Topological Hochschild cohomology and applications to Morava K-theories

A.Lazarev

1 Introduction

In this paper we prove that Morava K-theories admit a structure of an MUalgebra. By neglect of structure they also admit an A_{∞}- structure. This problem was first studied in [6]. However, in the cited reference the existence of an A_{∞} structure on K(n) was established with a weaker notion of an A_{∞} ring spectrum (without the unit condition) and only for odd p.

Using the obtained result we compute the topological Hochschild cohomology of K(n).

This paper is using the language of [2] routinely and relies heavily on the results of the previous paper of the author [3].

2 Morava K-theories as MU-algebras

Theorem 2.1 Let L_{n} be the MU-module obtained by killing he regular ideal generated by (p, x_{2}, x_{4}, …, x̂_{2n}, …) in the ring MU_{•} = Z[x_{2}, x_{4}, …]. Then L_{n} admits a unique MU-algebra structure (which extends the MU_{•}-algebra structure on L_{n•} = Z/pZ[x_{2n}]).

Proof. We will prove this theorem by induction up the Postnikov tower of L_{n}. The 1st Postnikov stage of L_{n} is HZ/p, the Eilenberg-MacLane spectrum mod p which is, of course, an MU-algebra. Suppose that we proved that L_{n}^{(i)}, the 2n(i − 1) + 1st Postnikov stage of L_{n}, is an MU-algebra and show that L_{n}^{(i+1)} is also such. Notice that the MU-module L_{n}^{(i)} is gotten from MU by killing the elements x_{2n}^{i} plus all the remaining polynomial generators in the ring MU_{•}, so that

L_{n}^{(i)} = Z/pZ[x_{2n}]/x_{2n}^{i}.

Lemma 2.2

T HH_{MU}^{•}(L_{n}^{(i)}, HZ/p) = Z/pZ[y_{2}, y_{4}, …]

where deg y_{2m} = 2m for 2m ≠ 2n + 2 and deg y_{2n+2} = 2ni + 2

Proof. The first step is to compute π_{•}(L_{n}^{(i)} ∧_{MU} L_{n}^{(i)}). Consider the Kunneth spectral sequence

T or_{Z[x_{2},x_{4},...]}^{st}(Z/pZ[x_{2n}]/x_{2n}^{i}, Z/pZ[x_{2n}]/x_{2n}^{i}) ⟹ π_{s+t}(L_{n}^{(i)} ∧_{MU} L_{n}^{(i)}).

To compute the E_{2}-term of this spectral sequence we introduce a differential graded algebra

A_{∗} = Z[x_{2}, x_{4}, …] ⊗ Λ(z_{1}, z_{3}, …),

where

d(z_{1}) = p, d(z_{3}) = x_{2}, …, d(z_{2n−1}) = x_{2n−2}, d(z_{2n+1}) = x_{2n}^{i}, d(z_{2n+2}) = x_{2n+2} …

Here degx_{2m} = 2m, degz_{2m+1} = 2m + 1 if m ≠ n and degz_{2n+1} = 2ni + 1. Then A_{∗} is quasiasomorphic to Z/pZ[x_{2n}]/x_{2n}^{i} and therefore

T or_{Z[x_{2},x_{4},...]}(Z/pZ[x_{2n}]/x_{2n}^{i}, Z/pZ[x_{2n}]/x_{2n}^{i} = A_{⋆} ⊗_{Z[x_{2},x_{4},...]} Z/pZ[x_{2p^{n}}]/x_{2p^{n}}^{i+1}

= Λ_{Z/pZ}(z_{1}, z_{3}, …) ⊗ Z/pZ[x_{2n}]/x_{2n}^{i}.

Since this is a spectral sequence of Hopf algebras all higher differentials are zero. We do not know apriori whether L_{n}^{(i)} is a homotopy commutative MUalgebra (if p=2 then it is just plain false). Therefore π_{•}(L^{(i)}∧_{MU} L^{(i)}) could be noncommutative and there would exist nontrivial multiplicative extensions in our spectral sequence. Let us show that this is not the case. Consider the (skew)-commutators of the elements z_{k}. On the one hand they should all lie in the bar filtration 2 or higher. On the other hand they should be primitive as commutators of primitive elements. This is possible only if they all equal zero and we see that in fact the algebra π_{•}(L_{n}^{(i)} ∧_{MU} L_{n}^{(i)}) is commutative, no mutiplicative extensions are present and

π_{•}(L)_{n}^{(i)} ∧_{MU} L_{n}^{(i)}) = Λ_{Z/pZ}(z_{1}, z_{3}, …) ⊗ Z/pZ[x_{2n}]/x_{2n}^{i}.

Next we have the following spectral sequence:

E_{2}^{st} = Ext_{Λ_{Z/pZ}(z_{1},z_{3},...)⊗Z/pZ[x_{2n}]/x_{2n}^{i}}^{st}(Z/pZ[x_{2n}]/x_{2n}^{i}, Z/pZ)
⟹ T HH_{MU}^{s+t}(L_{n}^{(i)}, HZ/p).

We have

E_{2}^{st} = Ext_{Λ_{Z/pZ}(z_{1},z_{3},...)}^{st}(Z/pZ, Z/pZ) = Z/pZ[y_{2}, y_{4}, …],

where deg y_{2m} = 2m for m ≠ n + 1 and degy_{2n+2} = 2ni + 2. This spectral sequence collapses since it is even. Again we do not know if T HH_{MU}(L_{n}^{(i)}, HZ/p) is a homotopy commutative MU-algebra, and therefore there might conceivably be nontrivial multiplicative extensions. To show that this does not happen consider the MU-algebra T HH_{MU}(L_{n}^{(i)}, L_{n}^{(i)}). The considerations analogous to foregoing show that

T HH_{MU}^{•}(L_{n}^{(i)}, L_{n}^{(i)}) = Z/pZ[y_{2}, y_{4}, …] ⊗ Z/pZ[x_{2n}]/x_{2n}^{i}.

(no multiplicative extensions are present in this case since the according to the results of J.McClure [4] T HH(A, A) is a homotopy commutative spectrum for any A). Next there is a map of algebras

T HH_{MU}(L_{n}^{(i)}, L_{n}^{(i)}) → T HH_{MU}(L_{n}^{(i)}, HZ/p)

induced by the canonical map

L_{n}^{(i)} → HZ/p

and comparing the spectral sequences we conclude that

T HH_{MU}(L_{n}^{(i)}, HZ/p) = Z/pZ[y_{2}, y_{4}, …].

Our lemma is proved.

The next lemma computes the cohomology operations from L_{n} to HZ/p in the category of MU-modules.

Lemma 2.3

[L_{n}^{(i)}, HZ/p]_{MU}^{•} = Λ_{Z/pZ}^{∗}(z_{1}, z_{3}, …)

where deg z_{2m+1} = 2m + 1 for m ≠ n and deg z_{2n+1} = 2ni + 1 and ∗ denotes the dual vector space (over Z/pZ)

Proof. Consider two hypercohomology spectral sequences

E_{2}^{st} = Ext_{Z/pZ[x_{2},x_{4},...]}^{st}(Z[x_{2n}]/x_{2n}^{i}, Z/pZ) ⟹ [L_{n}^{(i)}, HZ/p]_{MU}^{s+t}

and

\overline{E}_{2}^{st} = Ext_{Z/pZ[x_{2},x_{4},...]}^{st}(Z[x_{2n}]/x_{2n}^{i}, Z/pZ[x_{2n}/X_{2n}^{i}]) ⟹ [L_{n}^{(i)}, L_{n}^{(i)}]_{MU}^{s+t}

and a map between them

E_{2}^{••} = Λ_{Z/pZ}^{⋆}(z_{1}, z_{3}, …) → Λ_{Z/pZ}^{∗}(z_{1}, z_{3}, …) ⊗ Z/pZ[x_{2n}]/x_{2n}^{i} = \overline{E}_{2}^{••}

induced by the map L_{n}^{(i)} → HZ/p.

Just as before the spectral sequence \overline{E}_{2}^{••} is a spectral sequence of Hopf algebras and collapses for that reason. Therefore E_{2}^{••} also collapses and we have

E_{2}^{st} = E_{∞}^{st} = Λ_{Z/pZ}^{∗}(z_{1}, z_{3}, …).

The next step is to compute the module of derivations from L_{n}^{(i)} into HZ/p and its image upon the forgetful map into [L_{n}, HZ/p]_{MU}.

Lemma 2.4

Der_{MU}^{•}(L_{n}^{(i)}, HZ/p) = T HH_{MU}^{•−1}(L_{n}^{(i)}, HZ/p)/Z/pZ =
Σ^{−1}Z/pZ[y_{2}, y_{4}, …]/Z/pZ

where as before deg y_{2m} = 2m for m ≠ n and deg y_{2n+2} = 2ni + 2.

Proof. We have the following homotopy cofibre sequence

HZ/p → T HH_{MU}(L_{n}^{(i)}, HZ/p) → Der_{MU}(L_{n}^{(i)}, HZ/p)

from which the result follows.

Remark 2.5 Using this result we can talk unambigously about derivations and singular extensions corresponding to Hochschild cohomology classes of L_{n}^{(i)} with coefficients in HZ/p having positive degree.

Consider now the element Σ^{−1}(y_{2n+2}) ∈ Der_{MU}^{2ni+1}(L_{n}^{(i)}, HZ/p) and its image l^{∗}(Σ^{−1}(y_{2n+2}) in the group [L_{n}, HZ/p]_{MU}, where l is the canonical forgetful map

l : Der_{MU}(L_{n}^{(i)}, HZ/p) → [L_{n}, HZ/p]_{MU}.

Lemma 2.6

l^{∗}(Σ^{−1}(y_{2n+2})) = z_{2n+1}^{∗} ∈ Λ_{Z/pZ}^{∗}(z_{1}, z_{3}, …) = [L_{n}^{(i)}, HZ/p]_{MU}^{•}.

Here the element z_{2n+1}^{∗} denotes the primitive element in the coalgebra [L_{n}^{(i)}, HZ/p]_{MU}^{•} that is dual to z_{2n+1}.

Proof. This is analogous to the statement (ii) of Theorem 8.1 of [3] and is proved similarly. First we find by comparing the spectral sequences for Der_{MU}^{•}(L_{n}^{(i)}, HZ/p) and [L_{n}^{(i)}, HZ/p]_{MU} that the image of Σ^{−1}(y_{2n}) is as stated on the level of the E_{∞}-terms. Next, this image is contained in the subspace of primitive cohomology operations from L_{n}^{(i)} to HZ/p, i.e. those operations which are derivations up to homotopy and therefore no higher filtration terms will appear. The lemma is proved.

We now begin to perform the inductive step - proving that L_{n}^{(i+1)} is an MU-algebra. The cases i = 1 and i > 1 differ and will be considered separately.

1)When i = 1 L_{n}^{(i)} = HZ/p. Take the element

y_{2n+2} ∈ Z/pZ[y_{2}, y_{4}, …] = T HH_{MU}^{•}(L_{n}^{(i)}, HZ/p)

and consider the associated singular extension of MU-algebras:

Σ^{2ni}HZ/p → X → L_{n}^{(i)}  (1)

We want to prove that (1) is the 2n + 1th Postnikov stage of L_{n}. For this it suffices to show that the MU_{•}-module structure on X_{•} is given by the formula

X_{•} = MU_{•}/(p.x_{2}, x_{4}, …, x_{2n}^{2}, x_{2n+2}, …)

Notice that (1) gives rise to the following (algebraic) singular extension of MU_{•} - algebras:

Σ^{2n}Z/pZ → X_{•} → Z/pZ  (2)

If (2) is nontrivial (as an extension of MU_{•}-algebras) then we are done - it is elementary to verify that there is only one (up to a scalar) nontrivial singular graded extension of Z/pZ by Σ^{2n}Z/pZ, namely

Σ^{2n}Z/pZ → Z/pZ[x_{2n}]/x_{2n}^{2} → Z/pZ

which is what we need. Therefore, we have to eliminate the possibility for (2) to be trivial, i.e. split. Suppose, on the contrary, that this is the case. We will deduce from it that (1) is split in the homotopy category of MU-modules, which would be a contradiction by Lemma (2.6)

Indeed, consider the spectral sequence

E_{2}^{st} = Ext_{MU_{•}}^{st}(Z/pZ, Z/pZ ⊕ Σ^{2n}Z/pZ) = Ext_{MU_{•}}^{st}(L_{n}^{(i)}, X_{•}) ⇒ [L_{n}^{(i)}, X]_{MU}^{•}.

Then

E_{2}^{st} = (Λ_{Z/pZ}(z_{1}, z_{3}, …) ⊕ Σ^{2n}Λ_{Z/pZ}(z_{1}, z_{3}, …))^{∗}.

Comparing E_{2}^{st} with the spectral sequence for [X, X]_{MU}^{•} we see that the former collapses. The element 1 ∈ E_{2}^{00} corresponds the MU-module map L_{n}^{(i)} → X splitting the cofibre sequence (1) and we are done.

2) The case i > 1 is a little harder. We want to obtain the singular extension

Σ^{2ni}HZ/p → Y → L_{n}^{(i)}

where Y_{•} = MU_{•}/(p, x_{2}, …, x_{2n−2}, x_{2n}^{i+1}, x_{2n+2}, …).

However the relevant subspace in T HH^{•}(L_{n}^{(i)}, HZ/p) is 2-dimensional, it is spanned by the elements

y_{2n+2}, y_{2ni+2} ∈ T HH^{•}(L_{n}^{(i)}, HZ/p) = Z/pZ[y_{2}, y_{4}, …]

since y_{2n+2} and y_{2ni+2} both have degree 2ni + 2. Take any nonzero element y in this subspace, i.e. a linear combination of y_{2n+2} and y_{2ni+2}. Associated to y is a certain singular extension of L_{n}^{(i)} by Σ^{2ni}HZ/p which gives rise to the following algebraic singular extension of MU-algebras:

Σ^{2ni}Z/pZ → Y_{•} → L_{n•}^{(i)}  (3)

Just as before, we show that (3) is nontrivial. But now the space of extensions like (3) is also 2-dimensional. The basis in this space corresponds to two different MU_{•}-algebra structures on Y_{•}. One is given as

Y_{•} = MU_{•}/(p, x_{2}, …, x_{2n−2}, x_{2n}^{i+1}, x_{2n+2}, …)

and the other as

Y_{•} = MU_{•}/(p, x_{2}, …, x_{2n−2}, x_{2n}^{i}, x_{2n+2}, …, x_{2ni}^{2}, x_{2ni+2}, …).

Let us denote the corresponding algebraic extensions as E_{1} and E_{2} respectively. We need to choose the element y which gives rise to the extension E_{1}.

Claim. The element y_{2ni+2} ∈ T HH^{2ni+2}(L_{n}^{(i)}, HZ/p) gives rise to the extension E_{2} (or a multiple of it). Indeed, consider the canonical map L_{n}^{(i)} → L_{n}^{1} = HZ/p and the induced map on T HH:

f : T HH(L_{n}^{(1)}, HZ/p) = Z/pZ[\overline{y}_{2}, \overline{y}_{4}, …] → T HH(L_{n}^{(i)}, HZ/p) =
Z/pZ[y_{2}, y_{4}, …]

(here we put bars over the polynomial generators of T HH^{•}(L_{n}^{(1)}, HZ/p) to distinguish them from the corresponding elements in T HH^{•}(L_{n}^{(i)}, HZ/p). Then f(\overline{y}_{2ni+2}) = y_{2ni+2}. Indeed, the map f is is so on the level of E_{2}-terms of the corresponding spectral sequences and f(\overline{y}_{2ni+2}) should be primitive no higher degree correction terms will appear.

The considerations as in the case i = 1 tell us that the element \overline{y}_{2ni+2} gives rise to the algebraic extension (up to a scalar factor)

Σ^{2ni}Z/pZ → \overline{Y}_{•} → L_{n•}^{(1)} = Z/pZ  (4)

where

\overline{Y}_{•} = MU_{•}/(p, x_{2}, …, x_{2ni}, x_{2ni+2}^{2}, …).

Therefore the extension corresponding to \overline{y}_{2ni+2} is the extension induced from (4) by the augmentation map Z/pZ[x_{2n}]/x_{2n}^{i} → Z/p which is nothing but the extension E_{2}.

Next take an element y ∈ T HH^{•}(L_{n}^{(i)}, HZ/p) which is a nonzero linear combination of y_{2n+2} and y_{2ni+2} and not proportional to y_{2ni+2}. The corresponding algebraic extension will have the form αE_{1} + βE_{2}, where α and β are scalars with α ≠ 0 and “+” means the Baer sum of extensions. Altering y by adding to it a multiple of y_{2ni+2} and perhaps multiplying it by a scalar we obtain an element ỹ whose corresponding algebraic extension is E_{1}. (we leave it to the reader to check that addition of elements in T HH^{•}(L_{n}^{(i)}, HZ/p) corresponds to the Baer sum of associated extensions.)

We proved that the element ỹ descends to the 2ni + 1st k-invariant of L_{n}.Inspection shows that there was no freedom in picking the element ỹ, i.e. that the MU-algebra structure on L_{n}^{(i+1)} is unique. With this our theorem is proved.

Corollary 2.7 The spectrum k(n), the nth connective Morava K-theory admits a structure of an MU-algebra.

Remark 2.8 Notice, that we do not claim the uniqueness of this structure. The point is that k(n) can be obtained from MU by killing the ideal (p, x_{2}, x_{4}, …, x̂_{2}(p^{n} − 1), …) and the preceding theorem applies. However k(n) can be obtaibed by killing other ideals as well, which means that even the MU_{•}module structure on k(n)_{•} is not unique. Once we agree to call k(n) the MU-module corresponding to the ideal (p, x_{2}, x_{4}, …, x̂_{2(p^{n}−1),...}) we have the uniqueness of the MU-algebra structure on k(n).

3 Topological Hochschild cohomology of Morava K-theories

Now we know that Morava K-theory spectra are MU-algebras, and therefore, we can talk about their topological Hochschild cohomology. The computation of topological Hochschild (co)homology of spectra other than Eilenberg-MacLane was initiated by McClure and Staffeldt (cf. [5]). In this section we compute T HH_{MU}(K(n), K(n)) for all primes p and T HH_{S}(K(n), K(n)) for odd primes. The results are similar to the calculation of topological Hochschild cohomology of the Eilenberg-MacLane spectrum HZ/p in[3].Some additional care needs to be taken because K(n) is not a commutative MUalgebra like HZ/p, but this circumstance does not affect the final result. In the future paper we apply these results to realize topologically the Lubin-Tate universal deformation of height n formal groups.

Theorem 3.1 (i)

T HH_{MU}^{•}(K(n), K(n)) = K(n)_{•}[y_{2}, y_{4}, … ŷ_{2p^{n}}, …];
Der_{MU}^{•−1}(K(n), K(n)) = K(n)_{•}[y_{2}, y_{4}, … ŷ_{2p^{n}}, …]/K(n)_{•}.

(ii)

[K(n), K(n)]_{MU}^{•} = Λ_{K(n)_{•}}(z_{1}, z_{3}, …, ẑ_{2p^{n}−1}, …).

and the map

Der_{MU}^{•}(K(n), K(n)) → [K(n), K(n)]_{MU}^{•}

sends the elements y_{2i} ∈ Der_{MU}^{2i−1}(K(n), K(n)) to z_{2i−1} ∈ [K(n), K(n)]_{MU}^{2i−1}. (iii) The forgetful map

[K(n), K(n)]_{MU}^{•} ⟶ [K(n), K(n)]_{S}^{•}

maps the elements z_{2p^{k}−1} to the operations Q_{k} ∈ [K(n), K(n)]_{S}^{2p^{k}−1} (cf., for instance [1]) and all other z’s go to zero.

Proof. The proof is along the same lines as the proof of Theorem 8.1 of [3], and we will be somewhat sketchy therefore. First compute π_{•}(K(n) ∧_{MU} K(n)). We have the Kunneth spectral sequence

T or_{Z[x_{2},x_{4},...]}(K(n)_{•}, K(n)_{•}) = Λ_{K(n)_{•}}(z_{1}, z_{3}, …, ˆ z_{2p^{i}−1}, …)
⟹ π_{•}(K(n) ∧_{MU} K(n))

which collapses so we have multiplicatively

π_{•}K(n) ∧_{MU} K(n) = Λ_{K(n)_{•}}(z_{1}, z_{3}, …, ˆ z_{2p^{i}−1}, …).

Next consider the spectral sequence

E_{2}^{ij} = Ext_{π_{•}K(n)∧_{MU} K(n)}^{ij}(K(n)K(n)) ⟹ T HH_{MU}^{i+j}(K(n), K(n)).

We have

E_{2}^{ij} = Ext_{Λ_{K(n)•}(z_{1},z_{3},...)}^{ij}(K(n)K(n)) = K(n)_{•}[y_{2}, y_{4}, …].

This spectral sequence collapses, but apriori there could be some nontrivial additive extensions. Indeed, K(n) is not a commutative MU-algebra, and T HH_{MU}(K(n)K(n)) need not be a K(n)-module. In particular we don’t know whether T HH_{MU}^{•}(K(n), K(n)) is a Z/pZ-algebra. Let us show that this is, in fact, true.

To this end consider the connective Morava K-theory spectrum k(n). This is an MU-algebra and just as above we have a spectral sequence

Ẽ_{2}^{ij} = k(n)_{•}[ỹ_{2}, ỹ_{4}, …] ⟹ T HH_{MU}(k(n), k(n))

(we put tildes over y_{i}’s to distinguish them from the polynomial generators in T HH_{MU}^{•}(K(n), K(n))). Notice, that Ẽ_{2}^{ij} is a connected graded algebra whose 0th graded component is Z/pZ. Therefore, in this spectral sequence there is no room for additive extensions and we conclude that

T HH_{MU}^{•}(k(n), k(n) = k(n)_{•}[ỹ_{2}, ỹ_{4}, …].

Further, consider the following maps of MU-modules:

T HH_{MU}(K(n), K(n)) →^{f} T HH_{MU}(k(n), K(n)) ←^{g} T HH_{MU}(k(n), k(n)).

Both maps f and g are induced by the localization map k(n) → K(n). Comparing spectral sequences we see, that f induces an isomorphism

T HH_{MU}^{•}(K(n), K(n)) → T HH_{MU}^{•}(k(n), K(n))

and g induces a monomorphism

T HH_{MU}^{•}(k(n), k(n)) → T HH_{MU}^{•}(k(n), K(n)).

This shows that T HH_{MU}^{•}(k(n), K(n)) = T HH_{MU}^{•}(K(n), K(n)) is a Z/pZalgebra and part (i) is proved.

Part (ii) is proved similarly to Theorem 8.1 of [3]. An easy calculation shows that

[K(n), K(n)]_{MU} = Λ_{K(n)_{•}}(z_{1}, z_{3}, …, ẑ_{2p^{i}−1}, …).

Next we first show that the elements y_{2i} can map to either z_{2i−1} or zero and then considering the induced map on the corresponding spectral sequences we see, that the image of y_{2i−1} is not zero. Part (ii) is proved.

Part (iii) is also proved similar to the corresponding statement in [3]. The only nontrivial thing is to prove that the element z_{2p^{i}−1} ∈ [K(n), K(n)]_{MU}^{2p^{k}−1} considered as a K(n)-operation in the category of S-modules is not homotopic to zero. Denote MU(i, n) the MU-module obtained by killing the prime p and all polynomial generators in MU_{•} except for x_{2p^{i}−2} and x_{2p^{i}}. Consider the following homotopy cofibre sequence of MU-modules:

Σ^{2p^{i}−2}MU(i, n) ⟶^{x_{2p^{i}−2}} MU(i, n) ⟶^{p} K(n) ⟶^{∂} Σ^{2p^{i}−1}MU(i, n).

Now the map of MU-modules

p ∘ ∂ : K(n) → Σ^{2p^{i}−1}K(n)

is exactly the K(n)-operaton Q_{i} and we are done.

The next theorem describes topological Hochschild cohomology of K(n) as an S-algebra.

Theorem 3.2 Let p be an odd prime. Then

T HH_{S}^{•}(K(n), K(n)) = K(n)_{•}[α_{0}, α_{2}, …, α_{n−1}]

where α_{k} ∈ T HH^{2p^{i}}(K(n), K(n))

Proof. We have a spectral sequence

E_{2}^{ij} = Ext_{K(n)_{•}K(n)}(K(n), K(n)) ⟹ T HH_{S}^{•}(K(n), K(n))

and the calculation of Robinson in [6] shows that

E_{2}^{ij} = K(n)_{•}[α_{0}, α_{2}, …, α_{n−1}].

This spectral sequence collapses and to prove the theorem we only need to show that there are no nontrivial additive extensions, that is, that T HH_{S}^{•}(K(n), K(n)) is a Z/pZ-algebra.

There is an algebra map

T HH_{MU}^{•}(K(n), K(n)) → T HH_{S}^{•}(K(n), K(n)).

Since T HH_{MU}^{•}(K(n), K(n)) is a Z/pZ-algebra this is only possible if T HH_{S}^{•}(K(n), K(n)) is an Z/pZ-algebra and the theorem is proved.

References

[1] A.Baker, U.Wurgler. Bockstein operations in Morava K-theories. Forum Math. 3, no. 6, 543–56,1991

[2] A.D.Elmendorf, I.Kriz, M.A.Mandell, J.P.May. Rings, Modules and Algebras in Stable Homotopy Theory. Mathematical Surveys and Monographs. Vol. 47. AMS,1996

[3] A.Lazarev Topological Hochschild cohomology I. Submitted to Topology

[4] J.E.McClure. Preprint, 1999

[5] J.E.McClure, R.E.Staffeldt. On the topological Hochschild homology of bu I, Amer.J.Math.115, 1-45, 1993

[6] A.Robinson Obstruction theory and the strict associativity of Morava K-theories, London Math.Soc.Lecture Notes 139, 143-152, 1989