THH AND TAQ

John Rognes

I. Symmetric ring spectra (September 19th 2006)

We follow Hovey, Shipley and Smith (Symmetric spectra, JAMS, 1999).

I.1. Symmetric sequences.

Let Σ be the skeleton category of finite sets and bijections, with obΣ = ℕ_{0} (the non-negative integers), Σ(n, n) = Σ_{n} (the symmetric group on n letters) for n ≥ 0, and Σ(m, n) = ∅ (the empty set) for m ≠ n.

Disjoint union of finite sets defines a functor +: Σ × Σ → Σ that takes (m, n) to m + n and maps Σ(m, m) × Σ(n, n) = Σ_{m} × Σ_{n} to Σ(m + n, m + n) = Σ_{m+n} by the standard inclusion Σ_{m} × Σ_{n} → Σ_{m+n}. It makes (Σ, +, 0) a permutative (= strict symmetric monoidal) category.

A symmetric sequence of pointed simplicial sets, is a functor X: Σ → 𝒮_{∗}, where 𝒮_{∗} is the category of pointed simplicial sets. Equivalently, it is a sequence of pointed simplicial sets X_{n}, with a pointed left Σ_{n}-action on X_{n}, for each n ≥ 0. Here X_{n} = X(n) and π ∈ Σ_{n} acts on X_{n} by X(π): X(n) → X(n). We refer to X_{n} as the space at the n-th level of X.

A map X → Y of symmetric sequences is the same as a natural transformation of functors from Σ, or equivalently, a sequence of Σ_{n}-equivariant maps X_{n} → Y_{n}, for n ≥ 0. We write 𝒮_{∗}^{Σ} for the category of symmetric sequences. Similarly, we can consider symmetric sequences of pointed topological spaces, or in any other category.

I.2. The tensor product.

Given two symmetric sequences X and Y, we can form their external smash product X ¯⊗Y, which is the functor Σ × Σ → 𝒮_{∗} that takes (p, q) to the smash product X_{p} ∧ Y_{q} of pointed simplicial sets, with the obvious Σ_{p} × Σ_{q}-action.

The left Kan extension of this functor along +: Σ × Σ → Σ is, by definition, the tensor product X ⊗ Y of the two symmetric sequences.

\xymatrix{Σ × Σ \ar[r]^{X ¯⊗Y} \ar[d]_{+} & 𝒮_{∗} \\ Σ \ar[ur]_{X⊗Y}}

Its value at n ≥ 0 is given by the following colimit formed in 𝒮_{∗}, over the left fiber

Typeset by AMS-TEX

category +/n:

(X ⊗ Y)_{n} = colim_{p+q→n} X_{p} ∧ Y_{q}

= ⋁_{p+q=n} Σ_{n+} ∧_{Σ_{p}×Σ_{q}} X_{p} ∧ Y_{q} .

We can present a k-simplex in (X ⊗ Y)_{n} as (α, x ∧ y), where x ∈ X_{p} and y ∈ Y_{q} are k-simplices, and α ∈ Σ_{n} is a morphism p + q → n in Σ. For (σ, τ) ∈ Σ_{p} × Σ_{q} we have (α(σ × τ), x ∧ y) = (α, σ(x) ∧ τ(y)).

The tensor product is coherently associative and unital, having the symmetric sequence I with I_{0} = S^{0} and I_{n} = ∗ (a point) for n > 0 as a unit. The role of the symmetric groups is to also make the tensor product commutative, up to the following coherent isomorphism.

The twist isomorphism τ: X ⊗ Y ⟶^{≅} Y ⊗ X takes (α, x ∧ y) to (αρ, y ∧ x), where ρ: q + p → p + q is the (q, p)-shuffle bijection.

\xymatrix{x ∧ y & p + q \ar[r]^{α} & n \\ y ∧ x & q + p \ar[u]^{ρ} \ar[ur]_{αρ}}

It makes (𝒮_{∗}^{Σ}, ⊗, I) a symmetric monoidal category.

I.3. The sphere symmetric sequence.

Let S^{1} = Δ^{1}/∂Δ^{1} be the simplicial circle, and let S^{n} = S^{1} ∧ ⋯ ∧ S^{1} (n times) be the simplicial n-sphere, for n ≥ 0. The symmetric group Σ_{n} acts on S^{n} by permuting the smash factors. This defines the sphere symmetric sequence S, with S_{n} = S^{n}.

There is a unit map e: I → S that is the identity at level 0.

We now define a multiplication map m: S ⊗ S → S in 𝒮_{∗}^{Σ}. At the level of the n-th spaces, it is the Σ_{n}-equivariant map

⋁_{p+q=n} Σ_{n+} ∧_{Σ_{p}×Σ_{q}} S^{p} ∧ S^{q} → S^{n}

whose restriction to the (p, q)-summand is left adjoint to the identity map S^{p}∧S^{q} → S^{n}, viewed as a (Σ_{p} × Σ_{q} → Σ_{n})-equivariant map.

Key Lemma. (S, m, e) is a commutative monoid in 𝒮_{∗}^{Σ}.

Commutativity is the assertion that m = mτ: S ⊗ S → S.

((Also discuss function objects.))

I.4. Symmetric spectra.

By definition, a symmetric spectrum is a left S-module in the category of symmetric sequences. In other words, it is a symmetric sequence X together with a map S ⊗ X → X that is associative and unital. We write Sp^{Σ} = S-Mod for the category of symmetric spectra.

Equivalently, a symmetric spectrum is a symmetric sequence X together with structure maps

σ: S^{1} ∧ X_{n} → X_{1+n}

for each n ≥ 0, such that the m-fold composite map

σ^{m}: S^{m} ∧ X_{n} → X_{m+n}

is (Σ_{m} × Σ_{n} → Σ_{m+n})-equivariant, for each m, n ≥ 0. Here Σ_{m} acts on S_{m} = S^{m}, Σ_{n} acts on X_{n}, Σ_{m+n} acts on X_{m+n}, and Σ_{m} × Σ_{n} → Σ_{m+n} is the standard inclusion.

To see this, note that the map S ⊗ X → X is a sequence of Σ_{n}-equivariant maps

⋁_{p+q=n} Σ_{n+} ∧_{Σ_{p}∧Σ_{q}} S^{p} ∧ X_{q} → X_{n}

for n ≥ 0, which is equivalent to a sequence of (Σ_{p} × Σ_{q} → Σ_{p+q})-equivariant maps S^{p} ∧ X_{q} → X_{p+q}, for p, q ≥ 0. The associativity condition satisfied by the module action ensures that it is enough to specify these maps for p = 1, i.e., to give the structure maps σ.

I.5. The smash product.

Recall that for a commutative ring R, the tensor product ⊗ of abelian groups gives rise to an internal tensor product ⊗_{R} of left R-modules, given by the coequalizer diagram

\xymatrix{M ⊗ R ⊗ N \ar[r] \ar[r] & M ⊗ N \ar[r] & M ⊗_{R} N .}

Here M and N are left R-modules, but by commutativity M may also be regarded as a right R-module, and the two maps on the left are derived from the module actions M ⊗ R → M and R ⊗ N → N, respectively. By commutativity, again, the left R-module action on M also induces one on the coequalizer M ⊗_{R} N.

Since S is a commutative monoid in symmetric sequences, the category Sp^{Σ} = S-Mod can be given an internal pairing ⊗_{S} in the same way, which we prefer to denote by ∧. So for two symmetric spectra X and Y, we define their smash product X ∧Y by the coequalizer diagram

\xymatrix{X ⊗ S ⊗ Y \ar[r] \ar[r] & X ⊗ Y \ar[r] & X ∧ Y .}

In other words, X ∧Y is the colimit in symmetric sequences of the left hand part of the diagram, and this limit can be formed level-wise as the colimit of the diagram

\xymatrix{⋁_{i+j+k=n} Σ_{n+} ∧_{Σ_{i}×Σ_{j}×Σ_{k}} X_{i} ∧ S^{j} ∧ Y_{k} \ar[r] \ar[r] & ⋁_{p+q=n} Σ_{n+} ∧_{Σ_{p}×Σ_{q}} X_{p} ∧ Y_{q}}

of pointed Σ_{n}-spaces. As in the algebraic case, the resulting symmetric sequence X ∧ Y is naturally a symmetric spectrum.

The smash product of symmetric spectra is coherently associative and unital, having the sphere spectrum S as a unit. The twist isomorphism of symmetric sequences also induces a twist isomorphism τ: X ∧ Y ⟶^{≅} Y ∧ X of symmetric spectra.

Proposition. (Sp^{Σ}, ∧, S) is a symmetric monoidal category.

((Also discuss function spectra.))

I.6. Algebraic structures.

A symmetric ring spectrum is a symmetric spectrum R with a unit map e: S → R and a multiplication map m: R ∧ R → R, making the diagrams

\xymatrix{(R ∧ R) ∧ R \ar[rr]^{≅} \ar[d]_{m∧id} &  & R ∧ (R ∧ R) \ar[d]^{id∧m} \\ R ∧ R \ar[r]^{m} & R & R ∧ R \ar[l]_{m}}

(associativity) and

\xymatrix{S ∧ R \ar[r]^{e∧id} \ar[dr]_{≅} & R ∧ R \ar[d]^{m} & R ∧ S \ar[l]_{id∧e} \ar[dl]^{≅} \\  & R}

(unitality) commute. If furthermore the diagram

\xymatrix{R ∧ R \ar[rr]^{τ}_{≅} \ar[dr]_{m} &  & R ∧ R \ar[dl]^{m} \\  & R}

(commutativity) commutes, then R is a commutative symmetric ring spectrum.

Let R be a symmetric ring spectrum. A left R-module is a symmetric spectrum M with a map R∧M → M, such that the usual associativity and unitality diagrams commute. Similarly for right modules. We sometimes refer to R-modules as Rmodule spectra, for emphasis. Let R-Mod denote the category of R-modules and R-module maps.

If M is a right R-module and N is a left R-module, the smash product M ∧_{R} N is defined as the coequalizer

\xymatrix{M ∧ R ∧ N \ar[r] \ar[r] & M ∧ N \ar[r] & M ∧_{R} N}

in symmetric spectra.

If R is commutative (so that the distinction between left and right R-modules disappears), then M ∧_{R} N is also naturally an R-module. In this case, (R-Mod, ∧_{R}, R) is a symmetric monoidal category. An R-algebra A is a monoid in R-Mod. This amounts to a map R → A of symmetric ring spectra, such that R is central in A. A commutative R-algebra A is a commutative monoid in R-Mod. This amounts to a map R → A of commutative symmetric ring spectra.

I.7. Homotopy and homology.

There is a stable model structure on symmetric spectra, whose associated homotopy category Ho(Sp^{Σ}) is equivalent to Boardman’s stable homotopy category. Furthermore, the smash product of symmetric spectra induces a symmetric monoidal pairing on Ho(Sp^{Σ}) that agrees with the smash product in Boardman’s category, under the claimed equivalence.

The k-th homotopy group of a symmetric spectrum X is defined as the sequential colimit

π_{k}(X) = colim_{n} π_{n+k}(X_{n})

over the composite maps

π_{n+k}(X_{n}) ⟶^{E} π_{1+n+k}(S^{1} ∧ X_{n}) ⟶^{σ_{∗}} π_{1+n+k}(X_{n+1}) .

Taken together, these form a graded abelian group π_{∗}(X). The action by the symmetric groups plays no role here, so these are the homotopy groups of the underlying sequential spectrum of X. For a symmetric ring spectrum R, π_{∗}(R) is a graded ring. For a commutative symmetric ring spectrum R, π_{∗}(R) is a graded commutative ring.

A map of symmetric spectra X → Y that induces an isomorphism π_{∗}(X) → π_{∗}(Y) is called a π_{∗}-equivalence. Each π_{∗}-equivalence is a weak equivalence in the stable model structure on Sp^{Σ}, i.e., a stable equivalence, but the converse does not generally hold.

The stably fibrant symmetric spectra are the symmetric Ω-spectra, i.e., those X for which each X_{n} is a Kan (= fibrant) simplicial set, and each adjoint structure map σ̃: X_{n} → ΩX_{n+1} = \underline{𝒮_{∗}}(S^{1}, X_{n+1}) is a weak equivalence. For maps X → Y between symmetric Ω-spectra, the stable equivalences are the same as the π_{∗}-equivalences.

((Discuss orthogonal spectra. For symmetric spectra coming from orthogonal spectra, stable equivalences and π_{∗}-equivalences agree. So for these, the homotopy groups are the homotopy invariant ones.))

I.8. The sphere spectrum.

The sphere spectrum S is a commutative symmetric ring spectrum, with the unit map e: S → S being the identity and the multiplication map m: S ∧ S → S being the natural isomorphism.

An S-algebra is the same as a symmetric ring spectrum, and a commutative S-algebra is the same as a commutative symmetric ring spectrum.

The homotopy groups π_{∗}(S) = π_{∗}^{S} are the stable homotopy groups of spheres.

I.9 Eilenberg–Mac Lane spectra.

Let A be an abelian group. For each set U let A[U] = ⨁_{u∈U} A, and for each pointed set U let A(U) = A[U]/A[∗], where ∗ is the base point in U. This construction is natural in U, so that each pointed function f: U → V induces an abelian group homomorphism A(f): A(U) → A(V). By forgetting structure, we can view A(f) as a function of pointed sets.

For each simplicial set W_{•}, let A(W_{•}) be the pointed simplicial set [q] ↦ A(W_{q}). This makes sense by the naturality property just stated.

For example, when W_{•} = S^{1} = Δ^{1}/∂Δ^{1}, Δ_{q}^{1} consists of the (q + 2) morphisms [q] → [1] in Δ, so W_{q} = S_{q}^{1} consists of the q surjective morphisms [q] → [1], together with the base point. Hence A(S_{q}^{1}) = A ⊕ ⋯ ⊕ A = A^{q}, and A(S^{1}) = BA is the bar construction on A. We usually write [a_{1}| … |a_{q}] for an element of A(S_{q}^{1}) = BA_{q}.

Iterating n times, the Eilenberg–Mac Lane spectrum HA is defined as the symmetric spectrum with n-th space HA_{n} = A(S^{n}) = B^{n}A, where Σ_{n} permutes the smash factors in S^{n} = S^{1} ∧ ⋯ ∧ S^{1}, or equivalently, the ordering of the n bar constructions. The structure map σ: S^{1} ∧ HA_{n} → HA_{1+n} is given in degree q by the inclusion S_{q}^{1} ∧ A(S_{q}^{n}) → A(S_{q}^{1} ∧ S_{q}^{n}).

The n-fold bar construction HA_{n} = B^{n}A is a K(A, n)-complex, with homotopy groups π_{n+k}B^{n}A = A for k = 0 and = 0 for k ≠ 0. Being a simplicial (abelian) group, it is a Kan simplicial set. Furthermore the adjoint structure map σ̃: HA_{n} → ΩHA_{1+n} equals the weak homotopy equivalence B^{n}A → ΩB^{1+n}A, so HA is a symmetric Ω-spectrum (= stably fibrant). Hence the homotopy groups π_{k}HA = A for k = 0 and = 0 for k ≠ 0 are homotopically significant.

Let R be an associative ring. There is a natural multiplication map m: HR ∧ HR → HR and unit map e: S → HR that make HR a symmetric ring spectrum, alias an S-algebra. The multiplication map is derived from a map HR⊗HR → HR of symmetric sequences, which at the n-th level is a wedge sum over (p, q) with p + q = n of Σ_{n}-equivariant maps. These are left adjoint to the (Σ_{p} × Σ_{q} → Σ_{n})- equivariant maps

R(S^{p}) ∧ R(S^{q}) → R(S^{p} ∧ S^{q}) = R(S^{n})

given by

∑_{i} r_{i}x_{i} ∧ ∑_{j} r′_{j}y_{j} ↦ ∑_{i,j} r_{i}r′_{j}(x_{i} ∧ y_{j})

where r_{i}, r′_{j} ∈ R, the x_{i} are simplices in S^{p} and the y_{j} are simplices in S^{q}. The product r_{i}r′_{j} is formed in the ring R.

The opposite multiplication, mτ: HR ∧ HR → HR is likewise derived from the map given by

∑_{i} r_{i}x_{i} ∧ ∑_{j} r′_{j}y_{j} ↦ ρ(∑_{i,j} r′_{j}r_{i}(y_{j} ∧ x_{i})) = ∑_{i,j} r′_{j}r_{i}(x_{i} ∧ y_{j}) .

Hence HR is a commutative symmetric ring spectrum if and only if R is a commutative ring.

The Eilenberg–Mac Lane functor A ↦ HA embeds abelian groups into symmetric spectra, rings into symmetric ring spectra, and more generally embeds algebra into topology in the form of stable homotopy theory.

Proposition. Let R be a ring, M a right R-module and N a left R-module. Then HM is a left HR-module, HN is a left HR-module, and

π_{i}(HM ∧_{HR} HN) ≅ Tor_{i}^{R}(M, N) .

((Similarly, π_{i}F_{HR}(HM, HN) ≅ Ext_{R}^{−i}(M, N) for two left R-modules M and N, where F_{HR} denotes the HR-module function spectrum.))

I.10 Thom spectra.

To any Euclidean ℝ^{n}-bundle ξ: E → X, with principal O(n)-bundle P → X, we can associate the Thom complex T h(ξ) = P_{+} ∧_{O(n)} S^{n}. Now S^{n} = S^{ℝ^{n}} = ℝ^{n} ∪{∞} as a pointed topological space, still with S^{n} ≅ S^{1} ∧ ⋯ ∧ S^{1}.

((Universal case γ^{n} with principal bundle EO(n) → BO(n).))

((See also Chapter V.))

II. THH: structural properties (October 3rd 2006)

Recall the Hochschild–Kostant–Rosenberg theorem: for a smooth algebra A over a commutative ring k, the Hochschild homology HH_{∗}(A) is isomorphic to the exterior algebra Ω_{A}^{∗} of differential forms on A, generated by the Kähler differentials HH_{1}(A) ≅ Ω_{A}^{1}. Connes’ B-operator HH_{n}(A) → HH_{n+1}(A) then corresponds to the exterior derivation d: Ω_{A}^{n} → Ω_{A}^{n+1}. For commutative, non-smooth A there is still a map Ω_{A}^{∗} → HH_{∗}(A), and the Hochschild homology can even be defined for non-commutative A. It is therefore common to interpret HH_{∗}(A), with Connes’ operator, as a kind of non-commutative de Rham complex over A.

We will study the extension of this de Rham theory to brave new rings, i.e., to associative S-algebras, where S is the sphere spectrum. To be concrete, let B be a symmetric ring spectrum. Its topological Hochschild homology, denoted T HH(B), can be defined in several equivalent ways. We begin with one that is rather explicit.

II.1. The Hochschild complex.

Definition. Let T HH(B)_{•} be the simplicial symmetric spectrum

[q] ↦ T HH(B)_{q} = B ∧ B ∧ ⋯ ∧ B

((q + 1) copies of B), with face maps

d_{i}(b_{0} ∧ b_{1} ∧ ⋯ ∧ b_{q}) = \{ \begin{matrix}b_{q}b_{0} ∧ b_{1} ∧ ⋯ ∧ b_{q−1} & for i = 0, \\ b_{0} ∧ ⋯ ∧ b_{i}b_{i+1} ∧ ⋯ ∧ b_{q} & for 0 < i ≤ q, \end{matrix}

and degeneracy maps

s_{j}(b_{0} ∧ b_{1} ∧ ⋯ ∧ b_{q}) = b_{0} ∧ ⋯ ∧ b_{j} ∧ 1 ∧ b_{j+1} ∧ ⋯ ∧ b_{q}

for 0 ≤ j ≤ q. Let

T HH(B) = |T HH(B)_{•}|

be the geometric realization.

These formulas must be interpreted in terms of the product and unit maps μ: B ∧ B → B and η: S → B, and the cyclic twist map τ: B^{∧(q+1)} → B^{∧(q+1)}. Note that 1 ∧ b_{0} ∧ ⋯ ∧ b_{q} is not degenerate, unless some b_{j} = 1.

The explicit Hochschild complex can also be interpreted as the homology of B as a B-bimodule (= left B ∧ B^{op}-module):

T HH(B) = Tor^{B∧B^{op}}(B, B) := B ∧_{B∧B^{op}} B .

If B is resolved as a B-bimodule by the two-sided bar complex

β(B, B, B)_{•}: [q] ↦ B ∧ B ∧ ⋯ ∧ B ∧ B

((q + 2) copies of B), then T HH(B)_{•} = B ∧_{B∧B^{op}} β(B, B, B)_{•}.

II.2. Cyclic structure.

The simplicial construction admits the refined structure of being a cyclic object.

Definition. T HH(B)_{•} is a cyclic symmetric spectrum, with cyclic twist operators

t_{q}(b_{0} ∧ b_{1} ∧ ⋯ ∧ b_{q}) = b_{q} ∧ b_{0} ∧ ⋯ ∧ b_{q−1} .

Hence T HH(B) naturally admits a left S^{1}-action

α: S_{+}^{1} ∧ T HH(B) → T HH(B) .

For example, the S^{1}-action takes a 0-simplex b ∈ T HH(B)_{0} once around the geometric realization of the 1-simplex t_{1}s_{0}(b) = 1 ∧ b.

The inclusion of 0-simplices defines a map η: B → T HH(B). When combined with the S^{1}-action, we obtain a map

ω: S_{+}^{1} ∧ B → T HH(B) .

The retraction S_{+}^{1} → 1_{+} in the cofiber sequence

1_{+} → S_{+}^{1} → S^{1}

defines a preferred stable section σ: S^{1} → S_{+}^{1}. Combined with the S^{1}-action on T HH(B), it defines a stable map α(σ ∧ id) that we denote

d: ΣT HH(B) → T HH(B) .

((The author usually writes σ in place of d for this map, called the suspension operator.)) In terms of this preferred splitting of S_{+}^{1} ∧ T HH(B), we can write α as (id, d).

As observed by Hesselholt (1996, 1.4.4), the iterated map dd: Σ^{2}T HH(B) → T HH(B) satisfies

dd = dη = ηd ,

where now η ∈ π_{1}(S) denotes the stable Hopf map. For the composite stable map

S^{1} ∧ S^{1} ⟶^{σ∧σ} S_{+}^{1} ∧ S_{+}^{1} ≅ (S^{1} × S^{1})_{+} ⟶^{m_{+}} S_{+}^{1}

factors as the stable map

S^{1} ∧ S^{1} ⟶^{η} S^{1} ⟶^{σ} S_{+}^{1} ,

where η arises (after one suspension) as the Hopf construction on the multiplication m: S^{1} × S^{1} → S^{1}. See [Ha61] for details. Thus dd is null-homotopic whenever η acts trivially on T HH(B), up to homotopy.

II.3. Tensored structure.

When B is commutative, there is a more concise definition

T HH(B)_{•} = B ⊗ S_{•}^{1} ,

where S_{•}^{1} = Δ^{1}/∂Δ^{1} is the simplicial circle, with (q + 1) simplices in degree q.

The tensor product B ⊗ X_{•} for simplicial sets X_{•} is defined by prolongation of the smash power

B ⊗ Y = ⋀_{y∈Y} B ,

defined for (finite) sets Y. The two-sided bar construction can then be written as β(B, B, B)_{•} = B ⊗ Δ^{1}, where Δ^{1} has (q + 2) simplices in degree q.

The topological formula T HH(B) = B ⊗ S^{1} also exhibits the S^{1}-action α, by rotation in the S^{1}-term. Since these constructions take place in the category of commutative symmetric ring spectra, the unit map

η: B = B ⊗ 1 → B ⊗ S^{1} = T HH(B)

makes T HH(B) a commutative B-algebra. The collapse map S^{1} → 1 induces an augmentation

ϵ: T HH(B) = B ⊗ S^{1} → B ⊗ 1 = B .

The pinch map S^{1} → S^{1} ∨S^{1}, the fold map S^{1} ∨S^{1} → S^{1} and the flip map S^{1} → S^{1} induce maps

ψ: T HH(B) → T HH(B) ∧_{B} T HH(B)
ϕ: T HH(B) ∧_{B} T HH(B) → T HH(B)
χ: T HH(B) → T HH(B)

that make T HH(B) a commutative Hopf algebra over B.

Note that ψ takes values in the smash product over B, not in the smash product over S. Furthermore, it is only A_{∞} coassociative and counital. Similarly, χ is only a homotopy inverse.

II.4. E_{n}-algebras.

Intermediate between strictly associative ring spectra and strictly commutative ring spectra, we have the notion of an E_{n} ring spectrum, which comes with an action of an operad weakly equivalent to the little n-cubes operad. The E_{1} ring spectra are the same as A_{∞} ring spectra, which can be rigidified to strictly associative ring spectra, while the E_{∞} ring spectra can be rigidified to strictly commutative ring spectra. By restriction of the operad action, an E_{n+1} ring spectrum is in particular an E_{n} ring spectrum.

Fiedorowicz and Vogt show that an E_{n+1} ring spectrum B can be rigidified to an E_{n} algebra in strictly associative ring spectra, i.e., to a strictly associative ring spectrum with an action by an E_{n} operad, in that category. This E_{n} operad action carries over for topological functors from strictly associative ring spectra to other categories. Basterra and Mandell have announced similar results.

For example, T HH is a functor from symmetric ring spectra to symmetric spectra with S^{1}-action. Hence T HH of any E_{n+1} ring spectrum B produces an E_{n} algebra in symmetric spectra with S^{1}-action. In other words, T HH(B) is then an E_{n} ring spectrum with S^{1}-action. Similarly, the natural map η: B → T HH(B) is then an E_{n} ring spectrum map.

For n = ∞, this recovers the fact that T HH of a commutative symmetric ring spectrum B is again a commutative B-algebra. However, there are natural examples of spectra that admit an E_{n} ring spectrum structure for some finite n ≥ 2, that are not known to admit an E_{∞} structure.

II.5. Many objects.

There is also a more flexible definition, in terms of the category of finite cell B-modules, enriched in symmetric spectra.

Definition. Let 𝒞 be a small category enriched in symmetric spectra. Its topological Hochschild homology T HH(𝒞) is the geometric realization of the cyclic symmetric spectrum

T HH(𝒞)_{q} = ⋁_{c_{0},...,c_{q}∈ob𝒞} 𝒞(c_{0}, c_{q}) ∧ 𝒞(c_{1}, c_{0}) ∧ ⋯ ∧ 𝒞(c_{q}, c_{q−1}) ,

with simplicial face and degeneracy operators, and cyclic operators, suggested by composition, identity and twist in the diagram

\xymatrix{c_{0} \ar[rrr] & c_{1} \ar[l] & … \ar[l] & c_{q} \ar[l]}

When 𝒞 is the enriched category with only one object ∗, and 𝒞(∗, ∗) = B, then this definition simplifies to the original definition of T HH(B).

For another example, let 𝒞 = ℱ_{B} be the category of finitely generated free Bmodules, with objects ⋁^{n} B = B ∧ n_{+} for n ≥ 0, and morphisms

ℱ_{B}(⋁^{m} B, ⋁^{n} B) = ∏_{m} ⋁^{n} B

as symmetric spectra. The previous example embeds into this one, taking ∗ to the free B-module of rank 1, and the induced map

T HH(B) → T HH(ℱ_{B})

is an equivalence. See Dundas et al, Lemma 2.5.17, for the proof when B is connective. We refer to this equivalence as Morita equivalence.

Presumably, letting 𝒞 = 𝒞_{B} be the category of finite cell B-modules and Bmodule maps, the inclusion ℱ_{B} → 𝒞_{B} also induces an equivalence

T HH(ℱ_{B}) → T HH(𝒞_{B}) .

((Find a reference.))

It may also be useful to work with T HH(𝒞), where 𝒞 is a category of coherent crystals over B.

II.6. Étale maps.

Let A be a commutative S-algebra, and let B be an A-algebra, commutative or not. We can define the relative topological Hochschild homology T HH^{A}(B) as the geometric realization of the simplicial spectrum

[q] ↦ T HH^{A}(B)_{q} = B ∧_{A} B ∧_{A} ⋯ ∧_{A} B .

There is a commutative diagram

\xymatrix{A \ar[r] \ar[d] & T HH(A) \ar[r]^{ϵ} \ar[d] & A \ar[d] \\ B \ar[r] & T HH(B) \ar[r] & T HH^{A}(B) .}

By assumption, A is central in B, so T HH(B) is a T HH(A)-module. The induced map

T HH(B) ∧_{T HH(A)} A → T HH^{A}(B)

is an equivalence, so the left hand side may also be taken as the definition of the relative topological Hochschild homology. ((Check this for non-commutative B!)) If B is commutative, this map is an equivalence of commutative B-algebras.

Definition. We say that A → B is formally thh-étale if the relative unit map η: B → T HH^{A}(B) is an equivalence.

In that case, the space of associative A-algebra derivations of B with values in any symmetric bimodule M is contractible, and conversely. This uses the cofiber sequence

B ∧_{B∧_{A}B^{op}} I_{B/A} → B → T HH^{A}(B)

and the equivalences

𝒜Der_{A}(B, M) ≃ ℳ_{B∧_{A}B^{op}}(I_{B/A}, M) ≃ ℳ_{B}(B ∧_{B∧_{A}B^{op}} I_{B/A}, M)

for symmetric B-modules M. See Lazarev (2001) and Rognes (Galois, Proposition 9.2.5). Here I_{B/A} is the homotopy fiber of B ∧_{A} B → B.

II.7. Quasi-coherence.

Lemma. If the natural map

B ∧_{A} T HH(A) → T HH(B)

is a weak equivalence, then A → B is formally thh-étale.

Conversely, if A → B is formally thh-étale and either (1) ϵ: T HH(A) → A is faithful, (2) A and B are connective, or (3) A → B is separable, then B ∧_{A} T HH(A) → T HH(B) is a weak equivalence.

Proof. If the induced map

B ∧_{A} T HH(A) → T HH(B)

is an equivalence, then A → B is formally thh-étale, by base change along ϵ.

Conversely, if we assume that ϵ: T HH(A) → A is faithful, then B∧_{A}T HH(A) → T HH(B) is an equivalence for all formally étale A → B, so that T HH is a quasicoherent étale sheaf over A.

Similarly, if A and B are connective, and A → B is formally étale, then B ∧_{A} T HH(A) → T HH(B) is an equivalence. For its cofiber C is then connective and satisfies C ∧_{T HH(A)} A ≃ ∗, which implies C ≃ ∗ since π_{0}T HH(A) ≅ π_{0}A. These hypotheses are most relevant when A = HR and B = HT are both ordinary rings, with R → T (formally) étale, when they show that T HH is a quasi-coherent étale sheaf over A = HR.

Along the same lines, there are equivalences

B ∧_{A} T HH(A) = B ∧_{A} A ∧_{A∧A} A ≃ B ∧_{A∧A} A

≅ A ∧_{A∧A} B ≃ (B ∧_{A} B^{op}) ∧_{B∧B^{op}} B = T HH(B, B ∧_{A} B) ,

which is equivalent to T HH(B) if T HH(B, I_{B/A}) ≃ ∗. This condition holds if A → B is separable, so that μ: B ∧_{A} B → B admits a bimodule section σ, and the resulting idempotent δ = σμ in π_{0}(B∧_{A}B^{op}) = B_{0}^{A}(B) lifts to π_{0}(B∧B^{op}) = B_{0}(B). For then B ≃ B[δ^{−1}] as a B ∧ B^{op}-module, while I_{B/A}[δ^{−1}] ≃ ∗. Compare Geller and Weibel (1991). □

((In what generality is B_{0}^{A}(B) → B_{0}(B) surjective on idempotents?))

II.8. Galois descent (October 24th 2006).

Lemma. If A → B is a faithful G-Galois extension, with G finite, then

i: T HH(A) → T HH(B)^{hG}

is a weak equivalence.

Proof. It suffices to show that 1 ∧ i: B ∧_{A} T HH(A) → B ∧_{A} T HH(B)^{hG} is a weak equivalence, since B is faithful over A. To see this, we factor 1 ∧ i as the following chain of weak equivalences:

B ∧_{A} T HH(B)^{hG} ⟶ (B ∧_{A} T HH(B))^{hG}
⟵ (B ∧_{A} B ∧_{A} T HH(A))^{hG}
⟶ (F (G_{+}, B) ∧_{A} T HH(A))^{hG}
⟶ F (G_{+}, B ∧_{A} T HH(A))^{hG} ≃ B ∧_{A} T HH(A)

These use that B is dualizable over A, B is separable over A, B is Galois over A and G is finite, respectively. □

Are there similar quasi-coherence (resp. Galois descent) results for relative theories T HH(A|K) and T HH(B|L) (to be defined later), under weaker hypotheses (tame ramification, resp. something more general) on A → B?

References

[Ha61] Hardie, K.A., A generalization of the Hopf construction, Q. J. Math., Oxf. II. Ser. 12 (1961), 196–204.

III. THH: calculations (October 10th 2006)

To make calculations, we compare the spectrum level structures with the induced structures at the level of homotopy and homology, usually with mod p coefficients.

III.1. Homotopy.

For spectra X and Y we write [X, Y] for the abelian group of maps from X to Y in the stable homotopy category. More generally, [X, Y]_{n} = [Σ^{n}X, Y] is the abelian group of degree n maps, and [X, Y]_{∗} is the resulting graded abelian group.

The homotopy groups of a spectrum X are defined by

π_{∗}(X) = [S, X]_{∗} .

The stable homotopy groups of spheres is the graded commutative ring

π_{∗}(S) = [S, S]_{∗} ,

which naturally acts on π_{∗}(X) by composition:

π_{∗}(S) ⊗_{ℤ} π_{∗}(X) = [S, S]_{∗} ⊗_{ℤ} [S, X]_{∗} ⟶^{∘} [S, X]_{∗} = π_{∗}(X) .

However, the groups π_{∗}(S) are mostly unknown, and its homological properties are terrible, so it is usually not convenient to work with the π_{∗}(S)-module π_{∗}(X) as an algebraic invariant of X.

III.2. Homology.

Let p be a prime, and write H𝔽_{p} for the mod p Eilenberg–Mac Lane spectrum. It is a commutative S-algebra, with unit η: S → H𝔽_{p} and product μ: H𝔽_{p} ∧ H𝔽_{p} → H𝔽_{p}. We have graded 𝔽_{p}-modules

H_{∗}(X; 𝔽_{p}) = π_{∗}(H𝔽_{p} ∧ X)

and

H^{∗}(X; 𝔽_{p}) = [X, H𝔽_{p}]_{−∗}

for all spectra X. There are universal coefficient and Künneth isomorphisms

H^{∗}(X; 𝔽_{p}) ≅ Hom(H_{∗}(X; 𝔽_{p}), 𝔽_{p})

and

H_{∗}(X; 𝔽_{p}) ⊗ H_{∗}(Y; 𝔽_{p}) ≅ H_{∗}(X ∧ Y; 𝔽_{p})

for all spectra X and Y, with Hom and ⊗ formed over 𝔽_{p}. These extend the usual results for the reduced mod p (co-)homology of based spaces.

The mod p Steenrod algebra is the non-commutative algebra

A = H^{∗}(H𝔽_{p}; 𝔽_{p}) = [H𝔽_{p}, H𝔽_{p}]_{−∗}

of stable cohomology operations, with multiplication ϕ defined by composition. It admits a cocommutative coproduct, and the dual Steenrod algebra

A_{∗} = H_{∗}(H𝔽_{p}; 𝔽_{p}) = π_{∗}(H𝔽_{p} ∧ H𝔽_{p})

is the Hom-dual commutative algebra, with a non-cocommutative coproduct ψ. In other words, A and A_{∗} are dual Hopf algebras. The (canonical) coproduct χ on A_{∗} is induced by the twist map τ: H𝔽_{p} ∧ H𝔽_{p} → H𝔽_{p} ∧ H𝔽_{p}.

For each spectrum X, composition of stable maps defines a homomorphism

A ⊗ H^{∗}(X; 𝔽_{p}) = [H𝔽_{p}, H𝔽_{p}]_{−∗} ⊗ [X, H𝔽_{p}]_{−∗} ⟶^{∘} [X, H𝔽_{p}]_{−∗} = H^{∗}(X; 𝔽_{p})

that makes H^{∗}(X; 𝔽_{p}) a left A-module. In the special case X = H𝔽_{p}, this recovers the product ϕ. Dually, for each spectrum X the natural map

X ≅ S ∧ X ⟶^{η∧1} H𝔽_{p} ∧ X

induces a homomorphism

ν: H_{∗}(X; 𝔽_{p}) → H_{∗}(H𝔽_{p} ∧ X; 𝔽_{p}) ≅ A_{∗} ⊗ H_{∗}(X; 𝔽_{p}) .

In the special case X = H𝔽_{p}, this is the coproduct ψ: A_{∗} → A_{∗} ⊗ A_{∗}. In general, the coaction map ν makes H_{∗}(X; 𝔽_{p}) into an A_{∗}-comodule.

For spectra X of finite type, for which H_{∗}(X; 𝔽_{p}) is finite in each degree, we can go back and forth between the A-module H^{∗}(X; 𝔽_{p}) and the A_{∗}-comodule H_{∗}(X; 𝔽_{p}), via Hom-duality. However, when H_{∗}(X; 𝔽_{p}) not finite in each degree, it is better to work in homology. We therefore will view mod p homology as a functor from spectra to A_{∗}-comodules. This covariant point of view is also more convenient when discussing algebra structures, etc. ((Explain A_{∗}-comodule structure on tensor products and Hom-duals?))

To go back from homology to homotopy we can then use the Adams spectral sequence

E_{2}^{s,t} = Ext_{A_{∗}}^{s,t}(𝔽_{p}, H_{∗}(X; 𝔽_{p})) ⟹ π_{t−s}(X_{p}) .

When X is of finite type, π_{∗}(X_{p}) ≅ π_{∗}(X) ⊗_{ℤ} ℤ_{p}, otherwise the p-completion is more subtle.

III.3. The dual Steenrod algebra.

The structure of the dual Steenrod algebra was described by Milnor. For p odd, we have

A_{∗} = P (ξ_{k} | k ≥ 1) ⊗ E(τ_{k} | k ≥ 0)

where |ξ_{k}| = 2p^{k} − 2 and |τ_{k}| = 2p^{k} − 1. Here P (−) denotes the polynomial algebra over 𝔽_{p} on the listed generators, and E(−) denotes the exterior algebra over 𝔽_{p}. The coproduct is given by the formulas

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} .

In all such formulas, ξ_{0} is to be read as 1. The class τ_{0} is dual to the Bockstein operation β, while ξ_{1}^{k} is dual to the Steenrod reduced power operation P^{k}.

((Discuss relation to H_{∗}(K(𝔽_{p}, 1); 𝔽_{p}).))

It will be more convenient for us to work with the conjugate generators ξ̄_{k} = χξ_{k} and τ̄_{k} = χτ_{k}. We still have

A_{∗} = P (ξ̄_{k} | k ≥ 1) ⊗ E(τ̄_{k} | k ≥ 0)

with |ξ̄_{k}| = 2p^{k} − 2 and |τ̄_{k}| = 2p^{k} − 1, but

ψ(ξ̄_{k}) = ∑_{i+j=k} ξ̄_{i} ⊗ ξ̄_{j}^{p^{i}}

and

ψ(τ̄_{k}) = 1 ⊗ τ̄_{k} + ∑_{i+j=k} τ̄_{i} ⊗ ξ̄_{j}^{p^{i}} .

As before, ξ̄_{0} is to be read as 1.

((Include formulas for p = 2, too.))

III.4. Homology of THH.

Let B be a symmetric ring spectrum. The 0-simplex inclusion η: B → T HH(B) and the circle action α: S_{+}^{1} ∧ T HH(B) → T HH(B) induce A_{∗}-comodule homomorphisms η : H_{∗}(B; 𝔽_{p}) → H_{∗}(T HH(B); 𝔽_{p}) and

α: E(s_{1}) ⊗ H_{∗}(T HH(B); 𝔽_{p}) → H_{∗}(T HH(B); 𝔽_{p}) ,

respectively. Here H_{∗}(S_{+}^{1}; 𝔽_{p}) = E(s_{1}), with s_{1} in homological degree 1. The stable map d: ΣT HH(B) → T HH(B) induces the A_{∗}-comodule homomorphism

d: H_{∗}(T HH(B); 𝔽_{p}) → H_{∗+1}(T HH(B); 𝔽_{p})

given by d(x) = α(s_{1} ⊗ x).

The Hopf map η: S^{1} → S^{0} induces the zero map on homology, so dd = 0 at the level of homology, i.e., d is a differential on H_{∗}(T HH(B); 𝔽_{p}).

The adjoint map α̃: T HH(B) → F (S_{+}^{1}, T HH(B)) to the circle action, and the canonical stable equivalence ν: DS_{+}^{1} ∧ T HH(B) → F (S_{+}^{1}, T HH(B)), induce an A_{∗}-comodule homomorphism

ν^{−1}α̃: H_{∗}(T HH(B); 𝔽_{p}) → E(ι_{1}) ⊗ H_{∗}(T HH(B); 𝔽_{p})

given by

(ν^{−1}α̃)(x) = 1 ⊗ x + ι_{1} ⊗ dx .

Here DS_{+}^{1} = F (S_{+}^{1}, S) denotes the functional dual, and

H_{∗}(DS_{+}^{1}; 𝔽_{p}) ≅ H^{−∗}(S^{1}; 𝔽_{p}) = E(ι_{1}) ,

with ι_{1} in homological degree (−1), dual to s_{1}.

If B is a commutative symmetric ring spectrum, then α̃ and ν are commutative ring spectrum maps. Here we give F (S_{+}^{1}, T HH(B)) and DS_{+}^{1} = F (S_{+}^{1}, S) the pointwise multiplications, induced from the products on T HH(B) and S, respectively, and the diagonal coproduct on S_{+}^{1}. It follows that ν^{−1}α̃ is a H_{∗}(B; 𝔽_{p})- algebra homomorphism, so

(ν^{−1}α̃)(xy) = 1 ⊗ xy + ι_{1} ⊗ d(xy)

equals

(ν^{−1}α̃)(x) · (ν^{−1}α̃)(y) = (1 ⊗ x + ι_{1} ⊗ dx) · (1 ⊗ y + ι_{1} ⊗ dy)
= 1 ⊗ xy + ι_{1} ⊗ (dx · y + (−1)^{|x|}x · dy) .

Hence

d(xy) = dx · y + (−1)^{|x|}x · dy ,

and d acts as a derivation on H_{∗}(T HH(B); 𝔽_{p}).

III.5. The Bökstedt spectral sequence.

Proposition. Let B be a symmetric ring spectrum. (a) The skeleton filtration of T HH(B) induces a spectral sequence of A_{∗}-comodules, with

E_{q,∗}^{1}(B) = H_{∗}(B; 𝔽_{p}) ⊗ H̄_{∗}(B; 𝔽_{p}) ⊗ ⋯ ⊗ H̄_{∗}(B; 𝔽_{p})

(with q copies of H̄_{∗}(B; 𝔽_{p}) = coker(η_{∗}: 𝔽_{p} → H_{∗}(B; 𝔽_{p})) and

E_{q,∗}^{2}(B) = HH_{q}(H_{∗}(B; 𝔽_{p})) ,

converging strongly to H_{∗}(T HH(B); 𝔽_{p}).

(b) The edge homomorphism

E_{0,∗}^{1}(B) = H_{∗}(B; 𝔽_{p}) → HH_{0}(H_{∗}(B; 𝔽_{p})) = E_{0,∗}^{2}(B) → H_{∗}(T HH(B); 𝔽_{p})

equals the homomorphism induced by the inclusion η: B → T HH(B) of 0-simplices. (c) Connes’ operator

E_{q,∗}^{2}(B) = HH_{q}(H_{∗}(B; 𝔽_{p})) → HH_{q+1}(H_{∗}(B; 𝔽_{p})) = E_{q+1,∗}^{2}(B) abuts to the homomorphism d: H_{∗}(T HH(B); 𝔽_{p}) → H_{∗+1}(T HH(B); 𝔽_{p}) induced by the map d: ΣT HH(B) → T HH(B).

Proof. This is the spectral sequence associated to the unraveled exact couple of A_{∗}-comodules obtained by applying H_{∗}(−; 𝔽_{p}) to the cofiber sequences

|sk_{q−1}T HH(B)| → |sk_{q}T HH(B)| → S^{q} ∧ T HH(B)_{q}^{nd} ,

where T HH(B)_{q}^{nd} = B ∧ B̄ ∧ ⋯ ∧ B̄ models the non-degenerate q-simplices. Here B̄ denotes the cofiber of η: S → B, which we implicitly take to be a cofibration. Hence

E_{q,∗}^{1}(B) = H_{q+∗}(S^{q} ∧ T HH(B)_{q}^{nd}; 𝔽_{p})
≅ H_{∗}(B ∧ B̄ ∧ ⋯ ∧ B̄; 𝔽_{p})
≅ H_{∗}(B; 𝔽_{p}) ⊗ H̄_{∗}(B; 𝔽_{p}) ⊗ ⋯ ⊗ H̄_{∗}(B; 𝔽_{p}) .

By a standard inspection, the d^{1}-differential is induced by the alternating sum of the simplicial face maps, hence recovers the boundary operator in the normalized Hochschild complex for H_{∗}(B; 𝔽_{p}). ((Reference to Segal?)) Thus the E^{2}-term is given by the homology of that complex, i.e., by the indicated Hochschild homology groups.

((Circle action, edge homomorphism.)) □

When B is commutative, the Bökstedt spectral sequence reflects the multiplicative structure on T HH(B). The algebra structure on Hochschild homology is given by the shuffle product.

Proposition. Let B be a commutative symmetric ring spectrum. (a) The Bökstedt spectral sequence

E_{q,∗}^{2}(B) = HH_{q}(H_{∗}(B; 𝔽_{p})) ⟹ H_{∗}(T HH(B); 𝔽_{p})

is an augmented commutative A_{∗}-comodule H_{∗}(B; 𝔽_{p})-algebra spectral sequence.

(b) If each term E_{∗∗}^{r}(B) is flat over H_{∗}(B; 𝔽_{p}), then E_{∗∗}^{∗}(B) is a commutative A_{∗}-comodule H_{∗}(B; 𝔽_{p})-Hopf algebra spectral sequence.

((See Angeltveit–Rognes (2005).))

((There is a similar E-based spectral sequence converging to E_{∗}(T HH(B)), for each S-algebra E, see [EKMM].))

III.6. First calculations.

We write P (x) = 𝔽_{p}[x] and E(x) = P (x)/(x^{2} = 0) = 𝔽_{p}{1, x} for the polynomial algebra and the exterior algebra on a generator x, respectively. We also write P_{h}(x) = P (x)/(x^{h} = 0) for the truncated polynomial algebra of height h. Furthermore we write

Γ(y) = 𝔽_{p}{γ_{k}(y)) | k ≥ 0}

for the divided power algebra on y, with γ_{i}(y) · γ_{j}(y) = (i, j)γ_{i+j}(y), where (i, j) = (i + j)!/(i!j!). We identify γ_{0}(y) = 1 and γ_{1}(y) = y. There is an isomorphism

Γ(y) = ⨂_{e≥0} P_{p}(γ_{p^{e}}(y)) .

((Γ(y) is Hom dual to P (η), where y is dual to η.))

To compute the E^{2}-term of a Bökstedt spectral sequence, the following standard calculations are useful.

Lemma. (a)

HH_{∗}(P (x)) ≅ P (x) ⊗ E(dx)

for |x| even, with dx ∈ HH_{1}(P (x)) represented by 1 ⊗ x in the Hochschild complex. The coproduct is given by ψ(dx) = dx ⊗ 1 + 1 ⊗ dx, i.e., dx is P (x)-coalgebra primitive.

(b)

HH_{∗}(E(x)) ≅ E(x) ⊗ Γ(dx)

for |x| odd, with γ_{k}(dx) ∈ HH_{k}(E(x)) represented by 1 ⊗ x ⊗ ⋯ ⊗ x (with k copies of x) in the Hochschild complex. The coproduct is given by

ψ(γ_{k}(dx)) = ∑_{i+j=k} γ_{i}(dx) ⊗ γ_{j}(dx) ,

i.e., dx is E(x)-coalgebra primitive.

((What about p = 2?))

Proof. We can compute HH_{∗}(P (x)) as P (x)-bimodule Tor. There is a short free resolution

0 → P (x) ⊗ 𝔽_{p}{dx} ⊗ P (x) ⟶^{∂} P (x) ⊗ P (x) ⟶^{μ} P (x) → 0

of P (x), viewed as a P (x)-bimodule by the algebra multiplication μ, where ∂(dx) = 1 ⊗ x − x ⊗ 1. It maps to the two-sided bar resolution by taking dx to 1 ⊗ x ⊗ 1. Tensoring with P (x) over P (x) ⊗ P (x)^{op} we get the complex

0 → P (x) ⊗ 𝔽_{p}{dx} ⟶^{0} P (x) → 0

with homology P (x) ⊗ E(dx), mapping to the Hochschild complex by taking dx to 1⊗x. The induced map in homology is an isomorphism, and identifies P (x)⊗E(dx) with HH_{∗}(P (x)). ((Discuss product and coproduct.))

((Do E(x) too.)) □

((Reference to Cartan–Eilenberg. See Proposition 3.3 of Ausoni (2005) for a more complicated case.))

We now study the example B = H𝔽_{p}, which is a commutative symmetric ring spectrum. Its homology algebra is

H_{∗}(H𝔽_{p}; 𝔽_{p}) = A_{∗} = P (ξ̄_{k} | k ≥ 1) ⊗ E(τ̄_{k} | k ≥ 0) .

Thus the Bökstedt spectral sequence has E^{2}-term

E_{∗∗}^{2}(𝔽_{p}) = HH_{∗}(A_{∗}) ≅ A_{∗} ⊗ E(dξ̄_{k} | k ≥ 1) ⊗ Γ(dτ̄_{k} | k ≥ 0) .

It is free, thus flat, over A_{∗}.

The A_{∗}-algebra generators of this E^{2}-term are the exterior classes dξ̄_{k} in filtration q = 1, and the divided powers γ_{p^{e}}(dτ̄_{k}) in filtration q = p^{e}, for e ≥ 0.

The A_{∗}-coalgebra primitives constitute the free A_{∗}-module generated by the classes dξ̄_{k} and dτ̄_{k}, all in filtration q = 1. Among these, the A_{∗}-comodule primitives constitute the 𝔽_{p}-module generated by the same classes.

Suppose that the shortest nonzero differentials in the Bökstedt spectral sequence are of length r ≥ 2, i.e., are d^{r}-differentials, so that E^{2}(𝔽_{p}) = E^{r}(𝔽_{p}). The A_{∗}- comodule A_{∗}-Hopf algebra structure on the E^{2}-term then remains (unchanged) at the E^{r}-term, and the d^{r}-differentials are generated, by this structure, by differentials from A_{∗}-algebra generators to A_{∗}-coalgebra primitives that are also A_{∗}- comodule primitives.

Any such generating differentials must map from a class γ_{p^{e}}(dτ̄_{k}) with e ≥ 1, to a class dξ̄_{k} or dτ̄_{k}, not necessarily with the same index k. The total degrees of these classes are 2p^{k+e}, 2p^{k} − 1 and 2p^{k}, respectively, so the only possibilities are the differentials

d^{r}(γ_{p^{e}}(dτ̄_{k})) = dξ̄_{k+e}

where r = p^{e} − 1, for some e ≥ 1. More precisely, this formula might hold up to a unit in 𝔽_{p}, which we suppress. In fact, the differential structure is as rich as possible, i.e., for each k ≥ 0 the differential above for e = 1 occurs.

Proposition. (a) In the Bökstedt spectral sequence for B = H𝔽_{p}, with

E_{∗∗}^{2}(𝔽_{p}) = A_{∗} ⊗ E(dξ̄_{k} | k ≥ 1) ⊗ Γ(dτ̄_{k} | k ≥ 0)

there are nonzero differentials

d^{p−1}(γ_{p}(dτ̄_{k})) = dξ̄_{k+1}

for each k ≥ 0.

(b) By the A_{∗}-coalgebra structure, these imply differentials

d^{p−1}(γ_{j}(dτ̄_{k})) = dξ̄_{k+1} · γ_{j−p}(dτ̄_{k})

for each j ≥ p, k ≥ 0, which leave the E^{p}-term

E_{∗∗}^{p}(𝔽_{p}) = A_{∗} ⊗ P_{p}(dτ̄_{k} | k ≥ 0) .

For filtration reasons, this equals the E^{∞}-term.

To establish the differentials in (a), we appeal to the homology operations in H_{∗}(T HH(B); 𝔽_{p}) that are derived from its E_{∞}-ring spectrum structure. These also allow us to make the following deductions.

Proposition. (a) There are multiplicative extensions

(dτ̄_{k})^{p} = dτ̄_{k+1}

in H_{∗}(T HH(𝔽_{p}); 𝔽_{p}), for all k ≥ 0, so

H_{∗}(T HH(𝔽_{p}); 𝔽_{p}) ≅ A_{∗} ⊗ P (dτ̄_{0})

as a commutative A_{∗}-comodule A_{∗}-Hopf algebra, with |dτ̄_{0}| = 2.

(b) There is an 𝔽_{p}-algebra isomorphism

π_{∗}T HH(𝔽_{p}) ≅ P (μ_{0})

where μ_{0}, in degree 2, is represented by dτ̄_{0}.

III.7. Power operations.

A commutative symmetric ring spectrum B has a canonical structure as an E_{∞} ring spectrum. In terms of the positive model structure on commutative symmetric ring spectra, this can be seen as follows: A (positively) cofibrant B has the property that the Σ_{j}-action on the j-th smash power

B^{∧j} = B ∧ ⋯ ∧ B ,

permuting the smash factors, is free off the base point at each level. Hence the map

EΣ_{j+} ∧_{Σ_{j}} B^{∧j} → B^{∧j}/Σ_{j}

that collapses EΣ_{j} to a point, is a weak equivalence. The (strictly) commutative product μ: B ∧ B → B induces maps

B^{∧j}/Σ_{j} → B

and the composite maps

ξ_{j}: EΣ_{j+} ∧_{Σ_{j}} B^{∧j} → B

for j ≥ 0 provide the structure maps for an E_{∞} ring spectrum structure on B, for the Barratt–Eccles operad ℰ with j-th space ℰ(j) = EΣ_{j}. The point of replacing B^{∧j}/Σ_{j} with the extended power

D_{j}(B) = EΣ_{j+} ∧_{Σ_{j}} B^{∧j}

is that the homology of the latter can be readily computed. We focus on the case j = p, where D_{p}(B) extends the p-fold smash power B^{∧p}. By a transfer argument, there is a split surjection

H_{∗}(EC_{p+} ∧_{C_{p}} B^{∧p}; 𝔽_{p})) → H_{∗}(EΣ_{p+} ∧_{Σ_{p}} B^{∧p}; 𝔽_{p}) ,

where C_{p} ⊂ Σ_{p} is the cyclic subgroup generated by T = (12 … p). As a model for EC_{p} we can take S^{∞} = S(ℂ^{∞}), with its usual C_{p}-CW structure. The associated mod p cellular complex is W_{∗} = C_{∗}(EC_{p}; 𝔽_{p}), with W_{i} = 𝔽_{p}[C_{p}]{e_{i}} for each i ≥ 0, d(e_{i}) = (1 − T)e_{i−1} for i odd and d(e_{i}) = (1 + T + ⋯ + T^{p−1})e_{i−1} for i ≥ 2 even.

There is then an isomorphism

H_{∗}(EC_{p+} ∧_{C_{p}} B^{∧p}; 𝔽_{p}) ≅ H_{∗}(W_{∗} ⊗_{C_{p}} H_{∗}(B; 𝔽_{p})^{⊗p})

and these homology groups are generated by the cycles e_{i} ⊗ x^{⊗p} for i ≥ 0 (and e_{0}⊗x_{1}⊗⋯⊗x_{p}), where x (and the x_{1}, …, x_{p}) ranges through a basis for H_{∗}(B; 𝔽_{p}). It follows that

H_{∗}(D_{p}(B); 𝔽_{p}) = H_{∗}(EΣ_{p+} ∧_{Σ_{p}} B^{∧p}; 𝔽_{p})

is generated by the cycles e_{i} ⊗ x^{⊗p} for i ≡ −1, 0 mod 2p − 2 and |x| even, and for i ≡ p − 2, p − 1 mod 2p − 2 and |x| odd (and e_{0} ⊗ x_{1} ⊗ ⋯ ⊗ x_{p}). We define Q_{i}(x) in H_{∗}(B; 𝔽_{p}) to be the image under

(ξ_{p})_{∗}: H_{∗}(D_{p}(B); 𝔽_{p}) → H_{∗}(B; 𝔽_{p})

of the class of e_{i} ⊗ x^{⊗p}, for these i and x. We can write the degree of Q_{i}(x) as

i + p|x| = |x| + r(2p − 2) − ϵ

for unique r ≥ 0 and ϵ ∈ {0, 1}, and it is traditional to rewrite Q_{i} in terms of upper indices as follows:

β^{ϵ}Q^{r}(x) = Q_{i}(x) = (ξ_{p})_{∗}(e_{i} ⊗ x^{⊗p}) .

Here β = β^{1}Q^{0} is the mod p homology Bockstein operation, of degree (−1), and Q^{r} = β^{0}Q^{r} has degree r(2p − 2).

These extended power homology operations

β^{ϵ}Q^{r}: H_{∗}(B; 𝔽_{p}) → H_{∗}(B; 𝔽_{p})

(for commutative symmetric ring spectra B) are known as Dyer–Lashof operations. Dyer and Lashof first described these operations for p = 2, while the odd-primary case is due to Araki and Kudo. See Steinberger (1986) for a list of the formal properties of these operations.

For |x| = 2r even (with ϵ = 0 and i = 0), the operation Q^{r}(x) = Q_{0}(x) is the image under (ξ_{p})_{∗} of the class of e_{0} ⊗x^{⊗p}. Since ξ_{p} extends the usual p-fold product B^{∧p} → B, it follows that in this case

Q^{r}(x) = x^{p}

in the algebra structure on H_{∗}(B; 𝔽_{p}). The other Dyer–Lashof operations are of higher degree.

In the special case of B = H𝔽_{p}, the operations

β^{ϵ}Q^{r}: A_{∗} → A_{∗}

were first described by Leif Kristensen, see Steinberger’s paper for a published reference. We need the following facts:

Q^{p^{k}}(ξ̄_{k}) = ξ̄_{k+1}

for k ≥ 1,

Q^{p^{k}}(τ̄_{k}) = τ̄_{k+1}

for k ≥ 0, and

β(τ̄_{k}) = ξ̄_{k}

for k ≥ 0, where ξ̄_{0} = 1, as usual. These formulas are all forced by the formal properties of the Dyer–Lashof operations, such as the Nishida relations.

In the special case B = DX_{+} = F (X_{+}, S), for a finite CW complex X, the Dyer–Lashof operations on H_{∗}(B; 𝔽_{p}) are compatible with the Steenrod operations on H^{∗}(X; 𝔽_{p}) under the isomorphisms

H_{∗}(B; 𝔽_{p}) = π_{∗}(H𝔽_{p} ∧ F (X_{+}, S)) ≅ π_{∗}F (X, H𝔽_{p}) = H^{−∗}(X; 𝔽_{p}) .

More precisely, β^{ϵ}Q^{r} corresponds to β^{ϵ}P^{−r}.

The maps

T HH(B) ⟶^{α̃} F (S_{+}^{1}, T HH(B)) ⟵_{≃}^{ν} DS_{+}^{1} ∧ T HH(B)

are maps of commutative symmetric ring spectra, hence the induced homomorphism

ν^{−1}α̃: H_{∗}(T HH(B); 𝔽_{p}) → E(ι_{1}) ⊗ H_{∗}(T HH(B); 𝔽_{p})

in homology commutes with the Dyer–Lashof operations. In view of the formula (ν^{−1}α̃)(x) = 1 ⊗ x + ι_{1} ⊗ dx, this implies the relation

β^{ϵ}Q^{r}(dx) = d(β^{ϵ}Q^{r}x)

for all r ≥ 0, ϵ ∈ {0, 1}, x ∈ H_{∗}(T HH(B); 𝔽_{p}). The argument uses the Cartan formula for the Dyer–Lashof operations in the homology of DS_{+}^{1} ∧ T HH(B), and the relation with Steenrod operations mentioned above for X = S_{+}^{1}.

As a corollary, we can deduce the multiplicative relation

(dτ̄_{k})^{p} = Q^{p^{k}}(dτ̄_{k}) = dQ^{p^{k}}(τ̄_{k}) = dτ̄_{k+1}

for each k ≥ 0.

III.8 Differentials.

Following Ausoni (2005), we show that there are differentials

d^{p−1}(γ_{p}(dτ̄_{k})) = dξ̄_{k+1}

for k ≥ 0, in the Bökstedt spectral sequence for H_{∗}(T HH(𝔽_{p}); 𝔽_{p}). In the abutment, we can compute that

dξ̄_{k+1} = d(βτ̄_{k+1}) = β(dτ̄_{k+1}) = β((dτ̄_{k})^{p}) = 0

since β acts as a derivation, and we are working in characteristic p. Hence dξ̄_{k+1} must be hit by a differential.

In the lowest case, k = 0, the only possible source of this differential is γ_{p}(dτ̄_{0}). It follows from the coalgebra structure that there are also differentials

d^{p−1}(γ_{j}(dτ̄_{0})) = dξ̄_{1} · γ_{j−p}(dτ̄_{0})

for all j ≥ p. The homology of this differential acting on E(dξ̄_{1}) ⊗ Γ(dτ̄_{0}) is thus P_{p}(dτ̄_{0}).

Turning to the next case, k = 1, only possible sources of differentials hitting dξ̄_{2} were γ_{p}(dτ̄_{1}) and γ_{p^{2}}(dτ̄_{0}), but we have just seen that the latter class already supported a d^{p−1}-differential. Hence there is a differential d^{p−1}(γ_{p}(dτ̄_{1})) = dξ̄_{2}, and the coalgebra structure implies that there are also differentials d^{p−1}(γ_{j}(dτ̄_{1})) = dξ̄_{2}· γ_{j−p}(dτ̄_{1}) for all j ≥ p. The homology of this differential acting on E(dξ̄_{2}) ⊗ Γ(dτ̄_{1}) is P_{p}(dτ̄_{1}).

By induction on k it follows that we have precisely the d^{p−1}-differentials listed above, killing the classes dξ̄_{k+1}, together with their coalgebraic consequences

d^{p−1}(γ_{j}(dτ̄_{k})) = dξ̄_{k+1} · γ_{j−p}(dτ̄_{k})

for all k ≥ 0 and j ≥ p. By the Künneth formula, we are left with the E^{p}-term

E_{∗∗}^{p}(𝔽_{p}) = A_{∗} ⊗ P_{p}(dτ̄_{k} | k ≥ 1) .

Here all the algebra generators are in Hochschild filtration 1, so there is no room for further differentials, and this also equals the E^{∞}-term.

In view of the multiplicative relations (dτ̄_{k})^{p} = dτ̄_{k+1} in H_{∗}(T HH(𝔽_{p}); 𝔽_{p}), we deduce that

H_{∗}(T HH(𝔽_{p}); 𝔽_{p}) = A_{∗} ⊗ P (dτ̄_{0})

as a (bi-)commutative A_{∗}-comodule A_{∗}-Hopf algebra.

The class dτ̄_{0} is A_{∗}-comodule primitive, since the coaction can be calculated as

ν(dτ̄_{0}) = (1 ⊗ d)(ψ(τ̄_{0})) = 1 ⊗ dτ̄_{0} + τ̄_{0} ⊗ d1 = 1 ⊗ dτ̄_{0} .

Hence the Adams spectral sequence

E_{2}^{s,t} = Ext_{A_{∗}}^{s,t}(𝔽_{p}, H_{∗}(T HH(𝔽_{p}); 𝔽_{p})) ⟹ π_{t−s}T HH(𝔽_{p})

collapses to the s = 0-line, where π_{∗}T HH(𝔽_{p}) is identified with the subalgebra P (dτ̄_{0}) of A_{∗}-comodule primitives in H_{∗}(T HH(𝔽_{p}); 𝔽_{p}). We let

μ_{0} = dτ̄_{0} ∈ π_{2}T HH(𝔽_{p})

be the generating homotopy class, and conclude that

π_{∗}T HH(𝔽_{p}) = P (μ_{0}) .

Heuristically, we might think of H𝔽_{p} as an S-algebra built from S by attaching a 1-cell labeled τ̄_{0} by a degree p map, together with higher cells. The Kähler differentials of H𝔽_{p} over S then receive a contribution called dτ̄_{0}, in bidegree (1, 1). Going from the symmetric algebra on the Kähler differentials to T HH, the calculation above shows that this class freely generates all of T HH(𝔽_{p}).

IV. The ku-algebra structures on ku/p (October 11th)

((Discuss the Lazarev obstruction theory classifying A_{∞} ku-ring spectrum structures on ku/p, working up its Postnikov tower. The extensions of the (2m − 2)-th Postnikov section P = P^{2m−2}ku/p to the (2m)-th section are classified by the A_{∞} ku-algebra derivations

𝒜Der_{ku}(P, Σ^{2m+1}Hℤ/p) ≅ T HH_{ku}^{2m+2}(P, Hℤ/p) ≅ ℤ/p{y_{0}^{m+1}, y_{1,m}}

that map to the underlying ku-module extension in

H_{ku}^{2m+1}(P, Hℤ/p) ≅ ℤ/p{Q_{1,m}} .

Here y_{1,m} ↦ Q_{1,m}, so there are p choices of ku-algebra extensions for each Postnikov stage.))

V. Thom spectra (October 16th)

We focus on the real case. Complex Thom spectra can be handled in much the same way, taking MU_{n} = F (S^{n}, MU(n)) as the n-th space.

V.1. The Thom spectrum MO.

Let EO(n) be the simplical space with q-simplices Map([q], O(n)) ≅ O(n)^{q+1}. The group O(n) acts freely on the contractible space EO(n), both from the left and from the right, and these actions commute. Let BO(n) = EO(n)/O(n) be the orbit space for the right action. The remaining left O(n)-action on BO(n) is given by the adjoint (= conjugation) action of O(n) on O(n), together with functoriality of the bar construction. We will write BO(n)^{ad} for this left O(n)-space.

The Thom spectrum MO is the orthogonal spectrum

n ↦ MO_{n} = MO(n) = EO(n)_{+} ∧_{O(n)} S^{n}

with left O(n)-action induced from the left O(n)-action on EO(n). The n-th stucture map σ: S^{1} ∧ MO_{n} → MO_{1+n} equals the composite

S^{1} ∧ EO(n)_{+} ∧_{O(n)} S^{n} ≅ EO(n)_{+} ∧_{O(n)} (S^{1} ∧ S^{n}) → EO(1 + n)_{+} ∧_{O(1+n)} S^{1+n} .

The underlying symmetric spectrum is the same sequence of based spaces, with left Σ_{n}-action given by restricting the O(n)-action along the inclusion Σ_{n} ⊂ O(n). There are product maps

(EO(m)_{+} ∧_{O(m)} S^{m}) ∧ (EO(n)_{+} ∧_{O(n)} S^{n}) ≅ E(O(m) × O(n))_{+} ∧_{O(m)×O(n)} (S^{m} ∧ S^{n})

→ EO(m + n)_{+} ∧_{O(m+n)} S^{m+n}

that make MO a commutative orthogonal ring spectrum.

V.2. The suspension spectrum of BO.

Let S[BO]^{ad} be the orthogonal spectrum

n ↦ S^{n} ∧ BO(n)^{ad}

with O(n) acting diagonally on S(n) and BO(n)^{ad}. There are obvious structure maps

σ: S^{1} ∧ S^{n} ∧ BO(n)^{ad} → S^{1+n} ∧ BO(1 + n)^{ad} .

There are product maps

(S^{m} ∧ BO(m)^{ad}) ∧ (S^{n} ∧ BO(n)^{ad})

≅ S^{m} ∧ S^{n} ∧ B(O(m) × O(n))^{ad}
→ S^{m+n} ∧ BO(m + n)^{ad}

that make S[BO]^{ad} a commutative orthogonal ring spectrum.

Let BO = colim_{n} BO(n), and let Σ^{∞}BO_{+} be the unreduced orthogonal suspension spectrum of BO, with n-th space S^{n} ∧ BO_{+}. Here O(n) acts only on the S^{n}-factor, fixing BO.

The underlying sequential spectrum of S[BO]^{ad} maps to the underlying sequential spectrum of Σ^{∞}BO_{+}, via the inclusion

S^{n} ∧ BO(n)_{+} → S^{n} ∧ BO_{+}

at level n. The induced map of homotopy groups

colim_{n} π_{k+n}(S^{n} ∧ BO(n)_{+}) → colim_{n} π_{k+n}(S^{n} ∧ BO_{+})

is an isomorphism for each integer k, so these sequential spectra are stably equivalent. But the homotopy category of orthogonal spectra is equivalent to the homotopy category of sequential spectra, so it follows that also the orthogonal spectra S[BO]^{ad} and Σ^{∞}BO_{+} are stably equivalent.

Note that the orthogonal ring spectrum structure on S[BO]^{ad} is commutative and easy to describe, while that on Σ^{∞}BO_{+} is only an E_{∞} ring spectrum structure.

V.3. The Thom diagonal.

The Thom diagonal map

θ_{n}: MO(n) → MO(n) ∧ BO(n)_{+}^{ad}

extends the proper map

(id, π): E(γ^{n}) → E(γ^{n}) × BO(n)^{ad}

over E(γ^{n}) = EO(n) ×_{O(n)} ℝ^{n} ⊂ MO(n). It is O(n)-equivariant for the diagonal action in the target, when O(n) acts on BO(n) by the left adjoint action, as indicated.

We continue the Thom map with the usual inclusions

MO(n)∧BO(n)_{+}^{ad} → F (S^{n}, MO(n)∧S^{n} ∧BO(n)_{+}^{ad}) → F (S^{n}, (MO ∧S[BO]^{ad})_{2n})

to get a map of orthogonal ring spectra

θ: MO → {n ↦ F (S^{n}, (MO ∧ S[BO]^{ad})_{2n})} .

For any orthogonal spectrum X there is a map of orthogonal spectra

σ̃: X = {n ↦ X_{n}} → {n ↦ F (S^{n}, X_{2n})}

that at the n-th level is adjoint to the iterated structure map

σ^{n}: S^{n} ∧ X_{n} → X_{n+n} = X_{2n} .

The latter is O(n)-equivariant via the diagonal embedding O(n) → O(n) × O(n) ⊂ O(2n), so the adjoint map σ̃^{n}: X_{n} → F (S^{n}, X_{2n}) is also O(n)-equivariant. When X is an orthogonal ring spectrum, the product maps

F (S^{m}, X_{2m}) ∧ F (S^{n}, X_{2n}) → F (S^{m} ∧ S^{n}, X_{2m} ∧ X_{2n}) → F (S^{m+n}, X_{2(m+n)})

make {n ↦ F (S^{n}, X_{2n})} an orthogonal ring spectrum, too. ((Check!)) The map σ̃ induces an isomorphism

colim_{n} π_{k+n}(X_{n}) → colim_{n} π_{k+n}(F (S^{n}, X_{2n})) ≅ colim_{n} π_{k+2n}(X_{2n})

of homotopy groups for each integer k, hence is a stable equivalence.

There results a chain of maps of commutative orthogonal ring spectra

MO ⟶^{β} {n ↦ F (S^{n}, (MO ∧ S[BO]^{ad})_{2n})} ⟵_{≃}^{σ̃} MO ∧ S[BO]^{ad} ,

and a stable equivalence of orthogonal (E_{∞} ring-)spectra

MO ∧ S[BO]^{ad} ≃ MO ∧ Σ^{∞}BO_{+} ≅ MO ∧ BO_{+} .

V.4. A Hopf–Galois structure.

We wish to view β, or σ̃^{−1}β, as the coaction in a Hopf–Galois extension S → MO, with Hopf algebra H = S[BO]^{ad}.

((Is there a good coproduct ψ: H → H ∧H, lifting the diagonal map Σ^{∞}BO_{+} → Σ^{∞}(BO × BO)_{+} ≅ Σ^{∞}BO_{+} ∧ Σ^{∞}BO_{+}?))

((ETC))

VI. THH of the integers

((Consider the case B = Hℤ, with H_{∗}(Hℤ) = P (ξ̄_{k} | k ≥ 1) ⊗ E(τ̄_{k} | k ≥ 1) ⊂ A_{∗}. The Bökstedt E^{2}-term equals HH_{∗}(H_{∗}(Hℤ)) = H_{∗}(Hℤ) ⊗ E(dξ̄_{k} | k ≥ 1) ⊗ Γ(dτ̄_{k} | k ≥ 1). The differentials d^{p−1}(γ_{p}(dτ̄_{k})) = dξ̄_{k+1} for k ≥ 1, leave E_{∗∗}^{∞}(ℤ) = H_{∗}(Hℤ)⊗E(dξ̄_{1})⊗P_{p}(dτ̄_{k} | k ≥ 1) and H_{∗}(T HH(ℤ)) = H_{∗}(Hℤ)⊗E(dξ̄_{1})⊗P (dτ̄_{1}). We find generating A_{∗}-comodule primitives

λ_{1} = 1 ∧ dξ̄_{1}
μ_{1} = 1 ∧ dτ̄_{1} + τ_{0} ∧ dξ̄_{1}

in H_{∗}(V (0) ∧ T HH(ℤ)), where H_{∗}(V (0)) = E(τ_{0}), so

V (0)_{∗}T HH(ℤ) = E(λ_{1}) ⊗ P (μ_{1}) .

A Bockstein spectral sequence arguement implies

π_{∗}T HH(ℤ) = \{ \begin{matrix}ℤ & for ∗ = 0, \\ ℤ/n & for ∗ = 2n − 1 > 0, \\ 0 & otherwise. \end{matrix}

))

VII. THH for local fields

VII.1. Kähler forms.

((Define derivations, differentials, relation to HH_{1}, higher forms, map to HH_{∗}, étale, polynomial and smooth examples, HKR-theorem.))

VII.2. Rings of integers in local fields.

We follow Hesselholt–Madsen:

Let A be a complete DVR (discrete valuation ring) with perfect residue field k of characteristic p, and fraction field K of characteristic 0. For example, K can be a finite extension of ℚ_{p} and A its valuation ring. In that case k is a finite field.

Let π = π_{K} be a uniformizer, so v(π) = 1 where v = v_{K} is the normalized valuation, πA = (π) ⊂ A is the maximal ideal, k = A/(π) and K = A[π^{−1}]:

k ⟵^{i} A ⟶^{j} K .

Let W = W (k) be the Witt ring of k, with uniformizer p. We can write

A = W [x]/(ϕ(x)) ,

where the irreducible polynomial of π over W is an Eisenstein polynomial

ϕ(x) = x^{e} − pθ(x)

in W [x], where θ(x) is of degree < e and θ(0) is a unit in W. It follows that θ(π) is also a unit. Hence π^{e} = pθ(π) and p = π^{e}θ(π)^{−1} in A. The formal derivative is ϕ′(x) = ex^{e−1} − pθ′(x), so

ϕ′(π) = eπ^{e−1} − pθ′(π)

and

Ω_{A/W}^{1} ≅ A/(ϕ′(π)){dπ} .

If p ∤ e, we say that A is tamely ramified over W. Then v(ϕ′(π)) = v(π^{e−1}) = e − 1, (p) ⊊ (ϕ′(π)), and Ω_{A/W}^{1} ≅ k{1, π, …, π^{e−2}}.

Example 1: K = ℚ_{p}, A = W = ℤ_{p}, π = p, e = 1, ϕ(x) = x − p, θ(x) = 1, ϕ′(π) = 1.

Example 2: K = ℚ_{p}(ζ_{p}), A = ℤ_{p}[ζ_{p}], π = 1 − ζ_{p}, e = p − 1, ϕ(x) = x^{p−1} − px^{p−2} + ⋯ + (−1)^{p−1}p, θ(x) = x^{p−2} + ⋯ + (−1)^{−p}, ϕ′(π) = (p − 1)π^{p−2} − pθ′(π), Ω_{A/W}^{1} ≅ 𝔽_{p}^{p−1}.

If p | e we say that A is wildly ramified over W. Then v(ϕ′(π)) ≥ v(p) = e and (ϕ′(π)) ⊆ (p).

Example 3: K = ℚ_{p}(\sqrt[p]{p}), A = ℤ_{p}[\sqrt[p]{p}], π = \sqrt[p]{p}, e = p, ϕ(x) = x^{p} − p, θ(x) = 1, ϕ′(π) = pπ^{p−1} = π^{2p−1}.

Following Lindenstrauss–Madsen:

HH_{∗}(ℤ_{p}) → ℤ_{p} is a p-adic equivalence, so HH_{∗}(W) → HH_{∗}^{ℤ_{p}}(W) ≅ W is a p-adic equivalence, the Witt ring W is unramified over ℤ_{p}. Similarly, HH_{∗}(A) → HH_{∗}^{W} (A) is a p-adic equivalence.

From A = W [x]/(ϕ(x)) we get a 2-periodic resolution of A over A^{e} = A ⊗_{W} A,

A ⟵^{μ} A^{e} ⟵^{1⊗x−x⊗1} A^{e} ⟵^{\frac{1⊗ϕ(x)−ϕ(x)⊗1}{1⊗x−x⊗1}} A^{e} ⟵ …

and calculate

HH_{∗}^{W} (A) = \{ \begin{matrix}A & for ∗ = 0, \\ A/(ϕ′(π)) & for ∗ = 2n − 1 > 0, \\ 0 & otherwise. \end{matrix}

Thus this is also HH_{∗}(A), up to p-adic equivalence.

If p | e, then

V (0)_{∗}HH(A) ≅ A/p ⊗ E(dπ) ⊗ Γ(μ_{A}) .

Here μ_{A} ∈ V (0)_{2}HH(A) has Bockstein image a generator of the p-torsion in Ω_{A}^{1}, say (ϕ′(π)/p)dπ. Note that (Ω_{A}^{∗})/p ≅ A/p ⊗ E(dπ).

In view of the pushout square of commutative S-algebras

\xymatrix{T HH(ℤ) \ar[r] \ar[d] & ℤ \ar[d] \\ T HH(A) \ar[r] & HH(A)}

with V (0)_{∗}T HH(ℤ) = E(λ_{1})⊗P (μ_{1}), this fits well with the Lindenstrauss–Madsen calculation

V (0)_{∗}T HH(A) ≅ A/p ⊗ E(dπ) ⊗ P (μ_{A}) .

Note that (Ω_{A}^{∗})/p ≅ A/p ⊗ E(dπ). ((Does μ_{1} map to μ_{A}^{p}? Does λ_{1} suspend to γ_{p}(μ_{A})? Can one extract T HH(A) from the other three terms in the square above?))

For p ∤ e the answers are more complicated.

Get a weak equivalence

B ∧_{A} T HH(A) → T HH(B)

for B ⊂ L like A ⊂ K, when K → L is an unramified (thus étale) extension, but not more generally.

((What about HH_{∗}(k) (in the derived sense, over the ground ring ℤ) and HH_{∗}(A|K)?))

VII.3. Log poles.

Quillen’s K-theory localization theorem gives a cofiber sequence

K(k) ⟶^{i_{∗}} K(A) ⟶^{j^{∗}} K(K) .

In homotopy, we get the exact sequence

0 → A^{×} ⟶^{j^{∗}} K^{×} ⟶^{v} ℤ → 0

where v: π ↦ 1.

By analogy, Hesselholt and Madsen define the target T HH(A|K) for a trace map from K(K) to sit in a cofiber sequence:

\xymatrix{T HH(k) \ar[r]^{i_{∗}} & T HH(A) \ar[r]^{j} \ar[dr]_{j^{∗}} & T HH(A|K) \ar[d] \ar[r]^{∂} & ΣT HH(k) \\  &  & T HH(K)}

In homotopy we get an extension

0 → Ω_{A}^{1} ⟶^{j} Ω_{(A|K)}^{1} ⟶^{res} k → 0

where

Ω_{(A|K)/W}^{1} ≅ A/(πϕ′(π)){dlog π} ,

j: dπ ↦ π dlog π and res: dlog π ↦ 1.

Now

πϕ′(π) = eπ^{e} − pπθ′(π) = π^{e}(e − πθ′(π)θ(π)^{−1})

so (πϕ′(π)) ⊆ (π^{e}) = (p) for all values of e. Furthermore,

dlog p = dlog(π^{e}θ(π)^{−1}) = (e − πθ′(π)θ(π)^{−1}) dlog π ,

so dlog p generates the p-torsion in Ω_{(A|K)}^{1}. (The term πθ′(π)θ(π)^{−1} is known as the elasticity of θ(π).)

Thus V (0)_{1}T HH(A|K) ≅ A/p{dlog π}, while V (0)_{2}T HH(A|K) ≅ A/p{κ_{0}}, where the Bockstein image of κ_{0} equals dlog p.

Hesselholt–Madsen prove

V (0)_{∗}T HH(A|K) ≅ A/p ⊗ E(dlog π) ⊗ P (κ_{0}) .

Note that (Ω_{(A|K)}^{∗})/p ≅ A/p ⊗ E(dlog π), while V (0)_{∗}T HH(ℤ_{p}|ℚ_{p}) ≅ P (κ_{0}). This formula turns out to hold for all e, divisible by p or not.

One gets a weak equivalence

B ∧_{A} T HH(A|K) → T HH(B|L)

for B ⊂ L like A ⊂ K, when K → L is a tamely ramified extension, but not more generally.

((Check whether T HH(A|K) → T HH(B|L)^{hG} is a weak equivalence for a faithful G-Galois extension K → L, wildly ramified or not.))

((See how κ_{0} comes from V (0)_{2}T HH(ℤ_{p}|ℚ_{p}). Make the homotopy exact sequence for T HH(k), T HH(A), T HH(A|K) clear.))

((Also consider the case of

k ⟵ A = k[x] ⟶ K = k[x, x^{−1}]

with Ω_{A}^{1} = A{dx}, Ω_{(A|K)}^{1} = A{dlog x} and Ω_{K}^{1} = K{dx}.))

VII.4. The trace map.

The trace map tr: K(A) → T HH(A) → HH(A) extends the map BGL_{1}(A) → T HH(A) → HH(A) that takes [a] in the bar complex, representing the symbol {a} in K_{1}(A), to a^{−1} ⊗ a in the Hochschild complex. This equals the Kähler form a^{−1}da, so

tr: K_{1}(A) → Ω_{A}^{1} = T HH_{1}(A) = HH_{1}(A)

satisfies tr({a}) = a^{−1}da, for a ∈ A^{×}.

Let U_{A}^{1} ⊂ A^{×} be the kernel of A^{×} → k^{×}. The exponential map exp(x) = ∑_{n≥0} x^{n}/n! defines a group isomorphism exp: πA → U_{A}^{1} (for p odd), and the composite

πA ⟶_{≅}^{exp} U_{A}^{1} ⊂ A^{×} → K_{1}(A) ⟶^{tr} Ω_{A}^{1}

equals the composite

πA ⊂ A ⟶^{d} Ω_{A}^{1} ,

because exp(πx)^{−1}d exp(πx) = d(πx).

The extended trace map tr: K(K) → T HH(A|K) takes {π} to dlog π. ((Justify, almost by definition.))

We have U_{A}^{1}/(U_{A}^{1})^{p} ≅ A^{×}/A^{×p}, since k is perfect, and A^{×}/A^{×p} ≅ V (0)_{1}K(A), since A is local(?), so there is an isomorphism A/p ≅ V (0)_{1}K(A) given by x ↦ {exp(πx)}.

This provides a complete calculation of

tr: A/p ≅ V (0)_{1}K(A) → (Ω_{A}^{1})/p

with tr({exp(πx)}) = d(πx), and the extension tr: V (0)_{1}K(K) → (Ω_{(A|K)}^{1})/p ≅ A/p{dlog π}

with tr({π}) = dlog π. Note that the map A/p ≅ V (0)_{1}K(A) → (Ω_{A}^{1})/p, given by x ↦ d(πx), is not A-linear, but is a degree ≤ 1 differential operator.

VIII. Topological André–Quillen (co-)homology (November 22nd)

Let A → B be a map of commutative S-algebras, X a based space. Each square in

\xymatrix{A \ar[r] \ar[d] & B \ar[r] \ar[d] & X ⊗_{A} B \ar[d] \\ B \ar[r] & B ∧_{A} B \ar[r] \ar[d]_{μ} & X ⊗_{B} (B ∧_{A} B) \ar[d] \\  & B \ar[r] & X ˜⊗_{B}(B ∧_{A} B)}

is a pushout in commutative A-algebras. The composite in the middle column is the identity, so the composite in the right hand column is an equivalence

X ⊗_{A} B ≃ X ˜⊗_{B}(B ∧_{A} B) .

Thus the suspension spectrum of B ∧_{A} B in the category 𝒞_{B}/B of augmented commutative B-algebras

E_{B}^{∞}(B ∧_{A} B) = {n ↦ S^{n} ˜⊗_{B}(B ∧_{A} B)} ,

is equivalent to the sequential spectrum

{n ↦ S^{n} ⊗_{A} B} .

Here S^{1} ⊗_{A} B = T HH^{A}(B), and we think of S^{n} ⊗_{A} B as the n-th higher A-based T HH of B.

To compute, we use

Tor^{S^{n}⊗_{A}B}(B, B) = S^{n+1} ⊗_{A} B

to get a spectral sequence

E_{∗∗}^{2} = Tor_{∗∗}^{π_{∗}(S^{n}⊗_{A}B)}(π_{∗}(B), π_{∗}(B)) ⟹ π_{∗}(S^{n+1} ⊗_{A} B)

for n ≥ 0. Passing to augmentation ideals and their underlying B-modules,

T AQ^{A}(B) ≃ {n ↦ I_{B}(S^{n} ⊗_{A} B)}

as B-module (bi-)spectra.