Lubin-Tate Deformations in Algebraic Topology

M. J. Hopkins and H. R. Miller

Introduction

Part 1. Lubin-Tate spectra

Introduction to Part I

1. Formal groups, deformations, and the associated ring spectra

In this section we review the deformation theory for formal groups, due to J. Lubin and J. Tate [22], and explain how this gives rise, by means of the Landweber Exact Functor Theorem [20], to a diagram in the homotopy category of periodic ring spectra indexed by the category of formal groups of finite height over perfect fields. A more complete account of our approach to this fairly standard material can be found in the forthcoming book [?].

1.1. Formal groups. Our results find their most natural expression in terms of a coordinate-free theory of (one-dimensional smooth commutative) formal groups. In this section we will review one of the coordinate-free approaches to this subject.

Definition 1.1.1. Let A be a commutative ring. A nilpotent Aalgebra is an A-algebra-without-unit I such that I^{n} = 0 for some n.

Nilpotent A-algebras form a category Nil_{A}. It is equivalent to the full subcategory of the category Alg_{A} of augmented A-algebras generated by the objects with nilpotent augmentation ideal. We will consider functors from Nil_{A} to sets. One example is the formal affine line D, which sends I to the underlying set of I.

Definition 1.1.2. A formal group over A is a functor G : Nil_{A} → Ab which is isomorphic to D as a functor to sets.

The first two examples are the additive formal group G_{a} given by D with its addition and the multiplicative group G_{m} sending I to the multiplicative subset of A ⊕ I of elements of the form 1 − a for a ∈ I.

An isomorphism of set-valued functors x : G → D is a coordinate for G. The map x : a ↦ 1 − a is a coordinate for G_{m}, for example.

D is pro-represented by the augmented A-algebra A[[t]] (with augmentation defined by ϵt = 0), letting an element a ∈ I correspond to the unique map of augmented A-algebras A[[t]] → A ⊕ I sending t to a. If G comes to us prorepresented by some augmented A-algebra R, a coordinate x specifies an element of the augmentation ideal of R, also written x, making R ≅ A[[x]], and conversely such an element of R specifies a coordinate.

Formal groups over a given ring A form the objects of a category FG_{A} in which morphisms are natural transformations of groupvalued functors. Two homomorphisms φ, ψ : G → H can be added “pointwise” to obtain the sum φ + ψ, and the monoid of endomorphisms of G thereby becomes a ring. In particular there is a unit map [−]_{G} : ℤ → End_{A}(G).

A ring homomorphism f : A → B induces a functor f_{#} : Nil_{B} → Nil_{A} by regarding I ∈ Nil_{B} as a nilpotent A-algebra via f. Since the underlying set is unchanged by f_{#}, composition with it sends a formal group G/A to a formal group over B which we will write f^{∗}G. If g : SpecB → SpecA corresponds to f, we may think of f^{∗}G as the pullback of G along g. This defines a functor f^{∗} : FG_{A} → FG_{B}.

Remark 1.1.3. This definition is unsatisfactory from the perspective of algebraic geometry, since it is not local in the flat topology. From this point of view the correct definition of a formal group (over the affine scheme SpecA) is a functor G : Nil_{A} → Ab such that iG is isomorphic to D for some faithfully flat ring homomorphism i : A → S. However, there is no loss in generality for our purposes, since we consider only formal groups over fields and local rings.

The collection of all formal groups over all rings forms a category FG in which a morphism from G/A to H/B is a ring homomorphism f : A → B together with an isomorphism θ : f^{∗}G → H of formal groups over B. (If we adopt the extended definition proposed in Remark 1.1.3, we have to add the observation that if A → S is faithfully flat then so is B → B ⊗_{A} S.)

A formal group G/A determines a free (or, under the extended definition, projective) A-module of rank one, the module ω_{G} of invariant differentials. It may be defined as follows: The set of natural transformations of functors Nil_{A} → Set from G to the functor sending I to the set A ⊕ I has a natural structure of augmented A-algebra, since A ⊕ I has such a structure. Write O_{G} for this; it is the ring of functions on G. The module of invariant differentials is then the module of indecomposables of this augmented A-algebra:

ω_{G} = I_{G}/I_{G}^{2}, I_{G} = ker (ϵ : O_{G} → A).

Since ω_{G} is free of rank one, End_{A}(ω_{G}) is canonically isomorphic to A, and by naturality we obtain a natural ring homomorphism

d : End_{A}(G) → A.

An endomorphism α is invertible exactly when dα is a unit, and it is strict if dα = 1.

As usual, in characteristic p the Frobenius plays an important role. If pK = 0, let ϕ : K → K be the ring homomorphism x ↦ x^{p}. This induces a map ϕ_{#} : Nil_{K} → Nil_{K}. There is a natural transformation f : id → ϕ^{∗} given on a ∈ I ∈ Nil_{K} by f(a) = a^{p}, and it induces a homomorphism

f_{G} : G → ϕ^{∗}G

natural in the formal group G/K with pK = 0. Write f_{G}^{(n)} : G → (ϕ^{n})^{∗}G for the composite f_{(ϕ^{n−1})^{∗}G} ∘ ⋯ ∘ f_{G}.

Lemma 1.1.4. Let θ : G → H be a nonzero homomorphism of formal groups over a ring K of characteristic p ≠ 0. For some n ∈ ℕ there is a factorization

[figure: G; H; (ϕ^{n})^{∗}G; θ; f_{G}^{(n)}; θ_{(n)}]

in which dθ_{(n)} ≠ 0, and the pair (n, θ_{(n)}) is unique.

Proof. Choosing coordinates, this becomes a well-known calculation ([10]). Assume θ′(0) = 0 and differentiate the expression

θ(G(s, t)) = H(θ(s), θ(t))

with respect to t and set t = 0 to get

θ′(s)G_{2}(s, 0) = H_{2}(θ(s), 0)θ′(0) = 0,

using G(s, 0) = s = H(s, 0). Since G_{2}(s, 0) ≡ 1 mod (s), we can cancel it to find θ′(s) = 0, which implies that

θ(s) = θ_{(1)}(s^{p})

for a unique power series θ_{(1)}. Now proceed by induction.

Definition 1.1.5. The integer n is the height of the homomorphism θ. The height of the zero homomorphism is declared to be ∞.

Of particular importance is the case in which θ is the endomorphism [p] : G → G, the p-fold sum of the identity map.

Definition 1.1.6. Let G be a formal group over a ring K with pK = 0. The height of G/K is the height of the endomorphism [p].

Thus G has infinite height if and only if [p] = 0, and ([10]) is then isomorphic to the additive formal group. We restate Lemma 1.1.4 in this case:

Corollary 1.1.7. Let K be a ring of characteristic p ≠ 0 and G a formal group over K of finite height n. Then there is a unique factorization

[p] = v_{(n)} ∘ f_{G}^{(n)}, v_{(n)} : (ϕ^{n})^{∗}G → G,

and v_{(n)} is natural on the category of formal groups of height n over rings of characteristic p.

If K is a field, the fact that dv_{(n)} ≠ 0 implies that v_{(n)} is an isomorphism. It is the nth Verschiebung.

1.2. Formal group laws.

Definition 1.2.1. A formal group law is a formal group whose underlying set-valued functor \underline{is} D.

A coordinate x for a formal group G thus specifies a formal group law, written G_{x}, by transfer of structure.

Given a formal group law G/R, there is a unique power series in two variables, which will also be denoted by the symbol G, such that for any I ∈ Nil_{R} the group structure in G(I) is given by

a +_{G} b = G(a, b), a, b ∈ I.

The power series G(u, v) satifies G(s, 0) = 0 = G(0, t) and the identites

G(G(s, t), u) = G(s, G(t, u)), G(s, t) = G(t, s),

and a formal power series satisfying these conditions is equivalent to giving a formal group law.

For example, the coordinate for G_{m} given above leads to the formal group law u+v −uv. The additive formal group G_{a} is already a formal group law, with power series u + v.

A homomorphism between formal group laws, φ : G → H, is given by a formal power series φ(t) such that φ(0) = 0 and φ(s +_{G} t) = φ(s) +_{H} φ(t).

The module of invariant differentials of a formal group law is canonically trivialized by the class of x : G → D, which we write u ∈ ω_{G}. An endomorphism φ(x) of G induces an endomorphism dφ : ω_{G} → ω_{G}, given by multiplication by dφ = φ′(0).

Let G/R be a formal group and pick a coordinate x and a prime p. The p-series [p]_{G_{x}}(t) for the formal group law G_{x} depends up to conjugation only on G and not on the coordinate, so for each n ≥ 0 the ideal generated by the coefficients of [p]_{G_{x}}(t) through the coefficient of t^{p^{n}−1} depends only on G, and will be written I_{p,n}(G/R). If G and H are isomorphic formal groups then I_{p,n}(G) = I_{p,n}(H), since by appropriate choice of coordinate G and H can be represented by the same formal group law. It is well-known [1, 24] that for each n ≥ 0 there is an element u_{n} ∈ R such that I_{p,n+1}(G/R) = I_{p,n}(G/R) + Ru_{n} (where we define I_{p,0} = 0). Clearly the element u_{n} is well-defined modulo I_{p,n} up to multiplication by a unit in R/I_{p,n}. (For example, I_{p,1}(G) = pR and u_{0} = pu for some unit u ∈ R.) The following definition is thus independent of choices.

Definition 1.2.2. A formal group G/R is regular at p if the sequence p, u_{1}, u_{2}, … acts regularly on R. G/R is regular if it is regular at each prime.

The height of a formal group G/K over a ring K of characteristic p can also be expressed in terms of the sequence of ideals I_{p,n}(G): G has height n if and only if I_{p,n}(G) = 0 and I_{p,n+1}(G) ≠ 0.

We end this section with a brief recollection about representing the functor assigning to a ring the groupoid of formal group laws over that ring. There is clearly a universal formal group law,

G_{u}(s, t) = ∑_{i,j} a_{i,j}s^{i}t^{j}

over the ring L = ℤ[a_{i,j}]/I where I is the ideal generated by the relations among the a_{i,j}’s imposed by the identities making G_{u} a formal group law. Lazard proved that L is a polynomial algebra on countably many generators. Given a formal group law H/R and an arbitrary power series φ(t) ∈ R[[t]] with φ(0) = 0, there is a unique formal group law G/R such that φ : G → H—namely,

G(s, t) = H^{φ}(s, t) = φ^{−1}H(φ(s), φ(t)).

Thus

W = L[t_{0}^{±1}, t_{1}, …]

carries the universal isomorphism of formal group laws,

φ(s) = ∑_{i=0}^{∞} t_{i}s^{i+1} : G_{u}^{φ} → G_{u}.

There are structural maps η_{L}, η_{R} : L → W representing respectively target and source, ϵ : W → L representing the identity isomorphism, c : W → W representing the inverse, and Δ : W → W ⊗_{L} W representing composition. With this structure the pair (L, W) is a “Hopf algebroid,” and we may consider the category Comod of right comodules over it. In particular, for any L-module M, M ⊗_{L} W ∈ Comod.

1.3. Deformation theory. Here we give a formulation of the deformation theory of J. Lubin and J. Tate [22].

Definition 1.3.1. Let G_{0} be a formal group over a field K. A deformation of G_{0}/K consists of the following data:

(1) a Noetherian complete local ring R with maximal ideal 𝔪 and residue map ϵ : R → R/𝔪;

(2) a field embedding η : K → R/𝔪;

(3) a formal group G over R; and

(4) an isomorphism t : ηG_{0} → ϵG of formal groups over R/𝔪.

Deformations of G_{0}/K form the objects of a category L(G_{0}/K). A morphism from (G/R, η, t) to (G′/R′, η′, t′) consists of

(1) a local homomorphism f : R → R′ with reduction f̄: R/𝔪 → R′/𝔪 such that f̄η = η′, and

(2) an isomorphism θ : f^{∗}G → G′ of formal groups over R′ such that

[figure: f̄ηG_{0}; η′G_{0}; f̄ϵG; ϵ′f^{∗}G; ϵ′G′; f̄t; =; t′; =; ϵ′θ]

commutes. Composition is evident.

The fundamental theorem of Lubin and Tate is:

Theorem 1.3.2 (J. Lubin and J. Tate [22]). Let K be a perfect field of nonzero characteristic and let G_{0} be a formal group of finite height over K. Then the category L(G_{0}/K) of deformations of G_{0}/K contains an initial object (Ĝ_{0}/Λ, η, t).

They also establish the structure of the local ring supporting the universal deformation. An intrinsic characterization of the initial object (Ĝ_{0}/Λ, η, t), along with a coordinate free version of the proof of Theorem 1.3.2, can be found in [24]. The following characterization follows easily from the original work of Lubin and Tate. We caution the reader that this notion is not related to the specific formal group law constructed in [21].

Definition 1.3.3. A formal group G/R is a Lubin-Tate formal group if

(1) R is a complete local ring with maximal ideal 𝔪 and perfect residue field K of nonzero characteristic p,

(2) G reduces to a formal group of finite height n over K, and

(3) G/R is regular at p and I_{p,n}(G) = 𝔪.

In particular, any choice of elements u_{0}, u_{1}, … u_{n−1} as above forms a regular sequence generating the maximal ideal in R, which is thus a regular local ring, and since u_{0} and p are associates R is of characteristic zero. The ring of Witt vectors 𝕎(K) is universal among complete local rings with residue field K, and so maps to R uniquely over the identity on residue fields. The hypotheses imply that

𝕎(K)[[u_{1}, …, u_{n−1}]] ⟶^{≅} R.

Since any prime other than p is a unit in R, G/R is regular.

Lubin-Tate formal groups form a subcategory of the category FG of all formal groups, in which the ring homomorphisms are required to be local.

Lubin and Tate show that in the universal deformation (Ĝ_{0}/Λ, η, t) the map η is an isomorphism and Ĝ_{0}/Λ is a Lubin-Tate formal group. It is not hard to show that conversely if G/R is a Lubin-Tate formal group then (G/R, 1, 1) is a universal deformation of the reduction of G to R/𝔪.

Corollary 1.3.4. Reduction modulo the maximal ideal gives an equivalence of categories

\{ \begin{matrix}Lubin-Tate \\ formal groups \end{matrix} \} → \{ \begin{matrix}Formal groups of finite height \\ over perfect fields of nonzero characteristic \end{matrix} \} .

Proof. The theorem shows that reduction is representative. To see that it is fully faithful, let (g, φ) : G_{0}/K_{0} → G_{1}/K_{1} be a map of formal groups of finite height over perfect fields. Let Ĝ_{i}/Λ_{i} be the corresponding initial objects. Identify Λ_{i}/𝔪 with K_{i} and η_{i}G_{i} with ϵ_{i}Ĝ_{i}.

Then we may regard (Ĝ_{1}/Λ_{1}, g, φ) as an element of L(G_{0}/K_{0}). The universality of Ĝ_{0}/Λ_{0} thus gives a unique pair (f, θ) where f : Λ_{0} → Λ_{1} is a map of local rings covering g and θ : fĜ_{0} → Ĝ_{1} is an isomorphism of formal groups covering φ.

1.4. Periodic ring spectra. The work of this section occurs entirely in the stable homotopy category. This category is symmetric monoidal, so the notion of a commutative monoid is defined, and is here called a ring spectrum.

Definition 1.4.1. A periodic ring spectrum is a ring spectrum E such that

(1) π_{i}(E) = 0 for all odd i, and

(2) π_{2}(E) contains a unit.

Periodic ring spectra generate a full subcategory of the category of ring spectra and maps between them compatible with the unit and the multiplication.

Condition (1) alone implies that the Atiyah-Hirzebruch spectral sequence for E^{∗}(ℂP^{n}) collapses at its E_{2} term, and so E^{∗}(ℂP^{∞}) is a graded power series ring over E^{∗} on one generator in dimension 2. Condition (2) then implies that E^{0}(ℂP^{∞}) is a power series ring on one generator over the commutative ring E^{0} (but doesn’t determine a choice of generator). Similarly, E^{0}(ℂP^{∞} × ℂP^{∞}) is a power series ring on two generators over E^{0}. The multiplication in ℂP^{∞} thus induces an augmentation preserving map

E^{0}(ℂP^{∞}) → E^{0}(ℂP^{∞})ˆ⊗_{E^{0}}E^{0}(ℂP^{∞}).

The functor sending I ∈ Nil_{E^{0}} to

Hom_{Alg_{E^{0}}}(E^{0}(ℂP^{∞}), E^{0} ⊕ I)

is thus endowed with the structure of a formal group which we denote by G_{E}. It is natural in the periodic ring spectrum E. The augmented E^{0}-algebra E^{0}(ℂP^{∞}) identifies naturally with the ring of functions O_{G_{E}} on the formal group G_{E}. The filtration associated with the spectral sequence coincides under this identification with the I_{G_{E}}-adic filtration, and we have:

Lemma 1.4.2. There are natural isomorphisms of free E^{0}-modules of rank one

ω_{G_{E}} ≅ QE^{0}(ℂP^{∞}) ≅ E^{0}(S^{2}).

It follows that there are as well natural isomophisms of E^{0}-modules

E^{0}(S^{2m}) ≅ E^{0}(S^{2})^{⊗m} ≅ ω_{G_{E}}^{⊗m},

compatible (as m varies) with the ring structures, so the entire graded ring E^{∗} depends functorially on G_{E}/E^{0}.

These last modules have other aliases:

E^{0}(S^{2m}) = π_{2m}(E) = E_{2m} = E^{−2m}.

Complex bordism provides a canonical example of a periodic ring spectrum. The representing spectrum MU is not itself periodic, but it determines a periodic ring spectrum MP with

MP = ⋁_{k=−∞}^{∞} Σ^{2k}MU

and ring structure given on factors by the composite

Σ^{2i}MU ∧ Σ^{2j}MU ⟶^{T} Σ^{2(i+j)}MU ∧ MU ⟶^{μ} Σ^{2(i+j)}MU.

The fundamental theorem of D. G. Quillen (see [1]) amounts to the following theorem.

Theorem 1.4.3. G_{MP} /MP^{0} is universal in the sense of Lazard.

We are thus entitled to write

MP^{0} = L = MP_{0}.

J. F. Adams’s formalism of co-operations [1] translates in this setting to a natural structure on MP_{0}(X) of a comodule over the Hopf algebroid (L, W).

A choice of coordinate x for a formal group G/R determines a ring homomorphism L → R, and hence for each k a functor of spectra

X ↦ R ⊗_{L} MP_{k}(X).

These functors are independent, up to a canonical isomorphism, of choice of cooridnate, and so the coordinate can be safely ignored.

The Landweber exact functor theorem [20, 24] asserts that if G/R is regular then these functors together determine a homology theory. A more recent result of M. Hovey and N. Strickland ([19], especially Theorem 2.8 and Corollary 2.17), following J. Franke [9], finishes the proof of

Theorem 1.4.4. There is a fully faithful functor

E : \{ \begin{matrix}Regular \\ formal groups \end{matrix} \} → \{ \begin{matrix}Periodic \\ ring spectra \end{matrix} \}

together with natural isomorphisms

G/R ⟶^{≅} G_{E_{G/R}}/E_{G/R}^{0}.

Since any Lubin-Tate formal group is regular, we get, by Corollary 1.3.4, a faithful functor

EΛ : \{ \begin{matrix}Formal groups of finite height \\ over perfect fields of nonzero characteristic \end{matrix} \} → \{ \begin{matrix}Periodic \\ ring spectra \end{matrix} \}

together with natural isomorphisms

Ĝ_{0}/Λ(G_{0}/K) ⟶^{≅} G_{EΛ_{G_{0}/K}}/EΛ_{G_{0}/K}^{0}.

A ring spectrum map EΛ_{G_{0}/K} → EΛ_{H_{0}/L} is in the image of the functor EΛ if and only if it induces a local ring homomorphism on π_{0}. We remark that a proof of naturality in our setting is simpler than in the general case, because all our spectra are p-adically complete and hence phantom maps cannot occur.

The objective of the remainder of the paper is to show that this functor lifts to a functor into a suitably rigid topological category of ring spectra, by a functor which is a weak equivalence on mapping spaces, and to draw some consequences of this fact. The next section outlines the requisite theory of associative ring spectra.

2. Associative ring spectra

3. The Quillen spectral sequence

In this section we will review a few aspects of the homology and cohomology theories for universal algebras constructed by Quillen in [29]. These results are sketched in Quillen’s article, but we think the reader may appreciate a more detailed treatment.

The approach Quillen takes is based on his theory of Model Categories [28, 8, 18]. We will be interested in structures defined over a commutative ring A. Recall that in [28] Quillen defines a model category structure on the categories of simplicial A-algebras and of simplicial commutative A-algebras in which the acyclic fibrations are the surjective weak homotopy equivalences of the underlying simplicial sets, and cofibrations coincide with retracts of “almost free maps” [23]. A map f : B → C is almost free if there exists subsets S_{n} ⊆ C_{n}, n ≥ 0, such that (i) the given maps define an isomorphism to C_{n} from the coproduct (in the category of A-algebras or of commutative A-algebras, depending upon the case) of B_{n} with the free A-algebra (or commutative A-algebra) generated by S_{n}, and (ii) the sets S_{n} are stable under all the face and degeneracy maps except for d_{0}.

We will write sC for the category of simplicial objects over a category C.

3.1. Preliminaries on associative algebra homology. Let A be a commutative ring. By an A-algebra we mean a ring homomorphism A → B with central image, or, equivalently, a monoid in the symmetric monoidal category of A-modules. In the category A-Alg/B of A-algebras over B, the abelian objects are precisely those isomorphic to B ⊕ M, where M is a B-bimodule over A (i.e. for which the left and right action of A coincide), the product is given by (b, m)(b′, m′) = (bb′, bm′ + mb′), and the augmentation is the identity on B and kills M. Given an object (π : C → B) ∈ A-Alg/B,

Hom_{A-Alg/B}(C, B ⊕ M) ≅ Der_{A}(C, M)

via f ↦ pr_{2} ∘ f. The cohomology of B with coefficients in (the abelian object of A-Alg/B determined by) the B-bimodule M is thus [29]

D_{A}^{∗}(B; M) = π^{∗}(Der_{A}(P, M))

where p : P → B is an acyclic fibration from a cofibrant object in sA-Alg and π^{∗} denotes cohomotopy, i.e. the homology groups of the normalized cochain complex associated to the cosimplicial A-module Der_{A}(P, M). The map p is also an acyclic fibration from a cofibrant object in sA-Alg/B, so this can be regarded as the derived functors of Der_{A}(−, M) on this category, evaluated at the initial object B.

For any A-algebra C we may form the C-bimodule of noncommutative differentials

D_{C/A} = ker (μ : C^{e} → C),  μ : x ⊗ y ↦ xy.  (1)

where C^{e} = C ⊗_{A} C^{o}, and C^{o} is C with the product reversed. If we regard C as a C-bimodule by means of left and right multiplication then the map μ is the C-bimodule map sending 1 ⊗ 1 to 1, but it is not generally an algebra map; so D_{C/A} is a sub C-bimodule of C^{e} but not generally an ideal unless C is commutative. The map x ↦ 1⊗x−x⊗1 from C to C^{e} lands D_{C}/A. The resulting map

d : C → D_{C/A}

is a A-derivation, and is easily checked to be the universal A-derivation out of C:

Der_{A}(C, M) = Hom_{C^{e}}(D_{C/A}, M) = Hom_{B^{e}}(B^{e} ⊗_{C^{e}} D_{C/A}, M),

naturally in the C-bimodule M over A. This implies that if we identify the category Ab(A-Alg/B) of abelian objects in A-Alg/B with the category of B-bimodules then the abelianization functor is given by

Ab(C) = B^{e} ⊗_{C^{e}} D_{C/A}.

Thus the associative algebra Quillen homology object of the A-algebra B is

D_{B/A} = B^{e} ⊗_{P^{e}} D_{P/A}

regarded as an object in the homotopy category of simplicial B-bimodules; here P is a cofibrant simplicial A-algebra mapping by an acyclic fibration to B. The Quillen homology, with coefficients in a B-bimodule M, is

D_{s}^{A}(B; M) = π_{s}(D_{B/A} ⊗_{B^{e}} M).

The Quillen cohomology may be expressed in terms of this object as well:

D_{A}^{s}(B; M) = π^{s}Hom_{B^{e}}(D_{B/A}, M).

Theorem 3.1.1 (Quillen [29] Proposition 3.6). Let B be an Aalgebra and M a B-bimodule over A. Let P be a cofibrant replacement for B as an A-algebra, and let B^{he} = P ⊗_{A} B^{o}. The functors 𝕋or_{∗}^{B^{he}}(B, M) and 𝔼xt_{B^{he}}^{∗}(B, M) are independent of the choice of P up to canonical isomorphism, and there are canonical isomorphisms

D_{s}^{A}(B; M) = \{ \begin{matrix}D_{B/A} ⊗_{B^{e}} M & if s = 0 \\ 𝕋or_{s+1}^{B^{he}}(B, M) & if s > 0. \end{matrix}

D_{A}^{s}(B; M) = \{ \begin{matrix}Der_{A}(B, M) & if s = 0 \\ 𝔼xt_{B^{he}}^{s+1}(B, M) & if s > 0. \end{matrix} .

Proof. We will focus on the case of homology and leave the other case to the reader.

Let P → B be a cofibrant replacement for B as an A-algebra. By the lifting property there is an A-algebra map P → B^{he} over B, and any such map is a weak equivalence since the other two maps are. The Eilenberg-Moore spectral sequence shows that a weak-equivalence of simplicial algebras induces an isomorphism in 𝕋or_{∗}^{−}(B, M) and in 𝔼xt_{−}^{∗}(B, M).

The case s = 0 of the calculation is obvious.

Suppose that T is the tensor algebra over A on a set S. Then an A-derivation from T into a T -bimodule M is determined by a map of sets from S to M, which can be arbitrary. Thus

Der_{A}(T, M) = Hom_{T^{e}}(T^{e}⟨S⟩, M),

so

D_{T/A} ≅ T^{e}⟨S⟩,

where T^{e}⟨S⟩ denotes the free T^{e}-module generated by S.

It follows that if Q is an almost free simplicial A-algebra then D_{Q/A} is an almost free Q-bimodule, and consequently that D_{P/A} is a cofibrant P -bimodule. Thus

D_{s}^{A}(B; M) = π_{s}(D_{P/A} ⊗_{P^{e}} M) = 𝕋or_{s}^{P^{e}}(D_{P/A}, M).

Now the P -bimodule short exact sequence

0 ⟶ D_{P/A} ⟶ P^{e} ⟶^{μ} P ⟶ 0

induces a long exact sequence in 𝕋or_{∗}^{P^{e}}(−, M). The middle term vanishes in positive dimensions, so the boundary homomorphism induces an isomorphism

D_{s}^{A}(B, M) → 𝕋or_{s+1}^{P^{e}}(P, M)

for s > 0.

Since P → B is a weak equivalence of P^{e}-modules we may replace P by B here. Next, P^{o} → B^{o} is a weak equivalence of simplical Amodules, and P is levelwise projective as an A-module, so P^{e} = P ⊗_{A} P^{o} → P ⊗_{A} B^{o} = B^{he} is a weak eqivalence of A-algebras and hence induces an isomorphism 𝕋or_{∗}^{P^{e}}(B, M) → 𝕋or_{∗}^{B^{he}}(B, M).

3.2. Preliminaries on commutative algebra homology. This is the case Quillen focuses on in [29]. Let Ring be the category of commutative rings, let B ∈ Ring, and consider the category Ring/B of commutative rings over B. An object is abelian if it is of the form B⊕ M, where M is a left B-module, the product is given by (b, m)(b′, m′) = (bb′, bm′ + b′m), and the augmentation is the identity on B and kills M. B ⊕ M is If A is another commutative ring and A → B is a ring homomorphism, the B ⊕ M is an abelian commutative A-algebra over B as well. Once again

Hom_{A-ComAlg/B}(C, B ⊕ M) ≅ Der_{A}(C, M).

Identifying the abelian objects with B-modules, the abelianization functor is given by the B-module of Kähler differentials Ω_{B/A}, which satisfy

Der_{A}(C, M) = Hom_{C}(Ω_{C/A}, M) = Hom_{B}(B ⊗_{C} Ω_{C/A}, M)

naturally in C ∈ A-ComAlg/B. Ω_{C/A} is obtained from D_{C/A} by equalizing the left and right actions of C. Since in the ideal D_{C/A} ⊆ C ⊗_{A} C we have

(db)(dc) = b(dc) − (dc)b,

this amounts to dividing D_{C/A} by its square. If C is a free commutative A-algebra generated by a set S, then

Ω_{C/A} ≅ C⟨S⟩,

and it follows that if C is a retract of a free commutative A-algebra then Ω_{C/A} is projective as a C-module.

Now we may define the Quillen homology object of p : A → B by factorizing p in the category of simplicial commutative rings into a cofibration followed by an acyclic fibration, A → P → B, and forming the cotangent complex

L_{B/A} = B ⊗_{P} Ω_{P/A}

in the homotopy category of simplicial B-modules. Since a simplicial commutative A-algebra is cofibrant if and only if it is a retract of an almost free object, L_{B/A} is a projective B-module in each degree. Given a B-module M, the Quillen homology and cohomology groups are defined by

L_{s}^{A}(B; M) = π_{s}(L_{B/A} ⊗_{B} M)

and

L_{A}^{s}(B; M) = π^{s}Hom_{B}(L_{B/A}, M).

Among the properties of these functors which we will need let us mention

Proposition 3.2.1 (Quillen [29]). Let B and C be commutative A-algebras and assume that Tor_{s}^{A}(B, C) = 0 for s > 0. Let D = B ⊗_{A} C. Then

L_{D/C} ≅ C ⊗_{A} L_{B/A}

in the homotopy category of simplicial D-modules.

3.3. Preliminaries on the bar construction. Let B → C be a ring homomorphism. The functor C ⊗_{B} − participates in a triple on left B-modules, whose category of algebras is the category of left C-modules. For any left C-module M we obtain a natural simplicial left C-module

Bar_{B}(C, C, M) = (C ⊗_{B} M ⇐ C ⊗_{B} C ⊗_{B} M ⋯)

together with an augmentation to M. Its underlying simplicial augmented B-module admits a natural contracting homotopy.

If N is a right B-module we define

Bar_{B}(N, C, M) = N ⊗_{C} Bar_{B}(C, C, M);

it is augmented to N ⊗_{C} M.

If C and M are projective over B then Bar_{B}(C, C, M) is a cofibrant simplicial C-module mapping by an acyclic fibration to M, so in this case

π_{∗}(Bar_{B}(N, C, M)) = Tor_{∗}^{C}(N, M).

If we have an augmentation ϵ : C → B then B becomes a C-module and we can consider

Bar_{B}C = Bar_{B}(B, C, B).

If furthermore B and C are commutative then this is a simplicial commutative ring. If again C is B-projective then

Tor_{∗}^{C}(B, B) = H_{∗}(Bar_{B}C)

is a graded algebra which is strictly commutative in the sense that squares of elements of odd degree are zero.

Let I = ker ϵ, so that H_{1}(Bar_{B}C) = I/I^{2}.

Now let A be a commutative ring, let p : Q → B be a map of commutative A-algebras, and let C = Q ⊗_{A} B. Then C is a commutative A-algebra, and there a natural augmentation ϵ : C → B sending q ⊗ b to (pq)b. Assume that Q is projective over A. Then C is projective over B and we are in the situation studied above.

The left and right C-module structures on I/I^{2} coincide and factor through ϵ : C → B. There is a natural A-derivation d : Q → I/I^{2}, given by

x ↦ [x ⊗ 1 − 1 ⊗ px],

which induces a natural B-module map

B ⊗_{Q} Ω_{Q/A} → I/I^{2}.

Since Tor_{∗}^{C}(B, B) is naturally a strictly commutative graded Balgebra, we obtain a natural map of graded strictly commutative Balgebras

α : Λ_{∗}^{B}(B ⊗_{Q} Ω_{Q/A}) → Tor_{∗}^{C}(B, B)

from the free strictly commutative graded B-algebra generated by the B-module B ⊗_{Q} Ω_{Q/A}.

Lemma 3.3.1. This map is an isomorphism if Q is a free commutative A-algebra.

The functor Λ_{t} respects base-change:

Lemma 3.3.2. Let A → B be a ring homomorphism and let M be an A-module. The natural map

B ⊗_{A} Λ_{∗}^{A}(M) → Λ_{∗}^{B}(B ⊗_{A} M)

is an isomorphism.

Proof. This holds for M a free A-module. For a general A-module, pick a presentation 0 ← M ← F_{0} ← F_{1}. This determines a push-out square

[figure: F_{1}; F_{0}; 0; M] .

The left-adjoints B⊗_{A} and Λ_{∗}^{A}, Λ_{∗}^{B}, commute with each other and preserve this pushout. This completes the proof.

3.4. The spectral sequence. Let A → B be a map of commutative rings, and let Q be a cofibrant simplicial commutative A-algebra mapping by an acyclic fibration to B. Since the condition for a map to be an acyclic fibration is the same in the category of simplicial A-algebras as it is in the category of simplicial commutative rings, Q → B is an acyclic fibration of associative A-algebras. Moreover Q is A-projective in each degree, so by the first section above

Ext_{C}^{q+1}(B, M) ≅ D_{A}^{q}(B; M), q > 0,

where C = Q ⊗_{A} B. There is thus a natural isomorphism of simplicial graded B-modules (indeed, of simplicial graded commutative B-algebras)

Λ_{∗}^{B}(L_{B/A}) → 𝕋or_{∗}^{C}(B, B).

Theorem 3.4.1 (Quillen [29] Theorem 8.1). Let B be a commutative A-algebra and M a B-module, and let C = Q ⊗_{A} B where Q is a cofibrant simplicial associative A-algebra replacement for B. Then there are natural spectral sequences of B-modules

E_{s,t}^{2} = H_{s}(Λ_{t}^{B}L_{B/A} ⊗_{B} M) ⇒ 𝕋or_{s+t}^{C}(B, M)

and

E_{2}^{s,t} = H^{s}(Hom_{B}(Λ_{t}^{B}L_{B/A}, M)) ⇒ 𝔼xt_{C}^{s+t}(B, M).

Proof. We focus on the case of homology. Consider the bisimplicial B-module

Bar_{B}(B, C, C) ⊗_{C} M

Cartier’s Eilenberg-Zilber theorem ([4] Satz 2.9, [12] p. 205) asserts that the normalized chain complex of the diagonal simplical object is isomorphic to the total chain complex of the double complex obtained by normalizing the bisimplicial object in both directions. Now

diag (Bar_{B}(B, C, C) ⊗_{C} M) = (diag Bar_{B}(B, C, C)) ⊗_{C} M,

and diag Bar_{B}(B, C, C) is a right C-module mapping by an acyclic fibration to B. Since Q_{t} is projective over A for each t, C_{t} is projective over B for each t, and it follows that diag Bar_{B}(B, C, C) is a cofibrant C-module. Thus the homology of the total complex is 𝕋or_{∗}^{C}(B, M).

We obtain a spectral sequence by filtering by degree in C. The resulting E_{1}-term is obtained by computing the bar construction differential and ignoring the simplicial structure of C. Thus

E_{1}^{s,∗} = N^{∗}Tor_{s}^{C}(B, M),

the normalized chain complex associated to the simplicial B-module with t-th entry Tor_{s}^{C_{t}}(B, M).

To compute this Tor group we recall the Cartan-Eilenberg ([3], XVI, 4) universal coefficient spectral sequence

E_{s,t}^{2} = Tor_{s}^{B}(Tor_{t}^{C}(N, B), M) ⇒ Tor_{s+t}^{C}(N, M),

for any ring-homomorphism C → B, left C-module N, and left Bmodule M. We claim that in our situation

Tor_{t}^{C}(B, B) ≅ Λ_{t}^{B}L_{B/A}

and these B-modules are projective. Thus

Tor_{t}^{C}(B, M) ≅ Λ_{t}^{B}L_{B/A} ⊗_{B} M

as simplicial B-modules, so in the filtration spectral sequence

E_{s,t}^{2} = H_{s}(Λ_{t}^{B}L_{B/A} ⊗_{B} M)

as asserted.

The first of our claims follows from 3.3.1. The second then follows: If N is a free B-module then Λ_{t}^{B}N is a free module for each t. Consequently if N is projective then so is Λ_{∗}^{B}N. Since Ω_{Q/A} is free over Q if Q is a free commutative A-algebra, L_{B/A} is projective over B in each degree, so Tor_{t}^{C}(B, B) is projective over B for each t, and the Cartan-Eilenberg spectral sequence collapses as claimed.

4. Etale cooperation algebras

4.1. Perfection. In characteristic p, there is a very convenient condition guaranteeing that the cotangent complex is zero. To state it, recall first that the Frobenius of a commutative 𝔽_{p}-algebra A is the map ϕ : A → A given by x ↦ x^{p}. Write A^{ϕ} for A regarded as an A-algebra via ϕ. The Frobenius is a natural 𝔽_{p}-algebra endomorphism. Consequently there is a commutative diagram of 𝔽_{p}-algebras

[figure: A; B; A^{ϕ}; A^{ϕ} ⊗_{A} B; B; i; ϕ; ϕ; i; ϕ_{A}]

natural in the 𝔽_{p}-algebra homomorphism i : A → B. The induced map ϕ_{A} is the relative Frobenius. A^{ϕ} ⊗_{A} B will be regarded as an A-algebra by identifying A with A^{ϕ} rather than using the composite with ϕ. Then ϕ_{A} is an A-algebra map. On decomposable tensors we have the formula

ϕ_{A}(a ⊗ b) = (ia)b^{p}.

For example, let B be the commutative A-algebra freely generated by a set S. We have the structure map i : A → B, and a map f : B → B given by the Frobenius on A and the identity on S. These two maps express B as the pushout A^{ϕ} ⊗_{A} B, and using this identification the relative Frobenius is the map ϕ_{A} : B → B which is the identity on A and sends x to x^{p} for x ∈ S.

Lemma 4.1.1. For any map A → B of 𝔽_{p}-algebras, the relative Frobenius ϕ_{A} : A^{ϕ} ⊗_{A} B → B induces the zero map

Ω_{A^{ϕ}⊗_{A}B/A} → Ω_{B/A}.

Proof. Let M be any B-module, and consider

Hom_{B}(Ω_{B/A}, M) → Hom_{B}(Ω_{A^{ϕ}⊗_{A}B/A}, M).

The claim amounts to the assertion that for any A-derivation d : B → M, the composition d ∘ ϕ_{A} = 0. But we can compute:

dϕ_{A}(a ⊗ b) = d((ia)b^{p}) = d(ia) ⊗ b + (ia)d(b^{p}) = 0 + 0 = 0.

Proposition 4.1.2. For any homomorphism i : A → B of commutative 𝔽_{p}-algebras, the relative Frobenius ϕ_{A} induces the zero map

L_{A^{ϕ}⊗_{A}B/A} → L_{B/A}.

Proof. Let A → P → B be a factorization of i into a cofibration followed by an acyclic fibration. Since cofibrations are stable under cobase change ([8] 3.14), the map A^{ϕ} → A^{ϕ} ⊗_{A} P is a cofibration. By the Künneth spectral sequence

Tor_{∗}^{A}(π_{∗}P, B) ⇒ π_{∗}(P ⊗_{A} B)

the map A^{ϕ} ⊗_{A} P → A^{ϕ} ⊗_{A} B is a weak equivalence, and since A^{ϕ} ⊗_{A} − is right exact it is also surjective and hence is an acyclic fibration. Thus

L_{A^{ϕ}⊗_{A}B/A} = B ⊗_{A^{ϕ}⊗_{A}B} Ω_{A^{ϕ}⊗_{A}B/A}.

The map L_{A^{ϕ}⊗_{A}B/A} → L_{B/A} is induced from

[figure: A^{ϕ}; A^{ϕ} ⊗_{A} P; A^{ϕ} ⊗_{A} B; A; P; B; =; ϕ_{A}; ϕ_{A}] ,

so the result follows from Lemma 4.1.1.

Definition 4.1.3. A map i : A → B of 𝔽_{p}-algebras is perfect if the relative Frobenius ϕ_{A} : A^{ϕ} ⊗_{A} B → B is an isomorphism.

For example, if A = 𝔽_{p}, the relative Frobenius is the absolute Frobenius, so 𝔽_{p} → B is perfect if and only if B is perfect in the usual sense.

Corollary 4.1.4. If i : A → B is perfect then L_{B/A} = 0.

Proof. The map ϕ_{A} : L_{A^{ϕ}⊗_{A}B/A} → L_{B/A} is simultaneously zero and an isomorphism.

4.2. Cooperation algebras. Suppose G/R and H/S are formal groups, and consider the functor W_{G,H} : Ring → Set sending T to the set of triples (g, h, θ) where g : R → T and h : S → T are ring homomorphisms and θ : gG → hH is an isomorphism of formal groups over T. The functoriality is clear: if f : T → U then the induced map sends (g, h, θ) to (f ∘ g, f ∘ h, fθ). Denote by SpR the spectrum of R, namely the functor Ring(R, −) : Ring → Set. There are natural maps

SpR ⟵^{η_{L}} W_{G,H} ⟶^{η_{R}} SpS

sending (g, h, θ) to g and to h.

The functor W_{G,H} is representable by

W (G, H) = R ⊗_{L} W ⊗_{L} S.

We will denote also the two maps in the universal example by η_{L} : R → W (G, H) and η_{R} : S → W (G, H).

We can now state the main algebraic result of this section.

Theorem 4.2.1. Let H/S be a Lubin-Tate formal group. Write l = S/𝔪 for the residue field of the local ring S, and let p be the characteristic of l and n the height of the reduction of H to l. Let G/R be any formal group such that R is I-adically complete where I = I_{p,n}(G). Let f : S → R be any ring homomorphism sending 𝔪 to I, and let θ : G → fH be an isomorphism of formal groups over R. Regarding R as a W = W (G, H)-module via the map W → R classifying θ, we have:

D_{R}^{∗}(W; R) = 0.

To prove this we will need several properties of the “cooperation algebras” W (G, H).

Proposition 4.2.2. Let R, S be fields of nonzero characteristic. Let G be a formal group over R and H a formal group over S, both of finite height. Then the map η_{L} : R → W (G, H) is perfect provided that S is perfect.

Proof. We will show that

[figure: R; W (G, H); R; W (G, H); η_{L}; ϕ; ϕ; η_{L}]

is a pushout by checking the universal property. Let T be any commutative ring. By definition,

Ring(W (G, H), T) ≅ {(i : R → T, j : S → T, θ : iG ⟶^{≅} jH)}.

Thus

Ring(R, T) ×_{Ring(R,T)} Ring(W (G, H), T) ≅

{(i : R → T, j̄ : S → T, θ̄: iϕG ⟶^{≅} j̄H)},

and the map sends

(i, j, θ) ↦ (i, ϕj, ϕθ).

We wish to show that this is a bijection.

If R and S have different characteristics, or if G and H have different heights, then both sides are empty—A is the zero ring—and the result is true. So assume both fields have characteristic p and both formal groups have height n.

Pick (i, j̄, θ̄) in the target. Since S is perfect, there is a (unique) map j such that j̄ = jϕ. Thus θ : ϕiG ⟶^{≅} ϕjH, and we want to see that θ is in the image of ϕ.

Recall the nth Vershiebung v_{(n)} : ϕ^{n}G → G, an isomorphism natural in the formal group G of height n. The diagram

[figure: ϕ^{n}iG; ϕ^{n}jH; iG; jH; θ̄; v_{(n)}; v_{(n)}; v_{(n)}θ̄v_{(n)}^{−1}]

commutes. By naturality of v_{(n)}, the same diagram commutes with θ̄ replaced by ϕ^{n}(v_{(n)}θv_{(n)}^{−1}). Since v_{(n)} is an isomorphism, it follows that

θ̄ = ϕ^{n}(v_{(n)}θ̄v_{(n)}^{−1}),

so θ̄ is indeed in the image of ϕ.

For uniqueness, assume that (i, j, θ) and (i, j′, θ′) both have the same image. Then ϕj = ϕj′; but S is perfect, so j = j′. Then we have the commutative diagrams

[figure: ϕ^{n}iG; ϕ^{n}jH; iG; jH; ϕ^{n}θ=ϕ^{n}θ′; v_{(n)}; v_{(n)}; θ or θ′] .

Since v_{(n)} is an isomorphism, this forces θ = θ′.

Remark 4.2.3. This result extends to the general case a computation made by D. C. Ravenel [30] in the case of the Honda formal group H over the prime field 𝔽_{p}. He proved that then

W (H, H) = 𝔽_{p}[t_{0}^{±1}, t_{1}, …]/(t_{i}^{p^{n}} − t_{i}).

Proposition 4.2.4. Let G/R be any formal group and H/S a regular formal group. Then η_{L} : R → W (G, H) is flat.

Proof. Choose coordinates, and hence representing maps L → R and L → S from the Lazard ring L and an isomorphism W (G, H) ≅ R ⊗_{L} W ⊗_{L} S. The functor − ⊗_{R} W (G, H) : R-Mod → S-Mod a composite of two exact functors: first, the functor R-Mod → Comod given by M ↦ M ⊗_{L} W, which is exact since η_{L} : L → W is flat; and, second, the functor Comod → S-Mod given by N ↦ N ⊗_{L} S, which is exact by hypothesis.

Remark 4.2.5. In fact it is not hard to see that, conversely, if η_{L} : L → W (G_{u}, H) is flat, where G_{u}/L is the universal formal group law, then H/R is regular. A proof of the Landweber exact functor theorem (at least in case some I_{p,n}(G) = R for every prime p) has been built from this Lemma by the first author, using the fact that up to faithfully flat extension height classifies formal groups in characteristic p. A version of this proof can be found in [24].

Proposition 4.2.6. Let G/R and H/S be formal groups, let p be a prime number, and let n ∈ ℕ. Let G_{0} be the reduction of G to R/I_{p,n}(G) and H_{0} the reduction of H to S/I_{p,n}(H). Then

η_{L}(I_{p,n}(G))W (G, H) = W (G, H)η_{R}(I_{p,n}(H)),

so the quotients of W (G, H) by these ideals coincide, and they are naturally isomorphic with W (G_{0}, H_{0}).

Proof. Since the displayed ideals are, respectively, I_{p,n}(η_{L}G) and I_{p,n}(η_{R}H), the first clause follows from the fact that the formal groups η_{L}G and η_{R}H are isomorphic over W (G, H). Dividing by one is the same as dividing by both, which by associativity of the tensor product is simply W (G_{0}, H_{0}).

We can now prove the main theorem of this section, Theorem 4.2.1. Write H_{0} for the reduction of H to l and G_{0} for the reduction of G to k = R/I, and write W = W (G, H) and W_{0} = W (G_{0}, H_{0}). Since l is perfect, the map k → W_{0} is perfect and so, by Corollary 4.1.4, L_{W_{0}/k} = 0. By Proposition 4.2.6, W_{0} = k ⊗_{R} W. Now R → W is flat by Proposition 4.2.4 since H/S is regular, so by flat base-change (Proposition 3.2.1) k ⊗_{R} L_{W/R} ≅ L_{W_{0}/k} ≅ 0 in the homotopy category of simplicial k-modules.

Now it follows from Lemma 3.3.2 and Dold’s lemma that

k ⊗_{R} Λ_{∗}^{R}(L_{W/R}) ≅ 0

in the homotopy category of simplicial k-modules. Then the Quillen spectral sequence (Theorem 3.4.1) gives us that

D_{R}^{∗}(W; k) = 0.

Finally, filter R by the I-adic filtration. The long exact sequences in D_{R}^{∗}(W; −) lead by induction D_{R}^{∗}(W; R/I^{j}) = 0 for all j, and then the Milnor sequence implies that D_{R}^{∗}(W; R) = 0 as well.

Part 2. Higher real K-theories

In this Part we reverse strategies. Instead of aiming for maximal generality and invariance, as we did in Part I, we will now focus our attention on a single explicit family of formal groups over finite fields, and particular finite subgroups of their automorphism groups. Our work becomes interesting only when the group has order divisible by the characteristic, p, and as is well known this can only happen if the height is divisible by (p − 1); so we will focus attention on height n = p − 1. The case p = 2 is special and already well understood, but it serves as a model for the new and more complicated case of an odd prime, so we will begin by recalling that story. It concerns the Adams operations on 2-completed K-theory, and the construction of KO_{ˆ2} as the homotopy fixed point set of complex conjugation acting on K_{ˆ2}.

5. Precursors: K-theory and KO-theory

Fix p = 2 and look at the multiplicative formal group law G_{m}(s, t) = s + t − st over the prime field 𝔽_{2}. It has height 1. The natural map

[−] : ℤ_{2} → End_{𝔽_{2}}(G_{m})

is an isomorphism, and

[α](t) = 1 − (1 − t)^{α} = ∑_{k=1}^{∞} \binom{α}{k}t^{k},  α ∈ ℤ_{2}.  (2)

Aut_{𝔽_{2}}(G_{m}) is thus the group of units in ℤ_{2}, which is isomorphic to ℤ_{2} × C_{2}, where C_{2}, the cyclic group of order 2, is generated by −1.

The universal deformation of G_{m}/𝔽_{2} is G_{m}/ℤ_{2}, since it satisfies the conditions of Definition 1.3.3. The automorphism group of G_{m}/𝔽_{2} acts on G_{m}/ℤ_{2} by naturality of the Lubin-Tate construction: for any α ∈ Aut_{𝔽_{2}}(G_{m}), there is a unique morphism (α_{∗}, α̃) : G_{m}/ℤ_{2} → G_{m}/ℤ_{2} covering (1, α) and such that α_{∗} is a local ring homomorphism. For α ∈ ℤ_{2}^{×} the equation (2) defines an automorphism of G_{m}/ℤ_{2} lifting the automorphism of G_{m}/𝔽_{2} given by the same formula, so

α_{∗} = 1,  α̃ = [α].

The action of α on the free ℤ_{2}-module of rank one ω_{G_{m}} is given by d[α] = α ∈ ℤ_{2}^{×}.

Write E for the A_{∞} ring spectrum

E = E_{G_{m}/ℤ_{2}} = EΛ_{G_{m}/𝔽_{2}}.

According to Theorem ??, the monoid of A_{∞} selfmaps of E is homotopically discrete with component monoid given by the multiplicative monoid of End(G_{m}/𝔽_{2}) = ℤ_{2}. There is a unique maximal finite subgroup, namely the group of order 2 generated by [−1], and we will study the homotopy fixed point set E^{hC_{2}} of this subgroup.

In the spectral sequence for the homotopy fixed point set

E_{2}^{s,t} = H^{s}(C_{2}; E_{t}(X)) ⟹ E_{t−s}^{hC_{2}}(X).

With X = S^{0}, we have the elementary computations

H^{∗}(C_{2}; ℤ_{2}) = ℤ_{2}[x]/(2x), |x| = 2,
H^{∗}(C_{2}; ℤ_{2}^{−}) = H^{∗}(C_{2}; ℤ_{2})⟨η⟩/(2η), |η| = 1.

The ring structure of ℤ_{2}[u^{±1}] then leads to

E_{2}^{∗} = H^{∗}(C_{2}; ℤ_{2}[u^{±1}]) = ℤ_{2}[ω^{±1}, η]/(2η), ||ω|| = (0, 4), ||η|| = (1, 2),

where ω = u^{2} and x = ω^{−1}η^{2}.

Next, there is a differential, determined by

d^{3}ω = η^{3}.

This leaves

E_{4} = ℤ_{2}[p^{±1}, q, η]/(q^{2} − 2p, 2η, η^{3}, qη),
||p|| = (0, 8), ||q|| = (0, 4), ||η|| = (1, 2),

where p = ω^{2} and q = 2ω. E_{4}^{s,∗} = 0 for s > 3, so E_{4} = E_{∞}; and no extensions are possible, so

π_{∗}(E^{hC_{2}}) = ℤ_{2}[p^{±1}, q, η]/(q^{2}−2p, 2η, η^{3}, qη), |p| = 8, |q| = 4, |η| = 1.

The Conner-Floyd theorem implies that the periodic ring spectrum associated to G_{m}/ℤ_{2} is the 2-adic completion of complex K-theory,

E_{G_{m}/ℤ_{2}} ≅ K_{ˆ2}

as ring spectra. The claim is that what we have done is no more and no less than to construct the 2-adic completion of real K-theory,

E_{G_{m}/ℤ_{2}}^{hC_{2}} ≅ KO_{ˆ2}

Our next tasks are to determine the odd-primary analogue of the group C_{2} generated by complex conjugation. We will then compute the spectral sequence for the homotopy ring of the homotopy fixed point set, and discover the odd-primary analogue of the number 8 (the dimension of the periodicity operator in KO_{∗}).

6. Maximal Finite Subgroups

In this section we will study the automorphism groups of certain formal groups over finite fields, and their finite subgroups. As promised, we focus on a single specific example: the Honda group of height n over 𝔽_{q}, q = p^{n}.

6.1. The endomorphism ring of the Honda group. If A is a ℤ_{(p)}-module, then there is a natural way to lift a basis for ω_{G} to a coordinate for G, the p-typical coordinate. A formal group law is p-typical if the identity coordinate is p-typical. We need to know two things about this condition. Assume H is p-typical. (1) If

θ(t) = ∑_{k=0}^{∞}^{H}a_{k}t^{p^{k}},  a_{0} ∈ A^{×},  (1)

then H^{θ} is again p-typical. (2) The p-series [p](t) has this form, and H is determined by its p-series.

Fix a natural number n > 0. Recall that the Honda group of height n is the p-typical formal group H with p-series

[p](t) = t^{p^{n}}.  (2)

It is defined over 𝔽_{p}, but we consider it over 𝔽_{q}, since any endomorphism over any larger field is already defined over 𝔽_{q}, and q is the smallest power of p for which this is true. To see this, note that by ?? the endomorphisms of H are the power series of the form

θ(t) = ∑_{k=0}^{∞}^{H}a_{k}t^{p^{k}}

which commute with [p]. This condition expands to

∑_{k=0}^{∞}^{H}a_{k}^{p^{n}}(t^{p^{k}})^{p^{n}} = [p](θ(t)) = θ([p](t)) = ∑_{k=0}^{∞}^{H}a_{k}(t^{p^{n}})^{p^{k}},

using the fact that H is defined over 𝔽_{p}. Equating coefficients, we find that a_{k}^{p^{n}} = a_{k}, which is to say that a_{k} ∈ 𝔽_{q}.

On the other hand it is easy to construct endomorphisms of H. Again because H is defined over 𝔽_{p} (or by commutation with [p])

ϕ(t) = t^{p} ∈ End_{𝔽_{q}}(H).

Also, for any a ∈ 𝔽_{q}, at commutes with [p] and is of the form (6.1), so

at ∈ End_{𝔽_{q}}(H), a ∈ 𝔽_{q}.

Taking sums and limits, we have

Lemma 6.1.1. End_{𝔽_{q}}(H) = {θ(t) = ∑_{k=0}^{∞}^{H}a_{k}t^{p^{k}} : a_{k} ∈ 𝔽_{q}} .

This is a familiar ring. The subring of elements of the form

∑_{n|k}^{H}a_{k}t^{p^{k}}

is commutative and in fact is the ring 𝕎_{n} of Witt vectors of the perfect field 𝔽_{q}. The Teichmuller represtentative of a ∈ 𝔽_{q} is at. Another element of End_{𝔽_{q}}(H) is given by

ϕ(t) = t^{n}.

End_{𝔽_{q}}(H) is generated by ϕ over 𝕎_{n}, with relations

ϕ^{n} = p, ϕa = (σa)ϕ, a ∈ 𝕎_{n},

where σ is the Frobenius automorphism of 𝕎_{n}, the unique local automorphism of 𝕎_{n} lifting the pth power self-map of 𝔽_{q}.

This is the ring of integers in a division algebra D, namely the central division algebra of rank n and Brauer invariant 1/n over ℚ_{p}. We well accordingly denote it by O_{D}.

D admits a discrete valuation, i.e. a function v : D → ℚ ∪ {∞} such that

(i) v(ab) = v(a) + v(b). (ii) v(a) = ∞ ⇔ a = 0. (iii) v(a + b) ≥ min{v(a), v(b)}. (iv) v(D^{×}) is a nonzero discrete (i.e. infinite cyclic) subgroup of ℚ.

In terms of our parametrization, we may define v(θ) as k/n where k is the least j for which a_{j} ≠ 0. Thus for example

v(ϕ) = 1/n.

We prefer this normalization because then v restricts on any commutative subfield K in D to the unique valuation extending the p-adic valuation on ℚ_{p}.

The existence of a discrete valuation has strong implications on the structure of D and of O_{D}. Any one-sided ideal is a two-sided ideal. There is a unique maximal ideal, which is principal. An element generating it is a uniformizer, and uniformizers are characterized as those elements with minimal positive valuation 1/n. The units in O_{D} are given by

O_{D}^{×} = {θ ∈ D : v(θ) = 0},

and under the identification with End_{𝔽_{q}}(H) this is the automorphism group of the Honda group.

Much more information can be found in [31], §12.

Finally, we comment on the Galois action. Let σ ∈ G = Gal(𝔽_{q}/𝔽_{p}) be the Frobenius element, σa = a^{p}. Since H is defined over 𝔽_{p}, σH = H. The group G acts on End_{𝔽_{q}}(H) since a ring homomorphism A → B induces a functor FG_{A} → FG_{B}. In terms of the expression (??) this action is given by

σ (∑_{k=0}^{∞}^{H}a_{k}t^{p^{k}}) = ∑_{k=0}^{∞}^{H}a_{k}^{p}t^{p^{k}}.

σ defines an automorphism of the division algebra D, which by the Skolem-Noether Theorem is induced by conjugation by some element of D^{×}. In this case it is clear that a conjugating element is ϕ: σθ = ϕθϕ^{−1}.

6.2. A nonabelian finite subgroup. Let p be a prime, and fix the following notation for the rest of this section:

n = p − 1

Our next step will be to locate a certain finite subgroup Γ_{0} in Aut_{𝔽_{q}}(H) = O_{D}^{×}.

Let ζ be a primitive pth root of unity. z = ζ − 1 is a root of the Eisenstein polynomial

∑_{i=1}^{p} \binom{p}{i}X^{i−1} = X^{p−1} + pX^{p−2} + ⋯ + p,  (3)

so in the field ℚ_{p}(ζ) the valuation of ζ − 1 is 1/n. Consequently

Lemma 6.2.1. If ζ is any element of order p in D^{×},

v(ζ − 1) = 1/n.

Corollary 6.2.2. Any element ζ of order p in Aut_{𝔽_{q}}(H) is strict: dζ = 1.

The following lemmas occur in a manuscript [25] of Jack Morava dating from 1974, and were shown to us also by Thomas Hewett.

Lemma 6.2.3. ℚ_{p}(ζ) = ℚ_{p}(π), where π^{n} = −p.

Proof. Let L = ℚ_{p}(ζ). Since L = ℚ_{p}(z) and z is a root of the irreducible polynomial (3), [L : ℚ_{p}] = n. From (3), z^{n} = −pv with v ∈ O_{L} such that v ≡ 1 mod z. Now the equation X^{n} − v = 0 is separable, and the nonzero elements of the residue field 𝔽_{p} of O_{L} are solutions, so by Hensel’s lemma there is a solution u ∈ O_{L}. Then π = z/u satisfies the equation π^{n} = −p. Thus ℚ_{p}(π) ⊆ ℚ_{p}(ζ). On the other hand, X^{n} + p is irreducible because it is an Eisenstein polynomial, so [ℚ_{p}(π) : ℚ_{p}] = n as well k and the inclusion must be an isomorphism.

When p = 3 we can be more explicit: since (\frac{1+π}{2})^{3} = −1, we can take ζ = (\frac{1+π}{2})^{2}.

Lemma 6.2.4. Let ω ∈ D be a primitive (p^{n}−1)st root of 1. There exists an element π ∈ D^{×} such that (1) π^{n} = −p and (2) conjugation by π induces the Frobenius automorphism on ℚ_{p}(ω).

Proof. In the standard presentation of D we find an element ϕ such that (1) ϕ^{n} = p and (2) conjugation by ϕ induces the Frobenius automorphism on ℚ_{p}(ω). Let π = ω^{n/2}ϕ, and compute

π^{n} = ω^{n/2}ϕ ⋯ ω^{n/2}ϕ = ω^{m}ϕ^{n}.

Here m = (n/2)(1 + p + ⋯ + p^{n−1}) = (p^{n} − 1)/2, so ω^{m} = −1 and hence π^{n} = −p.

Proposition 6.2.5. Let ω and π be as in the lemma, let ζ be a primitive pth root of 1 in ℚ_{p}(π), and let η = ω^{k}, where k = (p^{n} −1)/n^{2}. Then η and ζ generate a group of order pn^{2} which is normalized by π.

Proof. From πωπ^{−1} = ω^{p} we deduce

ηπη^{−1} = η^{−n}π.

But η^{−n} is an nth root of 1, and hence lies in the center ℚ_{p}. Thus conjugation by η induces an automorphism of ℚ_{p}(π). Since any Galois automorphism permutes the roots of unity, the subgroup Γ_{0} generated by η and ζ sits in an extension

1 → μ_{pn} → Γ_{0} → Gal(L/ℚ_{p}) → 1.

The last arrow is surjective since the first power of η which lies in L is η^{n}, so the quotient has order n = |Gal(L/ℚ_{p})|.

This subgroup is normalized by π since πηπ^{−1} = η^{p} and π and ζ commute.

Remark 6.2.6. The subgroup Γ_{0} is unique up to conjugation in O_{D}^{×}. For a start, according to [14] any other subgroup of D^{×} isomorphic to Γ_{0} is conjugate to Γ_{0} in D^{×}. But since π is a uniformizer of D (it has valuation 1/n) which normalizes Γ_{0}, for any a ∈ D^{×} the element aπ^{−nv(a)} ∈ O_{D}^{×} also conjugates Γ_{0} to aΓ_{0}a^{−1}.

Now we extend the subgroup Γ_{0} ⊂ Aut_{𝔽_{q}}(H) by G = Gal(𝔽_{q}/𝔽_{p}) in the group Aut(H/𝔽_{q}) of automorphisms of H/𝔽_{q} which may involve a nontrivial automorphism of the underlying ring. We would like to construct a sub-extension

1 →  Γ_{0}  →  Γ  → G → 1
↓  ↓  ↓
1 → Aut_{𝔽_{q}}(H) → Aut(H/𝔽_{q}) → G → 1

The bottom sequence depends only upon the formal group H/𝔽_{q}, and not on the “form” of H over 𝔽_{p}. The Honda group over 𝔽_{p} determines a splitting of this sequence, sending σ to the automorphism of H/𝔽_{q} which is σ on 𝔽_{q} and the identity on the formal group law H = σH. A splitting of the sequence determines an action of G on Aut_{𝔽_{q}}(H), and as we know the action in this case is given by conjugation with ϕ. It can be shown, however, that no subgroup of Aut_{𝔽_{q}}(H) conjugate to Γ_{0} is stable under conjugation with ϕ.

It is important to remember however that since Aut_{𝔽_{q}}(H) is not abelian, different splittings may result in nonconjugate actions. And indeed, in our construction Γ_{0} came equipped with a uniformizer π which normalized it, one satisfying X^{n}+p = 0 rather than the equation X^{n} − p = 0 satisfied by ϕ. Galois cohomology shows that there is a different form of H over 𝔽_{p} for which the Frobenius element acts by conjugation by π rather than by ϕ. In fact, Serre’s formulation of Galois cohomology ([10]) puts 𝔽_{p^{s}}-forms of H in bijection with conjugacy classes of elements of D with valuation s/n in precisely this way.

7. Computation of the E_{2}-term

7.1. The action on the Lubin-Tate ring. Let H be the Honda formal group law over 𝔽_{q} of height n. We will study the action of Aut(H/𝔽_{q}) on the Lubin-Tate ring Λ = Λ(H/𝔽_{q}). We have the extension

1 ⟶ Aut_{𝔽_{q}}(H) ⟶ Aut(H/𝔽_{q}) ⟶ Gal(𝔽_{q}/𝔽_{p}) ⟶ 1.  (4)

Let Ĥ be the universal deformation of H over Λ, with

[p]Ĥ(x) = ∑_{i=0}^{n} Ĥu_{i}x^{p^{i}}, u_{0} = p, u_{n} = 1,

where Λ = 𝕎_{n}[[u_{1}, …, u_{n−1}]]. Let 𝔪 be the maximal ideal in Λ.

If α ∈ Aut(H/𝔽_{q}), there is a unique pair (α_{∗}, α̃), where α_{∗} is a local automorphism of Λ covering σ, the image of α in Gal(𝔽_{q}/𝔽_{p}), and α̃ : α_{∗}Ĥ → Ĥ is an isomorphism covering α : σH = H → H.

First, we have a splitting of (4) arising from the fact that H is defined over 𝔽_{p}. It sends σ to (σ, 1), where 1 here is the identity isomorphism σH → H. Let’s just write σ for this element of Aut(H/𝔽_{q}). A straightforward computation shows that the resulting action of Gal(𝔽_{q}/𝔽_{p}) on Aut_{𝔽_{q}}(H) simply applies the Galois automorphism to the coefficients.

Lemma 7.1.1. σ_{∗} acts by naturality on 𝕎_{n}, σ_{∗}u_{i} = u_{i}, and σ̃ = 1 : σ_{∗}Ĥ = Ĥ → Ĥ.

Let α ∈ Aut_{𝔽_{q}}(H). α ∈ x𝔽_{q}[[x]] and as such it lifts to α̃(x) ∈ xΛ[[x]]. The content of the Lubin-Tate theorem here is that there is a unique lift α̃ for which the twist by α̃ of Ĥ is p-typical and has p-series of the form

[p]Ĥα̃(x) = ∑_{i=0}^{n} Ĥα̃w_{i}x^{p^{i}}

where w_{n} = 1 (and of course w_{0} = p, w_{i} ∈ 𝔪 for i < n). The element w_{i} is then α_{∗}u_{i}.

Thus on the one hand

[p]Ĥα̃(x) = ∑_{i=0}^{n} Ĥα̃w_{i}x^{p^{i}} = α̃^{−1} (∑_{i=0}^{n} Ĥ α̃(w_{i}x^{p^{i}}))

where w_{i} = α_{∗}u_{i} ∈ Λ; on the other hand,

[p]Ĥα̃(x) = α̃^{−1} ([p]Ĥ(α̃(x))) = α̃^{−1} (∑_{i=0}^{n} Ĥu_{i}α̃(x)^{p^{i}}) .

Therefore

∑_{i=0}^{n} Ĥu_{i}α̃(x)^{p^{i}} = ∑_{i=0}^{n} Ĥ α̃(w_{i}x^{p^{i}}).  (5)

This is a sort of functional equation. The map α_{∗} is entirely determined from α̃, which is in turn determined by the requirement that there exist elements w_{i} satisfying (5) with w_{0} = p, w_{n} = 1, w_{i} ∈ 𝔪 for i < n, and that α̃ lie over α.

For example, let ω ∈ 𝕎_{n} satisfy ω^{p^{n}−1} = 1. Then ω̄x ∈ Aut_{𝔽_{q}}(H) where ω̄ is the reduction of ω to 𝔽_{q}. This is the automorphism corresponding to ω under the isomorphism O_{D}^{×} ≅ Aut_{𝔽_{q}}(H). I claim that ω̃(x) = ωx ∈ Aut_{Λ}(Ĥ). Certainly this lifts ω(x). The functional equation reads:

∑_{i=0}^{n} Ĥu_{i}(ωx)^{p^{i}} = ∑_{i=0}^{n} Ĥωw_{i}x^{p^{i}},

which holds with

w_{0} = p, w_{i} = ω^{p^{i}−1}u_{i}, 1 ≤ i ≤ n.

Thus

ω_{∗}u_{i} = ω^{p^{i}−1}u_{i} for 1 ≤ i ≤ n.

Return to the general α ∈ Aut_{𝔽_{q}}(H). Since Ĥ is p-typical, we may write

α̃(x) = ∑_{i=0}^{∞} Ĥ ã_{i}x^{p^{i}}, ã_{i} ∈ Λ, ã_{0} ∈ Λ^{×}.

Plugging this into the functional equation, we find that modulo x^{p+1},

pã_{0}x + pã_{1}x^{p} + u_{1}ã_{0}^{p}x^{p} = ã_{0}px + ã_{0}w_{1}x^{p} + ã_{1}p^{p}x^{p}.

This began as the formal sum but modulo x^{p+1} the formal sum is just the ordinary sum. Thus

ã_{1} = p^{−1}(1 − p^{p−1})^{−1}ã_{0}(w_{1} − ã_{0}^{p−1}u_{1}),

so in particular we must have

α_{∗}u_{1} ≡ ã_{0}^{p−1}u_{1} mod p.

Now suppose that α is strict, that is, a_{0} = 1. Since in general

α_{∗}u = ã_{0}u,

we may form the element

e = \frac{(α_{∗} − 1)u_{1}u}{p} ∈ Λu.

Evidently

(1 + α_{∗} + ⋯ + α_{∗}^{p−1})e = 0

if α^{p} = 1, so we get an equivariant map from the reduced regular representation of the cyclic group C_{p} with coefficients in 𝕎_{n} into Λu. We’ll show that any automorphism of order p is in fact strict.

We will study the representation of Aut_{𝔽_{q}}(H) on

Λu/(p, 𝔪^{2}) = 𝔽_{q}⟨u, u_{1}u, …, u_{n−1}u⟩.

Let

α(x) = ∑_{i=0}^{∞}^{H}a_{i}x^{p^{i}}, a_{i} ∈ 𝔽_{q}.

First we’ll determine the representation on the module of indecomosables of Λ,

𝔪/𝔪^{2} = 𝔽_{q}⟨u_{1}, …, u_{n−1}⟩.

Lemma 7.1.2. Modulo 𝔪^{2}

α_{∗}u_{1} ≡ a_{0}^{−1}(a_{0}^{p}u_{1} + a_{1}p)
α_{∗}u_{2} ≡ a_{0}^{−1}(a_{0}^{p^{2}}u_{2} + a_{1}^{p}u_{1} + a_{2}p)
⋮
α_{∗}u_{n−1} ≡ a_{0}^{−1}(a_{0}^{p^{n−1}}u_{n−1} + a_{1}^{p^{n−2}}u_{n−2} + ⋯ + a_{n−2}^{p}u_{1} + a_{n−1}p).

Proof. Only the last term on each side of the functional equation (5) fails to have coefficients in 𝔪. If we work modulo x^{p^{n}} we may ignore both of them. Thus modulo 𝔪^{2} and x^{p^{n}},

∑_{i=0}^{n−1} u_{i}α(x)^{p^{i}} ≡ ∑_{k=0}^{n−1} α(w_{k}x^{p^{k}}).

We have here replaced the formal sum with the ordinary sum, since the difference lies in 𝔪^{2}, and α̃(x) by α(x), by which we mean any lift of α(x) to xΛ[[x]]. Moreover, modulo x^{p^{n}}

α(x) ≡ ∑_{j=0}^{n−1} a_{i}x^{p^{j}}

since the formal group H is additive modulo degree p^{n}. Thus

∑_{i,j} u_{i}a_{j}^{p^{i}}x^{p^{i+j}} ≡ ∑_{i,k} a_{i}w_{k}^{p^{i}}x^{p^{i+k}} ≡ ∑_{k} a_{0}w_{k}x^{p^{k}}.

This is the desired result.

Lemma 7.1.3. In Λ, α̃_{0} ≡ (pa_{n} + a_{0}) + a_{1}^{p^{n−1}}u_{n−1} + ⋯ + a_{n−1}^{p}u_{1} modulo 𝔪^{2}.

Proof. Study the coefficients of x^{p^{n}} in the functional equation. Modulo 𝔪^{2} and x^{p^{n}+1} there are still no cross terms, and we find

ã_{0} ≡ ã_{0}^{p^{n}} + a_{1}^{p^{n−1}}u_{n−1} + ⋯ + a_{n−1}^{p}u_{1} + a_{n}p mod 𝔪^{2}.

But ã_{0}^{p^{n}} ≡ a_{0}^{p^{n}} mod 𝔪^{2}, and since a_{0} ∈ 𝔽_{q} this is just a_{0}. This is the result.

Thus, in terms of the ordered basis u, u_{n−1}u, …, u_{1}u of Λu/(p, 𝔪^{2}), the matrix of the action of an element α ∈ Aut_{𝔽_{q}}(H) with a_{0} = 1 is:

[ \begin{matrix}1 & 0 & ⋯ & 0 & 0 \\ a_{1}^{p^{n−1}} & 1 & ⋯ & 0 & 0 \\ a_{2}^{p^{n−2}} & a_{1}^{p^{n−2}} & ⋯ & 0 & 0 \\ ⋮ & ⋮ & & ⋮ & ⋮ \\ a_{n−1}^{p} & a_{n−2}^{p} & ⋯ & a_{1}^{p} & 1 \end{matrix} ]

Thus (with u_{n} = 1) for 0 ≤ i ≤ n − 1,

(α_{∗} − 1)(u_{n−i}u + ⋯) = a_{1}^{p^{n−i−1}}u_{n−i−1}u + · · · .

It follows that on this module (α_{∗} − 1)^{n} = 0, and that if a_{1} ≠ 0 then any element of the form u + ⋯ generates Λu/(p, 𝔪^{2}) over 𝔽_{q}[α_{∗}]. Our element e ∈ Λu has this form.

Since (by Lemma 6.2.1) any element ζ ∈ Aut_{𝔽_{q}}(H) of order p has a_{0} = 1 and a_{1} ≠ 0, it thus follows that the resulting element e determines a map

ρ̄_{𝕎_{n}} → Λu

which reduces to an isomorphism

ρ̄_{𝔽_{q}} → Λu/(p, 𝔪^{2}).

Work mod p. Let R = Λ/p = 𝔽_{q}[[u_{1}, …, u_{n−1}]]. We have a map ρ̄ → Ru, where ρ̄ denotes the reduced regular representation of C_{p} over 𝔽_{q}. Of course 𝔽_{q}[C_{p}] = F_{q}[t]/(t^{p}) where t = ζ − 1. Notice that in 𝔽_{q}[x],

(x − 1)^{p−1} = \frac{(x − 1)^{p}}{x − 1} = \frac{x^{p} − 1}{x − 1} = 1 + x + ⋯ + x^{p−1},

so in 𝔽_{q}[C_{p}] we see that the “trace element” τ = 1 + ζ + ⋯ + ζ^{p−1} is simply t^{p−1}. ρ̄ is 𝔽_{q}[C_{p}] modulo the trace element, so

ρ̄ = 𝔽_{q}[t]/(t^{p−1}).

This is an augmented 𝔽_{q}-algebra, with augmentation ϵ̄ sending t to 0. Filter ρ̄ by setting

\begin{aligned}F^{1}ρ̄ & = & ker (ϵ̄: ρ̄ → 𝔽_{q}), \\ F^{2}ρ̄ & = & 0. \end{aligned}

Filter R by powers of the augmentation ideal. Then the map ρ̄ → Ru extends to a filtration-preserving equivariant algebra map

ν^{−1}Sρ̄ → R[u^{±1}]

where ν is the “norm” element

ν = ∏_{g∈C_{p}} g ∈ Sρ̄.

Lemma 7.1.4. This map induces an isomorphism on associated graded algebras.

7.2. Cohomology modulo invariants. To understand ν^{−1}Sρ̄ note that the equivariant pushout

\begin{matrix}𝔽_{q} & ⟶ & 0 \\ ↓ & & ↓ \\ ρ & ⟶ & ρ̄ \end{matrix}

of vector spaces induces a pushout diagram of commutative algebras which amounts to the exact sequence

0 ⟶ Sρ ⟶^{τ·} Sρ ⟶ Sρ̄⟶ 0

where the map τ· multiplies by the trace regarded as an element of Sρ.

Lemma 7.2.1. The monomials of Sρ are permuted by the action of C_{p}, and the only fixed monomials are the powers of ν.

Corollary 7.2.2. Sρ splits as the direct sum of the fixed subalgebra S[ν] and a free C_{p}-module.

The monomials [0]^{i_{0}} ⋯ [p − 1]^{i_{p−1}} in S_{k}ρ are in one-to-one correspondance with (p − 1)-element subsets

0 < i_{0} + 1 < i_{0} + i_{1} + 2 < ⋯ < i_{0} + ⋯ + i_{p−2} + (p − 1) < k + p

of the (k + p − 1)-element set {1, 2, … k + p − 1}:

dim S_{k}ρ = \binom{k + p − 1}{p − 1}.

As a check note that this binomial coefficient is 0 mod p unless p|k when it is 1 mod p.

Notice that τ carries S[ν] to the sub-C_{p}-module S[ν]⟨1, [0], …, [p− 1]⟩. Thus we have a commutative diagram with exact rows and columns:

0  0
↓  ↓
0 ⟶  S[ν]  ⟶ Sρ ⟶ free ⟶ 0
↓ ^{τ}  ↓ ^{τ}  ↓ τ̄
0 ⟶ S[ν]⟨1, [0], …, [p − 1]⟩ ⟶ Sρ ⟶ free ⟶ 0
↓ ↓ ↓
S[ν]⟨1 ⊕ ρ̄⟩  ⟶ Sρ̄ ⟶ Q ⟶ 0
↓  ↓  ↓
0  0  0

We claim that the remaining two columns are also short exact. This amounts to the claim that the upper left corner is a pull-back. The consequence is that τ̄ is a split monomorphism (since free kC_{p}-modules are injective), and hence that Q is free. Therefore the bottom sequence splits, giving

Lemma 7.2.3. Sρ̄ = S[ν]⟨1 ⊕ ρ̄⟩ ⊕ free.

Proof To see that the upper left corner is a pull-back, start with x ∈ Sρ. Suppose first that |x| = pk − 1 and assume that τx = ν^{k}. The terms in τx in which [0] occurs to maximal power all arise from terms in x to which [0] occurs to maximal power times the term [0] of τ. By assumption, the maximal power of [0] in τx is [0]^{k}, the sub-sum of τx involving [0] to maximal power is the whole of τx, and the factor of this sum not involving [0] is [1]^{k} ⋯ [p − 1]^{k}. Thus the maximal power of [0] in any term of x is [0]^{k−1}, and the only such term is [0]^{k−1}[1]^{k} ⋯ [p−1]^{k}. But the same must be true of [1], which gives a contradiction. This shows that the pull-back is zero in this dimension.

Now suppose that |x| = pk and that τx = ν^{k}(b_{0}[0]+⋯+b_{p−1}[p−1]. Assume that b_{0} ≠ 0. Again consider the terms in τx in which [0] occurs to maximal power. They all arise from the terms in x in which [0] occurs to maximal power times the term [0] of τ, and by assumption they sum to b_{0}ν^{k}[0]. Thus the only term in x to which [0] occurs to maximal power is b_{0}ν^{k}. This same term contributes all terms of τx in which [j] occurs to maximal power, for any j, so it must be the sole term in x. This establishes the claim.

In general, the image of the trace map is an ideal in the ring of invariants, since the trace map is a module-homomorphism over the ring of invariants. In fact it seems that the image of the trace is an ideal in H^{∗}(G; V) (assuming V is an algebra with G-action; here we have in mind the map τ : V → H^{∗}(G; V)) since Tate cohomology is an algebra. But I’ll have to check the books for this.

Anyway, the objective is to compute H^{∗}(Γ)/Im τ. Work first mod p. Now

H^{∗}(C_{p}; k) = k[b] ⊗ E[a], |a| = 1, |b| = 2

and

H^{∗}(C_{p}; ρ̄) = H^{∗}(C_{p}; k)⟨ā, b̄⟩/(ab̄ = bā), |ā| = 0, |b̄| = 1.

Thus

H^{∗}(C_{p}; Sρ)/Im τ = S[ν][a, b, ā, b̄]/(a^{2}, ā^{2}, b̄^{2}, āb̄, ab̄ − bā).

Studying the filtration we find that this implies that

H^{∗}(C_{p}; Λ/p[u^{±1}]/Im τ = S[ν^{±1}][a, b, ā, b̄]/(a^{2}, ā^{2}, b̄^{2}, ab̄ − bā)

Now the Bockstein δ is determined by

δa = b, δā = b̄.

It follows that

H^{∗}(C_{p}; Λ[u^{±1}])/Im τ = S_{𝕎_{n}}[u^{±1}][α, β]/(pα, pβ, α^{2}),

where α comes from ν^{−1}b and β comes from ν^{1−p}b. Thus ||α|| = (1, 2p− 2) and ||β|| = (2, 2p(p − 1)).

8. Differentials

Part 3. Prospects

Gorbunov, Symonds; Goerss-Hopkins; Devinatz-Hopkins; Nave; elliptic things.

References

[1] J. F. Adams, Stable Homotopy and Generalized Homology, Univ. of Chicago Press, 1974.

[2] A. Baker, A_{∞} structures on some spectra related to Morava K-theories, Quart. J. Math. Oxford Ser. (2) 42 (1991) 403–419.

[3] H. Cartan and S. Eilenberg, Homological Algebra, Princeton University Press, 1956.

[4] A. Dold and D. Puppe, Homologie der nicht-additiver functoren, Ann. Inst. Fourier 11 (1961) 201–312.

[5] J. W. S. Cassels and A. Fröhlich, Algebraic Number Theory, Academic Press, 1967.

[6] E. S. Devinatz and M. J. Hopkins, The action of the Morava stabilizer group on the Lubin-Tate moduli space of lifts, Amer. J. Math. 117 (1995) 669–710.

[7] E. S. Devinatz and M. J. Hopkins, Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups, manuscript, 1999.

[8] W. G. Dwyer and J. Spalinski, Homotopy theories and model categories, Handbook of Algebraic Topology, Elsevier (1995) 73–126.

[9] Jens Franke, On the construction of elliptic cohomology, Math. Nachr. 158 (1992) 43–65.

[10] A. Fröhlich, Formal Groups, Springer Lecture Notes in Math. 74, 1968.

[11] P. G. Goerss and M. J. Hopkins, Realizing commutative ring spectra as E_{∞} ring spectra, manuscript, 1999.

[12] P. G. Goerss and J. F. Jardine, Simplicial Homotopy Theory, Progress in Math. 174, Birkhäuser, 1999.

[13] V. Gorbounov and P. Symonds, Toward the homotopy groups of the higher real K-theory EO_{2}, Contemp. Math. 220 (1998) 103–115.

[14] T. Hewett, Finite subgroups of division algebras over local fields, J. Algebra 173 (1995) 518–548.

[15] T. Hewett, Normalizers of finite subgroups of division algebras over local fields, Math. Res. Lett. 6 (1999) 271–286.

[16] M. J. Hopkins and M. E. Mahowald, The Hurewicz image of EO_{2}, Institut Mittag-Leffler Report 20, 1993/94.

[17] M. J. Hopkins and H. R. Miller, Elliptic moduli in algebraic topology, in preparation.

[18] M. Hovey, Model Categories, AMS Math. Surveys and Monographs 63, 1999.

[19] M. Hovey and N. Strickland, Morava K-theories and localisation, Mem. Amer. Math. Soc. 139, 1999.

[20] P. S. Landweber, Homological properties of comodules over MU_{∗}MU and BP_{∗}BP, Amer. J. Math. 98 (1976) 273–610.

[21] J. Lubin and J. Tate, Formal complex multiplication in local fields, Ann. of Math. 81 (1965) 380–387.

[22] J. Lubin and J. Tate, Formal moduli for one-parameter formal Lie groups, Bull. Soc. Math. France 94 (1966) 49–60.

[23] H. R. Miller, The Sullivan conjecture on maps from classifying spaces, Ann. of Math. 120 (1984) 39–87.

[24] H. R. Miller, Lectures on Formal Groups, in preparation.

[25] J. Morava, ???, 1974.

[26] J. Morava, The Weil group as automorphisms of the Lubin-Tate group, Astérisque 63 (1979) 169–178.

[27] L. S. Nave, On the nonexistence of Smith-Toda complexes, manuscript, 1999.

[28] D. G. Quillen, Homotopical Algebra, Springer Lecture Notes in Math. 43, 1967.

[29] D. G. Quillen, On the homology of commutative rings, Proc. Symp. Pure Math. 17 (1970) 65–87.

[30] D. C. Ravenel, The structure of Morava stabilizer algebras, Invent. Math. 37 (1976) 109–120.

[31] I. Reiner, Maximal Orders, London Math. Soc. Monograph 5, 1975.

[32] C. Rezk, Notes on the Hopkins-Miller theorem, Contemp. Math. 220 (1998) 313–366.

[33] A. Robinson, Obstruction theory and the strict associativity of Morava Ktheories, in Advances in Homotopy Theory, London Math. Soc. Lecture Note Series 139 (1989) 143–152.