ELLIPTIC COHOMOLOGY AND

TOPOLOGICAL MODULAR FORMS

John Rognes

MA 422, Autumn 2001

§1. Introduction

((Complex cobordism, nilpotence and periodicity, formal group laws, elliptic cohomology, topological modular forms.))

§2. Vector bundles, Thom spectra and bordism

Complex projective spaces ℂP^{k}, complex Grassmann manifolds Gr_{n}(ℂ^{n+k}).

Infinite complex Grassmannian Gr_{n}(ℂ^{∞}) ≃ BU(n) classifying space for ℂ^{n}- bundles. Canonical bundle γ^{n} ↓ BU(n). Map p_{n,m}: BU(n)×BU(m) → BU(n+m) classifying direct sum p_{n,m}^{∗}(γ^{n+m}) = γ^{n} × γ^{m}. BU(n) → BU(n + 1) pulls γ^{n+1} back to γ^{n} ⊕ ϵ^{1}.

Vector bundle ξ ↓ X, disc bundle D(ξ), sphere bundle S(ξ), Thom complex T h(ξ) = M(ξ), zero section, normal bundle. T h(ξ × η) ≅ T h(ξ) ∧ T h(η). T h(ξ ⊕ ϵ^{1}) ≅ Σ^{2}T h(ξ).

MU(n) = T h(γ^{n}). MU(n) ∧ MU(m) → MU(n + m). Σ^{2}MU(n) = MU(n) ∧ S^{2} → MU(n + 1). The ring spectrum MU.

Stably complex manifolds. Complex cobordism ring. Pontryagin–Thom construction. Thom–Milnor theorem.

H_{∗}(ℂP^{k}) ≅ ℤ{β_{i} | 0 ≤ i ≤ k}. H^{∗}(ℂP^{k}) ≅ ℤ[t]/(t^{k+1} = 0).

H_{∗}(ℂP^{∞}) = H_{∗}(B𝕋) ≅ ℤ{β_{i} | i ≥ 0}. H^{∗}(ℂP^{∞}) = H^{∗}(B𝕋) ≅ ℤ[t].

Unitary group U(n), diagonal torus 𝕋^{n} ⊂ U(n).

H_{∗}(BU(n)) = H_{∗}(𝕋^{n})_{Σ_{n}} ≅ ℤ{β_{i_{1}} … β_{i_{n}} | i_{1}, …, i_{n} ≥ 0}, β_{0} = 1 unit.

H^{∗}(BU(n)) = H^{∗}(𝕋^{n})^{Σ_{n}} ≅ ℤ[t_{1}, …, t_{n}]^{Σ_{n}} = ℤ[c_{1}, …, c_{n}] where c_{k} = σ_{k}(t_{1}, …, t_{n}) is the k-th symmetric polynomial.

Infinite unitary group U = colim_{n} U(n). H_{∗}(BU) ≅ ℤ[β_{k} | k ≥ 1]. H^{∗}(BU) ≅ ℤ[c_{k} | k ≥ 1]. (Coproducts, primitives.)

§3. Homology of the MU-spectrum

3.1. H_{∗}(BU) and H^{∗}(BU).

The infinite Grassmannian BU ≃ Gr_{∞}(ℂ^{∞}) is obtained by stabilizing the finite Grassmannians BU(n) ≃ Gr_{n}(ℂ^{∞}) via maps i_{n}: BU(n) → BU(n+1) representing the stabilization ξ ↦ ξ⊕ϵ^{1} of complex vector bundles. It admits an H-space pairing μ: BU ×BU → BU obtained by stabilizing maps p_{n,m}: BU(n)×BU(m) → BU(n+ m) representing the cartesian product ξ, η ↦ ξ×η of complex vector bundles. Hence

Typeset by AMS-TEX

its homology H_{∗}(BU) is a Hopf algebra, with product ϕ = μ_{∗} induced by μ and coproduct ψ = Δ_{∗} induced by the diagonal map Δ: BU → BU × BU. Dually, its cohomology H^{∗}(BU) is a Hopf algebra with product the cup product ϕ = Δ^{∗} and coproduct ψ = μ^{∗} induced by the pairing μ.

In homology, H_{∗}(BU) = ℤ[β_{k} | k ≥ 1] with |β_{k}| = 2k. Here β_{k} is the image of a generator under H_{2k}(BU(1)) → H_{2k}(BU). The coproduct is ψ(β_{k}) = ∑_{i+j=k} β_{i} ⊗ β_{j}. In cohomology, H^{∗}(BU) = ℤ[c_{k} | k ≥ 1] with |c_{k}| = 2k. Here the kth Chern class c_{k} restricts to the kth symmetric polynomial σ_{k}(t_{1}, …, t_{n}) via H^{2k}(BU) → H^{2k}(BU(1)^{n}). The coproduct is ψ(c_{k}) = ∑_{i+j=k} c_{i} ⊗ c_{j}. These Hopf algebra structures are dual and abstractly isomorphic, hence self-dual.

3.2. Thom isomorphism.

Let ξ ↓ B be a ℂ^{n}-bundle, with disc–sphere bundle pair (D(ξ), S(ξ)) and Thom complex T h(ξ) = D(ξ)/S(ξ). The Serre spectral sequence

E_{2}^{s,t} = H^{s}(B; H^{t}(D^{2n}, S^{2n−1})) ⟹ H^{s+t}(D(ξ), S(ξ)) ≅ H̃^{s+t}(T h(ξ))

consists of the single row t = 2n, where the orientation coefficients H^{t}(D^{2n}, S^{2n−1}) ≅ ℤ are untwisted since any complex vector bundle has a canonical orientation. Hence the spectral sequence collapses to the Thom isomorphism

Φ_{ξ}: H^{s}(B) ⟶^{≅} H̃^{s+2n}(T h(ξ)) .

The unit 1 ∈ H^{0}(B) corresponds to the orientation class U_{ξ} = Φ_{ξ}(1) ∈ H̃^{2n}(T h(ξ)). For any point b ∈ B this class restricts to the canonical generator of H̃^{2n}(T h(ξ|_{b})) ≅ H^{2n}(D^{2n}/S^{2n−1}) ≅ ℤ, and conversely this property characterizes U_{ξ}. The Thom isomorphism can be expressed in terms of the cup product x ↦ π^{∗}(x) ∪ U_{ξ}, where π: D(ξ) → B is the bundle projection. Dually the Thom isomorphism in homology

H̃_{s+2n}(T h(ξ)) ⟶^{≅} H_{s}(B)

is given by the cap product α ↦ π_{∗}(U_{ξ} ∩ α). Both of these maps can also be interpreted in terms of the Thom diagonal T h(ξ) → B_{+} ∧ T h(ξ).

In the universal case of γ^{n} ↓ BU(n) with T h(γ^{n}) = MU(n) the orientation class U_{γ^{n}} ∈ H̃^{2n}(MU(n)) is represented by a map u_{n}: MU(n) → K(ℤ, 2n). These are compatible with products and stabilization, in the sense that the diagrams

\xymatrix{MU(n) ∧ MU(m) \ar[r]^{u_{n}∧u_{m}} \ar[d]^{T h(p_{n,m})} & K(ℤ, 2n) ∧ K(ℤ, 2m) \ar[d]^{∪} \\ MU(n + m) \ar[r]^{u_{n+m}} & K(ℤ, 2(n + m))}

and

\xymatrix{Σ^{2}MU(n) \ar[r]^{Σ^{2}u_{n}} \ar[d] & Σ^{2}K(ℤ, 2n) \ar[d] \\ MU(n + 1) \ar[r]^{u_{n+1}} & K(ℤ, 2(n + 1))}

homotopy commute. Taken together the maps u_{n} determine a ring spectrum map u: MU → Hℤ.

The Thom isomorphisms are H^{s}(BU(n)) ≅ H̃^{s+2n}(MU(n)) and H̃_{s+2n}(MU(n)) ≅ H_{s}(BU(n)). The homology isomorphisms stabilize to an isomorphism

H_{s}(MU) = colim_{n} H̃_{s+2n}(MU(n)) ⟶^{≅} colim_{n} H_{s}(BU(n)) = H_{s}(BU)

which is compatible with products, so there is an algebra isomorphism H_{∗}(MU) ≅ H_{∗}(BU). We let b_{k} ∈ H_{2k}(MU) correspond to β_{k} ∈ H_{2k}(BU), and obtain

H_{∗}(MU) ≅ ℤ[b_{k} | k ≥ 1]

with |b_{k}| = 2k.

The sphere bundle S(γ^{1}) = S^{∞} is contractible, so the zero section s_{0} : BU(1) → MU(1) is a homotopy equivalence. In homology it takes β_{k+1} ∈ H_{2(k+1)}(BU(1)) to the generator of H̃_{2k+2}(MU(1)) that represents b_{k} ∈ H_{2k}(MU). It follows by dualization and multiplicativity that the orientation class u_{n} ∈ H̃^{2n}(MU(n)) restricts via the zero section s_{0} : BU(n) → MU(n) to the nth Chern class c_{n} = s_{0}^{∗}(u_{n}) ∈ H^{2n}(BU(n)).

3.3. The Steenrod algebra.

Let H𝔽_{p} be the mod p Eilenberg–Mac Lane spectrum {n ↦ K(𝔽_{p}, n)}. Its first space K(𝔽_{p}, 1) ≃ BC_{p} has homology H_{∗}(BC_{p}; 𝔽_{p}) ≅ 𝔽_{p}{α_{k} | k ≥ 0}.

Let A = H^{∗}(H𝔽_{p}; 𝔽_{p}) be the mod p Steenrod algebra of stable mod p cohomology operations, and let A_{∗} = H_{∗}(H𝔽_{p}; 𝔽_{p}) be the (Milnor) dual Steenrod algebra. For p = 2, A_{∗} = P (ξ_{k} | k ≥ 1) with |ξ_{k}| = 2^{k} − 1. Here ξ_{k} is the image of α_{2^{k}} under H̃_{2^{k}}(K(𝔽_{2}, 1); 𝔽_{2}) → H_{2^{k}−1}(H𝔽_{2}; 𝔽_{2}). The coproduct is ψ(ξ_{k}) = ∑_{i+j=k} ξ_{i}^{2^{j}} ⊗ ξ_{j}.

For p odd, A_{∗} = P (ξ_{k} | k ≥ 1) ⊗ E(τ_{k} | k ≥ 0), with |ξ_{k}| = 2(p^{k} − 1) and |τ_{k}| = 2p^{k} − 1. Here ξ_{k} and τ_{k} are the images of α_{2p^{k}−1} and α_{2p^{k}} under H̃_{∗+1}(K(𝔽_{p}, 1); 𝔽_{p}) → H_{∗}(H𝔽_{p}; 𝔽_{p}) = A_{∗}, respectively. The coproducts are ψ(ξ_{k}) = ∑_{i+j=k} ξ_{i}^{p^{j}} ⊗ ξ_{j} and ψ(τ_{k}) = τ_{k} ⊗ 1 + ∑_{i+j=k} ξ_{i}^{p^{j}} ⊗ τ_{j}.

Abstractly this determines the mod p Steenrod algebra A as the dual Hopf algebra. For p = 2 it has algebra generators the Steenrod squares Sq^{i} for i ≥ 1, |Sq^{i}| = i. Here Sq^{i} is dual to ξ_{1}^{i} in the monomial basis for A_{∗}. Conversely there are Milnor elements Q_{k} for k ≥ 0 inductively defined by Q_{0} = Sq^{1} and Q_{k} = [Sq^{2^{k}}, Q_{k−1}]. Then Q_{k} is dual to ξ_{k+1} in the same basis. The coproduct is ψ(Sq^{k}) = ∑_{i+j=k} Sq^{i} ⊗ Sq^{j}. Each Q_{k} is primitive, i.e., ψ(Q_{k}) = Q_{k} ⊗ 1 + 1 ⊗ Q_{k}.

For p odd let q = 2p − 2. The Steenrod algebra has generators the Steenrod powers P^{i} for i ≥ 1, |P^{i}| = qi, and the Bockstein β, |β| = 1. Here P^{i} is dual to ξ_{1}^{i} and β is dual to τ_{0} in the monomial basis for A_{∗}. Now there are Milnor elements M_{k} for k ≥ 1 and Q_{k} for k ≥ 0 inductively defined by M_{1} = P^{1}, M_{k+1} = P^{p^{k}}M_{k}, Q_{0} = β and Q_{k+1} = [P^{p^{k}}, Q_{k}]. Then M_{k} is dual to ξ_{k} and Q_{k} is dual to τ_{k}, with respect to the same basis. The coproducts are ψ(P^{k}) = ∑_{i+j=k} P^{i} ⊗ P^{j} and ψ(β) = β ⊗ 1 + 1 ⊗ β. Each Q_{k} is primitive.

For p = 2, let P_{∗} = P (ξ_{k}^{2} | k ≥ 1) ⊂ A_{∗}. Then P_{∗} is a sub Hopf algebra of A_{∗}, with quotient Hopf algebra E_{∗} = A_{∗}//P_{∗} = E(ξ_{k} | k ≥ 1). Note that P_{∗} is concentrated in even degrees, while E_{∗} is primitively generated. The dual Hopf algebra to E_{∗} is E = E(Q_{k} | k ≥ 0) ⊂ A. It is a primitively generated sub Hopf algebra of A. The quotient Hopf algebra P = A//E is dual to P_{∗}, hence is evenly graded and generated by the even Steenrod operations Sq^{2i}.

For p odd, let P_{∗} = P (ξ_{k} | k ≥ 1) ⊂ A_{∗}. Then P_{∗} is a sub Hopf algebra of A_{∗}, with quotient Hopf algebra E_{∗} = A_{∗}//P_{∗} = E(τ_{k} | k ≥ 1). Now P_{∗} is concentrated in degrees divisible by q = 2p − 2, and again E_{∗} is primitively generated. The dual Hopf algebra to E_{∗} is the primitively generated sub Hopf algebra E = E(Q_{k} | k ≥ 0) ⊂ A. The dual Hopf algebra to P_{∗} is P = A//E ≅ A/AβA, which is generated by the Steenrod powers P^{i}.

Note that for p = 2 the Hopf algebra extensions E → A → P and P_{∗} → A_{∗} → E_{∗} are nontrivial, while for p odd they are split.

Viewing elements of A as cohomology operations, there is a natural pairing A ⊗ H^{∗}(X; 𝔽_{p}) → H^{∗}(X; 𝔽_{p}) for any space X. This makes H^{∗}(X; 𝔽_{p}) a left A-module in a natural way. The dual map H_{∗}(X; 𝔽_{p}) → A_{∗} ⊗ H_{∗}(X; 𝔽_{p}) likewise makes H_{∗}(X; 𝔽_{p}) a left A_{∗}-comodule. (Check this for X of finite type first, then pass to colimits.) When X is an H-space with pairing μ: X × X → X, the Cartan formula ensures that the induced maps ψ = μ^{∗}: H^{∗}(X; 𝔽_{p}) → H^{∗}(X; 𝔽_{p}) ⊗ H^{∗}(X; 𝔽_{p}) and ϕ = μ_{∗}: H_{∗}(X; 𝔽_{p}) ⊗ H_{∗}(X; 𝔽_{p}) → H_{∗}(X; 𝔽_{p}) are A-module and A_{∗}-comodule maps, respectively.

Since the Steenrod operations are stable, these A-module and A_{∗}-comodule structures also extend naturally to the mod p cohomology and mod p homology of spectra, respectively. In the case X = H𝔽_{p}, the A-module action A ⊗ H^{∗}(H𝔽_{p}; 𝔽_{p}) → H^{∗}(H𝔽_{p}; 𝔽_{p}) equals the composition ϕ: A ⊗ A → A, while the A_{∗}-comodule action H_{∗}(H𝔽_{p}; 𝔽_{p}) → A_{∗} ⊗ H_{∗}(H𝔽_{p}; 𝔽_{p}) equals the coproduct ψ: A_{∗} → A_{∗} ⊗ A_{∗}. If X is a ring spectrum, the induced product ϕ: H_{∗}(X; 𝔽_{p}) ⊗ H_{∗}(X; 𝔽_{p}) → H_{∗}(X; 𝔽_{p}) is an A_{∗}-comodule map, so H_{∗}(X; 𝔽_{p}) is an A_{∗}-comodule algebra.

3.4. The Milnor–Moore comodule algebra structure theorem.

Comodule algebra structure theorem. Let Σ be a connected commutative graded Hopf algebra over a field k. Let M be a connected graded left Σ-comodule kalgebra, and let C = k□_{Σ}M = P_{Σ}(M) be the k-algebra of Σ-comodule primitives in M. If there is a surjective Σ-comodule k-algebra map f: M → Σ, then M ≅ Σ ⊗ C as left Σ-comodules and right C-modules.

Proof. Let i: C → M be the inclusion, and choose a retraction g: M → C, so gi = 1. Extend g to a Σ-comodule map g̃ = (1 ⊗ g)ν: M → Σ ⊗ M → Σ ⊗ C. Then g̃ is an isomorphism.

To prove this, filter Σ ⊗ C by {F_{s}Σ ⊗ C}_{s} where F_{s}Σ ⊂ Σ consists of the classes in degrees ≤ s. To show that g̃ is injective, let m ≠ 0 and write ν(m) ≡ σ ⊗ m̄ modulo terms of lower filtration. It follows from coassociativity that m̄ is primitive (ν(m̄) = 1 ⊗ m̄), so g(m̄) ≠ 0 and g̃(m) ≠ 0.

To show that g̃ is surjective, let σ⊗c ∈ Σ⊗C and choose m ∈ M with f(m) = σ. The leading term of ν(m) is σ ⊗ 1, since it maps by 1 ⊗ f to the leading term of ψ(σ), which is σ ⊗ 1, and f is an isomorphism in degree 0. Then g̃(m · i(c)) ≡ (1 ⊗ g)(σ · 1 ⊗ 1 · i(c)) = σ ⊗ gi(c) = σ ⊗ c. So g̃ is surjective modulo the filtration, hence surjective.

Next note that 1□f: C = k□_{Σ}M → k□_{Σ}Σ = k is surjective. Let j: k → C be a section to 1□f. Then h = (g̃)^{−1}(1⊗j) = Σ = Σ⊗k → Σ⊗C → M is a Σ-comodule section to f, with h(1) = 1. Extend h to a left Σ-comodule, right C-module map h̃ = ϕ(h ⊗ i): Σ ⊗ C → M ⊗ M → M. Then h̃ is the desired isomorphism.

To prove this, we show that g̃h̃: Σ ⊗ C → M → Σ ⊗ C is an isomorphism. We have g̃h̃(σ ⊗ c) = (1 ⊗ g)ν(h(σ) · i(c)) = (1 ⊗ g)(hψ(σ) · (1 ⊗ i(c))) ≡ (1 ⊗ g)((σ ⊗ h(1))·(1⊗i(c))) = σ ⊗gi(c) = σ ⊗c. So g̃h̃ is an isomorphism modulo the filtration, hence an isomorphism. □

3.5. The Steenrod action on H_{∗}(MU; 𝔽_{p}).

Lemma [R86, 3.1.6]. The natural A_{∗}-comodule algebra action on H_{∗}(MU; 𝔽_{p}) lifts uniquely to a P_{∗}-comodule algebra action H_{∗}(MU; 𝔽_{p}) → P_{∗} ⊗ H_{∗}(MU; 𝔽_{p}).

Proof. This uses that H_{∗}(MU; 𝔽_{p}) is concentrated in even degrees, and coassociativity of the A_{∗}-comodule action. □

Lemma. The algebra map u_{∗}: H_{∗}(MU; 𝔽_{p}) → H_{∗}(Hℤ; 𝔽_{p}) ⊂ A_{∗} has image P_{∗}.

Proof. To prove this we consider the commutative diagram

\xymatrix{H_{s+2n}(BU(n); 𝔽_{p}) \ar[r]^{s_{0∗}} \ar[dr]_{c_{n}∩(−)} & H̃_{s+2n}(MU(n); 𝔽_{p}) \ar[d]^{π_{∗}(u_{n}∩(−))}_{≅} \ar[r]^{u_{n∗}} & H̃_{s+2n}(K(ℤ, 2n); 𝔽_{p}) \\  & H_{s}(BU(n); 𝔽_{p}) .}

The diagonal cap product with c_{n} is surjective, since it is dual to the injective cup product with c_{n} on H^{∗}(BU(n); 𝔽_{p}). Hence also s_{0∗} is surjective.

The image of u_{∗} on H_{∗}(MU; 𝔽_{p}) in A_{∗} is the subalgebra generated by the images of the algebra generators b_{k} in H_{∗}(MU; 𝔽_{p}), which come from H_{∗}(MU(1); 𝔽_{p}). Hence the image of u_{∗} is generated by the image of u_{1∗}, followed by the stabilization map H_{∗+2}(K(ℤ, 2); 𝔽_{p}) → A_{∗}. Since s_{0∗} is surjective, this equals the image of the map

BU(1) ⟶^{s_{0}} MU(1) ⟶^{u_{1}} K(ℤ, 2)

which is a π_{2}-equivalence, hence a homotopy equivalence. So the image equals the image of the stabilization map, which is 𝔽_{p}{ξ_{k} | k ≥ 0}. The subalgebra it generates is P_{∗}. □

Lemma [R86, 3.1.7]. H_{∗}(MU; 𝔽_{p}) ≅ P_{∗} ⊗ C as a left A_{∗}-comodule and right Cmodule, where C = 𝔽_{p}□_{A_{∗}}H_{∗}(MU; 𝔽_{p}) ≅ P (x_{i} | i ≠ p^{r} − 1), with |x_{i}| = 2i.

Dually, H^{∗}(MU; 𝔽_{p}) ≅ A//E ⊗ C^{∗}, where C^{∗} is dual to C.

§4. The Adams spectral sequence

4.1. Setup.

We will work in the stable homotopy category of spectra. It admits a symmetric monoidal smash product X, Y ↦ X ∧ Y with unit the sphere spectrum S = S^{0}. One model for the weak homotopy type of X ∧ Y is the “naive” or “handicrafted” smash product with 2nth space X_{n} ∧ Y_{n} and the obvious structure maps. So H_{∗}(X ∧ Y; 𝔽_{p}) ≅ H_{∗}(X; 𝔽_{p}) ⊗ H_{∗}(Y; 𝔽_{p}), and so on. But this “naive” construction does not have the symmetric monoidal properties of the “genuine” smash product.

Let H𝔽_{p} denote the mod p Eilenberg–Mac Lane spectrum. It is a ring spectrum, with unit map η: S^{0} → H𝔽_{p} and product map μ: H𝔽_{p} ∧ H𝔽_{p} → H𝔽_{p}.

Let X be any spectrum. We consider diagrams of spectra:

[figure: X; X_{0}; f_{0}; X_{1}; f_{1}; g_{0}; X_{2}; f_{2}; g_{1}; …; g_{2}; K_{0}; K_{1}; K_{2}]

where

X_{s+1} ⟶^{g_{s}} X_{s} ⟶^{f_{s}} K_{s} ⟶^{∂_{s}} ΣX_{s+1}

is a (co-)fiber sequence for each s ≥ 0.

Definition [R86, 2.1.3]. Such a tower (X_{s}, g_{s}) is called a mod p Adams resolution for X if (i) each K_{s} is a wedge sum of suspensions of H𝔽_{p}, and (ii) each map f_{s}^{∗}: H^{∗}(K_{s}; 𝔽_{p}) → H^{∗}(X_{s}; 𝔽_{p}) is surjective.

There is a cofiber sequence

Σ^{−1}H̄𝔽_{p} ⟶ S^{0} ⟶^{η} H𝔽_{p} ⟶ H̄𝔽_{p}

where η is the unit map, and H̄𝔽_{p} is its mapping cone.

Definition. The canonical Adams resolution is inductively constructed by letting K_{s} = H𝔽_{p} ∧ X_{s}, g_{s} = η ∧ 1: X_{s} → K_{s}, and X_{s+1} is the homotopy fiber of g_{s}. Then Σ^{s}X_{s} = (H̄𝔽_{p})^{∧s} ∧ X and Σ^{s}K_{s} = H𝔽_{p} ∧ (H̄𝔽_{p})^{∧s} ∧ X for all s ≥ 0.

Note that the canonical Adams resolution for X is obtained from the canonical Adams resolution for S^{0} by smashing with X.

Lemma. The canonical Adams resolution is a mod p Adams resolution for X.

Proof. To check (i), let (ξ_{i})_{i} be a basis for H_{∗}(X_{s}; 𝔽_{p}) as 𝔽_{p}-vector space, with |ξ_{i}| = n_{i}, and choose representing maps x_{i}: S^{n_{i}} → H𝔽_{p} ∧ X_{s}. Let x: ⋁_{i} S^{n_{i}} → H𝔽_{p} ∧ X_{s} be their wedge sum. Then the composite

H𝔽_{p} ∧ ⋁_{i} S^{n_{i}} ⟶^{1∧x} H𝔽_{p} ∧ H𝔽_{p} ∧ X_{s} ⟶^{μ∧1} H𝔽_{p} ∧ X_{s}

is a weak equivalence. Thus K_{s} ≃ ⋁_{i} Σ^{n_{i}}H𝔽_{p} is a wedge sum of suspensions of H𝔽_{p}.

To check (ii), note that in homology the map f_{s∗}: H_{∗}(X_{s}; 𝔽_{p}) → H_{∗}(K_{s}; 𝔽_{p}) is split injective, since it is induced by the unit map 1 ∧ η ∧ 1: H𝔽_{p} ∧ S^{0} ∧ X_{s} → H𝔽_{p} ∧ H𝔽_{p} ∧ X, whose composite with μ ∧ 1: H𝔽_{p} ∧ H𝔽_{p} ∧ X → H𝔽_{p} ∧ X is homotopic to the identity. □

Applying π_{∗} to an Adams resolution we obtain an unraveled exact couple:

\xymatrix{π_{∗}(X_{0}) \ar[d]^{f_{0∗}} & π_{∗}(X_{1}) \ar[d]^{f_{1∗}} \ar[l]_{g_{0∗}} & π_{∗}(X_{2}) \ar[d]^{f_{2∗}} \ar[l]_{g_{1∗}} & … \ar[l]_{g_{2∗}} \\ π_{∗}(K_{0}) \ar@{-->}[ur]_{∂_{0∗}} & π_{∗}(K_{1}) \ar@{-->}[ur]_{∂_{1∗}} & π_{∗}(K_{2}) \ar[ur]_{∂_{2∗}}}

with groups A^{s,t} = π_{t−s}(X_{s}), E^{s,t} = π_{t−s}(K_{s}) and homomorphisms i_{s} = g_{s∗}, j_{s} = f_{s∗} and k_{s} = ∂_{s∗} of bidegrees (−1, −1), (0, 0) and (1, 0), respectively.

There results a spectral sequence E_{r}^{∗,∗}, called the mod p Adams spectral sequence for X. It has E_{1}^{s,t} = π_{t−s}(K_{s}) and

d_{1}^{s,t} = j_{s+1} ∘ k_{s} = f_{s+1∗} ∘ ∂_{s∗}: E_{1}^{s,t} → E_{1}^{s+1,t} .

For r ≥ 1 there are differentials d_{r}: E_{r}^{s,t} → E_{r}^{s+r,t+r−1}, and natural isomorphisms H^{s,t}(E_{r}^{∗,∗}, d_{r}^{∗,∗}) ≅ E_{r+1}^{s,t}. For r > s these are inclusions E_{r+1}^{s,t} ⊆ E_{r}^{s,t}, and E_{∞}^{s,t} = ⋂_{r>s} E_{r}^{s,t}.

Now suppose that H^{∗}(X; 𝔽_{p}) has finite type, i.e., is bounded below and finite (dimensional) in each degree. Then we may assume that each K_{s} is a locally finite wedge sum of suspensions of H𝔽_{p}, meaning it is bounded below and each suspension of H𝔽_{p} occurs only finitely often. The canonical resolution has this property.

The natural homomorphism π_{t−s}(K_{s}) ≅ π_{t}(Σ^{s}K_{s}) → Hom_{A}^{t}(H^{∗}(Σ^{s}K_{s}; 𝔽_{p}), 𝔽_{p}) taking the homotopy class [f] of a map f: S^{t} → Σ^{s}K_{s} to its induced A-module homomorphism f^{∗}: H^{∗}(Σ^{s}K_{s}; 𝔽_{p}) → H^{∗}(S^{t}; 𝔽_{p}) = Σ^{t}𝔽_{p} is an isomorphism for each s ≥ 0, since K_{s} is a wedge sum of suspensions of H𝔽_{p}. We let P_{s} = H^{∗}(Σ^{s}K_{s}; 𝔽_{p}), which is a free (hence projective) A-module of finite type.

The long exact sequence in cohomology induced by the cofiber sequence above breaks up into short exact sequences, since each map f_{s}^{∗} is surjective:

0 → H^{∗}(ΣX_{s+1}; 𝔽_{p}) ⟶^{∂_{s}^{∗}} H^{∗}(K_{s}; 𝔽_{p}) ⟶^{f_{s}^{∗}} H^{∗}(X_{s}; 𝔽_{p}) → 0 .

These splice together to a long exact sequence

⋯ → H^{∗}(Σ^{s+1}K_{s+1}; 𝔽_{p}) ⟶^{∂_{s}^{∗}∘f_{s+1}^{∗}} H^{∗}(Σ^{s}K_{s}; 𝔽_{p}) → …

⋯ → H^{∗}(K_{0}; 𝔽_{p}) ⟶^{f_{0}^{∗}} H^{∗}(X; 𝔽_{p}) → 0 .

With the notation above, this is a projective resolution ϵ: P_{∗} ⟶^{≃} H^{∗}(X; 𝔽_{p}). Hence there is a natural isomorphism of chain complexes

(E_{1}^{s,∗}, d_{1}^{s,∗}) = (π_{∗}(K_{s}), f_{s+1∗}∂_{s∗}) ≅ Hom_{A}^{∗}((P_{∗}, ∂_{s}^{∗}f_{s+1}^{∗}), 𝔽_{p}) .

The homology of the left hand side is E_{2}^{s,∗}. By definition the homology of the right hand side is Ext_{A}^{s,∗}(H^{∗}(X; 𝔽_{p}), 𝔽_{p}), which is independent up to canonical isomorphism of the choice of projective resolution P_{∗}.

Theorem (Adams 1958) [R86, 2.1.1]. Let X be a spectrum, such that H^{∗}(X; 𝔽_{p}) is bounded below and of finite type. The mod p Adams spectral sequence for X has E_{2}-term

E_{2}^{s,t} = Ext_{A}^{s,t}(H^{∗}(X; 𝔽_{p}), 𝔽_{p}) ,

which is independent of the choice of Adams resolution.

Given a map of spectra ϕ: X → X′ and Adams resolutions (X_{s}, g_{s}) and (X′_{s}, g′_{s}) for X and X′, there is a map (ϕ_{s}: X_{s} → X′_{s}) of Adams resolutions, inducing a morphism of Adams spectral sequences. The map of E_{2}-terms is that induced on Ext_{A} by ϕ^{∗}. Hence the Adams spectral sequence is natural in X.

It is traditional to display this spectral sequence with t − s on the horizontal axis and s on the vertical axis. We call t − s the homotopical degree, and s the (co-)homological degree. The d_{r}-differentials reduce t − s by 1 and increase s by r, so they map one column to the left and r rows up.

4.2. Convergence.

Let X_{∞} = holim_{s} X_{s}, A^{∞} = lim_{s} A^{s} and RA^{∞} = lim_{s}^{1} A^{s}. There is a Milnor exact sequence

0 → RA^{∞} → π_{∗}(X_{∞}) → A^{∞} → 0 .

By definition the spectral sequence arising from the unraveled exact couple is conditionally convergent (Boardman) if A^{∞} = 0 and RA^{∞} = 0. Hence the spectral sequence is conditionally convergent if and only if X_{∞} ≃ ∗.

Fix a spectrum map p: S^{0} → S^{0} inducing multiplication by p in integral homology. Let p^{m}: S^{0} → S^{0} be the mth iterate of p, and let the mod p^{m} Moore spectrum S^{0}/p^{m} be the mapping cone of p^{m}. There are maps of cofiber sequences

\xymatrix{S^{0} \ar[r]^{p^{m}} \ar[d]^{p} & S^{0} \ar[r]^{i_{m}} \ar[d]^{1} & S^{0}/p^{m} \ar[r]^{j_{m}} \ar[d]^{ρ_{m}} & S^{1} \ar[d]^{p} \\ S^{0} \ar[r]^{p^{m−1}} & S^{0} \ar[r]^{i_{m−1}} & S^{0}/p^{m−1} \ar[r]^{j_{m−1}} & S^{1}}

defining the inclusion maps i_{m}, pinch maps j_{m} and the reduction map ρ_{m} inducing the surjection ℤ/p^{m} → ℤ/p^{m−1} on zeroth integral homology. Smashing the upper row with X and applying homotopy gives a short exact sequence

0 → π_{n}(X)/p^{m} ⟶^{i_{m∗}} π_{n}(X/p^{m}) ⟶^{j_{m∗}} _{p^{m}}π_{n−1}(X) → 0 .

Here X/p^{m} = X ∧ S^{0}/p^{m} and _{k}A ⊂ A is the kernel of k: A → A.)

For any spectrum X, define its p-completion to be X_{p}^{∧} = holim_{m} X/p^{m}, i.e., the homotopy limit of the diagram

⋯ → X ∧ S^{0}/p^{m} ⟶^{1∧ρ_{m}} X ∧ S^{0}/p^{m−1} → ⋯ → X ∧ S^{0}/p .

There is a natural map i: X → X_{p}^{∧} lifting the maps 1 ∧ i_{m}: X = X → X/p^{m}.

Lemma [R86, 2.1.15]. (a) The induced map i∧1: X ∧S^{0}/p → X_{p}^{∧} ∧S^{0}/p is a weak equivalence, so i_{∗}: H_{∗}(X; 𝔽_{p}) ≅ H_{∗}(X_{p}^{∧}; 𝔽_{p}) and i^{∗}: H^{∗}(X_{p}^{∧}; 𝔽_{p}) ≅ H^{∗}(X; 𝔽_{p}). Thus i induces an isomorphism of mod p Adams spectral sequences, starting with the E_{2}-term.

(b) If X has finite type, then π_{∗}(X_{p}^{∧}) ≅ π_{∗}(X)_{p}^{∧} = π_{∗}(X) ⊗_{ℤ} ℤ_{p}, where ℤ_{p} = lim_{m} ℤ/p^{m} is the ring of p-adic integers.

(c) If X is bounded below of finite type, the natural map X ∧ (S^{0})_{p}^{∧} → X_{p}^{∧} is a weak equivalence.

Proof. (a) There is a cofiber sequence holim_{m} X/p → X/p ⟶^{i∧1} X_{p}^{∧} ∧ S^{0}/p where the homotopy limit is over maps p: X/p → X/p. The composite p^{2} : X/p → X/p induces the zero homomorphism on homotopy, so lim_{m} π_{∗}(X/p) = 0 and lim_{m}^{1} π_{∗}(X/p) = 0. Hence holim_{m} X/p ≃ ∗ and i ∧ 1 is a weak equivalence.

(b) If X has finite type the groups π_{∗}(X/p^{m}) are all finite, so lim_{m}^{1} π_{∗}(X/p^{m}) = 0 and π_{∗}(X_{p}^{∧}) ≅ lim_{m} π_{∗}(X/p^{m}). Similarly lim_{m}^{1} π_{∗}(X)/p^{m} = 0 and there is a short exact sequence 0 → lim_{m} π_{n}(X)/p^{m} → lim_{m} π_{∗}(X/p^{m}) → lim_{m} _{p^{m}}π_{n−1}(X) → 0. The p-torsion subgroup of π_{n−1}(X) is finite, and for m large the maps in the latter limit are given by multiplication by p, hence are nilpotent. So lim_{m} _{p^{m}}π_{n−1}(X) = 0 and lim_{m} π_{∗}(X/p^{m}) ≅ lim_{m} π_{∗}(X)/p^{m} ≅ π_{∗}(X) ⊗_{ℤ} ℤ_{p}.

(c) The natural map X ∧ (S^{0})_{p}^{∧} → X_{p}^{∧} is an equivalence for X = S^{0}, hence also for X = K a finite CW spectrum. If X is bounded below and of finite type, and k is any integer, there exists a finite CW spectrum K and a k-connected map K → X. In the commutative square

\xymatrix{K ∧ (S^{0})_{p}^{∧} \ar[r] \ar[d] & K_{p}^{∧} \ar[d] \\ X ∧ (S^{0})_{p}^{∧} \ar[r] & X_{p}^{∧}}

the vertical maps are at least k-connected and the top map is an equivalence, hence the lower map is at least k-connected. Letting k → ∞ proves (c). □

Lemma. For X bounded below, the p-nilpotent completion X_{p}^{∧} agrees with the Bousfield H𝔽_{p}-localization L_{H𝔽_{p}}(X).

Proof. By (a) above, i: X → X_{p}^{∧} induces an isomorphism on H_{∗}(−; 𝔽_{p}). It remains to show that X_{p}^{∧} is H𝔽_{p}-local, i.e., that [Z, X_{p}^{∧}]_{∗} = 0 for any Z with H_{∗}(Z; 𝔽_{p}) = 0. But X_{p}^{∧} is the homotopy limit of the bounded-below spectra X/p^{m}, which are the homotopy limits of their finite Postnikov sections. These are in turn finite extensions of suspensions of H𝔽_{p}, and [Z, H𝔽_{p}]_{∗} = 0 for all Z as above. The result follows since the class of H𝔽_{p}-local spectra is closed under finite extensions and homotopy limits. □

A Bousfield localization L_{E} is called smashing if the canonical map X∧L_{E}(S^{0}) → L_{E}(X) is always an equivalence. Examples show that p-completion is not smashing.

Lemma [R86, 2.1.16]. Let X be a bounded below spectrum with each π_{n}(X) a finite p-group, and (X_{s}, g_{s}) any mod p Adams resolution for X. Then X_{∞} ≃ ∗.

Proof. Construct a diagram

X = Y_{0} ⟵ Y_{1} ⟵ Y_{2} ⟵ …

(not an Adams resolution) by letting Y_{s+1} be the fiber in

Y_{s+1} ⟶^{h_{s}} Y_{s} ⟶ L_{s}

where L_{s} is a (finite) wedge sum of suspensions of H𝔽_{p} and the right hand map represents a basis for the bottom mod p cohomology group of Y_{s}. For each n the finiteness of π_{n}(X) implies that π_{n}(Y_{s}) = 0 for s sufficiently large. Hence lim_{s} π_{n}(Y_{s}) = 0. (Also lim_{s}^{1} π_{n+1}(Y_{s}) = 0, so π_{n}(holim_{s} Y_{s}) = 0 by the Milnor short exact sequence, and holim Y_{s} ≃ ∗.)

Let (X_{s}, g_{s}) be an Adams resolution for X. In the (co-)fiber sequence

X_{s+1} ⟶^{g_{s}} X_{s} ⟶^{f_{s}} K_{s}

the map g_{s} induces zero on cohomology. Assume we have a map X_{s} → Y_{s} for s ≥ 0. Then the composite X_{s+1} ⟶^{g_{s}} X_{s} → Y_{s} → L_{s} represents zero on cohomology, hence is null homotopic. Thus we can choose a lift X_{s+1} → Y_{s+1}. Starting with the identity X = X_{0} = Y_{0} we get a map of towers (X_{s}, g_{s}) → (Y_{s}, h_{s}). Hence the canonical map lim_{s} π_{∗}(X_{s}) → π_{∗}(X) factors through lim_{s} π_{∗}(Y_{s}) = 0, and must be zero.

Each X_{s} for s ≥ 0 also satisfies the hypothesis of the lemma, so likewise the image of lim_{s} π_{∗}(X_{s}) in each π_{∗}(X_{s}) is zero. Thus lim_{s} π_{∗}(X_{s}) = 0. Since each π_{∗}(X_{s}) is finite in each degree, lim_{s}^{1} π_{∗}(X_{s}) = 0. Thus π_{∗}(holim_{s} X_{s}) = 0 and X_{∞} = holim_{s} X_{s} ≃ ∗. □

Theorem (Adams 1958), [R86, 2.1.1]. Let X be a spectrum that is bounded below and of finite type. Then the mod p Adams spectral sequence

E_{2}^{s,t} = Ext_{A}^{s,t}(H^{∗}(X; 𝔽_{p}), 𝔽_{p}) ⟹ π_{t−s}(X_{p}^{∧}) ≅ π_{t−s}(X)_{p}^{∧}

is strongly convergent to π_{∗}(X_{p}^{∧}) ≅ π_{∗}(X)_{p}^{∧}.

Strong convergence means that there is a complete Hausdorff filtration

π_{n}(X_{p}^{∧}) = F_{n}^{0} ⊃ F_{n}^{1} ⊃ ⋯ ⊃ F_{n}^{s} ⊃ F_{n}^{s+1} ⊃ …

with filtration quotients F_{t−s}^{s}/F_{t−s}^{s+1} ≅ E_{∞}^{s,t} given by the spectral sequence E_{∞}-term for all s ≥ 0, t. That the filtration is complete Hausdorff means that lim_{s} F_{n}^{s} = 0 and lim_{s}^{1} F_{n}^{s} = 0 for all n. Thus the natural map π_{n}(X_{p}^{∧}) → lim_{s} F_{n}^{0}/F_{n}^{s+1} is an isomorphism.

Proof. Let (X_{s}, g_{s}) be the canonical Adams resolution for X. Smashing with S^{0}/p^{m} yields the canonical Adams resolution (X_{s}/p^{m}, g_{s} ∧ 1) for X/p^{m}. Passing to the homotopy limit over ρ_{m} yields an Adams resolution ((X_{s})_{p}^{∧}, (g_{s})_{p}^{∧}), which is equivalent to the canonical Adams resolution for X_{p}^{∧} obtained by smashing (X_{s}, g_{s}) with (S^{0})_{p}^{∧} since each X_{s} and K_{s} is bounded below and of finite type. By [R86, 2.1.16] holim_{s} X_{s}/p^{m} ≃ ∗, so

holim_{s}(X_{s})_{p}^{∧} = holim_{s} holim_{m} X_{s}/p^{m} ≃ holim_{m} holim_{s} X_{s}/p^{m} ≃ holim_{m} ∗ = ∗ .

Thus the Adams spectral sequence for X_{p}^{∧} is conditionally convergent.

To check that that the spectral sequence is strongly convergent, it is (necessary and) sufficient to check that Boardman’s obstruction group RE_{∞} = lim_{r}^{1} E_{r}^{∗,∗} vanishes. But E_{2}^{s,t} = Ext_{A}^{s,t}(H^{∗}(X; 𝔽_{p}), 𝔽_{p}) is finite in each bidegree. In fact, for the canonical Adams resolution E_{1}^{s,t} = π_{t−s}(K_{s}) = H_{t}((H̄𝔽_{p})^{∧s} ∧ X; 𝔽_{p}) is finite (dimensional) for each (s, t). Thus there can only be finitely many nonzero differentials originating at a given E_{2}^{s,t}, so RE_{∞} = 0.

Hence the Adams spectral sequence for X_{p}^{∧} is strongly convergent to π_{t−s}(X_{p}^{∧}). But this spectral sequence is isomorphic to the Adams spectral sequence for X, by [R86, 2.1.15a]. □

§5. Calculations with the Adams spectral sequence

5.1. The bar resolution.

Let k be a commutative ring and X_{•} a simplicial k-module. The associated chain complex Ch(X)_{∗} has Ch(X)_{q} = X_{q} in each degree, and boundary map d_{q} = ∑_{i=0}^{q}(−1)^{i}d_{q}^{i}: Ch(X)_{q} → Ch(X)_{q−1}. Let D(X)_{q} ⊂ Ch(X)_{q} be the submodule spanned by the degenerate simplices s_{q−1}^{j}(y) ∈ X_{q} for all y ∈ X_{q−1}, 0 ≤ j ≤ q. Then D(X)_{∗} is a chain contractible subcomplex of Ch(X)_{∗}. Let the normalized complex N(X)_{∗} = Ch(X)_{∗}/D(X)_{∗} be the quotient complex; the projection Ch(X)_{∗} → N(X)_{∗} is then a chain equivalence. Let M(X)_{q} ⊂ Ch(X)_{q} = X_{q} be the submodule of all x ∈ X_{q} such that d_{q}^{i}(x) = 0 for all 0 < i ≤ q. Then M(X)_{∗} ⊂ Ch(X)_{∗} is a subcomplex, with boundary map d_{q} = d_{q}^{0}. The inclusion M(X)_{∗} → Ch(X)_{∗} is also a chain equivalence.

Let k be a commutative ring, A a k-algebra, M a left A-module and N a right A-module. For example, consider k = 𝔽_{p}, A the mod p Steenrod algebra, M = H^{∗}(X; 𝔽_{p}) for a spectrum X and N = A. ((Assume that A is a flat k-module.))

The two-sided algebraic bar construction B(N, A, M)_{•} is the simplicial k-module with B(N, A, M)_{q} = N ⊗ A^{⊗q} ⊗ M. Writing n[a_{1}| … |a_{q}]m for a typical element here, the face maps are given by

d_{q}^{i}(n[a_{1}| … |a_{q}]m) = \{ \begin{matrix}na_{1}[a_{2}| … |a_{q}]m & i = 0 \\ n[a_{1}| … |a_{i}a_{i+1}| … |a_{q}]m & 0 < i < q \\ n[a_{1}| … |a_{q−1}]a_{q}m & i = q \end{matrix}

and the degeneracy maps by s_{q}^{j}(n[a_{1}| … |a_{q}]m) = n[a_{1}| … |a_{j}|1|a_{j+1}| … |a_{q}]m. The normalized chain complex is B(N, A, M)_{∗} = N(B(N, A, M))_{∗} with N ⊗J(A)^{⊗q}⊗M in degree q. Here J(A) = cok(η: k → A) is the cokernel of the unit map of A, i.e., the coaugmentation ideal. The bar construction is naturally augmented over N ⊗_{A} M, via the map ϵ: B(N, A, M)_{0} = N ⊗ M → N ⊗_{A} M that takes n[]m to n ⊗_{A} m.

Now consider the case when N = A is a (left and) right A-module of rank one. Then ϵ: B(A, A, M)_{•} → M is a map of simplicial left A-modules and a simplicial homotopy equivalence, with M considered as a simplicial k-module concentrated in degree 0. The simplicial contraction h is given by h_{q}(a[a_{1}| … |a_{q}]m) = 1[a|a_{1}| … |a_{q}]m. At the level of normalized chain complexes ϵ: B(A, A, M)_{∗} → M is a chain equivalence of left A-module chain complexes, with chain contraction h_{q}: B(A, A, M)_{q} → B(A, A, M)_{q+1} given by the same formula. So d_{q+1}h_{q} + h_{q−1}d_{q} = 1 for all q. We call B(A, A, M) the (normalized algebraic) bar resolution of M as a left A-module.

Now suppose that A and M are flat as k-modules and let L be a right Amodule. Then B(A, A, M)_{∗} is a flat left A-module resolution of M. The Torgroups Tor_{n}^{A}(L, M) can thus be computed as the homology of L ⊗_{A} B(A, A, M) ≅ B(L, A, M)_{∗} the (normalized, algebraic) bar complex for (L, A, M).

If A comes with an augmentation ϵ: A → k, and L = M = k via ϵ, the Torgroups Tor_{n}^{A}(k, k) = H_{n}^{k}(A) are called the homology groups of A/k, and are the homology of the bar complex B(k, A, k)_{∗} with J(A)^{⊗q} in degree q, and face maps d_{q} = ∑_{i=0}^{q}(−1)^{i}d_{q}^{i} where

d_{q}^{i}([a_{1}| … |a_{q}]) = \{ \begin{matrix}ϵ(a_{1})[a_{2}| … |a_{q}] & i = 0 \\ [a_{1}| … |a_{i}a_{i+1}| … |a_{q}]m & 0 < i < q \\ [a_{1}| … |a_{q−1}]ϵ(a_{q}) & i = q \end{matrix} .

Next suppose that A and M are projective as k-modules, and let L be a left Amodule. Then B(A, A, M)_{∗} is a projective left A-module resolution of M. The Extgroups Ext_{A}^{n}(M, L) can thus be computed as the homology of Hom_{A}(B(A, A, M)_{∗}, L). In degree q this is Hom_{k}(J(A)^{⊗q} ⊗ M, L), and is some form of cobar complex.

If A is augmented by ϵ: A → k and L = k via ϵ, then Hom_{k}(J(A)^{⊗q} ⊗ M, k) is k-dual to J(A)^{⊗q} ⊗ M. ((Discuss classes in Ext_{A}^{1}(M, k).))

The E_{1}-term of the canonical Adams resolution for a spectrum X such that M = H^{∗}(X; 𝔽_{p}) is bounded below and of finite type has the form Hom_{A}^{∗}(P_{∗}, 𝔽_{p}), with P_{s} = H^{∗}(Σ^{s}K_{s}; 𝔽_{p}) ≅ A ⊗ I(A)^{⊗s} ⊗ M. Here I(A) = H^{∗}(H̄𝔽_{p}; 𝔽_{p}) = ker(ϵ: A → 𝔽_{p}) is the augmentation ideal, and P_{s} is a left A-module with respect to the diagonal action. (These structures involve the counit and coproduct on A, i.e., the Hopf algebra structure.) The composite map I(A) ⊂ A → J(A) is an isomorphism.

The projective resolution P_{∗} → M arises inductively by tensoring the extension I(A) → A → k with I(A)^{⊗s} ⊗ M for s ≥ 0. So d_{s}: P_{s} → P_{s−1} is the composite

P_{s} = A ⊗ I(A)^{⊗s} ⊗ M ⟶^{ϵ⊗1} k ⊗ I(A)^{⊗s} ⊗ M

≅ I(A) ⊗ I(A)^{⊗s−1} ⊗ M → A ⊗ I(A)^{⊗s−1} ⊗ M = P_{s−1} .

Similarly the bar resolution B(A, A, M)_{∗} → M has B(A, A, M)_{q} = A⊗J(A)^{⊗q} ⊗ M, but here the left A-module action only involves the leftmost tensor factor. Nonetheless the two resolutions are isomorphic.

Proposition [R86, 2.2.11]. There is a natural isomorphism of left A-module resolutions B(A, A, M)_{∗} ≅ P_{∗} of M.

The proof of this seems to be routinely omitted, perhaps because the formulas are cumbersome. In degree zero the map B(A, A, M)_{0} = A ⊗ M ≅ A ⊗ M = P_{0} takes a[]m to ∑ a′ ⊗ a′′m, where ψ(a) = ∑ a′ ⊗ a′′. In degree one the map B(A, A, M)_{1} = A ⊗ J(A) ⊗ M ≅ A ⊗ I(A) ⊗ M = P_{1} takes a[a_{1}]m to ∑ a′ ⊗ a′′a′_{1} ⊗ a′′′a′′_{1}m − ∑ a′ ⊗ a′′ ⊗ a′′′a_{1}m, where (1 ⊗ ψ)ψ(a) = ∑ a′ ⊗ a′′ ⊗ a′′′ and ψ(a_{1}) = ∑ a′_{1} ⊗ a′′_{1}.

5.2. Homotopy of MU.

From H_{∗}(MU; 𝔽_{p}) ≅ P_{∗} ⊗ C we have H^{∗}(MU; 𝔽_{p}) ≅ A//E ⊗ C^{∗}. Here E = E(Q_{k} | k ≥ 0) with |Q_{k}| = 2p^{k} − 1 and C ≅ P (x_{i} | i ≠ p^{r} − 1) with |x_{i}| = 2i. Clearly MU is connective and of finite type. Hence the mod p Adams spectral sequence is

E_{2}^{s,t} = Ext_{A}^{s,t}(H^{∗}(MU; 𝔽_{p}), 𝔽_{p}) ⟹ π_{t−s}(MU)_{p}^{∧} .

Here Ext_{A}^{∗,∗}(H^{∗}(MU; 𝔽_{p}), 𝔽_{p}) ≅ Ext_{A}^{∗,∗}(A//E ⊗ C^{∗}, 𝔽_{p}) ≅ Ext_{A}^{∗,∗}(A//E, 𝔽_{p}) ⊗ C, since C has finite type. Applying the natural equivalence Hom_{A}^{∗}(A//E, N) ≅ Hom_{E}^{∗}(𝔽_{p}, N) to an injective resolution η: 𝔽_{p} ⟶^{≃} I_{∗} and taking cohomology yields equivalences Ext_{A}^{∗,∗}(A//E, 𝔽_{p}) ≅ Ext_{E}^{∗,∗}(𝔽_{p}, 𝔽_{p}). By the Künneth theorem

Ext_{E}^{∗,∗}(𝔽_{p}, 𝔽_{p}) ≅ ⨂_{k≥0} Ext_{E(Q_{k})}^{∗,∗}(𝔽_{p}, 𝔽_{p}) .

Finally Ext_{E(Q_{k})}^{∗,∗}(𝔽_{p}, 𝔽_{p}) ≅ P (v_{k}) with v_{k} in bidegree (s, t) = (1, 2p^{k} − 1), repre-

senting the dual of Q_{k} in Hom_{E(Q_{k})}^{|Q_{k}|}(Σ^{|Q_{k}|}E(Q_{k}), 𝔽_{p}). Thus

E_{2}^{∗,∗} ≅ P (v_{k} | k ≥ 0) ⊗ P (x_{i} | i ≠ p^{r} − 1) .

All classes sit in even columns, so the spectral sequence collapses with E_{2}^{∗,∗} = E_{∞}^{∗,∗}. Multiplication by the class v_{0} = a_{0} that is dual to the mod p Bockstein Q_{0} = β represents additive extensions in the E_{∞} term, so

π_{∗}(MU)_{p}^{∧} ≅ ℤ[v_{k}, x_{i} | k ≥ 1, i ≠ p^{r} − 1] ⊗ ℤ_{p} .

Here |v_{k}| = 2p^{k} − 2. Since MU has finite type, and this formula holds for every prime p, it follows that:

Theorem (Milnor 1960, Novikov). There are classes x_{i} ∈ π_{2i}(MU) for i ≥ 1 such that

π_{∗}(MU) ≅ ℤ[x_{i} | i ≥ 1] .

When i = p^{k} −1 we have x_{i} ≡ v_{k} up to a p-adic unit, where v_{k} has Adams filtration s = 1 and is dual to the indecomposable Q_{k} ∈ E.

5.3. Homotopy of ko.

Let ko be the connective real K-theory spectrum. Its mod 2 cohomology is H^{∗}(ko; 𝔽_{2}) ≅ A//A_{1}, where A_{1} = ⟨Sq^{1}, Sq^{2}⟩ ⊂ A is the finite sub Hopf algebra of A generated by Sq^{1} and Sq^{2}. The Adams spectral sequence for ko is

E_{2}^{s,t} = Ext_{A}^{s,t}(A//A_{1}, 𝔽_{2}) ≅ Ext_{A_{1}}^{s,t}(𝔽_{2}, 𝔽_{2}) ⟹ π_{t−s}(ko)_{2}^{∧} .

There are cofiber sequences bo → ko → Hℤ, bso → bo → ΣHℤ/2, bspin → bso → Σ^{2}Hℤ/2 and Σ^{8}ko → bspin → Σ^{4}Hℤ, by real Bott periodicity. The associated long exact sequences in cohomology split as extensions Σ^{2}A/A(Sq^{2}) ≅ ΣH^{∗}(bo; 𝔽_{2}) → A/ASq^{1} → H^{∗}(ko; 𝔽_{2}), Σ^{3}A/A(Sq^{3}) ≅ ΣH^{∗}(bso; 𝔽_{2}) → ΣA → H^{∗}(bo; 𝔽_{2}), Σ^{5}A/A(Sq^{1}, Sq^{2}Sq^{1}Sq^{2}) ≅ ΣH^{∗}(bspin; 𝔽_{2}) → Σ^{2}A → H^{∗}(bso; 𝔽_{2}) and ΣH^{∗}(Σ^{8}ko; 𝔽_{2}) → Σ^{4}A/ASq^{1} → H^{∗}(bspin; 𝔽_{2}). Hence there is an exact complex

0 → Σ^{12}H^{∗}(ko; 𝔽_{2}) →

⟶^{Sq^{2}Sq^{1}Sq^{2}} Σ^{7}A/ASq^{1} ⟶^{Sq^{1}Sq^{2}} Σ^{4}A ⟶^{Sq^{2}} Σ^{2}A ⟶^{Sq^{2}} A/ASq^{1} ⟶^{1}

→ H^{∗}(ko; 𝔽_{2}) → 0

that is induced up from the exact complex of A_{1}-modules

0 → Σ^{12}𝔽_{2} ⟶^{Sq^{2}Sq^{1}Sq^{2}} Σ^{7}A_{1}//A_{0} ⟶^{Sq^{1}Sq^{2}} Σ^{4}A_{1} ⟶^{Sq^{2}} Σ^{2}A_{1} ⟶^{Sq^{2}} A_{1}//A_{0} ⟶^{1} 𝔽_{2} → 0 .

There result long exact sequences of Ext-groups

→ Ext_{A_{1}}^{s−1,t−2}(A_{1}/A_{1}(Sq^{2}), 𝔽_{2}) → Ext_{A_{1}}^{s,t}(𝔽_{2}, 𝔽_{2}) → Ext_{A_{1}}^{s,t}(A_{1}//A_{0}, 𝔽_{2}) →
→ Ext_{A_{1}}^{s−1,t−2}(A_{1}/A_{1}(Sq^{3}), 𝔽_{2}) → Ext_{A_{1}}^{s,t}(A_{1}/A_{1}(Sq^{2}), 𝔽_{2}) → Ext_{A_{1}}^{s,t}(A_{1}, 𝔽_{2}) →

→ Ext_{A_{1}}^{s−1,t−3}(A_{1}/A_{1}(Sq^{1}, Sq^{2}Sq^{1}Sq^{2}), 𝔽_{2}) → Ext_{A_{1}}^{s,t}(A_{1}/A_{1}(Sq^{3}), 𝔽_{2}) → Ext_{A_{1}}^{s,t}(A_{1}, 𝔽_{2}) →

→ Ext_{A^{1}}^{s−1,t−5}(𝔽_{2}, 𝔽_{2}) → Ext_{A_{1}}^{s,t}(A_{1}/A_{1}(Sq^{1}, Sq^{2}Sq^{1}Sq^{2}), 𝔽_{2}) → Ext_{A_{1}}^{s,t}(A_{1}//A_{0}, 𝔽_{2}) → .

All of these decompose into short exact sequences, and yield an injective periodicity operator of bidegree (s, t) = (4, 12):

w: Ext_{A_{1}}^{s−4,t−12}(𝔽_{2}, 𝔽_{2}) → Ext_{A_{1}}^{s,t}(𝔽_{2}, 𝔽_{2})

with cokernel P (h_{0}){1, v} ⊕ 𝔽_{2}{h_{1}, h_{1}^{2}}, with v in bidegree (s, t) = (3, 7). Thus

Ext_{A_{1}}^{∗,∗}(𝔽_{2}, 𝔽_{2}) ≅ P (h_{0}, h_{1}, v, w)/(h_{0}h_{1}, h_{1}^{3}, h_{1}v, v^{2} = 4w) .

Likewise π_{∗}(ko) ≅ ℤ[η, α, β]/(2η, η^{3}, ηα, α^{2} = 4β) with η, α, β in degrees 1, 4 and 8, respectively. The complexification map ko → ku takes α to 2v_{1}^{2} and β to v_{1}^{4}.

5.4. Preview of eo_{2}.

14 JOHN ROGNES

6. Hopf algebroids and flat ring spectra

The Atiyah–Hirzebruch spectral sequence. MU_{∗}MU. Hopf algebroids. Cogroup and cogroupoid objects in commutative rings. Flat ring spectra. The Hopf algebroid (E_{∗}, E_{∗}E).

7. Formal group laws and complex oriented ring spectra Formal group laws. Examples. Lazard’s theorem. The Hopf algebroid (L, LB).

Complex oriented ring spectra. E^{∗}(ℂP^{∞}). Ring spectrum maps MU → E. Quillen–Landweber (MU_{∗}, MU_{∗}MU) ≅ (L, LB).

8. The Adams–Novikov spectral sequence

E_{∗}E-comodules. Homological algebra. The cobar complex. The Adams–Novikov spectral sequence.

Landweber exact theories

Invariant prime ideals. Landweber filtration theorem. Landweber exact functor theorem. Johnson–Wilson E(n). Morava K(n).

Chromatic theory

FGLs in characteristic p. Height. Chromatic filtration. Morava stabilizer groups.

The nilpotence theorem Thick subcategories The periodicity theorem Chromatic convergence 9. Elliptic curves

Weierstrass form. Elliptic formal group law. Elliptic curve Hopf algebroid (A, Λ).

10. Elliptic cohomology

Jacobi quartic. Euler formal group law. Landweber–Ravenel–Stong elliptic cohomology.

11. Lubin–Tate deformation theory Hopkins–Miller theory

E_{∞} ring spectra E_{n} with (discrete) Galois and Morava stabilizer actions.

The Hopkins–Mahowald paper

Deuring normal form. Automorphisms of supersingular curve over 𝔽_{4}. EO_{2} and eo_{2}.

Topological modular forms

Adams–Novikov E_{2}-term for EO_{2} is cohomology of the elliptic curve Hopf algebroid (A, Λ), suitably localized and completed.