU. Note that in this context the object \underline{ℕ} is just the discrete space ℕ and similarly for \underline{ℤ}, so ∐_{d} BU(d) is augmented over \underline{ℕ} and ℤ × BU is augmented over \underline{ℤ}.

Now let X be a formal scheme, and consider the category X̂_{X} of category of formal schemes over X in the sense of [8]. For simplicity we will assume that X is solid, which means that X = spf(O_{X}) for some formal ring O_{X}. Let A be the category of discrete O_{X}-algebras, and let F be the category of functors from A to sets. The category X̂_{X} can be regarded as a subcategory of F (compare [8, Remark 2.1.5]), and the inclusion X̂_{X} ⟶ F preserves products.

We will refer to Λ-(semi)ring objects in X̂_{X} as Λ-(semi)ring schemes (suppressing the words “formal” and “over X” for brevity).

Let 𝔾 be an ordinary formal group over X, in other words a commutative group object in X̂_{X} that is isomorphic in X̂_{X} to 𝔸̂_{X}^{1} = spf(O_{X}[[x]]). We can then define the schemes

Div_{d}^{+}(𝔾) = 𝔾^{d}/Σ_{d}
Div^{+}(𝔾) = ∐_{d∈ℕ} Div_{d}^{+}(𝔾)
Div_{0}(𝔾) = lim_{-→}_{d} Div_{d}^{+}(𝔾)
Div(𝔾) = \underline{ℤ} × Div_{0}(𝔾)
Div_{d}(𝔾) = {d} × Div_{0}(𝔾) ⊂ Div(𝔾).

More detailed definitions are given in [8, Section 5], where it is also explained how these formal schemes relate to the theory of divisors on 𝔾. In [8, Proposition 6.2.7] it is observed that

1. Div^{+}(𝔾) is the free commutative monoid object in C generated by 𝔾.

2. Div(𝔾) is the free commutative group object in C generated by 𝔾.

3. Div_{0}(𝔾) is the free commutative monoid object generated by 𝔾 considered as a based object in C, which is the same as the free commutative group object generated by 𝔾 considered as a based object in C.

Moreover, all these universal properties are stable under base change: if X′ is a formal scheme over X then Div^{+}(𝔾) ×_{X} X′ is the free commutative monoid in X̂_{X′} generated by 𝔾 ×_{X} X′ and so on.

Recall that O_{𝔾} = O_{X}[[x]] and thus O_{𝔾^{d}} = O_{X}[[x_{1}, …, x_{d}]]. If c_{k} denotes the coefficient of t^{d−k} in ∏_{i}(t−x_{i}) then O_{Div_{d}^{+}(𝔾)} = O_{X}[[c_{1}, …, c_{d}]] and O_{Div^{+}(𝔾)} = ∏_{d≥0} O_{X}[[c_{1}, …, c_{d}]]. There are also isomorphisms

O_{Div_{0}(𝔾)} = O_{X}[[c_{1}, c_{2}, …]]
O_{Div(𝔾)} = ∏_{d∈ℤ} O_{X}[[c_{1}, c_{2}, …]].

Using these, one sees that Div_{d}^{+}(𝔾) is a closed subscheme of Div_{d}(𝔾), and Div^{+}(𝔾) is a closed subscheme of Div(𝔾).

If E is an even periodic ring spectrum, X = spec(π_{0}E) and 𝔾 = spf(E^{0}ℂP^{∞}) then there are natural isomorphisms

spf(E^{0}BU(d)) = Div_{d}^{+}(𝔾)
spf(E^{0}BU) = Div_{0}(𝔾)
spf(E^{0}(ℤ × BU)) = Div(𝔾).

This is just a translation of well-known calculations; details are given in [8, Section 8].

Proposition 3.2. Let 𝔾 be an ordinary formal group over a scheme X. Then Div^{+}(𝔾) has a natural structure as a Λ-semiring scheme, and Div(𝔾) has a natural structure as a Λ-ring scheme. Moreover, there is a canonical homomorphism dim: Div(𝔾) ⟶ \underline{ℤ} of Λ-ring schemes, which sends Div_{d}(𝔾) to d.

Proof. Recall that F is the category of functors from discrete O_{X}-algebras to sets. Define R^{+}, R ∈ F by R^{+}(A) = ℕ[𝔾(A)] and R(A) = ℤ[𝔾(A)]. It is clear that R^{+} is a Λ-semiring object in F, and R is a Λ-ring object.

There is an evident inclusion j: 𝔾 = Div_{1}^{+}(𝔾) ⟶ Div^{+}(𝔾). As Div^{+}(𝔾) is a commutative monoid scheme, the set Div^{+}(𝔾)(A) is a commutative monoid for all A ∈ A. As R^{+}(A) is the free commutative monoid generated by the set 𝔾(A), there is a unique homomorphism ϕ^{+} : R^{+}(A) ⟶ Div^{+}(𝔾)(A) extending j. These maps are natural in A so we get a map ϕ^{+} : R^{+} ⟶ Div^{+}(𝔾) in F. If we interpret the colimits in X̂_{X} then we have Div^{+}(𝔾) = ∐_{d} 𝔾^{d}/Σ_{d}; this translates to the statement that Div^{+}(𝔾) is the initial example of a formal scheme in X̂_{X} equipped with a map R^{+} ⟶ Div^{+}(𝔾) in F. By similar arguments, we find that Div(𝔾) is the initial example of a formal scheme over X with a map ϕ: R ⟶ Div(𝔾) in F. Moreover, one can check that the schemes Div^{+}(𝔾)^{k} and Div(𝔾)^{k} enjoy the evident analogous universal properties for all k ≥ 0.

It now follows that there is a unique map ×: Div^{+}(𝔾) × Div^{+}(𝔾) ⟶ Div^{+}(𝔾) making the following diagram commute:

R^{+} × R^{+}  R_{+} Div^{+}(𝔾) × Div^{+}(𝔾)  Div^{+}(𝔾)._{_{ϕ^{+}×ϕ^{+}} u}  ^{×}  w _{u_{ϕ^{+}}} w_{×}

Similarly, all the other structure maps for the Λ-semiring structure on R^{+} induce operations on Div^{+}(𝔾), and one checks easily that this makes Div^{+}(𝔾) into a Λ-semiring scheme. A similar argument works for Div(𝔾). It is clear that there is a map dim: Div(𝔾) ⟶ \underline{ℤ} as described.

The above Λ-semiring structure can be made more explicit as follows. Let c_{d,k} ∈ O_{X}[[x_{1}, …, x_{d}]] be defined by

∏_{i=1}^{d}(t − x_{i}) = ∑_{i=0}^{d} c_{d,i}x^{d−i}.

Let p_{d,e,k}(c_{d,1}, …, c_{d,d}, c′_{e,1}, …, c′_{e,e}) be defined by

∏_{i=1}^{d} ∏_{j=1}^{e}(t − (x_{i} +_{F} x′_{j})) = ∑_{k=0}^{de} p_{d,e,k}t^{de−k}.

Suppose d, r ∈ ℕ and put N = ( \begin{matrix}d \\ r \end{matrix}). Let q_{d,r,k}(c_{N,1}, …, c_{N,N}) be defined by

∏_{I}(t − ∑_{j}^{F} x_{i_{j}}) = ∑_{k=0}^{N} q_{d,r,k}t^{N−k},

where the sum on the left runs over all lists I = (i_{1}, …, i_{r}) such that 1 ≤ i_{1} < … < i_{r} ≤ d. Then the multiplication map

×: Div_{d}^{+}(𝔾) × Div_{e}^{+}(𝔾) ⟶ Div_{de}^{+}(𝔾)

corresponds to the map

O_{X}[[c_{de,1}, …, c_{de,de}]] ⟶ O_{X}[[c_{d,1}, …, c_{d,d}, c′_{1,e}, …, c′_{e,e}]]

(of formal O_{X}-algebras) sending c_{de,k} to p_{d,e,k}. Similarly, the map corresponding to λ^{r}: Div_{d}^{+}(𝔾) ⟶ Div_{N}^{+}(𝔾) sends c_{N,k} to q_{d,r,k}.

4. Adams operations

We now recall the theory of Adams operations in Λ-semirings; for a more detailed exposition see [6], for example.

Let R be a Λ-ring. For any a ∈ R we can form the power series

λ_{t}(a) = ∑_{k≥0} λ^{k}(a)(−t)^{k} ∈ R[[t]].

This is equal to 1 mod t and thus is invertible in R[[t]]. It is easy to check that λ_{t}(0) = 1 and λ_{t}(a + b) = λ_{t}(a)λ_{t}(b).

We next define

ψ_{t}(a) = −tλ_{−t}(a)^{−1} dλ_{−t}(a)/dt ∈ R[[t]],

and let ψ^{k}(a) be the coefficient of t^{k} in ψ_{t}(a). This defines an additive map ψ^{k}: R ⟶ R, called the k’th Adams operation.

Now consider the case R = ℤ[A] for some Abelian group A. It is not hard to see that

ψ^{k}(∑_{i} n_{i}[a_{i}]) = ∑_{i} n_{i}[a_{i}]^{k} = ∑_{i} n_{i}[ka_{i}],

so ψ^{k} is just the map induced by the homomorphism k.1_{A}: A ⟶ A. Thus, if A is actually a ℤ_{(p)}-module or a ℤ_{p}-module, then there is a natural way to define ψ^{k}: ℤ[A] ⟶ ℤ[A] for all k ∈ ℤ_{(p)} or k ∈ ℤ_{p} as appropriate. Moreover, we see that ψ^{k} is a ring homomorphism which preserves the semiring ℤ[A]^{+} and the subsets ℤ[A]_{d}^{+}, and that ψ^{k}ψ^{j} = ψ^{kj} and ψ^{k}λ^{j} = λ^{j}ψ^{k}.

Now consider the Λ-ring scheme Div(𝔾). As our original definition of ψ^{k} is natural, we evidently get morphisms

ψ^{k}: Div(𝔾) ⟶ Div(𝔾)

of schemes. It is well-known that 𝔾 is actually a ℤ_{p}-module scheme, or in terms of our coordinate, that one can define the series [k]_{F} (x) in a sensible way for all k ∈ ℤ_{p}. This means that each ring ℤ[𝔾(A)] admits Adams operations ψ^{k} for all k ∈ ℤ_{p}, with properties as above. The argument of Proposition 3.2 shows that

• We can define operations ψ^{k} on Div(𝔾) for all k ∈ ℤ_{p}, extending the definition given previously.

• These maps are maps of ring schemes, induced by the maps k: 𝔾 ⟶ 𝔾.

• We have ψ^{j}ψ^{k} = ψ^{jk} for all j, k ∈ ℤ_{p}, and ψ^{k}λ^{j} = λ^{j}ψ^{k} for all k ∈ ℤ_{p} and j ∈ ℕ.

• The map ψ^{k} preserves Div_{d}(𝔾), Div^{+}(𝔾) and Div_{d}^{+}(𝔾) = 𝔾^{d}/Σ_{d} for all d.

Lemma 4.1. For any discrete O_{X}-algebra A, the group 𝔾(A) is a p-torsion group.

Proof. Our coordinate x gives an isomorphism x: 𝔾(A) ⟶ 𝔸̂^{1}(A) = Nil(A) (the set of nilpotents in A). As p lies in the maximal ideal of E^{0} = O_{X} and A is a discrete O_{X}-algebra, we see that p^{r} = 0 ∈ A for some r, and thus [p^{r}](x) is divisible by x^{2} in A[[x]]. It follows that [p^{rs}](x) is divisible by x^{2^{s}}. For any a ∈ 𝔾(A) we have x(a)^{2^{s}} = 0 for large s, so x(p^{rs}a) = 0 for large s, so p^{m}a = 0 for large m as required.

Lemma 4.2. Let A be a discrete O_{X}-algebra, and suppose we have a divisor D ∈ Div(𝔾)(A). Then ψ^{k}(D) = dim(D)[0] whenever the p-adic valuation v_{p}(k) is sufficiently large.

Proof. First suppose that D ∈ Div_{d}^{+}(𝔾)(A). We can then choose a faithfully flat map A ⟶ A′ such that the image of D in Div_{d}^{+}(𝔾)(A′) has the form ∑_{i=1}^{d}[a_{i}]. The map Div_{d}^{+}(𝔾)(A) ⟶ Div_{d}^{+}(𝔾)(A′) is automatically injective, so it suffices to show that for large m we have ψ^{p^{m}}(∑_{i}[a_{i}]) = ∑_{i}[p^{m}a_{i}] = d[0], which is immediate from the previous lemma.

Now suppose that D ∈ Div_{d}(𝔾)(A). We can then write D in the form D′−e[0] for some D′ ∈ Div_{d+e}^{+}(𝔾)(A) and we reduce easily to the previous case.

Finally, consider a general divisor D ∈ Div_{d}(𝔾)(A), which need not have constant dimension. Instead, we have a splitting A = A_{1} × … × A_{r} giving a bijection Div(𝔾)(A) = ∏_{i} Div(𝔾)(A_{i}) under which D becomes an r-tuple (D_{1}, …, D_{r}) with D_{i} ∈ Div_{d_{i}}^{+}(𝔾)(A_{i}) for some integers d_{i}. This means that dim(D) becomes (d_{1}, …, d_{r}) under the bijection \underline{ℤ}(A) = ∏_{i} \underline{ℤ}(A_{i}). The cases considered previously imply that

ψ^{k}D = (ψ^{k}D_{1}, …, ψ^{k}D_{r}) = (d_{1}[0], …, d_{r}[0]) = dim(D)[0]

when v_{p}(k) ≫ 0, as required.

Now consider instead the Λ-ring R(G). In this case we can define Adams operations ψ^{k} for k ∈ ℕ, and it is well-known that in terms of characters we have

χ_{ψ^{k}V} (g) = χ_{V} (g^{k}).

As a virtual representation is determined by its character and dim(V) = χ_{V} (1), it follows easily that ψ^{k} is a degree-preserving map of Λ-rings and that ψ^{j}ψ^{k} = ψ^{jk}. Moreover, if e is the exponent of G (in other words, least common multiple of the orders of the elements) then ψ^{k} depends only on the congruence class of k modulo e. If k is coprime to e (or equivalently, to |G|), it follows that ψ^{k}ψ^{j} = 1 for some j, so ψ^{k} is an isomorphism. In this case the map g ↦ g^{k} is a bijection, and it follows easily that ψ^{k} preserves the usual inner product on R(G). A virtual representation V is an irreducible honest representation iff χ_{V} (1) > 0 and ⟨V, V ⟩ = 1, and it follows that ψ^{k} sends irreducibles to irreducibles and thus sends R_{d}^{+}(G) to R_{d}^{+}(G). (Compare [7, Exercise 9.4].)

However, if k is not coprime to e then ψ^{k} need not preserve R^{+}(G). For example, take G = Σ_{3}, let ϵ be the nontrivial one-dimensional representation, and let ρ be the irreducible two-dimensional representation. We then have ψ^{2}(ρ) = ρ + 1 − ϵ ∉ R^{+}(G).

Lemma 4.3. Let p^{v} be the p-part of the exponent of G. Then for any homomorphism f: R(G) ⟶ Div(𝔾)(A) of Λ-rings and any V ∈ R_{d}(G) we have f(V) ∈ Div_{d}(𝔾) and ψ^{p^{v}}f(V) = d[0].

Proof. Let the exponent of G be e = p^{v}e′, where e′ is coprime to p. The map ψ^{e′}: Div(𝔾) ⟶ Div(𝔾) is an isomorphism and fixes d[0], and ψ^{e′}ψ^{p^{v}} = ψ^{e} so it suffices to show that ψ^{e}f(V) = d[0]. To see this note that ψ^{e}f(V) = f(ψ^{e}V) and χ_{ψ^{e}V} (g) = χ_{V} (g^{e}) = χ_{V} (1) = d for all g, so ψ^{e}V is the trivial representation of rank d. As f is a ring map, we have f(ψ^{e}V) = f(d) = d[0], as required.

This implies that ψ^{p^{k}}f(V) = d[0] for k ≫ 0 but Lemma 4.2 says that ψ^{p^{k}}f(V) = dim(f(V))[0] for k ≫ 0, so dim(f(V)) = d, so f(V) ∈ Div_{d}(𝔾) as claimed.

5. Chern approximations

Definition 5.1. Let G be a finite group, and let A be a discrete O_{X}-algebra. We define a functor X_{Ch}(G) from discrete O_{X}-algebras to sets by

X_{Ch}(G)(A) = { homomorphisms R^{+}(G) ⟶ Div^{+}(𝔾)(A) of Λ-seimirings }.

We write C(E, G) for the ring O_{X_{Ch}(G)} of natural transformations from X_{Ch}(G) to the forgetful functor 𝔸^{1}. We also put X(G) = spf(E^{0}BG). We refer to C(E, G) as the Chern approximation to E^{0}BG, and to X_{Ch}(G) as the Chern approximation to X(G).

Remark 5.2. We say that a homomorphism f: R(G) ⟶ Div(𝔾)(A) of Λ-rings is positive if f(R^{+}(G)) ⊆ Div^{+}(𝔾)(A). It is clear from Remark 2.7 that X_{Ch}(G)(A) bijects naturally with the set of positive homomorphisms R(G) ⟶ Div(𝔾)(A), and we will implicitly use this identification where convenient. We also see from Lemma 4.3 that positive homomorphisms satisfy f(R_{d}^{+}(G)) ⊆ Div_{d}^{+}(𝔾)(A).

Proposition 5.3. The functor X_{Ch}(G) is a formal scheme over X. The ring C(E, G) = O_{X_{Ch}(G)} is a quotient of a formal power series ring in finitely many variables over O_{X} (and thus is a complete Noetherian local ring).

Proof. Let V_{1}, …, V_{h} be the irreducible representations of G, and let d_{1}, …, d_{h} be their degrees. We assume that these are ordered so that V_{1} is the trivial representation of rank one. There are then natural numbers m_{ijk} and l_{ij}^{r} for r ≥ 0 and 1 ≤ i, j, k ≤ h such that

V_{i} ⊗ V_{j} ≃ ⨁_{k} m_{ijk}.V_{k}
λ^{r}V_{i} ≃ ⨁_{j} l_{ij}^{r}.V_{j}

(Here m.W means the direct sum of m copies of W.)

To give a homomorphism f: R^{+}(G) ⟶ Div^{+}(𝔾)(A) is the same as to give divisors D_{i} = f(V_{i}) ∈ Div_{d_{i}}^{+}(𝔾) for i = 1, …, h such that

D_{i}D_{j} = ∑_{ijk} m_{ijk}D_{k}
λ^{r}D_{i} = ∑_{j} l_{ij}^{r}D_{j}

This exhibits X_{Ch}(G)(A) as the equaliser of a pair of maps from ∏_{i=1}^{h} Div_{d_{i}}(𝔾) to

∏_{i,j} Div_{d_{i}d_{j}}(𝔾) × ∏_{r,i} Div_{( \begin{matrix}d_{i} \\ r \end{matrix})}(𝔾).

In particular, this is a pair of maps between formal schemes over X, so the equaliser is a formal scheme over X.

More explicitly, we have X_{Ch}(G) = spf(C(E, G)), where C(E, G) is defined as follows. We start with O_{X} and adjoin power series variables c_{ik} for i = 1, …, h and k = 1, …, d_{i}, and put c_{i0} = 1. We then put f_{i}(t) = ∑_{k=0}^{d_{i}} c_{ik}t^{d_{i}−k} and impose the relations obtained by equating coefficients in the following identities between polynomials:

∑_{a=0}^{d_{i}d_{j}} p_{d_{i},d_{j},a}(c_{i∗}, c_{j∗})t^{d_{i}d_{j}−a} = ∏_{k} f_{k}(t)^{m_{ijk}}

\begin{matrix}( & \begin{matrix}d_{i} \\ r \end{matrix} & ) \\ ∑ \\ a=0 \end{matrix} q_{d_{i},r,a}(c_{i∗})t^{( \begin{matrix}d_{i} \\ r \end{matrix})−a} = ∏_{j} f_{j}(t)^{l_{ij}^{r}}

The resulting quotient ring is C(E, G).

We next explain how to compare X_{Ch}(G) to X(G). Let \overline{G} be the category whose objects are Lie groups, and whose morphisms are the conjugacy classes of continuous homomorphisms. We then have a natural map

R^{+}(G) = ∐_{d} \overline{G}(G, U(d)) ⟶^{B} \overline{h}T (BG, ∐_{d} BU(d)) ⟶^{spf(E^{0}(−))} X̂_{X}(X(G), Div^{+}(𝔾)).

By taking adjoints, we obtain a map X(G) ⟶ Map(R^{+}(G), Div^{+}(𝔾)), and one checks easily that this actually lands in the subscheme X_{Ch}(G) ⊂ Map(R^{+}(G), Div^{+}(𝔾)) of Λ-semiring homomorphisms. We thus have a natural map

θ_{G}: X(G) ⟶ X_{Ch}(G).

In terms of our explicit description of C(E, G), the map θ^{∗}: C(E, G) ⟶ E^{0}BG sends c_{ik} to the k’th Chern class of the representation V_{i}.

It is natural to ask whether a homomorphism f: R(G) ⟶ Div(𝔾)(A) of Λ-rings is automatically positive. We next show that we always have f(R_{1}^{+}(G)) ⊆ Div_{1}^{+}(𝔾) ≃ 𝔾, but the corresponding claim for d > 1 seems to be false.

Proposition 5.4. If D ∈ Div_{1}(𝔾)(A) and λ^{k}(D) = 0 for all k > 1 then D ∈ Div_{1}^{+}(𝔾)(A).

Proof. We can write D = E − e[0] for some e ≥ 0 and E ∈ Div_{e+1}^{+}(𝔾)(A). Put D′ = λ^{e+1}E ∈ Div_{1}^{+}(𝔾). We have E = D + e[0] so

D′ = ∑_{i+j=e+1} λ^{i}(D)λ^{j}(e[0]) = λ^{1}(D)λ^{e}(e[0]) = D,

so D ∈ Div_{1}^{+}(𝔾) as claimed.

Corollary 5.5. If L ∈ R_{1}^{+}(G) and f: R(G) ⟶ Div(𝔾)(A) is a map of Λ-rings then f(L) ∈ Div_{1}^{+}(𝔾).

Proof. Clearly λ^{k}L = 0 for k > 1 so λ^{k}f(L) = 0 for k > 1, and f(L) ∈ Div_{1}(𝔾)(A) by Lemma 4.3 so f(L) ∈ Div_{1}^{+}(𝔾) by the proposition.

Proposition 5.6. For suitable formal groups 𝔾 and rings A, there exist divisors D ∈ Div_{2}(𝔾)(A) such that λ^{k}D = 0 for k > 2 but D ∉ Div_{2}^{+}(𝔾).

Proof. We will assume that O_{X} = 𝔽_{2}, so p = 2. Suppose that a, b ∈ 𝔾(A) and 2a = 2b = 0. Put c = a + b so 2a = 2b = 2c = a+b+c = 0, and put E = [a]+[b]+[c] and D = E −[0]. Then λ^{2}E = [a+b]+[b+c]+[c+a] = [c]+[a]+[b] = E and λ^{3}E = [0] = 1 so λ_{t}(E) = 1+tE +t^{2}E +t^{3} = (1+tD+t^{2})(1+t), so λ_{t}(D) = 1+tD+t^{2}. Thus λ^{k}D = 0 for k > 2. If D is in Div_{2}^{+}(𝔾)(A) we must have x(a)x(b)x(c) = c_{3}(E) = c_{3}(D + [0]) = 0. Note also that x(c) = x(a−b) = x(a)−_{F} x(b), which is a unit multiple of x(a)−x(b), so the condition is equivalent to x(a)^{2}x(b) = x(a)x(b)^{2}. The universal example for A is O_{X}[[y, z]]/([2](y), [2](z)) = 𝔽_{2}[y, z]/(y^{2^{n}}, z^{2^{n}}) (where y = x(a), z = x(b)). Clearly in this case we have y^{2}z ≠ yz^{2} so D ∉ Div_{2}^{+}(𝔾).

6. The group Σ_{3}

In this section we work through the case where G = Σ_{3} and E is the 2-periodic version of Morava K-theory at the prime 3 with height 2. Many constructions discussed here will be generalised later. Recall that the coefficient ring is E^{∗} = 𝔽_{3}[u, u^{−1}], where |u| = −2.

We have a coordinate x on 𝔾 such that

x(−a) = [−1](x(a)) = −x(a)
x(3a) = [3](x(a)) = x(a)^{9}
x(a + b) = x(a) +_{F} x(b) = x(a) + x(b) (mod x(a)^{3}x(b)^{3})

for all a, b ∈ 𝔾. (The first equation is true because the formal group law F associated to E has an integral lift whose logarithm log_{F} (x) = ∑_{k} x^{9^{k}}/3^{k} satisfies log_{F} (−x) = − log_{F} (x). The second is well-known, and the third follows from [9, Lemma 80].)

Define y, z: Div_{2}^{+}(𝔾) ⟶ 𝔸^{1} by y([a] + [b]) = x(a)x(b) and

z([a] + [b]) = x(λ^{2}([a] + [b])) = x(a + b) = x(a) +_{F} x(b)

(which is a unit multiple of x(a) + x(b)). One checks that that O_{Div_{2}^{+}(𝔾)} = 𝔽_{3}[[y, z]]. If we let Z = SDiv_{2}^{+}(𝔾) be the scheme of divisors D ∈ Div_{2}^{+}(𝔾) such that λ^{2}(D) = [0] then it follows that O_{Z} = 𝔽_{3}[[y, z]]/z = 𝔽_{3}[[y]]. There is an evident map δ: 𝔾 ⟶ Z defined by δ(b) = [b] + [−b], and y(δ(b)) = x(b)x(−b) = −x(b)^{2} so the map δ^{∗}: 𝔽_{3}[[y]] ⟶ 𝔽_{3}[[x]] sends y to −x^{2}. In particular, we see that δ is finite and faithfully flat, with degree two.

Next, note that

δ(b)^{2} = [2b] + [−2b] + 2[0] = ψ^{2}(δ(b)) + 2[0];

as δ is faithfully flat, it follows that D^{2} = ψ^{2}D + 2[0] for any D ∈ Z.

Let Y be the scheme of divisors D ∈ Z such that ψ^{2}(D) = D. To analyse this, note that

x(2b) = x(−b + 3b) = [−1](x(b)) +_{F} [3](x(b)) = −x(b) + x(b)^{9} (mod x(b)^{12}),

so

−x(2b)^{2} = −x(b)^{2} − x(b)^{10} (mod x(b)^{12}),

or in other words

y(ψ^{2}δ(b)) = y(δ(b)) − y(δ(b))^{5} (mod y(δ(b))^{6}).

As δ is faithfully flat, we deduce that

y(ψ^{2}(D)) = y(D) − y(D)^{5} (mod y(D)^{6})

for all D ∈ Z. It follows that (ψ^{2})^{∗}y − y is a unit multiple of y^{5} in 𝔽_{3}[[y]] and thus that O_{Y} = 𝔽_{3}[y]/y^{5}. The character table of G = Σ_{3} is

1 ϵ σ
1^{3} 1 1 2
1.2 1 −1 0
3 1 1 −1

From this we see that

R(G) = ℤ[ϵ, σ]/(ϵ^{2} − 1, ϵσ − σ, σ^{2} − σ − 1 − ϵ).

The only interesting λ-operation is that λ^{2}(σ) = ϵ.

Let f: R^{+}(G) ⟶ Div^{+}(𝔾)(A) be a Λ-semiring homomorphism, in other words a point of X_{Ch}(G). As Div_{1}^{+}(𝔾) ≃ 𝔾, there is a unique point a ∈ 𝔾 such that f(ϵ) = [a]. We also write D = f(σ) ∈ Div_{2}^{+}(𝔾). As f is a map of Λ-semirings, these satisfy

[2a] = [a]^{2} = f(ϵ^{2}) = f(1) = [0]
[a]D = f(ϵσ) = f(σ) = D
D^{2} = f(σ^{2}) = f(σ + 1 + ϵ) = D + [0] + [a]
λ^{2}D = f(λ^{2}σ) = f(ϵ) = [a].

As we work mod 3, the map 2: 𝔾 ⟶ 𝔾 is an isomorphism so the first equation gives a = 0, so the second equation is automatic and the last equation says that D ∈ Z. Thus, the third equation becomes

ψ^{2}D + 2[0] = D^{2} = D + 2[0].

The semiring Div^{+}(𝔾) embeds in the ring Div(𝔾) so we can cancel to see that ψ^{2}(D) = D, so D ∈ Y. We can thus define a map χ: X_{Ch}(G) ⟶ Y by χ(f) = f(σ). One can check that the whole argument is reversible, so χ is an isomorphism and

C(E, G) = O_{X_{Ch}(G)} ≃ O_{Y} = 𝔽_{3}[y]/y^{5}.

We also have a short exact sequence C_{3} ⟶ G ⟶ C_{2} leading to an Atiyah-Hirzebruch-Serre spectral sequence

H^{p}(C_{2}; E^{q}BC_{3}) ⟹ E^{p+q}BG.

We have E^{∗}BC_{3} = 𝔽_{3}[u^{±1}][x]/x^{9} (where |u| = 2 and |x| = 0), and C_{2} acts on this by u ↦ u and x ↦ [−1](x) = −x. Because C_{2} has order coprime to 3 we see that the spectral sequence collapses to an isomorphism

E^{∗}BG = (E^{∗}BC_{3})^{C_{2}} = E^{∗}[y]/y^{5},

where y = −x^{2}. After some comparison of definitions we see that the map θ_{G}: X(G) ⟶ X_{Ch}(G) is an isomorphism.

7. The Abelian case

Theorem 7.1. If G is Abelian then θ_{G}: X(G) ⟶ X_{Ch}(G) is an isomorphism.

Proof. Put G^{∗} = Hom(G, S^{1}), so R^{+}(G) = ℕ[G^{∗}], and let A be an O_{X}-algebra. If f: ℕ[G^{∗}] ⟶ Div^{+}(𝔾)(A) is a Λ-semiring homomorphism, then f induces a group homomorphism f′: G^{∗} = R_{1}^{+}(G) ⟶ 𝔾(A) = Div_{1}^{+}(𝔾)(A). Conversely, given a group homomorphism f′: G^{∗} ⟶ 𝔾(A) we get a map R^{+}(G) = ℕ[G^{∗}] ⟶ ℕ[𝔾(A)] of Λ-semirings. We can compose this with the map ℕ[𝔾(A)] ⟶ Div^{+}(𝔾)(A) in the proof of Proposition 3.2 to get a map R^{+}(G) ⟶ Div^{+}(𝔾)(A), or in other words a point of X_{Ch}(G)(A). One checks that these constructions give a bijection Hom(G^{∗}, 𝔾(A)) ≃ X_{Ch}(G)(A), or equivalently an isomorphism Hom(G^{∗}, 𝔾) ≃ X_{Ch}(G). There is also an isomorphism X(G) ≃ Hom(G^{∗}, 𝔾) (see [4, Proposition 2.9]), so we have an isomorphism X(G) ≃ X_{Ch}(G). A straightforward comparison of definitions shows that this is the same as θ_{G}.

8. The height one case

In this section we choose a prime p and let E be the p-complete complex K-theory spectrum. We thus have X = spf(ℤ_{p}), so discrete O_{X}-algebras are just p-torsion rings. We also have 𝔾 = 𝔾̂_{m}, so

𝔾(A) = {u ∈ A^{×} | 1 − u is nilpotent }.

Theorem 8.1. If E is the p-adic complex K-theory spectrum and G is a p-group then the map θ_{G}: X(G) ⟶ X_{Ch}(G) is an isomorphism.

The rest of this section constitutes the proof. We fix a p-group G and write θ = θ_{G} for brevity.

The first ingredient is the Atiyah-Segal completion theorem. In the case of a p-group, this says that

O_{X(G)} = E^{0}BG = ℤ_{p} ⊗ R(G).

We know that Div(𝔾̂_{m}) is the free ring scheme generated by the group scheme 𝔾̂_{m}. Recall that the affine line 𝔸^{1} is just the forgetful functor from p-torsion rings to sets. This is a ring scheme in a natural way, and it contains 𝔾̂_{m} as a subgroup of its group of units. We thus have a ring map

ξ: Div(𝔾̂_{m}) ⟶ 𝔸^{1}

extending the inclusion of 𝔾̂_{m} in 𝔸^{1}. If D = ∑_{i} n_{i}[u_{i}] ∈ Div(𝔾̂_{m})(A) then ξ(D) = ∑_{i} n_{i}u_{i} ∈ 𝔸^{1}(A) = A.

Next recall that Div_{d}^{+}(𝔾̂_{m}) = 𝔾̂_{m}^{d}/Σ_{d}, so to describe a function f: Div_{d}^{+}(𝔾̂_{m}) ⟶ 𝔸^{1} it suffices to give the symmetric function \overline{f}: 𝔾̂_{m}^{d} ⟶ 𝔸^{1} such that

f(∑_{i=1}^{d}[u_{i}]) = \overline{f}(u_{1}, …, u_{d}).

Let σ_{j}: 𝔸^{d} ⟶ 𝔸^{1} be the j’th elementary symmetric function and define

c_{j}(∑_{i}[u_{i}]) = σ_{j}(1 − u_{1}, …, 1 − u_{d})
a_{j}(∑_{i}[u_{i}]) = σ_{j}(u_{1}, …, u_{d})
a′_{j}(∑_{i}[u_{i}]) = a_{j}(∑_{i}[u_{i}]) − ( \begin{matrix}d \\ j \end{matrix} ).

Recall that O_{Div_{d}^{+}(𝔾̂_{m})} is the set of all maps Div_{d}^{+}(𝔾̂_{m}) ⟶ 𝔸^{1}, so c_{j}, a_{j} and a′_{j} can be viewed as elements of

this ring. The function u ↦ 1 − u is a coordinate on the formal group 𝔾̂_{m} so a well-known argument gives an isomorphism

O_{Div_{d}^{+}(𝔾̂_{m})} = ℤ_{p}[[c_{1}, …, c_{d}]].

It is not hard to deduce that

O_{Div_{d}^{+}(𝔾̂_{m})} = ℤ_{p}[[a′_{1}, …, a′_{d}]],

which is the completion of the ring ℤ_{p}[a_{1}, …, a_{d}] at the ideal generated by the elements a′_{i} = a_{i} − ( \begin{matrix}d \\ i \end{matrix}). Note also that a_{1}(D) = ξ(D) for D ∈ Div_{d}^{+}(𝔾̂_{m}).

Lemma 8.2. For any divisor D ∈ Div_{d}^{+}(𝔾̂_{m}) we have a_{j}(D) = ξ(λ^{j}(D)).

Proof. We may assume that D = ∑_{i}[u_{i}] for some elements u_{1}, …, u_{d} ∈ 𝔾̂_{m}. For any I ⊆ {1, …, d} with |I| = j we put u_{I} = ∏_{i∈I} u_{i}. We then have λ^{j}D = ∑_{I}[u_{I}] and thus

ξλ^{j}(D) = ∑_{I} u_{I} = σ_{j}(u_{1}, …, u_{d}) = a_{j}(D).

Let V be a complex vector bundle over a space Z with associated projective bundle P V, and let 𝔻(V) = spf(E^{0}P V) be the corresponding divisor on 𝔾 over spf(E^{0}Z). If V = ⨁_{i=1}^{d} L_{i} for some complex line bundles L_{i} then each L_{i} can be regarded as an element of E^{0}Z = K^{0}(Z; ℤ_{p}), and one sees easily that 𝔻(V) = ∑_{i}[L_{i}] so a_{j}(𝔻(V)) = σ_{j}(L_{1}, …, L_{d}) = λ^{j}(V). By the splitting principle we see that

ξλ^{j}(𝔻(V)) = a_{j}(𝔻(V)) = λ^{j}(V)

even when V does not split as a sum of line bundles.

Now consider the case Z = BG and suppose that V comes from a representation of G, which we also call V. Suppose we have a point x ∈ X(G)(A) for some p-torsion ring A, corresponding to a ring map x̂: E^{0}BG ⟶ A. One can now check from the definitions that θ(x)(V) = x̂_{∗}(𝔻(V)), so f(θ(x)(V)) = x̂(f(𝔻(V))) for any f ∈ O_{Div_{d}^{+}(𝔾̂_{m})}. In particular, we have

a_{j}(θ(x)(V)) = x̂(a_{j}(𝔻(V))) = x̂(λ^{j}(V)).

We can now construct the map ζ: X_{Ch}(G) ⟶ X(G) that will turn out to be inverse to θ. Let A be a p-torsion ring. A positive homomorphism f ∈ X_{Ch}(G)(A) gives rise to a homomorphism ξ_{A} ∘ f: R(G) ⟶ 𝔸^{1}(A) = A, which factors canonically through ℤ_{p} ⊗ R(G) = E^{0}BG = O_{X(G)} because A is a p-torsion ring. This gives a continuous homomorphism O_{X(G)} ⟶ A, or in other words a point of X(G)(A), which we call ζ(f). This construction gives a natural map ζ: X_{Ch}(G) ⟶ X(G), as required.

Suppose we start with a point x ∈ X(G)(A), corresponding to a ring map x̂: ℤ_{p} ⊗ R(G) = E^{0}BG ⟶ A. We need to check that ζθ(x) = x, or equivalently that ξ_{A}(θ(x)(V)) = x̂(V) for all V ∈ ℤ_{p} ⊗ R(G), and it will suffice to do this for all honest representations V ∈ R_{d}^{+}(G) for all d. In that context we have ξ = a_{1} so

ξ_{A}(θ(x)(V)) = a_{1}(θ(x)(V)) = x̂(λ^{1}(V)) = x̂(V)

as required. Thus ζθ = 1_{X(G)}.

Suppose instead that we start with a point f ∈ X_{Ch}(G)(A), in other words a positive homomorphism f: R^{+}(G) ⟶ Div^{+}(𝔾̂_{m})(A). We need to check that (θ(ζ(f)))(V) = f(V) ∈ Div_{d}^{+}(𝔾̂_{m})(A) for all V ∈ R_{d}^{+}(G). To see this, note that

a_{j}(θ(ζ(f))(V)) = ζ̂(f)(λ^{j}(V))
= ξ(f(λ^{j}(V)))
= ξ(λ^{j}(f(V)))
= a_{j}(f(V)).

It follows that a′_{j}(θ(ζ(f))(V)) = a′_{j}(f(V)) and the functions a′_{j} generate O_{Div_{d}^{+}(𝔾̂_{m})} so θ(ζ(f))(V) = f(V), as required. This shows that θζ = 1_{X_{Ch}(G)}, so θ is an isomorphism as claimed.

9. Reduction to the Sylow subgroup

By a well-known transfer argument, if P is a Sylow p-subgroup of G then the restriction map E^{0}BG ⟶ E^{0}BP is injective, and similar methods give some control over the image. In this section we develop some analogous results for the approximation C(E, G) to E^{0}BG. Let I be the kernel of the restriction map R(G) ⟶ R(P) (which is independent of the choice of P). Note that R(G)/I is isomorphic to the image of the restriction map, so it is a free Abelian group of finite rank and it inherits a Λ-ring structure.

Proposition 9.1. Any Λ-ring homomorphism f: R(G) ⟶ Div(𝔾)(A) factors through R(G)/I.

Proof. As usual, we let the exponent of G be e = p^{v}e′, where e′ is coprime to p. If V ∈ I and g ∈ G has p-power order then g is conjugate to an element of P and thus χ_{V} (g) = 0. If g is an arbitrary element of g then the order of g divides p^{v}e′ so the order of g^{e′} divides p^{v}, so χ_{V} (g^{e′}) = 0. This proves that ψ^{e′}(V) = 0, so ψ^{e′}(f(V)) = 0 in Div(𝔾)(A). However, the action of ψ^{e′} on Div(𝔾)(A) is induced by the action of e′ on 𝔾, which is invertible because e′ is coprime to p. This implies that f(V) = 0 as claimed.

Corollary 9.2. If |G| is prime to p then X_{Ch}(G) = X(G) = X.

Corollary 9.3. X_{Ch}(G) is a closed subscheme of the scheme of Λ-ring maps R(G)/I ⟶ Div(G).

Remark 9.4. The disadvantage of this point of view is that the positivity conditions f(R^{+}(G)) ⊆ Div^{+}(𝔾) become less visible when we work with R(G)/I. However, the situation simplifies again if we assume that P is normal in G. In that case, it is known that the map R^{+}(G) ⟶ R^{+}(P)^{G} is surjective; this was first proved by Gallagher [2], and we will give an alternative proof in Section 17. The sums of G-orbits of irreducibles in R^{+}(P) give a canonical system of generators for R^{+}(P)^{G}, which we can lift to get representations ρ_{1}, …, ρ_{t} of G say. Let d_{i} be the degree of ρ_{i}. If ρ ∈ R_{d}^{+}(G) then res_{P}^{G}(ρ) ∈ R_{d}^{+}(P)^{G} so res_{P}^{G}(ρ) ≃ ∑_{i} m_{i} res_{P}^{G}(ρ_{i}) for some integers m_{i} ≥ 0 with ∑_{i} m_{i}d_{i} = d, so ρ − ∑_{i} m_{i}ρ_{i} ∈ I. Thus, for any map f: R(G) ⟶ Div(𝔾) of Λ-rings we have f(ρ) = ∑_{i} m_{i}f(ρ_{i}), so if f(ρ_{i}) ∈ Div_{d_{i}}^{+}(𝔾) for all i then f(ρ) ∈ Div_{d}^{+}(𝔾). Using this, we see that X_{Ch}(G) is the scheme of Λ-ring maps R(G)/I ⟶ Div(𝔾) such that f(ρ_{i}) ∈ Div_{d_{i}}^{+}(𝔾) for i = 1, …, t.

We next give two results that help us to understand R(G)/I without computing R(G).

Proposition 9.5. Let h be the number of conjugacy classes of elements g ∈ G whose order is a power of p. Then R(G)/I ≃ ℤ^{h} as Abelian groups.

Proof. We already know that R(G)/I is a free Abelian group, so we just need to determine its rank, so it is enough to show that ℂ ⊗ R(G)/I ≃ ℂ^{h}. Let C be the set of conjugacy classes of p-power order, and let C′ be the set of all other conjugacy classes. Let U(G) be the ideal of virtual representations V ∈ R(G) whose character is zero on C′. It is well-known that ℂ ⊗ R(G) ≃ F (C ⨿ C′, ℂ). This isomorphism carries U(G) to F (C, ℂ) and I to F (C′, ℂ) so the map ℂ ⊗ U(G) ⟶ ℂ ⊗ R(G)/I is an isomorphism. Clearly dim_{ℂ} ℂ ⊗ U(G) = |C| = h, and the claim follows.

Remark 9.6. The proof shows that U(G) is a subgroup of finite index in R(G)/I. This index need neither be a power of p nor coprime to p, as one sees by taking G = Σ_{3} and p = 2 or 3; the index is 2 in both cases.

Proposition 9.7. There is a natural isomorphism ℤ_{p} ⊗ R(G)/I = KU_{p}^{0}(BG) (where KU_{p} is the p-adic completion of the complex K-theory spectrum.)

Proof. Let KU_{G} be the usual G-spectrum for equivariant K-theory, and let L_{G} be its p-completion. It is wellknown that KU_{G}^{0}(S^{0}) = R(G), which is free Abelian of finite rank, and it follows that L_{G}^{0}(S^{0}) = R(G)_{p}^{∧} = R(G) ⊗ ℤ_{p}. We give this ring and all its quotients the p-adic topology, or equivalently the profinite topology, which is compact. The argument of the Atiyah-Segal completion theorem shows that KU_{p}^{0}BG = L^{0}EG is the completion of R(G)_{p}^{∧} at the augmentation ideal J_{G}. By a compactness argument, we deduce that the map R(G)_{p}^{∧} ⟶ KU_{p}^{0}BG is surjective; the kernel is J_{G}^{∞} := ⋂_{k} J_{G}^{k}. Now let P be a Sylow p-subgroup, so by the same arguments KU_{p}^{0}BP = R(P)_{p}^{∧}/J_{P}^{∞}. It is well-known that J_{P}^{N} ≤ pR(P)_{p}^{∧} for N ≥ 0 (use the fact that x^{p} − ψ^{p}(x) ∈ pR(P) for all x ∈ R(P), for example) and it follows that J_{P}^{∞} = 0, so KU_{0}^{p}BP = R(P)_{p}^{∧}. We now have a diagram as follows.

J_{G}^{∞}  J_{P}^{∞} = 0 I_{p}^{∧}  R(G)_{p}^{∧}  R(P)_{p}^{∧} KU_{p}^{0}BG  KU_{p}^{0}BP. w v u  u v  w  w u u  _{u_{≃}} v  w

We have seen that the columns are short exact. As ℤ_{p} is flat over ℤ and I, R(G) and R(P) are finitely generated Abelian groups, we see that the middle row is left exact. The bottom horizontal map is injective by a transfer argument. By a diagram chase we deduce that I_{p}^{∧} = J_{G}^{∞}, so KU_{p}^{0}BG = R(G)_{p}^{∧}/I_{p}^{∧} = (R(G)/I)⊗ℤ_{p} as claimed.

10. Finiteness

It is a fundamental fact that the scheme X(G) is finite over X, or equivalently that E^{0}BG is a finitely generated module over E^{0}. This is proved in the present generality as [4, Corollary 4.4]; the argument is the same as in [5]. In this section we show that X_{Ch}(G) is also finite over X. We also study some auxiliary schemes that come up in the proof, as they turn out to be useful in specific computations.

Definition 10.1. For any v ≥ 0 we put

𝔾(v) = ker(p^{v}: 𝔾 ⟶ 𝔾).

In terms of our coordinate, we have 𝔾(v) = spf(E^{0}[[x]]/[p^{v}](x)). Next, recall that 𝔾(v) ≤ 𝔾 is a divisor of degree p^{nv}, where n is the height of 𝔾. For any m ≥ 0 we let 𝔾(v, m) be m times this divisor, considered as a subscheme of 𝔾; in other words 𝔾(v, m) = spf(E^{0}[[x]]/[p^{v}](x)^{m}). For any formal scheme Y over X such that O_{Y} is a free module of finite rank r over O_{X}, we define Y^{d}/Σ_{d} to be spf of the d’th symmetric tensor power of O_{Y} over O_{X}. If {e_{1}, …, e_{r}} is a basis for O_{Y} over O_{X} then the monomials e^{α} := ∏_{i=1}^{r} e_{i}^{α_{i}} with ∑_{i} α_{i} = d form a basis for O_{Y^{d}/Σ_{d}}, so this ring is free of rank ( \begin{matrix}r+d−1 \\ d \end{matrix} ) over O_{X}. We will use this construction in the cases Y = 𝔾(v) and Y = 𝔾(v, m). Finally, we define Z(v, d) to be the scheme of divisors D ∈ Div_{d}^{+}(𝔾) such that ψ^{p^{v}}D = d[0].

Theorem 10.2. The scheme Z(d, v) is finite and flat over X, of degree p^{ndv}. There are closed inclusions

𝔾(v)^{d}/Σ_{d} ⟶ Z(v, d) ⟶ 𝔾(v, d)^{d}/Σ_{d} ⟶ Div_{d}^{+}(𝔾).

The first two of these are infinitesimal thickenings, in other words the corresponding maps of rings are surjective with nilpotent kernel. If O_{X} is a field (necessarily of characteristic p) then Z(v, d) is the fibre of the nv-fold relative Frobenius map

F_{Div_{d}^{+}(𝔾)/X}^{nv}: Div_{d}^{+}(𝔾) ⟶ Div_{d}^{+}((F_{X}^{nv})^{∗}𝔾),

so

O_{Z(v,d)} = O_{X}[[c_{1}, …, c_{d}]]/(c_{i}^{p^{nv}}).

Proof. First suppose that O_{X} is a complete regular local ring. (In the topological context, this occurs when E is Landweber exact.) Consider the following diagram:

Z(v, d)  Div_{d}^{+}(𝔾)  𝔾_{d} X  Div_{d}^{+}(𝔾)  𝔾^{d} u v  w_{_{ψ^{p^{v}}} u u}  _{u u_{p^{v}}} u u^{π} w_{ζ}  _{u u_{π}}

In the right hand square, all the corresponding rings are complete regular local rings. A finite injective map of such rings always makes the target into a free module over the source [1, 2.2.7 and 2.2.11]. The maps π^{∗} and (p^{v})^{∗} are finite injective maps of degrees d! and p^{ndv}. It follows that (ψ^{p^{v}})^{∗} is finite and injective, and thus (as deg(fg) = deg(f) deg(g) in this context) that ψ^{p^{v}} is flat of degree p^{ndv}. The left hand square is a pullback by definition, and it follows that Z(v, d) is flat of degree p^{ndv} over X. Using [4, Proposition 5.2], it is not hard to deduce that this result remains true even if O_{X} is not regular.

Next, let x be a coordinate on 𝔾. Then {x^{i} | i < p^{nv}} is a basis for O_{𝔾(v)} over O_{X}, and {x^{i} | i < p^{nmv}} is a basis for O_{𝔾(v,m)}. Using this we obtain bases for the rings

A = O_{𝔾(v)^{d}} = O_{X}[[x_{1}, …, x_{d}]]/([p^{v}](x_{i}))

and

A′ = O_{𝔾(v,m)^{d}} = O_{X}[[x_{1}, …, x_{d}]]/([p^{v}](x_{i})^{m})

that are permuted by Σ_{d}, and the orbit sums give bases for the rings

B = O_{𝔾(v)^{d}/Σ_{d}} = A^{Σ_{d}}

and

B′ = O_{𝔾(v,m)/Σ_{d}} = (A′)^{Σ_{d}}.

Using these, it is easy to see that the map B′ ⟶ B is surjective, so the map 𝔾(v)^{d}/Σ_{d} ⟶ 𝔾(v, m)^{d}/Σ_{d} is a closed inclusion. A similar argument shows that 𝔾(v, m)^{d}/Σ_{d} is a closed subscheme of Div_{d}^{+}(𝔾).

Next, put

J = ker(A′ ⟶ A) = ([p^{v}](x_{1}), …, [p^{v}](x_{d})).

This is clearly a nilpotent ideal, and ker(B′ ⟶ B) = B′ ∩ J which is a nilpotent ideal in B′. Thus our map 𝔾(v)^{d}/Σ_{d} ⟶ 𝔾(v, m)^{d}/Σ_{d} is an infinitesimal thickening.

It is clear that 𝔾(v)^{d}/Σ_{d} is contained in Z(v, d). Next, let W (v, d) be the preimage of Z(v, d) in 𝔾^{d}, or equivalently the scheme of d-tuples \underline{a} = (a_{1}, …, a_{d}) ∈ 𝔾^{d} such that ∑_{i}[p^{v}a_{i}] = d[0]. If \underline{a} ∈ W (v, d) then for each i we have p^{v}a_{i} ∈ d[0] so a_{i} ∈ d.(p^{v})^{−1}[0] = d.𝔾(v) = 𝔾(v, d). This means that \underline{a} ∈ 𝔾(v, d)^{d} and thus ∑_{i}[a_{i}] ∈ 𝔾(v, d)^{d}/Σ_{d}, so the map W (v, d) ⟶^{π} Z(d, v) ⟶ Div_{d}(𝔾) factors through 𝔾(v, d)^{d}/Σ_{d}. As π is faithfully flat, it follows that Z(v, d) ⊆ 𝔾(v, d)^{d}/Σ_{d} as claimed.

We now have maps

𝔾(v)^{d}/Σ_{d} ⟶^{i} Z(v, d) ⟶^{j} 𝔾(v, d)^{d}/Σ_{d} ⟶^{k} Div_{d}^{+}(𝔾).

We know that ji, k and kj are closed inclusions and that ji is an infinitesimal thickening. It follows easily that i, j and k are closed inclusions and i and j are infinitesimal thickenings.

Now suppose that X is a field of characteristic p. We then have an iterated Frobenius map F_{X}^{nv}: X ⟶ X corresponding to the ring map a ↦ a^{p^{nv}} and thus a formal group 𝔾′ = (F_{X}^{nv})^{∗}𝔾 over X. The map F_{𝔾}^{nv} gives

rise to a map f = F_{𝔾/X}^{nv}: 𝔾 ⟶ 𝔾′. As 𝔾 has height n, the map p^{v}: 𝔾 ⟶ 𝔾 factors as 𝔾 ⟶^{f} 𝔾′ ⟶^{g} 𝔾, where g is an isomorphism. This is just the geometric statement of the fact that [p^{v}](x) = γ(x^{p^{nv}}) for some invertible power series γ. By definition Z(v, d) is the fibre of the map Div_{d}^{+}(𝔾) ⟶ Div_{d}^{+}(𝔾) induced by p^{v}: 𝔾 ⟶ 𝔾, and it follows easily that it is also the fibre of the map Div_{d}^{+}(𝔾) ⟶ Div_{d}^{+}(𝔾′) induced by f. It is easy to identify this with the map F_{Div_{d}^{+}(𝔾)}^{nv}. If we use the usual generators for the coordinate rings of Div_{d}^{+}(𝔾) and

Div_{d}^{+}(𝔾′) then the corresponding ring map sends c_{k} to c_{k}^{p^{nv}}, so O_{Z(v,d)} = O_{X}[[c_{1}, …, c_{d}]]/(c_{k}^{p^{nv}}).

Corollary 10.3. The scheme X_{Ch}(G) is finite over X.

Proof. Let V_{1}, …, V_{h} be the irreducible representations of G, and let d_{1}, …, d_{h} be their degrees. As in the proof of Proposition 5.3, we see that X_{Ch}(G) is a closed subscheme of ∏_{i} Div_{d_{i}}^{+}(𝔾). Now let p^{v} be the p-part of the exponent of G. We see from Lemma 4.3 that X_{Ch}(G) is actually contained in ∏_{i} Z(v, d_{i}), which is finite over X by the theorem. It follows that X_{Ch}(G) itself is finite, as claimed.

Corollary 10.4. For any divisor D ∈ Div(𝔾)(A) there exist w ≥ 0 such that ψ^{k}(D) = ψ^{l}(D) whenever k, l ∈ ℤ_{p} with k = l (mod p^{w}).

Proof. We can reduce easily to the case where D ∈ Div_{d}^{+}(𝔾) for some d ≥ 0. Lemma 4.3 tells us that ψ^{p^{v}}D = d[0] for some v, so D ∈ Z(v, d)(A) ≤ 𝔾(v, d)^{d}/Σ_{d}(A). Next note that A is a discrete O_{X}-algebra so p is nilpotent in A, say p^{r} = 0. It follows that [p^{r}](x) = f(x^{p}) for some power series f with f(0) = 0, and thus that [p^{rs}](x) is divisible by x^{p^{s}}. Thus, for large u we have [p^{u}](x) = 0 (mod x^{d}), so [p^{u+v}](x) = 0 (mod [p^{v}](x)^{d}), so 𝔾(v, d) ≤ 𝔾(u + v). If we put w = u + v this tells us that D ∈ 𝔾(w)^{d}/Σ_{d}, and the action of ψ^{k} on 𝔾(w)^{d}/Σ_{d} clearly depends only on the congruence class of k mod p^{w}, as required.

11. Generalised character theory

In [5], Hopkins, Kuhn and Ravenel describe ℚ ⊗ E^{0}BG in terms of “generalised characters”. In this section we will give an analogous but less precise description of ℚ ⊗ C(E, G).

To explain the HKR theory, write Θ = (ℚ_{p}/ℤ_{p})^{n} and Θ^{∗} = Hom(Θ, ℚ_{p}/ℤ_{p}) = ℤ_{p}^{n}. (Elsewhere these are denoted by Λ and Λ^{∗}, but there are enough Λ’s in this paper already.) Let Θ(v) be the subgroup of Θ killed by p^{v}, and let Level(v, 𝔾) be the scheme of maps ϕ: Θ(v) ⟶ 𝔾 such that ∑_{a∈Θ(v)}[ϕ(a)] ≤ 𝔾(v) in Div^{+}(𝔾). See [9] for more information about these schemes. Put D_{m} = O_{Level(m,𝔾)} and D = lim_{-→}_{m} D_{m} and

L = ℚ ⊗ D. This is a free module of countable rank over ℚ ⊗ O_{X}. If 𝔾 is a universal deformation (as in the case considered by HKR) then it can be described more explicitly: the Weierstrass preparation theorem implies that [p^{v}](x) is a unit multiple of a monic polynomial g_{v}(x) of degree p^{nv}, and L is obtained from ℚ ⊗ O_{X} by adjoining full set of roots for g_{v}(x) for all v.

Now let Ω(G) be the set of G-conjugacy classes of homomorphisms Θ^{∗} ⟶ G, and let F (Ω(G), L) be the ring of all functions u: Ω(G) ⟶ L (with pointwise operations). HKR construct an isomorphism

τ: L ⊗_{O_{X}} E^{0}BG ⟶ F (Ω(G), L).

They work with a particular admissible cohomology theory E, but it is not hard to extend their result to all admissible theories; see [4, Proposition 5.2 and Appendix B] for some pointers.

Now consider the Λ-semiring ℕ[Θ] = ∐_{d} Θ^{d}/Σ_{d} and the Λ-ring ℤ[Θ]. If we give Θ^{∗} its p-adic topology then every subgroup of finite index is open and any continuous homomorphism Θ^{∗} ⟶ GL_{n}(ℂ) factors through a finite quotient of Θ^{∗}. Using this we can identify ℕ[Θ] with the semiring of continuous representations of Θ^{∗}, and ℤ[Θ] with the corresponding ring of virtual representations.

Definition 11.1. We say that a Λ-ring homomorphism f: R(G) ⟶ ℤ[Θ] is positive if f(R^{+}(G)) ⊆ ℤ[Θ]^{+}. We write Ω_{Ch}(G) for the set of Λ-semiring homomorphisms R^{+}(G) ⟶ ℤ[Θ]^{+}, or equivalently the set of positive Λ-ring homomorphisms R(G) ⟶ ℤ[Θ].

Remark 11.2. The arguments of Lemma 4.3, Corollary 5.5 and Proposition 9.1 show that any positive homomorphism f: R(G) ⟶ ℤ[Θ] of Λ-rings automatically sends R_{d}^{+}(G) to ℤ[Θ]_{d}^{+} and I to 0 (where I is the kernel of the restriction map to a Sylow subgroup). There is also an evident analogue of Remark 9.4 for Ω_{Ch}(G).

Remark 11.3. From the definitions we know that the Λ-operations determine the Adams operations. Conversely, it is well-known and easy to check that the Adams operations determine the Λ-operations rationally. As ℤ[Θ] is torsion-free, it follows that a ring map f: R(G) ⟶ ℤ[Θ] preserves the Λ-operations iff it preserves the Adams operations. The corresponding statement for homomorphisms R(G) ⟶ Div(𝔾) is false, however.

Theorem 11.4. There are natural maps κ: Ω(G) ⟶ Ω_{Ch}(G) and τ_{Ch} : L ⊗ C(E, G) ⟶ L ⊗ E^{0}BG making the following diagram commute:

L ⊗ C(E, G)  L ⊗ E^{0}BG F (Ω_{Ch}(G), L)  F (Ω(G), L)._{_{τ_{Ch}} u}  ^{1⊗θ^{∗}} w  _{u_{τ}} w_{κ^{∗}}

(Here the tensor products are taken over E^{0}.) Moreover, the map τ_{Ch} is surjective with nilpotent kernel.

Proof. For brevity we will write C(L, G) = L ⊗ C(E, G) and L^{0}BG = L ⊗ E^{0}BG. This is a slight abuse because these functors do not arise from a spectrum L. We also let v be any integer greater than or equal to the p-adic valuation of the exponent of G.

Any homomorphism u: Θ^{∗} ⟶ G factors through Θ^{∗}/p^{v} = Θ(v)^{∗} and thus is automatically continuous (for the discrete topology on G). It thus gives a positive homomorphism u^{∗}: R(G) ⟶ ℤ[Θ], and it is well-known that this depends only on the conjugacy class of u, so this construction gives a natural map κ: Ω(G) ⟶ Ω_{Ch}(G).

It is easy to see using Adams operations that any positive homomorphism f: R(G) ⟶ ℤ[Θ] actually lands in the subring ℤ[Θ(v)]. Suppose we have a level structure ϕ: Θ(v) ⟶ 𝔾(A). As Θ(v) is a finite Abelian group, this gives rise as in Theorem 7.1 to a positive homomorphism R(Θ^{∗}/p^{v}) = ℤ[Θ(v)] ⟶ Div(𝔾)(A), which we can compose with f to get a positive homomorphism R(G) ⟶ Div(𝔾)(A), or in other words a point of X_{Ch}(G)(A), which we call ρ_{Ch}(f, ϕ). This construction produces a map ρ_{Ch} : Ω_{Ch}(G) × Level(v, 𝔾) ⟶ X_{Ch}(G) of formal schemes over X, corresponding to a map ρ_{Ch}^{∗}: C(E, G) ⟶ F (Ω_{Ch}(G), D_{v}) ⊂ F (Ω_{Ch}(G), L). After tensoring by L we obtain the required map τ_{Ch} : C(L, G) ⟶ F (Ω_{Ch}(G), L).

We next recall the definition of τ. Suppose that u: Θ^{∗}/p^{v} ⟶ G and ϕ ∈ Level(v, 𝔾) ⊂ Hom(Θ(v), 𝔾) = X(Θ^{∗}/p^{v}). We then have a point ρ(u, ϕ) := X(u)(ϕ) ∈ X(G). This construction gives a map ρ: Ω(G) × Level(v, 𝔾) ⟶ X(G) and thus a map ρ^{∗}: E^{0}BG ⟶ F (Ω(G), D_{v}) ⊂ F (Ω(G), L). After tensoring by L we obtain the required map τ.

One can check from the definitions that the following diagram commutes:

X_{Ch}(G)  X(G) Ω_{Ch}(G) × Level(m, 𝔾)  Ω(G) × Level(m, 𝔾). u  ^{θ} u_{ρ_{Ch}}  _{u_{κ×1}}  u_{ρ}

It follows easily that the diagram in the statement of the theorem commutes.

To understand τ_{Ch} more explicitly, let ϕ ∈ Level(v, 𝔾)(D_{v}) be the universal example of a level structure. For any element a ∈ Θ(v) we then have a point ϕ(a) ∈ 𝔾(D_{v}) and thus an element x_{a} := x(ϕ(a)) ∈ D_{v}. These elements satisfy x_{a+b} = x_{a} +_{F} x_{b} and 𝔾(v) = ∑_{a∈Θ(v)}[ϕ(a)] as divisors, or equivalently [p^{v}](t) is a unit multiple of ∏_{a}(t−x_{a}) in D_{v}[[t]]. It is also known that x_{a} −x_{b} is invertible in L whenever a ≠ b (because it is a unit multiple of x_{a−b}, which divides ∏_{c≠0} x_{c} = ±[p^{v}]′(0) = ±p^{v}). Let the representations V_{i} and the elements c_{ik} ∈ C(E, G) be as in the proof of Proposition 5.3. If f ∈ Ω_{Ch}(G) and f(V_{i}) = [a_{1}] + … + [a_{d}] ∈ ℕ[Θ(v)] then τ_{Ch}(c_{ik})(f) is the k’th symmetric function in the variables x_{a_{1}}, …, x_{a_{d}}, and this characterises τ_{Ch}.

We next show that τ_{Ch} is surjective. For any u ∈ Ω_{Ch}(G) we define ϵ_{u}: C(L, G) ⟶ L by ϵ_{u}(c) = τ_{Ch}(c)(u), and we put I_{u} = ker(ϵ_{u}) ≤ C(L, G). By the Chinese Remainder Theorem, it will suffice to show that I_{u} + I_{v} = C(L, G) whenever u ≠ v. If u ≠ v we can choose V ∈ R^{+}(G) such that u(V) ≠ v(V) ∈ ℕ[Θ]. If u(V) = ∑_{a} m_{a}[a] and v(V) = ∑_{a} n_{a}[a] then we must have m_{b} ≠ n_{b} for some b, and without loss we may assume m_{b} > n_{b}. Define

k_{a} = min(n_{a}, m_{a})
C = ∑_{a} k_{a}[a]
A = ∑_{a}(n_{a} − k_{a})[a]
B = ∑_{a}(m_{a} − k_{a})[a].

We can write A in the form [a_{1}]+…+[a_{d}] with a_{i} ≠ b for all i. We also write f_{A}(t) = ∏_{i}(t−x_{a_{i}}), so f_{A}(x_{b}) is invertible in L. On the other hand, the representation V ∈ R_{d}^{+}(G) gives rise in a tautological way to a divisor D_{V} ∈ Div_{d}^{+}(𝔾)(C(E, G)) with equation f_{D_{V}} (t) ∈ C(E, G)[t], say. We have ϵ_{u}f_{D_{V}} (t) = f_{A+C}(t) = f_{A}(t)f_{C}(t) and ϵ_{v}f_{D_{V}} (t) = f_{B}(t)f_{C}(t). The polynomial f_{C}(t) is monic and thus is not a zero-divisor, so f_{A}(t) = f_{B}(t) (mod I_{u} + I_{v}). We evidently have f_{B}(x_{b}) = 0 so f_{A}(x_{b}) = 0 (mod I_{u} + I_{v}). As f_{A}(x_{b}) is invertible in L, we deduce that I_{u} + I_{v} = 1 as required.

Finally, we must show that the kernel of τ_{Ch} is nilpotent. This kernel is the intersection of the ideals I_{u}, so by well-known arguments it suffices to show that every prime ideal in C(L, G) contains I_{u} for some u. To see this, put R = C(L, G) and let 𝔭 ≤ R be a prime ideal. If D ∈ Div_{d}^{+}(𝔾)(C(E, G)) is a divisor satisfying ψ^{p^{v}}(D) = d[0], then Theorem 10.2 implies that D ∈ 𝔾(v, d)^{d}/Σ_{d} and thus that D ≤ d^{2}.𝔾(v) as divisors, or equivalently f_{D}(t) divides [p^{v}](t)^{d^{2}}, which is a unit multiple in D_{v}[t] of ∏_{a∈Θ(v)}(t − x_{a})^{p^{v}}. Now let K be the field of fractions of R/𝔭, and note that x_{a} − x_{b} is invertible in L and thus in K when a ≠ b. As K[t] is a unique factorisation domain, we see that f_{D}(t) = ∏_{a}(t − x_{a})^{m_{a}} in K[t] for a unique system of integers m_{a}. We define w(D) = ∑_{a} m_{a}[a] ∈ ℤ[Θ(v)]_{d}^{+}. In particular, if V ∈ R_{d}^{+}(G) we can let D_{V} be the tautologically associated divisor over C(E, G) and put u(V) = w(D_{V}). One can check that this gives a homomorphism u: R^{+}(G) ⟶ ℤ[Θ] of Λ-semirings, or in other words an element u ∈ Ω_{Ch}(G). From the construction it is automatic that I_{u} ≤ 𝔭.

Example 11.5. If G is Abelian, it is easy to see that Ω_{Ch}(G) = Hom(G^{∗}, Θ) ≃ Hom(Θ^{∗}, G) = Ω(G) and that κ is an isomorphism.

Example 11.6. Consider the symmetric group G = Σ_{k}. This acts in an obvious way on ℂ^{k}, and we call this representation π. It is known that π generates R(G) as a Λ-ring. Thus, an element f ∈ Ω_{Ch}(G) is determined by the value f(π) ∈ ℤ[Θ]_{k}^{+}.

As discussed in [12], the set Ω(G) can be identified with the set of isomorphism classes of sets of order k with an action of Θ^{∗}. For any finite subgroup A < Θ we have a homomorphism Θ^{∗} ⟶ A^{∗} and thus an action of Θ^{∗} on A^{∗}. Note that 0^{∗} is just a single point with trivial action. We write m.A^{∗} for the disjoint union of m copies of A^{∗}. If T is a finite Θ^{∗}-set, then T can be written in an essentially unique way in the form ∐_{i} m_{i}.A_{i}^{∗}.

If we write [A] = ∑_{a∈A}[a] ∈ ℤ[Θ] then by working through the definitions we find that κ(∐_{i} m_{i}.A_{i}^{∗})(π) = ∑_{i} m_{i}[A_{i}], which effectively determines κ.

It is now easy to exhibit cases in which κ is not injective. For example, suppose that p = 2 and n > 1 and k = 6. We can then find two distinct, nonzero elements a, b ∈ Θ(1) and put c = a + b. Let A, B and C be the groups generated by a, b and c respectively, and put V = A + B = {0, a, b, c}. Then

κ(V^{∗} ⨿ 2.0^{∗})(π) = κ(A^{∗} ⨿ B^{∗} ⨿ C^{∗})(π) = 3[0] + [a] + [b] + [c],

so κ is not injective. In Section 15 we will give examples where κ is not surjective.

12. Calculating Ω_{Ch}(G)

In this section we define sets Ω′(G) and Ω′′(G) which in some cases may be easier to compute than Ω(G) or Ω_{Ch}(G), and we define natural maps

Ω(G) ww Ω′(G) v w Ω_{Ch}(G) v w Ω′′(G).

Definition 12.1. Let C be the set of conjugacy classes of elements of p-power order in G. We let the multiplicative monoid ℤ act on Θ^{∗} in the obvious way, and on C by k.[g] = [g^{k}]. We say that two homomorphisms u, v: Θ^{∗} ⟶ G are pointwise conjugate if u(a) is conjugate to v(a) for all a ∈ Θ^{∗}. We recall the definitions of Ω(G) and Ω_{Ch}(G) and define new sets Ω′(G) and Ω′′(G) as follows:

Ω(G) = Hom(Θ^{∗}, G)/conjugacy
Ω′(G) = Hom(Θ^{∗}, G)/pointwise conjugacy
Ω′′(G) = {ℤ-equivariant continuous maps Θ^{∗} ⟶ C}
Ω_{Ch}(G) = { positive Λ-ring homomorphisms R(G) ⟶ ℤ[Θ]}.

Proposition 12.2. There are natural maps as follows:

Ω(G) ww Ω′(G) v w Ω_{Ch}(G) v w Ω′′(G).

Proof. There are evident natural maps

Ω(G) ww Ω′(G) v w Ω′′(G).

We have also already constructed a map κ: Ω(G) ⟶ Ω_{Ch}(G). If u, v: Θ^{∗} ⟶ G are pointwise-conjugate then the induced maps from class functions on G to class functions on Θ^{∗} are evidently the same, so the induced maps R(G) ⟶ ℤ[Θ] are the same, so κ(u) = κ(v). This shows that κ factors through the projection Ω(G) ⟶ Ω′(G).

We next define a map ξ: Ω_{Ch}(G) ⟶ Ω′′(G). Suppose that u ∈ Ω_{Ch}(G) and a ∈ Θ^{∗} = Hom(Θ, S^{1}). Then a extends in a natural way to give a ℂ-algebra map â: ℂ[Θ] ⟶ ℂ and thus a ring map (1_{ℂ}⊗u)â: ℂ⊗R(G) ⟶ ℂ. Using the fact that ℂ⊗R(G) is the set of ℂ-valued class functions on G, we see that Hom_{ℂ−Alg}(ℂ⊗R(G), ℂ) can be identified with the set of conjugacy classes in G. Thus there exists h ∈ G (unique up to conjugation) such that (1 ⊗ u)(â(V)) = χ_{V} (h) for all V ∈ R(G). We can choose m so that u(V) ∈ ℤ[Θ(m)] for all V, and then we have χ_{V} (h^{p^{m}}) = χ_{ψ^{p^{m}} V} (h) = χ_{dim(V)}(h) = dim(V) for all V, so h^{p^{m}} = 1. This means that the conjugacy class [h] lies in C, so we can define ξ(u)(a) = [h]. We leave it to the reader to check that this gives a map ξ: Ω_{Ch}(G) ⟶ Ω′′(G) as claimed. The maps â: ℤ[Θ] ⟶ ℂ (as a runs over Θ^{∗}) are jointly injective, an it follows that ξ is injective. One can also check that the composite Ω′(G) ⟶ Ω_{Ch}(G) ⟶ Ω′′(G) is just the obvious inclusion, which implies that the map Ω′(G) ⟶ Ω_{Ch}(G) is injective.

13. Special divisors

In this section we study “special” divisors, which are related to the special unitary group in the same way that arbitrary divisors are related to the full unitary group.

Definition 13.1. A divisor D ∈ Div_{d}^{+}(𝔾) is special if λ^{d}D = [0]. We write SDiv_{d}^{+}(𝔾) for the scheme of special divisors.

Proposition 13.2. We have O_{SDiv_{d}^{+}(𝔾)} = O_{X}[[c_{2}, …, c_{d}]]. In the topological situation this can be identified with E^{0}BSU(d).

Proof. Put A = O_{𝔾^{d}} = O_{X}[[x_{1}, …, x_{d}]] and A′ = O_{Div_{d}^{+}(𝔾)} = A^{Σ_{d}} = O_{X}[[c_{1}, …, c_{d}]]. Here c_{i} is the i’th

elementary symmetric function, and in particular c_{1} = ∑_{i} x_{i}. Put c′_{1} = ∑_{i}^{F} x_{i} ∈ A′. If we regard λ^{d} as a map Div_{d}^{+}(𝔾) ⟶ Div_{1}^{+}(𝔾) = 𝔾 then c′_{1} = x ∘ λ^{d}, so we see that O_{ℍ_{3}} = A/c′_{1} and O_{SDiv_{d}^{+}(𝔾)} = A′/c′_{1}. Next, observe that the inclusion A′ ⟶ A induces an inclusion A′/(c_{1}^{2}, c_{2}, …, c_{d}) ⟶ A/(x_{1}, …, x_{d})^{2}. We have c′_{1} = c_{1} (mod (x_{1}, …, x_{d})^{2}) so c′_{1} = c_{1} (mod c_{1}^{2}, c_{2}, …, c_{d}). It follows easily that A′ = O_{X}[[c′_{1}, c_{2}, …, c_{d}]] and thus that A′/c′_{1} = O_{X}[[c_{2}, …, c_{d}]].

We next put ℍ_{d} = ker(𝔾^{d} ⟶^{+} 𝔾). If we let q: 𝔾^{d} ⟶ Div_{d}^{+}(𝔾) be the usual projection (which is finite and faithfully flat, with degree d!) then ℍ_{d} = q^{−1} SDiv_{d}^{+}(𝔾). It follows that the map q: ℍ_{d} ⟶ SDiv_{d}^{+}(𝔾) is also finite and faithfully flat, with the same degree. It clearly factors through ℍ_{d}/Σ_{d} := spf(O_{ℍ_{d}}^{Σ_{d}}), and one would like the induced map q: ℍ_{d}/Σ_{d} ⟶ SDiv_{d}^{+}(𝔾) to be an isomorphism. However, quotient constructions in algebraic geometry are never as simple as one would like, and we do not know whether this is true in general; certainly it becomes false if we remove our assumption that 𝔾 has finite height. For example, consider the case where 𝔾 is the additive group over 𝔽_{2} and d = 2; then 2a = 0 for all a ∈ 𝔾 so Σ_{2} acts trivially on ℍ_{2} = {(a, −a) | a ∈ 𝔾} so the map q: ℍ_{d}/Σ_{d} ⟶ SDiv_{d}^{+}(𝔾) has degree two. However, we do have the following partial result.

Proposition 13.3. If d is invertible in O_{X} then SDiv_{d}^{+}(𝔾) = ℍ_{d}/Σ_{d}.

Proof. As d is invertible in O_{X}, multiplication by d is an automorphism of 𝔾. Define maps 𝔾×ℍ_{d} ⟶^{f} 𝔾^{d} ⟶^{g} 𝔾 by f(a, b_{1}, …, b_{d}) = (a + b_{1}, …, a + b_{d}) and g(b_{1}, …, b_{n}) = ∑_{i} b_{i}/d, and then define h: 𝔾^{d} ⟶ 𝔾 × ℍ_{d} by h(\underline{b}) = (g(\underline{b}), b_{1}−g(\underline{b}), …, b_{d}−g(\underline{b})). Clearly h is inverse to f, so f is an isomorphism, giving an isomorphism O_{𝔾^{d}} = O_{G} ˆ⊗O_{ℍ_{d}} = O_{ℍ_{d}}[[x]] of rings. If we let Σ_{d} act trivially on 𝔾 then everything is equivariant, so we have O_{𝔾^{d}}^{Σ_{d}} = O_{ℍ_{d}}^{Σ_{d}}[[x]], so Div_{d}^{+}(𝔾) = 𝔾^{d}/Σ_{d} = 𝔾 × (ℍ_{d}/Σ_{d}).

14. The group Σ_{4}

We now consider the case where G = Σ_{4} and E is the 2-periodic Morava E-theory spectrum of height 2 at the prime 2. We shall show that the map C(E, G) ⟶ E^{0}BG is an isomorphism. To be more explicit, we need to name some representations. Note that Σ_{4} acts on ℂ^{4} with a one-dimensional fixed subspace; we let ρ be the representation of G on the quotient space. We also write ϵ for the sign representation. We let K = E/I_{2} denote the 2-periodic Morava K-theory spectrum.

Theorem 14.1. Let c_{2}, c_{3} ∈ E^{0}BΣ_{4} be the Chern classes of the representation ϵρ, and let w be the Euler class of ϵ. Then C(E, Σ_{4}) = E^{0}BΣ_{4}, and this is a free module of rank 17 over E^{0}, with the following monomials as a basis:

1  c_{2}  c_{2}^{2}  c_{2}^{3}  c_{3}
w  wc_{2}  wc_{2}^{2}  wc_{3}
w^{2}  w^{2}c_{2}  w^{2}c_{2}^{2}  w^{2}c_{3}
w^{3}  w^{3}c_{2}  w^{3}c_{2}^{2}  w^{3}c_{3}.

Moreover, we have

C(K, Σ_{4}) = K^{0}BΣ_{4} = C(E, Σ_{4})/I_{2} = K^{0}[w, c_{2}, c_{3}]/J,

where J is generated by the following elements:

w^{4} , c_{3}^{2} , c_{2}c_{3} ,
c_{2}^{4} + w^{2}c_{2}^{3} + wc_{2}^{2} + w^{2}c_{3},
wc_{2}^{3} + w^{2}c_{2} + wc_{3}.

We will prove this in a number of stages. In Section 14.1 we assemble the facts that we need about the representations of Σ_{4}, and in Section 14.2 we deduce that the map κ: Ω(Σ_{4}) ⟶ Ω_{Ch}(Σ_{4}) is a bijection. We then recall some formulae for the relevant formal group law, and in Section 14.4 we use them to analyse the structure of an auxiliary scheme denoted SDiv_{3}^{+}(𝔾_{0})^{C}. This allows us to complete our determination of C(K, Σ_{4}) in Section 14.5, with the help of some theory of Gröbner bases. We find in particular that C(K, Σ_{4}) is a Gorenstein ring, which enables us to use the inner products defined in [10] to show that the map θ: C(K, Σ_{4}) ⟶ K^{0}BΣ_{4} is injective; this is explained in Section 14.6. We know from [5] that K(n)^{∗}BΣ_{4} is concentrated in even degrees, and it follows that E^{0}BΣ_{4} is a free module over E^{0} of rank |Ω(Σ_{4})| = 17; see [11] for more details. In Section 14.7 we combine these various facts to prove the theorem.

14.1. Representation theory. Our first task is to understand the structure of R(Σ_{4}). We have already defined the characters ϵ and ρ. It is a standard calculation that there is another irreducible character σ of dimension 2 such that the character table is as follows:

class  size  1  ϵ  σ  ρ  ϵρ
1^{4}  1  1  1  2  3  3
1^{2}2  6  1  −1  0  1  −1
2^{2}  3  1  1  2  −1  −1
13  8  1  1  −1  0  0
4  6  1  −1  0  −1  1

The ring structure, Adams operations and Λ-operations are described in the following table.

ϵ^{2} = 1  ψ^{k}(ϵ) = ϵ^{k}  λ^{2}(σ) = ϵ
ϵσ = σ  ψ^{2}(σ) = 1 − ϵ + σ  λ^{2}(ρ) = ϵρ
σ^{2} = 1 + ϵ + σ  ψ^{3}(σ) = 1 + ϵ  λ^{3}(ρ) = ϵ
σρ = ρ + ϵρ  ψ^{2}(ρ) = 1 + σ + ρ − ϵρ
ρ^{2} = 1 + σ + ρ + ϵρ  ψ^{3}(ρ) = 1 + ϵ − σ + ρ.

(The first two columns are easily checked by looking at the characters, and the last column follows using the standard formulae relating Adams operations to Λ-operations.)

Let P be a Sylow 2-subgroup (a dihedral group of order 8) and I be the kernel of the restriction map R(Σ_{4}) ⟶ R(P); one checks that I = (σ − 1 − ϵ). Put τ = ϵρ ∈ R_{3}^{+}(Σ_{4}); one checks that λ^{k}(τ) = ϵ^{k}λ^{k}(ρ) and so λ^{2}(τ) = τ and λ^{3}τ = 1. We have

R(Σ_{4})/I = ℤ{1, ϵ, τ, ϵτ} = ℤ[ϵ, τ]/(ϵ^{2} − 1, τ^{2} − 1 − (1 + ϵ)(1 + τ)).

The operation ψ^{k} acts as the identity on this ring when k is odd, and we have

ψ^{2}(ϵ) = 1
ψ^{2}(τ) = 2 + ϵ + ϵτ − τ.

Proposition 14.2. The set Ω_{Ch}(Σ_{4}) can be identified with the set of pairs (d, u) ∈ Θ(1) × ℤ[Θ]_{3}^{+} such that

2d = 0
λ^{3}(u) = [0]
ψ^{−1}(u) = u
ψ^{2}(u) + u = 2[0] + [d] + [d]u.

Similarly, X_{Ch}(Σ_{4}) can be identified with the scheme of pairs (d, D) ∈ 𝔾(1) × Div_{3}^{+}(𝔾) such that

2d = 0
λ^{3}(D) = [0]
ψ^{−1}(D) = D
ψ^{2}(D) + D = 2[0] + [d] + [d]D.

Proof. Given a positive homomorphism f: R(Σ_{4}) ⟶ ℤ[Θ], let d ∈ Θ be the element such that f(ϵ) = [d] and put u = f(τ). We know from Remark 11.2 that f(I) = 0 and it follows easily from our description of R(Σ_{4})/I that d and u have the properties listed. Conversely, given d and u as described, we can define a homomorphism f: R(Σ_{4})/I ⟶ ℤ[Θ] of additive groups by

f(1) = [0]
f(ϵ) = [d]
f(τ) = D
f(ϵτ) = [d]D.

It is straightforward to check that this gives a homomorphism of Λ-rings, and that these constructions give the required bijection. The argument for X_{Ch}(Σ_{4}) is essentially the same.

14.2. Generalised character theory. We next work out the generalised character theory (as recalled in Section 11) of Σ_{4}. The set Ω(Σ_{4}) can be described in terms of Θ^{∗}-sets as in Example 11.6. We can thus write Ω(Σ_{4}) as the disjoint union Φ_{0} ⨿ … ⨿ Φ_{4}, where

• Φ_{0} consists of the set 4.0^{∗} := 0^{∗} ⨿ 0^{∗} ⨿ 0^{∗} ⨿ 0^{∗}.

• Φ_{1} consists of the sets 2.0^{∗} ⨿ A^{∗}, where A ≃ ℤ/2 (so |Φ_{1}| = 2^{n} − 1).

• Φ_{2} consists of the sets A^{∗} ⨿ B^{∗}, where A ≃ B ≃ ℤ/2, and A may be equal to B. We have |Φ_{2}| = \frac{1}{2}|Φ_{1}|(|Φ_{1}| + 1) = 2^{n−1}(2^{n} − 1).

• Φ_{3} consists of the sets A^{∗} where A ≃ (ℤ/2)^{2}. We have |Φ_{3}| = (2^{n} − 1)(2^{n−1} − 1)/3 (by counting the number of linearly independent pairs in (ℤ/2)^{n} and dividing by |GL_{2}(ℤ/2)| = 6).

• Φ_{4} consists of the sets A^{∗} where A ≃ ℤ/4. There are 2^{2n} − 2^{n} points in Θ of order exactly 4, and each subgroup in Φ_{4} contains precisely two of these, so |Φ_{4}| = (2^{2n} − 2^{n})/2 = 2^{n−1}(2^{n} − 1).

Proposition 14.3. The map κ: Ω(Σ_{4}) ⟶ Ω_{Ch}(Σ_{4}) is a bijection.

Proof. Define

Ψ_{i} = κ(Φ_{i}) ⊆ Ω_{Ch}(Σ_{4}).

Recall from Example 11.6 that κ(∐_{i} m_{i}.A_{i}^{∗})(π) = ∑_{i} m_{i}[A_{i}]. An easy case-by-case check shows that the sets Ψ_{i} are disjoint and that the maps κ: Φ_{i} ⟶ Ψ_{i} are bijections. It will thus be enough to show that the union of the sets Ψ_{i} is the whole of Ω_{Ch}(Σ_{4}).

Suppose we have an element f ∈ Ω_{Ch}(Σ_{4}), with f(ϵ) = [d] and f(τ) = u = [a] + [b] + [c] say. Let A, B and C be the cyclic subgroups generated by a, b and c respectively. Put v = f(π) = [0] + [d]u = [0] + [a + d] + [b + d] + [c + d], and recall that this determines f, because π generates R(Σ_{4}) as a Λ-ring. As ψ^{4}τ = 3[0] we have 4a = 4b = 4c = 0. By Proposition 14.2, we have

2d = 0
a + b + c = 0
[a] + [b] + [c] = [−a] + [−b] + [−c]
[2a] + [2b] + [2c] + [a] + [b] + [c] = 2[0] + [d] + [a + d] + [b + d] + [c + d].

Suppose that ψ^{2}(u) ≠ 3[0]. Without loss of generality we may assume that 2a ≠ 0 so −a ≠ a. The third equation implies that −a ∈ {b, c}, so we may assume that −a = b. As a + b + c = 0 we must have c = 0. Recall also that 4a = 0 so 2a = −2a. Putting all this in the last equation and cancelling 2[0] gives

2[2a] + [a] + [−a] = 2[d] + [d + a] + [d − a].

Note that 2a and d have order 2, but a, −a, d + a and d − a do not. It follows that we must have 2a = d and thus v = [0] + [a] + [2a] + [3a]. We conclude that f = κ(A^{∗}) ∈ Ψ_{4}.

We may thus assume that ψ^{2}(u) = 3[0], so 2a = 2b = 2c = 0. Suppose that d = 0. As a + b + c = 0 we see that D := {0, a, b, c} is a subgroup of Θ, of order 2^{e} say (so e ∈ {0, 1, 2}). This implies that v = 2^{2−e}[A] = κ(2^{2−e}.D^{∗})(π), so f = κ(2^{2−e}.D^{∗}) ∈ Ψ_{0} ⨿ Ψ_{2} ⨿ Ψ_{3}.

We may thus assume that 2a = 2b = 2c = 2d = 0 and d ≠ 0. The equation ψ^{2}(u) + u = 2[0] + [d] + [d]u then reduces to

[0] + [a] + [b] + [c] = [d] + [a + d] + [b + d] + [c + d].

It follows that d ∈ {a, b, c} and without loss we may assume that d = a. Note that c = a + b = d + b (because a + b + c = 0). If b = 0 this gives c = a = d so v = 3[0] + [a], so f = κ(2.0^{∗} ⨿ A^{∗}). The same argument works if c = 0, so we reduce to the case where a = d ≠ 0 and b and c are also nonzero. We then have

v = [0] + [a + d] + [b + d] + [c + d] = 2[0] + [c] + [b] = [B] + [C],

so f = κ(B^{∗} ⨿ C^{∗}) ∈ Φ_{2}.

14.3. The formal group law. Let 𝔾 be the formal group associated to E, and let 𝔾_{0} be its restriction to the special fibre X_{0} ⊂ X, or equivalently the formal group associated to K. This has a standard coordinate giving rise to a formal group law F over O_{X_{0}} = K^{0} = 𝔽_{4}, which is in fact defined over 𝔽_{2}. We will need the following formulae:

[2](x) = x^{4}
[−1](x) = x + x^{4} + x^{10} + x^{16} + x^{22} (mod x^{32})
x +_{F} y = x + y + x^{2}y^{2} (mod x^{4}y^{4}).

The first of these is well-known and the second can be proved by straightforward computation; for the third, one can adapt the method of [9, Section 15] to the case p = 2.

14.4. The scheme SDiv_{3}^{+}(𝔾_{0})^{C}. Let C be the group (of order 2) generated by ψ^{−1}, so

SDiv_{3}^{+}(𝔾_{0})^{C} = {D ∈ Div_{3}^{+}(𝔾_{0}) | λ^{3}D = [0] and ψ^{−1}D = D}.

We have seen that O_{SDiv_{3}^{+}(𝔾_{0})} = 𝔽_{4}[[c_{2}, c_{3}]]; our next task is to determine the quotient ring O_{SDiv_{3}^{+}(𝔾_{0})^{C}}.

Proposition 14.4. We have

O_{SDiv_{3}^{+}(𝔾_{0})^{C}} = 𝔽_{4}[[c_{2}, c_{3}]]/(c_{2}c_{3}, c_{3}^{2}) = 𝔽_{4}[[c_{2}]] ⊕ 𝔽_{4}.c_{3}.

Proof. Put A = 𝔽_{4}[[x, y, z]] and

d = x +_{F} y +_{F} z
c_{1} = x + y + z
c_{2} = xy + yz + zx
c_{3} = xyz.

Put A′ = A^{Σ_{3}} = 𝔽_{4}[[d, c_{2}, c_{3}]], so A is free of rank 6 over A′. Put B′ = A′/dA′ and

B = A/dA = B′ ⊗_{A′} A = 𝔽_{4}[[c_{2}, c_{3}]] = O_{SDiv_{3}^{+}(𝔾_{0})}.

For any element u ∈ A we write \overline{u} = (ψ^{−1})^{∗}(u), so u ↦ \overline{u} is a ring map and \overline{u} = [−1](u) for u ∈ \{x, y, z, d\}. Put C = B/(\overline{c}_{2} − c_{2}, \overline{c}_{3} − c_{3})B and C′ = B′/(\overline{c}_{2} − c_{2}, \overline{c}_{3} − c_{3})B′ = O_{SDiv_{3}^{+}(𝔾_{0})^{C}}. The claim is that the ideal in C′ generated by c_{3} is free of rank one over 𝔽_{4}, and that C′/c_{3}C′ = 𝔽_{4}[[c_{2}]], so that C′ = 𝔽_{4}[[c_{2}]] ⊕ 𝔽_{4}.c_{3}.

We will think of Div_{2}^{+}(𝔾_{0}) as being embedded in Div_{3}^{+}(𝔾_{0}) by the map D ↦ D + [0], so

O_{Div_{2}^{+}(𝔾_{0})} = O_{Div_{3}^{+}(𝔾_{0})}/c_{3} = 𝔽_{4}[[d, c_{2}]].

There is a faithfully flat map 𝔾_{0} ⟶ SDiv_{2}^{+}(𝔾_{0}) sending a to [a] + [−a], and clearly ψ^{−1}([a] + [−a]) = [a] + [−a] so SDiv_{2}^{+}(𝔾_{0}) ⊆ SDiv_{3}^{+}(𝔾_{0})^{C}. It follows that Div_{2}(𝔾_{0}) ∩ SDiv_{3}^{+}(𝔾_{0})^{C} = SDiv_{2}^{+}(𝔾_{0}), and thus that C′/c_{3}C′ = 𝔽_{4}[[c_{2}]] as claimed.

This implies that we must have \overline{c}_{2} − c_{2} = c_{3}r_{2} and \overline{c}_{3} − c_{3} = c_{3}r_{3} for some r_{2}, r_{3} ∈ B′.

Now work in B/(x, y, z)^{7}. We have

z = \overline{x} +_{F} \overline{y} = x + y + x^{4} + x^{2}y^{2} + y^{4}
\overline{x} = x + x^{4}
\overline{y} = y + y^{4}
\overline{z} = x + y + x^{2}y^{2}
c_{2} = x^{2} + xy + y^{2} + x^{5} + x^{4}y + x^{3}y^{2} + x^{2}y^{3} + xy^{4} + y^{5}
c_{3} = x^{2}y + xy^{2} + x^{5}y + x^{3}y^{3} + xy^{5}
\overline{c}_{2} − c_{2} = c_{2}c_{3} = x^{4}y + xy^{4}
\overline{c}_{3} − c_{3} = c_{3}^{2} = x^{4}y^{2} + x^{2}y^{4}.

We also find that the ideal c_{3}.(c_{2}, c_{3})^{2} maps to zero in this ring. Using this, we find that r_{2} = c_{2} (mod (c_{2}, c_{3})^{2}) and r_{3} = c_{3} (mod (c_{2}, c_{3})^{2}), so B′ = 𝔽_{4}[[r_{2}, r_{3}]] and B′/(r_{2}, r_{3}) = 𝔽_{4}. It follows that O_{SDiv_{3}^{+}(𝔾_{0})^{C}} = B′/(r_{2}c_{3}, r_{3}c_{3}) = B′/(c_{2}c_{3}, c_{3}^{2})3 as claimed.

Now put Y = {D ∈ SDiv_{3}^{+}(𝔾_{0})^{C} | ψ^{4}D = 3[0]}, and let U ⊂ 𝔾_{0} be the divisor 32[0]. We know that c_{k}(ψ^{4}D) = c_{k}(D)^{16} so

O_{Y} = O_{SDiv_{3}^{+}(𝔾_{0})^{C}} /(c_{1}^{16}, c_{2}^{16}, c_{3}^{16}) = 𝔽_{4}[c_{2}, c_{3}]/(c_{2}^{16}, c_{3}^{2}, c_{2}c_{3}).

We can also study Y using the maps

α: U ⟶ Y α(a) = [a] + [−a] + [0]
β: 𝔾_{0}(1)^{2} ⟶ Y β(a, b) = [a] + [b] + [a + b].

The map α gives a ring map α^{∗}: O_{Y} ⟶ 𝔽[x]/x^{32}, with

α^{∗}(c_{1}) = x + \overline{x} = x^{4} + x^{10} + x^{16} + x^{22}
α^{∗}(c_{2}) = x\overline{x} = x^{2} + x^{5} + x^{11} + x^{17} + x^{23}
α^{∗}(c_{3}) = 0.

The map β gives a ring map β^{∗}: O_{Y} ⟶ 𝔽[x, y]/(x^{4}, y^{4}). If we put z = x +_{F} y = x + y + x^{2}y^{2} then

β^{∗}(c_{1}) = x + y + z = x^{2}y^{2}
β^{∗}(c_{2}) = xy + yz + zx = x^{2} + xy + y^{2} + x^{2}y^{2}(x + y)
β^{∗}(c_{3}) = xyz = xy(x + y) + x^{3}y^{3}.

Proposition 14.5. The maps α^{∗} and β^{∗} are jointly injective (in other words, ker(α^{∗}) ∩ ker(β^{∗}) = 0). Moreover, we have c_{1} = c_{2}^{2} + c_{2}^{8} in O_{Y}.

Proof. Recall that O_{Y} = 𝔽_{4}[c_{2}, c_{3}]/(c_{2}^{16}, c_{3}^{2}, c_{2}c_{3}), so {c_{2}^{i} | 0 ≤ i < 16} ⨿ {c_{3}} is a basis for O_{Y} over 𝔽_{4}. As α^{∗}(c_{2}) = x^{2} (mod x^{3}), it is easy to see that α^{∗}(c_{2}^{i}) = x^{2i} (mod x^{2i+1}) and that these elements are linearly independent in O_{U} = 𝔽_{4}[x]/x^{32}. Moreover, we have

β^{∗}(1) = 1
β^{∗}(c_{2}) = x^{2} + xy + y^{2} + x^{2}y^{2}(x + y)
β^{∗}(c_{2}^{2}) = x^{2}y^{2}
β^{∗}(c_{2}^{3}) = x^{3}y^{3}
β^{∗}(c_{2}^{i}) = 0  for i > 3.

It is easy to check that β^{∗}(c_{3}) does not lie in the span of these elements, and to deduce that α^{∗} and β^{∗} are jointly injective as claimed. Thus, to show that c_{1} = c_{2}^{2} + c_{2}^{8} we need only check that α^{∗}(c_{1}) = α^{∗}(c_{2}^{2} + c_{2}^{8}) and β^{∗}(c_{1}) = β^{∗}(c_{2}^{2} + c_{2}^{8}), which is a straightforward computation.

14.5. The ring C(K, Σ_{4}). Consider a pair (d, D) ∈ 𝔾_{0}(1) × Y. This gives us a divisor [d]D ∈ Div_{3}^{+}(𝔾_{0}) defined over the ring

O_{𝔾_{0}(1)} ⊗ O_{Y} = 𝔽_{4}[w, c_{2}, c_{3}]/(w^{4}, c_{2}^{16}, c_{3}^{2}, c_{2}c_{3}).

Here of course c_{2} and c_{3} are the usual invariants of the divisor D, but the divisor [d]D also has invariants c_{k}([d]D) lying in the above ring. In order to apply the description of X_{Ch}(Σ_{4}) in Proposition 14.2, we will need to understand these invariants.

Proposition 14.6. We have

c_{1}([d]D) = c_{2}^{2} + c_{2}^{8} + w + c_{2}^{4}w^{2}
c_{2}([d]D) = c_{2} + (1 + c_{2}^{3} + c_{2}^{9} + c_{3})w^{2}
c_{3}([d]D) = c_{3} + c_{2}w + (c_{2}^{2} + c_{2}^{8})w^{2} + (1 + c_{3} + c_{2}^{3} + c_{2}^{9})w^{3}.

Proof. First recall that u +_{F} v = u + v + u^{2}v^{2} (mod u^{4}v^{4}) and w^{4} = 0 so w +_{F} v = w + v + w^{2}v^{2} for any v.

Next note that [d]α(a) = [d]([a]+[−a]+[0]) = [d+a]+[d−a]+[d], so α^{∗}(c_{k}([d]D)) = c_{k}([d+a]+[d−a]+[d]) is the k’th elementary symmetric function of \{w +_{F} x, w +_{F} \overline{x}, w\}, for example

α^{∗}c_{1}([d]D) = w + (w + x + w^{2}x^{2}) + (w + \overline{x} + w^{2}\overline{x}^{2})
= x^{4} + x^{10} + x^{16} + x^{22} + w + x^{8}w^{2} + x^{20}w^{2}
= α^{∗}(c_{2}^{2} + c_{2}^{8}) + w + α^{∗}(c_{2}^{4})w^{2}.

By similar computations, our other two equations also become true when we apply α^{∗}.

In the same way, we have [d]β(a, b) = [d + a] + [d + b] + [d + a + b], so β^{∗}c_{k}([d]D) is the k’th elementary symmetric function of the list {w +_{F} x, w +_{F} y, w +_{F} x +_{F} y}, or equivalently the list

{w + x + w^{2}x^{2}, w + y + w^{2}y^{2}, w + x + y + w^{2}x^{2} + w^{2}y^{2} + x^{2}y^{2}}.

We thus have

β^{∗}c_{3}([d]D) = (w + x + w^{2}x^{2})(w + y + w^{2}y^{2})(w + x + y + w^{2}x^{2} + w^{2}y^{2} + x^{2}y^{2})
= (x^{2}y + xy^{2} + x^{3}y^{3}) + (x^{2} + xy + y^{2} + x^{3}y^{2} + x^{2}y^{3})w +
x^{2}y^{2}w^{2} + (1 + x^{2}y + xy^{2})w^{3}
= β^{∗}(c_{3}) + β^{∗}(c_{2})w + β^{∗}(c_{2}^{2} + c_{2}^{8})w^{2} + β^{∗}(1 + c_{3} + c_{2}^{3} + c_{2}^{9})w^{3}.

By similar computations, our other two equations also become true when we apply β^{∗}. As α^{∗} and β^{∗} are jointly injective, it follows that our equations hold in O_{𝔾_{0}(1)×Y} as claimed.

Proposition 14.7. Let J be the ideal in 𝔽_{4}[w, c_{2}, c_{3}] generated by the elements

w^{4} , c_{3}^{2} , c_{2}c_{3} ,
c_{2}^{4} + w^{2}c_{2}^{3} + wc_{2}^{2} + w^{2}c_{3},
wc_{2}^{3} + w^{2}c_{2} + wc_{3}.

Then C(K, Σ_{4}) = 𝔽_{4}[w, c_{2}, c_{3}]/J. Moreover, the following monomials form a basis for this ring over 𝔽_{4}, so it has dimension 17.

1  c_{2}  c_{2}^{2}  c_{2}^{3}  c_{3}
w  wc_{2}  wc_{2}^{2}  wc_{3}
w^{2}  w^{2}c_{2}  w^{2}c_{2}^{2}  w^{2}c_{3}
w^{3}  w^{3}c_{2}  w^{3}c_{2}^{2}  w^{3}c_{3}

Proof. Proposition 14.2 is equivalent to the statement that

X(Σ_{4}) = {(d, D) ∈ 𝔾_{0}(1) × Y | D + ψ^{2}(D) = 2[0] + [d] + [d]D}.

This means that C(K, Σ_{4}) is the largest quotient of O_{𝔾_{0}(1)×Y} over which we have g(t) = 0, where

g(t) = f_{D}(t)f_{ψ^{2}D}(t) − t^{2}(t + w)f_{[d]D}(t).

Here we write f_{D}(t) = t^{3} + c_{1}(D)t^{2} + c_{2}(D)t + c_{3}(D) and similarly for our other divisors. As usual we write c_{k} for c_{k}(D), and we recall from Proposition 14.5 that c_{1} = c_{2}^{2} + c_{2}^{8}. We also recall that c_{k}(ψ^{2}D) = c_{k}(D)^{4}, so that

f_{ψ^{2}D}(t) = t^{3} + c_{1}^{4}t^{2} + c_{2}^{4}t + c_{3}^{4} = t^{3} + c_{2}^{8}t^{2} + c_{2}^{4}t.

The polynomial f_{[d]D}(t) = ∑_{k=0}^{3} c_{k}([d]D)t^{3−k} can be read off from Proposition 14.6. Putting all this together and expanding it out, we find that g(t) = ∑_{k=1}^{4} r_{k}t^{6−k}, where

r_{1} = c_{2}^{8} + c_{2}^{4}w^{2}
r_{2} = c_{2}^{4}w^{3} + (c_{2}^{9} + c_{2}^{3} + c_{3})w^{2} + (c_{2}^{8} + c_{2}^{2})w + (c_{2}^{10} + c_{2}^{4})
r_{3} = (c_{2}^{8} + c_{2}^{2})w^{2} + (c_{2}^{12} + c_{2}^{9} + c_{2}^{6})
r_{4} = (c_{2}^{8} + c_{2}^{2})w^{3} + c_{2}w^{2} + c_{3}w + c_{2}^{5}.

We thus have

C(K, Σ_{4}) = 𝔽_{4}[w, c_{2}, c_{3}]/(w^{4}, c_{2}^{16}, c_{3}^{2}, c_{2}c_{3}, r_{1}, r_{2}, r_{3}, r_{4}).

As 1 + c_{2}^{6} + c_{2}^{12} + w^{3} is invertible, we can replace r_{2} by

r′_{2} := (1 + c_{2}^{6} + c_{2}^{12} + w^{3})r_{2} = c_{2}^{4} + w^{2}c_{2}^{3} + wc_{2}^{2} + w^{2}c_{3},

which is one of the relations in the statement of the theorem. As w^{4} = 0 we have (r′_{2})^{2} = r_{1} and (r′_{2})^{4} = c_{2}^{16}, so the relations r_{1} and c_{2}^{16} are redundant. Similarly, we can replace r_{4} by the relation

r′_{4} := r_{4} + (c_{2} + w^{2} + c2^{4}w^{3})r′_{2} = wc_{2}^{3} + w^{2}c_{2} + wc_{3},

which is another of the relations in the statement of the theorem. One can check that

r_{3} = (1 + c_{2}^{3} + c_{2}^{6})(c_{2}^{2}(1 + (1 + c_{2}^{6})(w^{3} + c_{2}^{2}w^{2}))r′_{2} + c_{2}r′_{4}),

so r_{3} is redundant. We deduce that

C(K, Σ_{4}) = 𝔽_{4}[w, c_{2}, c_{3}]/(w^{4}, c_{3}^{2}, c_{2}c_{3}, r′_{4}, r′_{2})

as claimed.

We next show that the 17 monomials listed form a basis for this quotient ring. We order the set of monomials in w, c_{2} and c_{3} by saying that c_{2}^{i}c_{3}^{j}w^{k} < c_{2}^{i′}c_{3}^{j′}w^{k′} iff i < i′ or (i = i′ and j < j′) or (i = i′ and j = j′ and k < k′). We claim that our relations form a Gröbner basis for J with respect to this ordering. We first recall briefly what this means. The list of leading terms of our relations is (w^{4}, c_{3}^{2}, c_{2}c_{3}, c_{2}^{3}w, c_{2}^{4}). A polynomial is said to be top-reducible if any of its monomials is divisible by one of these leading terms; if so, we can subtract off a multiple of the corresponding relation to cancel the monomial, a process called top-reduction. Clearly, if a polynomial can be reduced to zero by iterated top-reduction then it must lie in J, but the converse need not hold for an arbitrary list of generators of an arbitrary ideal. Let a and b be any two of our relations, let a′ and b′ be their leading terms, and let c′ be the greatest common divisor of a′ and b′. The corresponding syzygy is the element c := (a′/c′)b − (b′/c′)a ∈ J. To say that our relations form a Gröbner basis means precisely that all these syzygies can be reduced to zero by iterated top-reduction. This can be checked by direct computation. For example, the syzygy of r′_{4} and r′_{2} is the element c_{2}r′_{4} − wr′_{2} = c_{2}^{3}w^{3} + c_{2}c_{3}w + c_{3}w^{3}. The first monomial is divisible by the leading term of r′_{4}, so we can top-reduce by subtracting w^{2}r′_{4} to get c_{2}c_{3}w + c_{2}w^{4}. We can then do two more top-reductions by subtracting w times the relation c_{2}c_{3} and c_{2} times the relation w^{4} to get 0, as required. Now observe that the 17 monomials listed in the statement of the theorem are precisely those that are not top-reducible. It follows from the theory of Gröbner bases that they form a basis for C(E, Σ_{4}), as claimed.

Corollary 14.8. C(K, Σ_{4}) is a Gorenstein ring, and the element w^{3}c_{3} generates the socle.

Proof. One sees easily from the relations listed that w, c_{2} and c_{3} annihilate w^{3}c_{3}, so w^{3}c_{3} lies in the socle. Now let a be an arbitrary element of the socle. It will be convenient to put e = c_{2}^{3} + wc_{2} + c_{3} (so that we = c_{3}e = 0) and to use the basis given in the Proposition but with c_{2}^{3} replaced by e. Using the equation wa = 0 we see immediately that a lies in the span of {w^{3}, w^{3}c_{2}, w^{3}c_{2}^{2}, e, w^{3}c_{3}}. Using the equation c_{3}a = 0 and the fact that c_{3}c_{2} = c_{3}e = c_{3}^{2} = 0 we find that the coefficient of w^{3} is zero, so a = αw^{3}c_{2} +βw^{3}c_{2}^{2} +γe+δw^{3}c_{3} say. One can check that w^{3}c_{2}^{3} = w^{3}c_{3} and c_{2}e = w^{3}c_{2}, so

0 = c_{2}a = αw^{3}c_{2}^{2} + βw^{3}c_{3} + γw^{3}c_{2},

so α = β = γ = 0, so a = δw^{3}c_{3}. This shows that the socle is one-dimensional, so the ring is Gorenstein as claimed.

14.6. A transfer argument.

Proposition 14.9. The map θ: C(K, Σ_{4}) ⟶ K^{0}BΣ_{4} is injective.

Proof. Note that every nontrivial ideal in C(K, Σ_{4}) contains the socle, so it will suffice to show that the socle is not contained in ker(θ), or equivalently that w^{3}c_{3} ≠ 0 in K^{0}BΣ_{4}. Let P be the Sylow subgroup in Σ_{4}; it will be enough to show that w^{3}c_{3} has nontrivial image in K^{0}BP. Put V = P ∩ A_{4}; one can check that this consists of the identity and the three transposition pairs, so it is isomorphic to C_{2}^{2}. Recall that the series ⟨2⟩(x) is defined to be [2](x)/x, which in our case is just x^{3}. As w is the Euler class of ϵ and V = ker(ϵ: P ⟶ C_{2}), standard arguments show that tr_{V}^{P} (1) = ⟨2⟩(w) = w^{3}. This means that w^{3}c_{3} = tr_{V}^{P} (c_{3}). To see that this is nonzero, we use the canonical bilinear form on K^{0}BP defined in [10]. This satisfies Frobenius reciprocity, so (tr_{V}^{P} (c_{3}), 1)_{P} = (c_{3}, 1)_{V}. If we let x and y be the Euler classes of two of the nontrivial characters of V, then K^{0}BV = 𝔽_{4}[x, y]/(x^{4}, y^{4}) and the Euler class of the third character is x +_{F} y = x + y + x^{2}y^{2}. One checks that the restriction of ρ to V is the regular representation minus the trivial representation, which is the sum of the three nontrivial characters. This implies that the restriction of c_{3} to V is xy(x +_{F} y) = x^{2}y + xy^{2} + x^{3}y^{3}. Using [10, Corollary 9.3] we see that (x^{i}y^{j}, 1)_{V} is 1 if i = j = 3 and 0 otherwise, so (c_{3}, 1)_{V} = 1. As (w^{3}c_{3}, 1)_{P} = (c_{3}, 1)_{V} = 1 we see that w^{3}c_{3} ≠ 0, as claimed.

14.7. The proof of Theorem 14.1. This is now easy. We know from [11] that E^{0}BΣ_{4} is a free module of finite rank over E^{0}. It follows by well-known arguments that K^{0}BΣ_{4} = (E^{0}BΣ_{4})/I_{2}, which is free of the same rank over K^{0} = E^{0}/I_{2} = 𝔽_{4}. The rank is also the same as the rank of L ⊗ E^{0}BΣ_{4} over L, and generalised character theory tells us that this is equal to |Ω(Σ_{4})| = 17. Thus, the source and target of the map θ: C(K, Σ_{4}) ⟶ K^{0}BΣ_{4} both have rank 17 over 𝔽_{4} and Proposition 14.9 tells us that the map is injective, so it must be an isomorphism. Now consider the map θ: C(E, Σ_{4}) ⟶ E^{0}BΣ_{4}. This is an isomorphism modulo I_{2}, so by Nakayama’s lemma it is surjective. As E^{0}BΣ_{4} is free it is a split surjection, so C(E, Σ_{4}) = E^{0}BΣ_{4} ⊕ N say. This implies that C(K, Σ_{4}) = K^{0}BΣ_{4} ⊕ N/I_{2}N, so by counting ranks we see that N/I_{2}N = 0, so by Nakayama again we see that N = 0. Thus C(E, Σ_{4}) = E^{0}BΣ_{4} as claimed. We know from Proposition 14.7 that our list of 17 monomials is a basis for K^{0}BΣ_{4} over 𝔽_{4}, and it now follows that it is also a basis for E^{0}BΣ_{4} over E^{0}.

15. Extraspecial p-groups

In this section we define a class of “extraspecial” p-groups (where p is an odd prime), and show that for these groups the map κ: Ω(G) ⟶ Ω_{Ch}(G) is injective but not surjective. It follows using Theorem 11.4 that the map θ: C(E, G) ⟶ E^{0}BG cannot be an isomorphism. We have not investigated the situation more deeply than this.

Let V be an elementary Abelian p-group of rank 2d equipped with a nondegenerate alternating form b: V × V ⟶ 𝔽_{p}. We will say that a subspace W ≤ V is isotropic if b(u, v) = 0 for all u, v ∈ W.

Let G be the set 𝔽_{p} × V with the group operation (x, u).(y, v) = (x + y + b(u, v), u + v). This has order p^{2d+1} and exponent p, and it fits in a central extension

Z = 𝔽_{p} ⟶^{j} G ⟶^{q} V.

In fact Z is the centre of G, and the non-central conjugacy classes are the fibres of q over V \ {0}, so they all have order p. This gives p + p^{2d} − 1 conjugacy classes altogether.

We can evidently view R(V) = ℤ[V^{∗}] as a sub Λ-ring of R(G).

Definition 15.1. For any nontrivial character ζ: Z ⟶ S^{1}, let ϕ(ζ) be the class function on G defined by

ϕ(ζ)(g) = \begin{cases}p^{d}ζ(g) & if g ∈ Z \\ 0 & otherwise. \end{cases}

We also write ρ_{V} for the regular representation of V, and ρ_{G} for the regular representation of G.

The following result is standard, but we give a proof for completeness.

Proposition 15.2. For each ζ ∈ Z^{∗} \ {1}, the class function ϕ(ζ) is an irreducible character. Moreover, we have

R(G) = ℤ[V^{∗}] ⊕ ℤ{ϕ(ζ) | ζ ∈ Z^{∗} \ {1}}.

Proof. Choose a maximal isotropic subspace W ≤ V, so W ≃ 𝔽_{p}^{d}. Put H = q^{−1}W ≤ G, which is isomorphic to Z × W as a group because W is isotropic. Let r: H ⟶ Z be the projection and put σ = ind_{H}^{G} π^{∗}ζ. We claim that σ = ϕ(ζ). To see this, first note that H is normal in G, so σ(g) = 0 for g ∉ H. Next, suppose that g ∈ H \ Z, say g = (x, w) with w ∈ W \ {0}. Let U be such that V = W ⊕ U, so U ≃ 𝔽_{p}^{d} and (0, u)^{−1}(x, w)(0, u) = (x − 2b(u, w), w). From the definitions we see that σ(x, w) = ∑_{u} ζ(x − 2b(u, w), 0). The map u ↦ (x − 2b(u, w), 0) is a surjection from U to Z, each of whose fibres has the same order, and ζ: Z ⟶ S^{1} is a nontrivial homomorphism; it follows easily that σ(x, w) = 0, as required. Finally, suppose that g ∈ Z, say g = (x, 0). Then (0, u)^{−1}(x, 0)(0, u) = (x, 0) so σ(x, 0) = ∑_{u} ζ(x, 0) = p^{d}ζ(x, 0). This shows that σ = ϕ(ζ) as claimed, so ϕ(ζ) is a character. One checks easily that ⟨ϕ(ζ), ϕ(ζ)⟩ = |G|^{−1} ∑_{z∈Z} p^{2d} = 1, so ϕ(ζ) is irreducible. As ζ runs over Z^{∗} \ {1} this gives p − 1 distinct irreducibles of degree p^{d}, and V^{∗} gives a further p^{2d} distinct irreducibles of degree 1. We have seen that G has p^{2d} +p−1 conjugacy classes and thus p^{2d}+p−1 irreducible characters, so our list is complete. It follows that R(G) = ℤ[V^{∗}]⊕ℤ{ϕ(ζ) | ζ ∈ Z^{∗}\{1}} as claimed.

Lemma 15.3. Let C be cyclic of order p. Then

λ^{k}(p^{d−1}ρ_{C}) = \begin{cases}( \begin{matrix}p^{d−1} \\ k/p \end{matrix} ) + \frac{1}{p} (( \begin{matrix}p^{d} \\ k \end{matrix} ) − ( \begin{matrix}p^{d−1} \\ k/p \end{matrix} )) ρ_{C} & if p|k \\ \frac{1}{p} ( \begin{matrix}p^{d} \\ k \end{matrix} ) ρ_{C} & otherwise \end{cases}

Proof. Let χ be a generator of C^{∗}, so R(C) = ℤ[χ]/(χ^{p} − 1) and ρ_{C} = ∑_{j=0}^{p−1} χ^{j}. We have χρ_{C} = ρ_{C} and so λ^{k}(ρ_{C}) = λ^{k}(χρ_{C}) = χ^{k}λ^{k}(ρ_{C}). If 0 < k < p then χ^{k} is also a generator, and it follows that λ^{k}(ρ_{C}) is an integer multiple of ρ_{C}. On the other hand, it is easy to check that λ^{0}(ρ_{C}) = λ^{p}(ρ_{C}) = 1. If we put A = ℤ{1, ρ_{C}} then A is a subring of R(C) (with ρ_{C}^{2} = pρ_{C}) and λ_{t}(ρ_{C}) ∈ A[t] so λ_{t}(p^{d−1}ρ_{C}) = λ_{t}(ρ_{C})^{p^{d−1}} also lies in A[t], say λ^{k}(ρ_{C}) = n_{k} + m_{k}ρ_{C}. Moreover, if we work mod ρ_{C} we have λ_{t}(ρ_{C}) ≅ 1 + t^{p} so λ_{t}(p^{d−1}ρ_{C}) ≅ (1 + t^{p})^{p^{d−1}}. Thus, if p divides k then n_{k} = ( \begin{matrix}p^{d−1} \\ k/p \end{matrix} ), and if p does not divide k then n_{k} = 0.

Moreover, by counting dimensions we see that n_{k} +pm_{k} = ( \begin{matrix}p^{d} \\ k \end{matrix} ) for all k. The lemma now follows easily.

Proposition 15.4.

χϕ(ζ) = ϕ(ζ)
ϕ(ζ)ϕ(ξ) = \begin{cases}ρ_{V} & if ζξ = 1 \\ p^{d}ϕ(ζξ) & otherwise \end{cases}
ψ^{k}χ = χ^{k}
ψ^{k}ϕ(ζ) = \begin{cases}p^{d} & if p|k \\ ϕ(ζ^{k}) & otherwise \end{cases}

λ^{k}(ϕ(ζ)) = \begin{cases}( \begin{matrix}p^{d−1} \\ k/p \end{matrix} ) + \frac{1}{p^{2d}} (( \begin{matrix}p^{d} \\ k \end{matrix} ) − ( \begin{matrix}p^{d−1} \\ k/p \end{matrix} )) ρ_{V} & if p|k \\ \frac{1}{p^{d}} ( \begin{matrix}p^{d} \\ k \end{matrix} ) ϕ(ζ^{k}) & otherwise \end{cases}

Proof. Everything except for λ^{k}(ϕ(ζ)) can be done by easy manipulation of characters. For the remaining case, it suffices to check that the claimed equations hold when restricted to any cyclic subgroup C ≤ G. First consider the case C = Z, so ρ_{V} restricts on C to the trivial representation of degree p^{2d}. Then ϕ(ζ) becomes p^{d}ζ, so λ^{k}ϕ(ζ) becomes ( \begin{matrix}p^{d} \\ k \end{matrix} )ζ^{k}. Using this, it is easy to check that the equations hold when restricted to Z.

Now suppose instead that C ≤ G is a cyclic group not contained in Z (which implies that |C| = p). Then ρ_{V} |_{C} = p^{2d−1}ρ_{C} and ϕ(ξ)|_{C} = p^{d−1}ρ_{C} for all ξ ∈ Z^{∗} \ {1}. Using Lemma 15.3 we deduce that our equations for λ^{k}(ϕ(ζ)) are correct when restricted to C, as required.

Definition 15.5. For any homomorphism α: V^{∗} ⟶ Θ, put

c_{α} = ∑_{χ∈V^{∗}}[α(χ)] ∈ ℤ[Θ(1)]_{p^{2d}}^{+}

and

U_{α} = {u ∈ ℤ[Θ(1)]_{p^{d}}^{+} | uψ^{p−1}(u) = c_{α}}.

We also put U = {(α, u) | u ∈ U_{α}}.

Theorem 15.6. There is a natural bijection Ω_{Ch}(G) = U. The map κ: Ω(G) ⟶ U is injective, and the image is the set of pairs (α, u) ∈ U such that the image of the dual map α^{∗}: Θ^{∗} ⟶ V is isotropic.

The proof will follow after a lemma.

Lemma 15.7. Let α: V^{∗} ⟶ Θ be a homomorphism with image A of order p^{e}. If e > d then U_{α} = ∅. If e ≤ d then U_{α} is the set of elements of the form u = p^{d−e} ∑_{c∈C}[c], where C runs over the cosets of A in Θ(1).

Proof. Put c′_{α} = ∑_{a∈A}[a] ∈ ℤ[Θ(1)]_{p^{e}}^{+} so that c_{α} = p^{2d−e}c′_{α}. Suppose u ∈ U_{α} and that b ∈ u. Put v = [−b]u, so v ∈ U_{α} and 0 ∈ v. Thus 0 ∈ ψ^{p−1}(v) also, so v ≤ vψ^{p−1}(v) = c_{α} = p^{2d−e}c′_{α}, so we can write v = ∑_{a∈A} n_{a}[a] for suitable natural numbers n_{a}. By looking at the multiplicity of [0] in the equation vψ^{p−1}(v) = p^{2d−e}c′_{α} we see that ∑_{a} n_{a}^{2} = p^{2d−e}. On the other hand, as v ∈ ℤ[Θ(1)]_{p^{d}}^{+} we have ∑_{a} n_{a} = p^{d}. It follows that

∑_{a}(n_{a} − p^{d−e})^{2} = ∑_{a} n_{a}^{2} − 2p^{d−e} ∑_{a} n_{a} + p^{2d−2e} ∑_{a} 1 = p^{2d−e} − 2p^{2d−e} + p^{2d−e} = 0,

so n_{a} = p^{d−e} for all a. If we now let C be the coset b + A we find that u = p^{d−e} ∑_{c∈C}[c]. Conversely, it is trivial to check that any element of this form lies in U_{α}.

Proof of Theorem 15.6. Let ζ be the usual character x ↦ e^{2πix/p} of Z = ℤ/p. Given f: R(G) ⟶ ℤ[Θ] it is clear that the restriction of f to R(V) = ℤ[V^{∗}] gives a homomorphism α: V^{∗} ⟶ Θ(1) ≤ Θ, and we put u = f(ϕ(ζ)) ∈ ℤ[Θ]_{p^{d}}^{+}. As f is a Λ-ring homomorphism we have

uψ^{p−1}(u) = f(ϕ(ζ)ϕ(ζ^{p−1})) = f(ρ_{V}) = ∑_{χ}[α(χ)],

so (α, u) ∈ U.

Conversely, suppose we start with (α, u) ∈ U. Let e and C be as in Lemma 15.7. We define a homomorphism f: R(G) ⟶ ℤ[Θ] of additive groups by f(χ) = [α(χ)] for χ ∈ V^{∗} and

f(ϕ(ζ^{k})) = ψ^{k}(u) = p^{d−e} ∑_{c∈kC}[c]

for k ∈ ℤ \ pℤ. It is easy to check that this is a ring homomorphism that sends R_{k}^{+}(G) to ℤ[Θ]_{k}^{+} and commutes with the Adams operations. As ℤ[Θ] is torsion free it follows that f commutes with Λ-operations as well, so f ∈ Ω_{Ch}(G). Clearly these constructions give the required bijection Ω_{Ch}(G) = U.

Now suppose we have a homomorphism μ: Θ^{∗} ⟶ G. Then μ(u) = (ω(u), σ(u)) for some functions ω: Θ^{∗} ⟶ 𝔽_{p} and σ: Θ^{∗} ⟶ V. As μ and the projection q: G ⟶ V are homomorphisms we see tht σ is a homomorphism. Let W ≤ V be the image of σ, and put e = dim_{𝔽_{p}} W. As the image of μ must be commutative, it is not hard to see that W is isotropic, so e ≤ d. As q^{−1}W ≃ 𝔽_{p} × W as groups, we see that ω is also a homomorphism. If we conjugate (ω, σ) by (x, u) ∈ G we get the homomorphism (ω + τ, σ) where τ(t) = 2b(u, σ(t)). As b is a perfect pairing, τ can be any map Θ^{∗} ⟶ 𝔽_{p} that factors through σ, so (ω, σ) is conjugate to (ω′, σ) if and only if ω|_{ker(σ)} = ω′|_{ker(σ)}. Now let σ^{∗}: V^{∗} ⟶ Θ be the dual of σ and put A = σ^{∗}(V^{∗}), so |A| = p^{e}. We also have a map ω^{∗}: 𝔽_{p}^{∗} ⟶ Θ and thus a point t = ω^{∗}(ζ) ∈ Θ(1). In R(𝔽_{p} × W) = ℤ[𝔽_{p}^{∗}] ⊗ ℤ[W^{∗}] we have

ϕ(ζ)|_{𝔽_{p}×W} = p^{d−e}ζ ⊗ ρ_{W} = p^{d−e} ∑_{ξ∈W^{∗}} ζ ⊗ ξ,

and it follows that μ^{∗}ϕ(ζ) = p^{d−e} ∑_{a∈A}[t + a] ∈ ℤ[Θ]. Thus, if we write [μ] for the conjugacy class of μ then κ[μ] = (σ^{∗}, p^{d−e} ∑_{a∈A}[t + a]) ∈ U. It follows that κ[μ] determines σ, and it also determines t modulo A, so it determines ω modulo σ^{∗}(Hom(V, 𝔽_{p})), so it determines the conjugacy class [μ]. This proves that κ is injective as claimed. We leave it to the reader to check that the image is as described.

16. An apparently more precise approach

There are some senses in which the Λ-operations do not capture all possible information about the representation theory of G, and it is reasonable to wonder whether a more accurate approximation to X(G) could be defined by taking more information into account. In this section we show that this is not the case: we construct an approximation Y (G) using all possible operations, and show that it is the same as X_{Ch}(G).

Definition 16.1. Let G be the category of Lie groups and continuous homomorphisms, and let \overline{G} be the quotient category in which conjugate homomorphisms are identified. Let N be the set of finite sequences \underline{n} = (n_{1}, …, n_{r}) with n_{i} ∈ ℕ. For \underline{n} ∈ N we put GL(\underline{n}) = ∏_{i} GL(n_{i}, ℂ) and

R(\underline{n}, G) = ∏_{i} R_{n_{i}}^{+}(G) = \overline{G}(G, GL(\underline{n})).

We make N into a category by putting N (\underline{n}, \underline{m}) = G(GL(\underline{n}), GL(\underline{m})), and we let \overline{N} be the category with the same objects and with morphisms \overline{N} (\underline{n}, \underline{m}) = \overline{G}(GL(\underline{n}), GL(\underline{m})); clearly this gives a covariant functor \underline{n} ↦ R(\underline{n}, G) from \overline{N} to sets.

Next, let T (\underline{n}) be the evident maximal compact torus in GL(\underline{n}), so T (\underline{n}) ≃ ∏_{j=1}^{N} S^{1} where N = ∑_{i=1}^{r} n_{i}. Let W (\underline{n}) be the Weyl group of T (\underline{n}), so W (\underline{n}) ≃ ∏_{i} Σ_{n_{i}}. We can thus form the scheme

D(\underline{n}) = Hom(T (\underline{n})^{∗}, 𝔾)/W (\underline{n}) = ∏_{i} Div_{n_{i}}^{+}(𝔾).

By elementary arguments in representation theory we see that any homomorphism f: GL(\underline{n}) ⟶ GL(\underline{m}) is conjugate to one that sends T (\underline{n}) into T (\underline{m}), and that the resulting map T (\underline{n}) ⟶ T (\underline{m}) is unique up to the action of W (\underline{m}). Using this, it is easy to make the assignment \underline{n} ↦ D(\underline{n}) into a functor \overline{N} ⟶ X̂_{X}.

Finally, we define a functor Y (G) from discrete O_{X}-algebras to sets by putting

Y (G)(A) = [\overline{N} , Sets](R(−, G), D(−)(A)).

Theorem 16.2. There is a natural isomorphism Y (G) ≃ X_{Ch}(G).

Before proving this, we relate Y (G) to an auxiliary model involving unitary groups rather than general linear groups.

Definition 16.3. Let G̃ be the quotient of G in which homomorphisms u, v: U ⟶ V are identified if u|_{K} is conjugate to v|_{K} for every compact subgroup K ≤ U. (For example, the homomorphism u: GL(1) ⟶ GL(1) given by u(z) = |z| becomes trivial in G̃.) Let Ñ be the category with the same objects as N and morphisms Ñ(\underline{n}, \underline{m}) = G̃(GL(\underline{n}), GL(\underline{m})). As G and T (\underline{n}) are compact, it is clear that the functors R(−, G) and D(−) factor through Ñ, and thus that

Y (G)(A) = [Ñ, Sets](R(−, G), D(−)(A)).

Lemma 16.4. Let K be a compact Lie group, and let v, w: K ⟶ U(d) be continuous homomorphisms. If v and w are conjugate in GL(d), then they are conjugate in U(d).

Proof. The statement can easily be translated as follows: Let V and W be finite-dimensional vector spaces over ℂ equipped with actions of G and invariant Hermitian inner products. Then if there exists an equivariant isomorphism f: V ⟶ W, then f can be chosen to preserve the inner products.

To see this, we first recall some facts about invariant Hermitian products. For any complex vector space V we let \overline{V} be the same set with the conjugate action of ℂ, and let \overline{V}^{∗} be the dual of \overline{V}. The set of Hermitian products β on V bijects with the set of isomorphisms β′: V ⟶ \overline{V}^{∗} satisfying certain symmetry and positivity conditions. For any representation V one can always choose an invariant Hermitian product so V is equivariantly isomorphic to \overline{V}^{∗}. For each irreducible representation S we fix a Hermitian product β_{S} on S; Schur’s lemma implies that Hom_{K}(S, \overline{S}^{∗}) = ℂβ′_{S} and that any other invariant Hermitian product is a positive scalar multiple of β_{S}.

Now let β be a Hermitian product on V and suppose that V = V_{0} ⊕ V_{1} and Hom_{K}(V_{1}, V_{0}) = 0. Then Hom_{K}(V_{1}, \overline{V} _{0}^{∗}) = 0 and

Hom_{K}(V_{0}, \overline{V} _{1}^{∗}) = Hom_{K}(\overline{V} _{1}^{∗}, V_{0})^{∗} = Hom_{K}(V_{1}, V_{0})^{∗} = 0

so the equivariant isomorphism β′: V ⟶ \overline{V}^{∗} must have the form β′_{0} ⊕ β′_{1} for some β′_{i}: V_{i} ⟶ \overline{V} _{i}^{∗}. This implies that V_{0} and V_{1} are orthogonal with respect to β.

Now let S_{1}, …, S_{t} be the distinct irreducible representations that occur in V. Then there is a unique decomposition V = V_{1} ⊕…⊕V_{t}, where V_{i} ≃ ℂ^{d_{i}} ⊗S_{i} for some d_{i} and thus Hom_{K}(V_{i}, V_{j}) = 0 when i ≠ j. By the previous paragraph, the subspaces V_{i} are orthogonal to each other. As Hom_{K}(S_{i}, \overline{S}_{i}^{∗}) = ℂβ_{S_{i}}, we find that the restriction of β to V_{i} has the form γ_{i} ⊗β_{S_{i}} for some Hermitian product γ_{i} on ℂ^{d_{i}}. By Gram-Schmidt, the space (ℂ^{d_{i}}, γ_{i}) is isomorphic to ℂ^{d_{i}} with its usual Hermitian product, so (V_{i}, β|_{V_{i}}) is equivariantly and isometrically isomorphic to the orthogonal direct sum of d_{i} copies of (S_{i}, β_{S_{i}}). This means that the numbers d_{i} determine the isometric isomorphism type of V, and the lemma follows immediately.

Lemma 16.5. There are natural bijections

Ñ(\underline{n}, \underline{m}) = \overline{G}(U(\underline{n}), GL(\underline{m})) = \overline{G}(U(\underline{n}), U(\underline{m})),

where U(\underline{n}) = ∏_{i} U(n_{i}) ≤ GL(\underline{n}).

Proof. It is easy to reduce to the case where the list \underline{m} has length 1, say \underline{m} = (d). As any representation of U(\underline{n}) admits a Hermitian inner product, we see that the map \overline{G}(U(\underline{n}), U(d)) ⟶ \overline{G}(U(\underline{n}), GL(d)) is surjective. It is also injective by Lemma 16.4. Similarly, by considering invariant Hermitian products we see that if K is compact and u: K ⟶ GL(\underline{n}) then u is conjugate to a homomorphism K ⟶ U(\underline{n}). By applying this to the inclusion map, we see that any compact subgroup of GL(\underline{n}) is conjugate to a subgroup of U(\underline{n}). It follows that any two homomorphisms v, w: GL(\underline{n}) ⟶ GL(d) are identified in G̃(GL(\underline{n}), GL(d)) iff their restrictions to U(\underline{n}) are conjugate, so we have a well-defined and injective restriction map Ñ(\underline{n}, d) ⟶ \overline{G}(U(\underline{n}), GL(d)). It is an easy consequence of the theory of roots and so on that any representation of U(\underline{n}) extends uniquely to a complex-analytic representation of GL(\underline{n}), so our restriction map is also surjective.

Proof of Theorem 16.2. Consider a point g ∈ Y (G)(A), in other words a natural transformation g\underline{n}: R(\underline{n}, G) ⟶ D(\underline{n})(A) for \underline{n} ∈ N. By putting together the maps

g_{d}: R_{d}^{+}(G) = R(d, G) ⟶ D(d)(A) = Div_{d}^{+}(𝔾)(A),

we get a function f: R^{+}(G) ⟶ Div^{+}(𝔾)(A). Next, for any d, e ≥ 0 we have projections GL(d) ⟵ GL(d, e) ⟶ GL(e) and we can use the resulting maps to identify R((d, e), G) with R_{d}^{+}(G) × R_{e}^{+}(G) and D(d, e)(A) with Div_{d}^{+}(𝔾)(A) × Div_{e}^{+}(𝔾)(A) and g_{(d,e)} with g_{d} × g_{e}. There are evident maps ⊕: GL(d, e) ⟶ GL(d + e) and ⊗: GL(d, e) ⟶ GL(de), and using the naturality of g with respect to these maps we find that f is a semiring homomorphism. Similarly, we have maps λ^{k}: GL(d) ⟶ GL(( \begin{matrix}d \\ k \end{matrix} )) in \overline{G} and the naturality of g with respect to these maps implies that f commutes with Λ-operations. It is clear that f(R_{d}^{+}(G)) ⊆ Div_{d}^{+}(𝔾)(A), so f ∈ X_{Ch}(G)(A). We define a map ρ: Y (G) ⟶ X_{Ch}(G) by ρ(g) = f. Because g\underline{n} = g_{n_{1}} × … × g_{n_{r}} we see that ρ is injective.

Now suppose we start with a point f ∈ X_{Ch}(G)(A). Let g_{d}: R(d, G) ⟶ D(d)(A) be the restriction of f: R(G) ⟶ Div(𝔾)(A), and put

g\underline{n} = g_{n_{1}} × … g_{n_{r}}: R(\underline{n}, G) ⟶ D(\underline{n})(A).

We need to check that this gives a natural transformation. As R(\underline{m}, G) = ∏_{i} R(m_{i}, G) and D(\underline{m})(A) = ∏_{i} D(m_{i})(A), it suffices to check naturality for maps u: \underline{n} ⟶ d in Ñ, or equivalently (by Lemma 16.5) for homomorphisms u: U(\underline{n}) ⟶ GL(d). We need to show that the left hand square in the following diagram commutes:

R(\underline{n}, G)  R_{d}^{+}(G)  R(G) D(\underline{n})(A)  Div_{d}^{+}(𝔾)  Div(𝔾)(A).^{u_{∗}} w_{_{g\underline{n}} u}  _{_{g_{d}} u} v  w _{u_{f}} w_{u_{∗}}  v  w

The right hand square commutes and the two right hand horizontal maps are injective so it suffices to show that the two composite maps R(\underline{n}, G) ⟶ Div(𝔾)(A) are the same. We call these two maps α(u) and β(u). Let F be the set of all functions from R(\underline{n}, G) to Div(𝔾)(A), thought of as a ring with pointwise operations. It is formal to check that α(u + v) = α(u) + α(v) and α(uv) = α(u)α(v), so α is a homomorphism of semirings from R^{+}(U(\underline{n})) to Div(𝔾)(A). It can thus be extended to a ring map R(U(\underline{n})) ⟶ Div(𝔾)(A), and the same applies to β. It is well-known that R(U(\underline{n})) = ⨂_{i} R(U(n_{i})) so it suffices to check that α = β on R(U(n_{i})) for all i. This reduces us to the case where \underline{n} = (e) say. It is also well-known that R(U(e)) = ℤ[λ^{1}, …, λ^{e}][(λ^{e})^{−1}], so it suffices to check that α(λ^{j}) = β(λ^{j}), which is true because f is a homomorphism of Λ-rings.

This shows that g ∈ Y (G)(A), and clearly ρ(g) = f. Thus ρ is surjective and hence an isomorphism.

17. A result on restrictions of characters

Theorem 17.1. Let G be a finite group with a normal subgroup N such that |N| is coprime to |G/N|. Then the restriction map R^{+}(G) ⟶ R^{+}(N)^{G} is surjective.

The proof will follow after some preliminary results.

Lemma 17.2. Let H be a group, and let W, X, Y be H-sets, with equivariant maps W ⟶^{f} X ⟵^{q} Y. Then there is an equivariant map f̃: W ⟶ Y with qf̃ = f iff for each w ∈ W there exists y ∈ Y with q(y) = f(w) and stab_{H}(y) ≥ stab_{H}(w).

Proof. Write W as a disjoint union of orbits.

Lemma 17.3. Let G and N be as above, and let ρ: N ⟶ GL(V) be an irreducible representation of N whose character is stable under G. Then there is a homomorphism σ: G ⟶ GL(V) extending ρ.

Proof. Suppose g ∈ G, and define ρ^{g}: N ⟶ GL(V) by ρ^{g}(x) = ρ(gxg^{−1}). By hypothesis, this has the same character as ρ, so there exists an intertwining operator θ: V ⟶ V such that ρ^{g}(x) = θ^{−1}ρ(x)θ for all x ∈ N. As V is an irreducible representation of N we see that Aut_{N}(V) = ℂ and thus θ is unique up to multiplication by a scalar matrix. We can thus define a map ϕ: G ⟶ P GL(V) by ϕ(g) = [θ]; this is a homomorphism making the following diagram commute.

N  G GL(V)  P GL(V)._{_{ρ} u} v  w_{u_{ϕ}} ww_{π}

Put n = |N| and d = dim_{ℂ}(V). As V is irreducible we know that d divides n. Put Y = {α ∈ GL(V) | det(α)^{n} = 1}, and note that π: Y ⟶ P GL(V) is surjective and ρ(N) ≤ Y. Let N^{2} act on G by (x, y).g = xgy^{−1} and on GL(V) by (x, y).α = ρ(x)αρ(y)^{−1}.

We claim that there is an N^{2}-equivariant map ζ: G ⟶ Y such that πζ = ϕ and ζ = ρ on N. Clearly G = N ⨿ (G \ N) as N^{2}-sets and ρ: N ⟶ Y is N^{2}-equivariant, so it suffices to define ζ on G \ N. Fix g ∈ G \ N, and choose θ as before. After multiplying by a suitable scalar, we may assume that det(θ) = 1 so θ ∈ Y. By Lemma 17.2, it will suffice to show that stab_{N^{2}}(g) ≤ stab_{N^{2}}(θ). Suppose that (x, y) stabilises g, so xgy^{−1} = g, so y = g^{−1}xg. By the definition of θ we have ρ(y) = θ^{−1}ρ(x)θ, or in other words ρ(x)θρ(y)^{−1} = θ, so (x, y) stabilises θ, as required.

Now define ξ: G^{2} ⟶ Y by ξ(g, h) = ζ(h)ζ(gh)^{−1}ζ(g). Clearly πξ(g, h) = 1, and the kernel of π: Y ⟶ GL(V) is the group C_{nd} of nd’th roots of unity, so we can regard ξ as a map G^{2} ⟶ C_{nd}. As ζ is equivariant, it is easy to check that ξ(xg, hy) = ξ(g, h) for x, y ∈ N, so we get an induced map \overline{ξ}: (G/N)^{2} ⟶ C_{nd}. One also sees directly that for g, h, k ∈ G/N we have

\overline{ξ}(h, k)\overline{ξ}(gh, k)^{−1}\overline{ξ}(g, hk)\overline{ξ}(g, h)^{−1} = 1,

so \overline{ξ} is a 2-cocycle. On the other hand nd divides n^{2} and thus is coprime to |G/N|, so we have H^{2}(G/N; C_{nd}) = 0. We can thus choose a function ω: G/N ⟶ C_{nd} such that ξ(g, h) = ω(h)ω(gh)^{−1}ω(g) for all g, h ∈ G. By putting g = h = 1 we see that ω(1) = 1 and thus ω(x) = 1 for x ∈ N. We define σ(g) = ω(g)^{−1}ζ(g); this clearly gives a homomorphism G ⟶ GL(V) with σ|_{N} = ρ, as required.

Proof of Theorem 17.1. For each irreducible representation ρ of N, let ρ′ denote the sum of the inequivalent G-conjugates of ρ. Any G-invariant character is a direct sum of copies of the characters of the representations ρ′, so it suffices to show that ρ′ extends to a representation of G. Let H be the stabiliser of χ_{ρ}, so N ≤ H ≤ G. Lemma 17.3 implies that ρ can be extended to a representation σ of H, and one sees from the Mackey formula that res_{N}^{G} ind_{H}^{G}(σ) ≃ ρ′, so ind_{H}^{G}(σ) is the required extension of ρ′.

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