POSTNIKOV TOWERS AND k-INVARIANTS FOR S-MODULES,

S-ALGEBRAS AND COMMUTATIVE S-ALGEBRAS

John Rognes

May 15th 2009

Let A be a connective S-module, (associative) S-algebra or commutative Salgebra. We will discuss its Postnikov tower

A → ⋯ → P^{n}A ⟶^{p_{n}} P^{n−1}A → ⋯ → P^{0}A = Hπ_{0}(A)

in each of these three categories. Here A → P^{n}A induces an isomorphism on π_{i} for i ≤ n, while π_{i}(P^{n}A) = 0 for i > n. Hence there is a homotopy cofiber sequence

Σ^{n}Hπ_{n}(A) → P^{n}A ⟶^{p_{n}} P^{n−1}A

for each n ≥ 1. We call P^{n}A the n-th Postnikov section of A.

S-module k-invariants.

Suppose first that A is a connective S-module. In the category of S-modules, we can extend the homotopy cofiber sequence to the right by a map

k_{n−1} : P^{n−1}A → Σ^{n+1}Hπ_{n}(A)

called the (n − 1)-st (additive) k-invariant of A. Conversely, we can recover P^{n}A from P^{n−1}A and k_{n−1} as the homotopy fiber of k_{n−1}. By the theorem of Eilenberg and Mac Lane,

[P^{n−1}A, Σ^{n+1}Hπ_{n}(A)] ≅ H^{n+1}(P^{n−1}A; π_{n}(A))

so we may view the S-module k-invariant k_{n−1} as an element in the spectrum cohomology group on the right.

For example, when A = ku, π_{∗}(ku) = ℤ[u] with |u| = 2, we get P^{1}ku = P^{0}ku = Hℤ, and P^{2}ku is determined by the homotopy cofiber sequence

Σ^{2}Hℤ → P^{2}ku ⟶^{p_{2}} Hℤ ⟶^{k_{1}} Σ^{3}Hℤ

where the first S-module k-invariant lies in

k_{1} ∈ H^{3}(Hℤ; ℤ) ≅ ℤ/2{Q̃_{1}} .

Typeset by AMS-TEX

Here Q̃_{1} is the composite operation

Hℤ ⟶^{π} Hℤ/2 ⟶^{Sq^{2}} Σ^{2}Hℤ/2 ⟶^{β} Σ^{3}Hℤ .

In the case of ku, k_{1} = Q̃_{1} is nontrivial.

((Proof using SU as third space in the Ω-spectrum of ku?))

The existence of the S-module Postnikov tower can be proved by induction on n. Assume for an n ≥ 1 that we have the n-connected map A → P^{n−1}A. Its homotopy cofiber P^{n−1}A/A is then n-connected, with π_{n+1}(P^{n−1}A/A) ≅ π_{n}(A). By attaching cells of dimension ≥ (n+2), we may kill all the higher homotopy of P^{n−1}A/A, so as to get an (n + 2)-connected map P^{n−1}A/A → Σ^{n+1}Hπ_{n}(A). The composite map

k_{n−1} : P^{n−1}A → P^{n−1}A/A → Σ^{n+1}Hπ_{n}(A)

is then the (n − 1)-st k-invariant of A. We can define P^{n}A as the homotopy fiber of k_{n−1}. The composite map A → P^{n−1}A → P^{n−1}A/A is null-homotopic, hence A → P^{n−1}A factors through P^{n}A. Since P^{n−1}A/A → Σ^{n+1}Hπ_{n}(A) is a π_{n+1}- isomorphism, A → P^{n}A must be a π_{n}-isomorphism, since π_{i}(P^{n−1}A) = 0 for i ≥ n.

S-algebra k-invariants.

Suppose next that A is a connective S-algebra. In the category of S-algebras, we shall see that we can express P^{n}A as a homotopy pullback

\xymatrix{P^{n}A \ar[r] \ar[d]_{p_{n}} & P^{n−1}A \ar[d]^{in_{1}} \\ P^{n−1}A \ar[r]^{a_{n−1}} & P^{n−1}A ∨ Σ^{n+1}Hπ_{n}(A)}

where the composite of a_{n−1} with pr_{1} : P^{n−1}A ∨ Σ^{n+1}Hπ_{n}(A) → P^{n−1}A is the identity.

For each S-algebra B and B-bimodule M, let the square-zero extension B ∨ M be the S-algebra with product

(B∨M)∧(B∨M) ≅ (B∧B)∨(B∧M)∨(M ∧B)∨(M ∧M) → B∨M ∨M ∨∗ → B∨M

where the middle map uses the multiplication in B, the left and right B-module actions on M, and the zero map M ∧ M → ∗. The right hand map folds the two copies of M together. The inclusion and projection maps

B ⟶^{in_{1}} B ∨ M ⟶^{pr_{1}} B

are maps of S-algebras. We write pr_{2} : B ∨ M → M for the other projection. A map a: B → B ∨ M of S-algebras that makes the diagram

\xymatrix{ & B ∨ M \ar[d]^{pr_{1}} \\ B \ar[ur]^{a} \ar[r]_{=} & B}

POSTNIKOV TOWERS FOR S-MODULES AND S-ALGEBRAS 3

commute is called an associative derivation of B with values in M. The space of such maps a is denoted ADer(B, M).

In the pullback diagram above, P^{n−1}A ∨ Σ^{n+1}Hπ_{n}(A) is the S-algebra given as a square-zero extension of P^{n−1}A by the P^{n−1}A-bimodule Σ^{n+1}Hπ_{n}(A). The map a_{n−1} is an associative derivation of P^{n−1}A with values in Σ^{n+1}Hπ_{n}(A). The bimodule structure is given by the projection P^{n−1}A → P^{0}A = Hπ_{0}(A), followed by the left and right module actions of π_{0}(A) on π_{n}(A).

When B = 𝕋X ≃ ⋁_{j≥0} X^{∧j} is the free S-algebra generated by an S-module X, the space ADer(𝕋X, M) is equivalent to the space of S-module maps

a|X: X → 𝕋X ∨ M

with pr_{1} ∘ (a|X): X → 𝕋X equal to the inclusion for j = 1, which in turn is equivalent to the space of S-module maps pr_{2}∘(a|X): X → M. These are equivalent to the space Mod_{𝕋X−𝕋X}(𝕋X ∧ X ∧ 𝕋X, M) of 𝕋X-bimodule maps

𝕋X ∧ X ∧ 𝕋X → M .

Here the 𝕋X-bimodule 𝕋X ∧ X ∧ 𝕋X also arises as the homotopy fiber of the multiplication map on 𝕋X, giving the homotopy cofiber sequence

𝕋X ∧ X ∧ 𝕋X → 𝕋X ∧ 𝕋X ⟶^{μ} 𝕋X .

For a general S-algebra B, let I_{B} be the B-bimodule defined by the homotopy cofiber sequence

I_{B} → B ∧ B ⟶^{μ} B .

Then there is a homotopy equivalence

(*)  ADer(B, M) ≃ Mod_{B−B}(I_{B}, M)

between the (derived) space of B-bimodule maps I_{B} → M and the space of associative derivations of B with values in M. Andrei Lazarev [L] proves this by resolving B by a simplicial S-algebra that is free in each degree, and using the equivalence above in each simplicial degree.

Topological Hochschild (co-)homology.

We can view Mod_{B−B}(I_{B}, M) as the underlying space of the bimodule function spectrum F_{B−B}(I_{B}, M). The topological Hochschild cohomology of B with values in M is defined as the (derived) bimodule function spectrum

THH^{•}(B, M) = F_{B−B}(B, M) = Ext_{B−B}(B, M) ,

with homotopy groups

THH^{q}(B, M) = π_{−q} THH^{•}(B, M) .

The homotopy cofiber sequence above the gives rise to the homotopy cofiber sequence

THH^{•}(B, M) → M → F_{B−B}(I_{B}, M) .

In particular, when B = P^{n−1}A and M = Σ^{n+1}Hπ_{n}(A) with n ≥ 1 the connecting map

π_{0} ADer(P^{n−1}A, Σ^{n+1}Hπ_{n}(A)) ≅ π_{0}F_{B−B}(I_{B}, M)

⟶^{≅} π_{−1} THH^{•}(B, M) = THH^{n+2}(P^{n−1}A, π_{n}(A))

is a bijection. Hence we may view the S-algebra k-invariant a_{n−1} as an element in the topological Hochschild cohomology group on the right.

The topological Hochschild homology of B with values in M is defined as the (derived) bimodule smash product

THH_{•}(B, M) = M ∧_{B−B} B = Tor^{B−B}(M, B) .

Replacing the right hand B with the two-sided bar construction β(B, B, B), with B ∧ B^{∧q} ∧ B in simplicial degree q, we get the usual Hochschild complex model for THH_{•}(B, M), with M ∧ B^{∧q} in simplicial degree q. When M = B, we abbreviate THH_{•}(B, B) to THH(B).

When B is a commutative S-algebra and M a B-module, we may view B as an S-algebra and M as a symmetric B-bimodule. In that case there is a base change equivalence

F_{B−B}(B, M) ≃ F_{B}(B ∧_{B−B} B, M) ≃ F_{B}(THH(B), M) .

For example, when A = ku, n = 2, B = P^{1}A = Hℤ and M = Σ^{3}Hℤ, the first S-algebra k-invariant lies in

a_{1} ∈ THH^{4}(Hℤ; ℤ) ≅ Ext_{ℤ}^{1}(THH_{3}(Hℤ), ℤ) ≅ ℤ/2 ,

as we shall compute later.

Existence of the S-algebra Postnikov tower.

To express P^{n}A as the homotopy pullback along an associative derivation a_{n−1}, we need an extension of Lazarev’s result to S-algebras B under a fixed S-algebra A. For each map ϕ: A → B of S-algebras define the B-bimodule Ω_{B/A} by the homotopy fiber sequence

I_{B/A} → B ∧_{A} B ⟶^{μ} B

where μ is the multiplication map. By an associative derivation of B over A with values in M we mean a map a: B → B ∨ M of S-algebras, under A and over B, meaning that the S-algebra diagram

\xymatrix{A \ar[r]^{in_{1}ϕ} \ar[d]_{ϕ} & B ∨ M \ar[d]^{pr_{1}} \\ B \ar[ur]^{a} \ar[r]_{=} & B}

commutes. Let ADer(B/A, M) be the space of such maps a. Then there is a homotopy equivalence

ADer(B/A, M) ≃ Mod_{B−B}(I_{B/A}, M) .

The identity map of I_{B/A} corresponds to a universal associative derivation d: B → B ∨ I_{B/A}.

When A → B = P^{n−1}A is the standard n-connected map, with homotopy cofiber B/A, we compute that I_{B/A} ≃ B ∧_{A} (B/A) is n-connected with

π_{n+1}(I_{B/A}) ≅ π_{0}(B) ⊗_{π_{0}(A)} π_{n+1}(B/A) ≅ π_{n}(A) .

Hence there is a B-bimodule map

I_{B/A} → Σ^{n+1}Hπ_{n}(A)

that corresponds to the associative derivation

a_{n−1} : P^{n−1}A → P^{n−1}A ∨ Σ^{n+1}Hπ_{n}(A)

under A. The homotopy pullback of a_{n−1} and in_{1} is then readily seen to be a model for P^{n}A.

Commutative S-algebra k-invariants.

Suppose now that A is a connective commutative S-algebra. Also in the category of commutative S-algebras, we can express P^{n}A as a homotopy pullback

\xymatrix{P^{n}A \ar[r] \ar[d]_{p_{n}} & P^{n−1}A \ar[d]^{in_{1}} \\ P^{n−1}A \ar[r]^{d_{n−1}} & P^{n−1}A ∨ Σ^{n+1}Hπ_{n}(A)}

where the composite of d_{n−1} with pr_{1} : P^{n−1}A ∨ Σ^{n+1}Hπ_{n}(A) → P^{n−1}A is the identity.

For each commutative S-algebra B and B-module M, let the square-zero extension B ∨ M be defined as before. The maps

B ⟶^{in_{1}} B ∨ M ⟶^{pr_{1}} B

are maps of commutative S-algebras. Let pr_{2} : B∨M → M be the other projection. A map d: B → B ∨ M of commutative S-algebras that makes the diagram

\xymatrix{ & B ∨ M \ar[d]^{pr_{1}} \\ B \ar[ur]^{d} \ar[r]_{=} & B}

commute is called a (commutative) derivation of B with values in M. The space of such maps d is denoted Der(B, M).

In the pullback diagram above, d_{n−1} is a derivation of P^{n−1}A with values in Σ^{n+1}Hπ_{n}(A).

When B = ℙX ≃ ⋁_{j≥0} X_{hΣ_{j}}^{∧j} is the free commutative S-algebra generated by an S-module X, Der(B, M) is equivalent to the space of S-module maps

d|X: X → ℙX ∨ M

with pr_{1} ∘(d|X) equal to the inclusion, which in turn is equivalent to the space of Smodule maps pr_{2}∘(d|X): X → M, and to the space of B-module maps B∧X → M. The B-module B ∧ X can be obtained from the commutative S-algebra B = ℙX by a process of stabilization, which we discuss next Topological André–Quillen (co-)homology.

First we pass to a based situation, by base changing along S → B, taking B to the augmented commutative B-algebra

B ∧ B .

The B-algebra unit map is the composite B ≅ B ∧ S → B ∧ B. The augmentation is the multiplication μ: B ∧ B → B.

In the pointed category C_{B}/B of commutative B-algebras over B, there is a suspension operator E taking C to the (derived) smash product

E(C) = B ∧_{C} B = Tor^{C}(B, B) .

This is the homotopy pushout in C_{B}/B of the diagram B ← C → B, and can be computed by replacing the right hand B by the two-sided bar construction β(B, B, B). When C = B ∧ B, we see that E(B ∧ B) = THH(B).

For comparison, the suspension ΣY of a based space Y is the homotopy pushout in based spaces of the diagram ∗ ← Y → ∗, which is the pushout of ∗ ← Y → CY.

To stabilize, we iterate the suspension process, forming a suspension E-spectrum

{m ↦ E^{m}(B ∧ B)} .

Such E-spectra form a stable category Spt(C_{B}/B). By a theorem of Maria Basterra and Mike Mandell [BM], there are equivalences

Spt(C_{B}/B) ⟶^{≃} Spt(Mod_{B}) ⟵^{≃} Mod_{B} .

The first equivalence is given by taking an E-spectrum

{m ↦ D_{m}}

in C_{B}/B to the Σ-spectrum

{m ↦ D_{m}/B}

in Mod_{B}. Here D_{m}/B denotes the unit cofiber, or equivalently, the augmentation fiber, of D_{m}.

((Discuss how ED_{m} → D_{m+1} induces Σ(D_{m}/B) → D_{m+1}/B.))

The second equivalence takes a B-module M to its suspension Σ-spectrum {m ↦ Σ^{m}M}.

The composite construction takes a commutative S-algebra B first to B ∧ B in C_{B}/B, then to its E-suspension spectrum {m ↦ E^{m}(B ∧ B)}, then to its unit cofibers {m ↦ E^{m}(B ∧ B)/B}, and then to the B-module

TAQ(B) = hocolim_{m} Σ^{−m}E^{m}(B ∧ B)/B .

This provides one definition of the topological André–Quillen homology of B, also denoted Ω_{B}^{1}.

Another definition is given by the (left derived) commutative B-algebra indecomposables Q_{B} of the (right derived) augmentation ideal I_{B} of the (left derived) smash product B ∧ B in C_{B}/B:

TAQ(B) = Q_{B}I_{B}(B ∧ B) .

When B = ℙX is free, we get that B ∧ B = B ∧ ℙX. The free functor ℙ takes homotopy pushouts of S-modules to homotopy pushouts of augmented commutative S-algebras, so there are equivalences S ∧_{ℙX} S ≃ ℙΣX and E(B ∧ ℙX) ≃ B ∧ ℙΣX. Hence

E^{m}(B ∧ B) ≃ B ∧ ℙΣ^{m}X

for all m ≥ 0. When X is connective, the inclusion

B ∧ (S ∨ Σ^{m}X) → B ∧ ℙΣ^{m}X

of the 0 ≤ j ≤ 1 summands is about 2m-connected, so

B ∧ X = Σ^{−m}B ∧ Σ^{m}X → Σ^{−m}E^{m}(B ∧ B)/B

is about m-connected. In the colimit, we get an equivalence

B ∧ X ≃ TAQ(ℙX)

of B-modules, for B = ℙX.

For general B, there is a homotopy equivalence

(**)  Der(B, M) ≃ Mod_{B}(TAQ(B), M)

between the (derived) space of B-module maps TAQ(B) → M and the space of derivations of B with values in M. Basterra [B] proved this by means of the second definition of TAQ(B), which is equivalent to the first one by the joint paper of Basterra and Mandell.

Let

TAQ_{q}(B, M) = π_{q}(M ∧_{B} TAQ(B))
TAQ^{q}(B, M) = π_{−q}(F_{B}(TAQ(B), M)

be the topological André–Quillen homology and cohomology groups of B with coefficients in M. We abbreviate TAQ_{q}(B, B) to TAQ_{q}(B) = π_{q} TAQ(B).

In particular, when B = P^{n−1}A and M = Σ^{n+1}Hπ_{n}(A) with n ≥ 1 the commutative S-algebra k-invariant d_{n−1} is an element in the topological André–Quillen cohomology group on the right.

π_{0} Der(P^{n−1}A, Σ^{n+1}Hπ_{n}(A))

≅ π_{0}F_{B}(TAQ(B), M) = TAQ^{n+1}(P^{n−1}A, π_{n}(A)) .

For example, when A = ku, n = 2, B = P^{1}A = Hℤ and M = Σ^{3}Hℤ, the first commutative S-algebra k-invariant lies in

d_{1} ∈ TAQ^{3}(Hℤ; ℤ) ≅ Ext_{ℤ}^{1}(TAQ_{2}(Hℤ), ℤ) ≅ ℤ/2 ,

as we shall soon compute.

Comparison maps.

We may view a commutative S-algebra A as an (associative) S-algebra or just an S-module. The extension P^{n}A → P^{n−1}A = B is the classified by an element in one of the sets

π_{0} Der(B, Σ^{n+1}Hπ_{n}(A)) → π_{0} ADer(B, Σ^{n+1}Hπ_{n}(A)) → [B, Σ^{n+1}Hπ_{n}(A)]

where the maps correspond to forgetting some of the structure. As discussed above, this diagram is isomorphic to

TAQ^{n+1}(B, π_{n}(A)) → THH^{n+2}(B, π_{n}(A)) → H^{n+1}(B, π_{n}(A)) .

These groups consist of the homotopy classes of B-module maps to Σ^{n+1}Hπ_{n}(A) from the diagram

B ∧ B → Σ^{−1} THH(B) → TAQ(B) ,

in reverse order. Since n ≥ 1, we may replace B ∧ B by (B ∧ B)/B, and THH(B) by THH(B)/B, so that this diagram consists of the terms m = 0, m = 1 and the homotopy colimit as m → ∞ of the diagram

⋯ → Σ^{−m}E^{m}(B ∧ B)/B → ….

By an inspection of the various identifications made, the forgetful maps above correspond to the stabilization map

ΣE^{m}(B ∧ B)/B → E^{m+1}(B ∧ B)/B

for m = 0, and their composites as m → ∞.

Calculations for Hℤ.

We consider the case B = Hℤ and n = 2, and show that the homomorphisms

TAQ^{3}(Hℤ, ℤ) ⟶^{≅} THH^{4}(Hℤ, ℤ) ⟶^{≅} H^{3}(Hℤ, ℤ) ≅ ℤ/2{Q̃_{1}}

are all isomorphisms, by showing that the maps

(Hℤ ∧ Hℤ)/Hℤ → Σ^{−1} THH(Hℤ)/Hℤ → TAQ(Hℤ)

induce isomorphisms on π_{2} (where all groups are ℤ/2) and on π_{3} (where all groups are 0).

Write H_{∗}(X) for H_{∗}(X; 𝔽_{p}) and consider p = 2. Then

H_{∗}(Hℤ) ≅ P (ξ̄_{1}^{2}, ξ̄_{k} | k ≥ 2)

with |ξ̄_{1}^{2}| = 2 and |ξ̄_{k}| = 2^{k} − 1. Hence

H_{∗}(Hℤ ∧ Hℤ) ≅ H_{∗}(Hℤ) ⊗ H_{∗}(Hℤ) ≅ H_{∗}(Hℤ){1, ξ̄_{1}^{2}, ξ̄_{2}, … }

with the remaining terms in dimensions ≥ 4. Here β(ξ̄_{2}) = ξ̄_{1}^{2}, so

π_{∗}(Hℤ ∧ Hℤ) ≅ (ℤ{1}, 0, ℤ/2{ξ̄_{1}^{2}}, 0, …) .

We use the bar spectral sequence

E_{∗∗}^{2} = Tor_{∗∗}^{H_{∗}(C)}(H_{∗}(B), H_{∗}(B)) ⟹ H_{∗}(E(C))

to compute the homology of the E-suspension E(C) = B ∧_{C} B. As before, C is an augmented commutative B-algebra. Each algebra indecomposable x in H_{i}(C) contributes a term [x] ∈ E_{1,i}^{2}. For dimension reasons it is an infinite cycle (but it may be hit by a differential). It therefore represents an element σ(x) ∈ H_{i+1}(E(C)), where

σ: S^{1} ∧ (C/B) → E(C)/B

is the stabilizing B-module map.

For B = Hℤ and C = Hℤ ∧ Hℤ we get E(C) = THH(Hℤ). The bar spectral sequence

E_{∗∗}^{2} = Tor^{H_{∗}(Hℤ∧Hℤ)}(H_{∗}(Hℤ), H_{∗}(Hℤ))
= HH_{∗}(H_{∗}(Hℤ)) ⟹ H_{∗}(THH(Hℤ))

was studied by Marcel Bökstedt. Here

E_{∗∗}^{2} = H_{∗}(Hℤ) ⊗ E(σξ̄_{1}^{2}, σξ̄_{k} | k ≥ 2) .

There are no differentials, for bidegree reasons, but there are multiplicative extensions (σξ̄_{k})^{2} = σξ̄_{k+1} for k ≥ 2, while (σξ̄_{1}^{2})^{2} = 0. Hence

H_{∗}(THH(Hℤ)) = H_{∗}(Hℤ) ⊗ E(σξ̄_{1}^{2}) ⊗ P (σξ̄_{2})

with |σξ̄_{1}^{2}| = 3 and |σξ̄_{2}| = 4, so

π_{∗}(THH(Hℤ)) ≅ (ℤ{1}, 0, 0, ℤ/2{σξ̄_{1}^{2}}, 0, …) .

The stabilization map takes ξ̄_{1}^{2} to σξ̄_{1}^{2}, hence

(Hℤ ∧ Hℤ)/Hℤ → Σ^{−1} THH(Hℤ)/Hℤ

is a π_{i}-isomorphism for all i ≤ 3.

In the next step, we consider E(C) = THH(Hℤ) as an augmented commutative Hℤ-algebra. The bar spectral sequence

E_{∗∗}^{2} = Tor_{∗∗}^{H_{∗}(THH(Hℤ))}(H_{∗}(Hℤ), H_{∗}(Hℤ))
⟹ H_{∗}(E^{2}(Hℤ ∧ Hℤ))

begins with

E_{∗∗}^{2} = H_{∗}(Hℤ) ⊗ Γ(σ^{2}ξ̄_{1}^{2}) ⊗ E(σ^{2}ξ̄_{2})

where Γ denotes the divided power algebra. The generator σ^{2}ξ̄_{1}^{2} is in bidegree (1, 3), its k-th divided power γ_{k}σ^{2}ξ̄_{1}^{2} is in bidegree (k, 3k), and the generator σ^{2}ξ̄_{2} is in bidegree (1, 4). Since Hℤ splits off from E^{2}(Hℤ ∧ Hℤ), there cannot be any differentials on σ^{2}ξ̄_{1}^{2}, γ_{2}σ^{2}ξ̄_{1}^{2} or σ^{2}ξ̄_{2}, but perhaps on γ_{4}σ^{2}ξ̄_{1}^{2}.

((More must be known!))

Hence

π_{∗}(E^{2}(Hℤ ∧ Hℤ)) ≅ (ℤ{1}, 0, 0, 0, ℤ/2{σ^{2}ξ̄_{1}^{2}}, 0, 0, 0, …)

(modulo odd torsion).

Assume inductively, for m ≥ 1, that

H_{∗}(E^{m}(C)) ≅ H_{∗}(Hℤ){1, σ^{m}ξ̄_{1}^{2}, σ^{m}ξ̄_{2}, … } ,

where |σ^{m}ξ̄_{1}^{2}| = m + 2, |σ^{m}ξ̄_{2}| = m + 3 and the remaining terms start in dimension m + 6.

The m-th bar spectral sequence

E_{∗∗}^{2} = Tor^{H_{∗}(E^{m}(C))}(H_{∗}(Hℤ), H_{∗}(Hℤ)) ⟹ H_{∗}(E^{m+1}(C))

starts with

E_{∗∗}^{2} = H_{∗}(Hℤ){1, σ^{m+1}ξ̄_{1}^{2}, σ^{m+1}ξ̄_{2}, … }

with σ^{m+1}ξ̄_{1}^{2} in bidegree (1, m + 2) and σ^{m+1}ξ̄_{2} in bidegree (1, m + 3), while and the remaining terms start in dimension m + 7. Here 2(m + 3) ≥ m + 7 since m ≥ 1. For bidegree reasons there is no room for differentials in this range of total degrees, hence the inductive hypothesis is verified for m + 1.

It follows that

H_{∗}(TAQ(Hℤ)) ≅ H_{∗}(Hℤ){ξ̄_{1}^{2}, ξ̄_{2}, … }

where the remaining terms start in dimension 6. Hence

TAQ_{∗}(Hℤ) ≅ (0, 0, ℤ/2{ξ̄_{1}^{2}}, 0, 0, 0, …) .

Since the stabilization maps take σ^{m}ξ̄_{1}^{2} to σ^{m+1}ξ̄_{1}^{2} for all m ≥ 1, it is also clear that

Σ^{−1} THH(Hℤ)/Hℤ → TAQ(Hℤ)

is a π_{i}-isomorphism for all i ≤ 5.

((Discuss where odd torsion enters?))

References

[B] Basterra, M., André-Quillen cohomology of commutative S-algebras, J. Pure Appl. Algebra 144 (1999), 111–143.

[BM] Basterra, Maria; Mandell, Michael A., Homology and cohomology of E_{∞} ring spectra, Math. Z. 249 (2005), 903–944.

[L] Lazarev, A., Homotopy theory of A_{∞} ring spectra and applications to MU-modules, K-Theory 24 (2001), 243–281.