Towers of MU-algebras and the generalized Hopkins-Miller theorem

A.Lazarev

Department of Mathematics, Univ. of Bristol, Bristol, BS8 1TW, UK. email A.Lazarev@bristol.ac.uk

Abstract

Our results are of three types. First we describe a general procedure of adjoining polynomial variables to A_{∞}-ring spectra whose coefficient rings satisfy certain restrictions. A host of examples of such spectra is provided by killing a regular ideal in the coefficient ring of MU, the complex cobordism spectrum. Second, we show that the algebraic procedure of adjoining roots of unity carries over in the topological context for such spectra. Third, we use the developed technology to compute the homotopy types of spaces of strictly multiplicative maps between suitable K(n)-localizations of such spectra. This generalizes the famous Hopkins-Miller theorem and gives strengthened versions of various splitting theorems.

Key words: S-algebras, topological derivations, Morava K-theories,Witt vectors.

1 Introduction

Our goals in this paper are three-fold. First, we generalize and extend the methods of [6] of constructing A_{∞} ring spectra by adjoining “polynomial variables”. Namely, suppose that R is a commutative S-algebra in the sense of [4] such that its coefficient ring R_{∗} has no elements of odd degree and the graded ideal I_{∗} in R_{∗} is generated by a regular sequence u_{1}, u_{2}, . . .. Then it is known from work of Strickland, [13] that the R-module R/I obtained by killing the ideal I has a structure of an R-ring spectrum. We assume that R/I is actually an R-algebra. Then it turns out that, informally speaking, all R-modules “between” R/I and R also possess structures of R-algebras.

More precisely, there are R-algebras R[u_{k}]/u_{k}^{l} whose coefficient rings are obtained from R_{∗} by killing the sequence (u_{1}, u_{2}, …, u_{k−1}, u_{k}^{l}, u_{k+1}, . . .. There are also natural “reduction maps” between these R-algebras.

The basic example is provided by taking R = MU, the complex cobordism S-algebra and MU/I be the Eilenberg-MacLane S-algebra H/p. It follows by induction that all MU-algebras obtained by killing any sequence of polynomial generators and/or a prime p have structures of MU-modules.

Second, we consider the question of adjoning roots of unity to an Salgebra. This problem was also treated in the recent preprint [12] by Schwanzl, Vogt and Waldhausen. Their definition of a topological extension has better formal properties than ours but it applies in a far less general situation. We show that one can adjoin roots of unity to such spectra as Morava Ktheories K(n), Johnson-Wilson theories E(n) and many other algebras over the complex cobordism S-algebra MU.

Third, we address the problem of computing S-algebra maps between a certain completion Ê(n) of E(n) and an MU-algebra E which is assumed to be “strongly K(n)-complete” in some precise sense explained later on in the paper. Examples of such MU-algebras include Ê(n), K(n) and the Artinian completion of the v_{n}-localization of the Brown-Peterson spectum BP. It turns out that the space of S-algebra maps between Ê(n) and a strongly K(n)-complete MU-algebra is homotopically discrete with the set of connected components being equal to the set of multiplicative cohomology operations Ê(n) → E. We call the results of this type the generalized Hopkins-Miller theorem because the original Hopkins-Miller theorem (cf.[10]) asserts that the space of S-algebra self-maps of the spectrum E_{n} is homotopically equivalent to the (discrete) space of multiplicative operations from E_{n} to itself. Here E_{n} is a 2-periodic version of the completed Johnson-Wilson theory Ê(n) which came to be popularly known as the Morava E-theory.

We chose to work with the 2(p^{n} − 1)-periodic theory E(n) rather than with E_{n}. The relation of E(n) to E_{n} is the same as the relation of the p-local Adams summand of the complex K-theory spectrum KU to KU itself. The advantage of E(n) is that it is smaller than E_{n}, however it does not admit the action of the full Morava stabilzer group.

One consequence of our generalized Hopkins-Miller theorem is that Ê(n) admits a unique S-algebra structure, the result previously obtained in [1].

Another consequence is that Ê(n) splits off E as an S-algebra for a certain class of strongly K(n)-complete MU-algebras E. Such splittings were previously known to be multiplicative only up to homotopy.

Throughout the paper the symbol F_{p} will denote the prime field with p elements, L/F_{p} an arbitrary (but fixed) finite separable extension of F_{p}.

Further W (L) and W_{n}(L) denote the ring of Witt vectors and the ring of Witt vectors of length n respectively.

2 Adjoining polynomial variables to S-algebras

In this section we assume that R is a fixed commutative S-algebra such that R_{∗} = π_{∗}R is a graded commutative ring concentrated in even degrees. We will also assume without loss of generality that R is q-cofibrant in the sense of [4]. All objects under consideration will be R-modules or R-algebras and smash products and homotopy classes of maps will be understood to be taken in the category of R-modules.

We begin by reminding the reader the notions of topological derivations and topological singular extensions of R-algebras. A detailed account can be found in [6]. Let A be an R-algebra and M an A-bimodule. Then the R-module A ∨ M has the obvious structure of an R-algebra (“square-zero extension” of A). Consider the set [A, A ∨ M]_{R−alg/A} of homotopy classes of R-algebra maps from A to A ∨ M which commute with the projection onto A. Then there exists an A-bimodule Ω_{A} and a natural in M isomorphism

[A, A ∨ M]_{R−alg/A} ≅ [Ω_{A}, M]_{A−bimod}

where the right hand side denotes the homotopy classes of maps in the category of A-bimodules.

Definition 2.1 The topological derivations R-module of A with values in M is the function R-module F_{A∧A^{op}}(Ω_{A}, M). We denote it by Der(A, M) and its ith homotopy group by Der^{−i}(A, M).

The A-bimodule Ω_{A} is constructed as the homotopy fibre of the multiplication map A ∧ A → A. There exists the following homotopy fibre sequence of R-modules:

THH(A, M) → M → Der(A, M)  (1)

Here THH(A, M) is the topological Hochschild cohomology spectrum of A with values in M, THH(A, M) := F_{A∧A^{op}}(A, A).

We will frequently use the notion of a primitive operation from A to M. Denote by m : A ∧ A → A the multiplication map and by m_{l} : A ∧ M → M and m_{r} : M ∧A → M the left and right actions of A in M respectively. Then a map p : A → M is called primitive if p is a “derivation up to homotopy”, i.e. the following diagram is homotopy commutative:

A ∧ A  ⟶^{m} A
1 ∧ p ∨ p ∧ 1 ↓  ↓ p
A ∧ M ∨ M ∧ A  ⟶^{m_{l}∨m_{r}} M

There is a forgetful map l : Der^{∗}(A, M) → [A, M]^{∗} defined as follows. For any topological derivation d : A → A ∨ M let l(d) be the composite map A → A ∨ M → M where the last map is just the projection onto the wedge summand. Then it is easy to see that the image of l is contained in the set of primitive operations from A to M.

Suppose we are given a topological derivation d : A → A ∨ M. Consider the following homotopy pullback diagram

X →  A
↓  ↓
A →^{d} A ∨ M

Here the rightmost downward arrow is the canonical inclusion of a retract. Then we have the following homotopy fibre sequence of R-modules:

Σ^{−1}M → X → A  (2)

Definition 2.2 The homotopy fibre sequence (2) is called the topological singular extension associated with the derivation d : A → A ∨ M.

For an element x ∈ R we will denote by R/x the cofibre of the map R →^{x} R Let I_{∗} be a graded ideal generated by (possibly infinite) regular sequence of elements (u_{1}, u_{2}, …) ∈ R_{∗} of degrees r_{1}, r_{2}, . . .. Then we can form the R-module R/I as the infinite smash product of R/x_{i}. By [13], Proposition 4.8 there is a structure of an R-ring spectrum on R/I. Clearly the coefficient ring of R/I is isomorphic to R_{∗}/I_{∗} where R_{∗}/I_{∗} is understood to be the direct limit of R_{∗}/(u_{1}, u_{2}, …, u_{k}).

Our standing assumption is that R/I has a structure of an R-algebra (i.e. strictly associative). This may seem a rather strong condition but, as we see shortly such a situation is rather typical. In fact P.Goerss proved in [5] that any spectrum obtained by killing a regular sequence in MU, the complex cobordism S-algebra, has a structure of an MU-algebra.

The construction we are about to describe allows one to construct new R-algebras by “adjoining” the indeterminates u_{k} to the R-algebra R/I. The basic idea is the same as in [6] where Morava K-theories at an odd prime were shown to possess MU-algebra structures. However the arguments we use here are considerably more general, in particular we make no assumption that the prime 2 is invertible. Let us introduce the notation R/I[u_{k}]/u_{k}^{l}_{∗} for the R_{∗}-algebra

lim_{n→∞}R_{∗}/(u_{1}, u_{2}, …, u_{k−1}, u_{k}^{l}, u_{k+1}, …, u_{n}).

This notation emphasizes the analogy with the truncated polynomial algebra even though R/I[u_{k}]/u_{k}^{l}_{∗} is not always one. (Take, for instance R_{∗} to be the ring of integers and I_{∗} = (u_{1}) = p, a prime number. Then R/I[u_{k}]/u_{1}^{l}_{∗} = Z/p^{l}Z.) For each l reduction modulo u_{k}^{l+1} determines a map of R_{∗}-algebras

R/I[u_{k}]/u_{k}^{l+1}_{∗} → R/I[u_{k}]/u_{k}^{l}_{∗}.

Now we can formulate our main theorem in this section.

Theorem 2.3 For each k and l there exist R-algebras R/I[u_{k}]/u_{k}^{l} with coefficient rings R/I[u_{k}]/u_{k}^{l}_{∗} and R-algebra maps

R_{i,k} : R/I[u_{k}]/u_{k}^{l+1} → R/I[u_{k}]/u_{k}^{l}

which give the reductions mod u_{k}^{l+1} on the level of coefficient rings.

Proof. Suppose by induction that the R-algebras R/I[u_{k}]/u_{k}^{l} with the required properties were constructed for l ≤ i. We will show that there exists an appropriate Bokstein operation from R/I[u_{k}]/u_{k}^{i} to Σ^{|u_{k}|i+1}R/I which allow us to build the next stage.

Consider the cofibre sequence

Σ^{|u_{k}|l}R →^{u_{k}^{l}} R →^{ρ_{l,k}} R/u_{k}^{l} →^{β_{l,k}} Σ^{|u_{k}|l+1}R.

According to [13] the R-module R/u_{k}^{l} admits an associative product ϕ : R/u_{k}^{l} ∧ R/u_{k}^{l} → R/u_{k}^{l} and for any other product ϕ′ there exists a unique element u ∈ π_{2|u_{k}|l+2}R/u_{k}^{l} for which ϕ′ = ϕ + u ∘ (β_{l,k} ∧ β_{l,k}).

Lemma 2.4 There exists a map of R-ring spectra r_{i,k} : R/u_{k}^{i+1} → R/u_{k}^{i} realizing the reduction map mod u_{k}^{i+1} on coefficient rings. Moreover in the cofibre sequence

R/u_{k}^{i+1} →^{r_{i,k}} R/u_{k}^{i} →^{β̄_{i,k}} Σ^{|u_{k}|i+1}R/u_{i}  (3)

the second map β̄_{k,i} : R/u_{k}^{i} → Σ^{|u_{k}|(i+1)}R/u_{k} is a primitive operation.

Proof. Use induction on i (in fact we only need the inductive assumption in order that R/u_{k} be a (bi)module spectrum over R/u_{k}^{i}.) Consider the following diagram of R-modules:

Σ^{|u_{k}|(i+1)}R →^{u_{k}} Σ^{|u_{k}|i}R → Σ^{|u_{k}|i}R/u_{k}
u_{k}^{i+1} ↓  ↓ u_{k}^{i}  ↓
R  =  R  →  pt
↓  ↓  ↓
R/u_{k}^{i+1}  →^{r_{i,k}} R/u_{k}^{i} → Σ^{|u_{k}|i+1}R/u_{k}
↓  ||
Σ^{|u_{k}|i+1}R → Σ^{|u_{k}|i+1}R/u_{k}

Here the rows and columns are homotopy cofibre sequences of R-modules. The map r_{i,k} : R/u_{k}^{i+1} → R/u_{k}^{i} is determined uniquely by the requirement that the diagram commute in the homotopy category of R-modules.

We now show that it is possible to choose a product on R/u_{k}^{i+1} so that r_{i,k} becomes an R-ring spectrum map. Take first any associative product ϕ : R/u_{k}^{i+1} ∧ R/u_{k}^{i+1} → R/u_{k}^{i+1} which exists by [13], Proposition 3.1. As in [13], Proposition 3.15 there is an obstruction d(ϕ) ∈ π_{2|u_{k}|(i+1)+2}(R/u_{k}^{i}) for the map r_{i,k} to be homotopy multiplicative. If the product ϕ is changed the obstruction changes according to the formula

d(ϕ + u ∘ (β_{i,k} ∧ β_{i,k})) = d(ϕ) + r_{i,k∗}u.

Since the map r_{i,k∗} : R_{∗}/u_{k}^{i+1} → R_{∗}/u_{k}^{i} is just the reduction modulo u_{k}^{i+1} it is surjective and we conclude that there exists a product on R/u_{k}^{i+1} for which the obstruction vanishes.

Further the map β̄_{i,k} : R/u_{k}^{i} → Σ^{|u_{k}|i+1}R/u_{k} coincides with the composition

R/u_{k}^{i} →^{β_{i,k}} Σ^{|u_{k}|i+1}R →^{ρ_{i,k}} Σ^{|u_{k}|i+1}R/u_{k}^{i} → Σ^{|u_{k}|i+1}R/u_{k}.

Here the last map is (a suspension of) the canonical R-ring map R/u_{k}^{i} → R/u_{k} which exists by the inductive assumption. The composition ρ_{i} ∘ β_{i} is the Bokstein operation which is primitive by [13], Proposition 3.14. It follows that β_{k,i} is also primitive and our lemma is proved.

Now taking the smash product of (3) with R/u_{1} ∧ R/u_{2} ∧ … ∧ R/u_{k−1} ∧ ∧R/u_{k+1}∧… we get the following homotopy cofibre sequence of R-modules:

R/I[u_{k}]/u_{k}^{i+1} ⟶^{R_{i,k}} R/I[u_{k}]/u_{k}^{i} ⟶^{Q_{i,k}} Σ^{|u_{k}|i+1}R/I.

It follows that Q_{i,k} is a primitive operation and R/I[u_{k}]/u_{k}^{i+1} has an R-ring spectrum structure such that R_{i,k} is an R-ring map. (Notice our abuse of notations here in using the symbol R_{i,k} even though R/I[u_{k}]/u_{k}^{i+1} is not yet proved to be an R-algebra).

Next we will describe the set of all primitive operations R/I[u_{k}]/u_{k}^{i} → Σ^{∗}R/I. Consider the cofibre sequence

Σ^{|u_{l}|}R →^{u_{l}} R →^{ρ_{l}} R/u_{l} →^{β_{l}} Σ^{|u_{l}|+1}R.

For l ≠ k introduce the operation

Q_{l} : R/I[u_{k}]/u_{k}^{i} = R/u_{1}∧R/u_{2}∧…∧R/u_{k−1}∧R/u_{k}^{i}∧R/u_{k+1}∧… → Σ^{|u_{l}|+1}R/I

obtained by smashing ρ_{l} ∘ β_{l} : R/u_{l} → Σ^{|u_{l}|+1}R/u_{l} with the identity map on the remaining smash factors. Clearly the operations Q_{l} are primitive. The next result shows that Q_{l} and Q_{i,k} are essentially all primitive operations from R/I[u_{k}]/u_{k}^{i} into (suspensions of) R/I.

Lemma 2.5 Any primitive operation R/I[u_{k}]/u_{k}^{i} → Σ^{∗}R/I can be written uniquely as an infinite sum a_{k}Q_{i,k} + Σ_{i≠k}a_{i}Q_{i} for a_{i} ∈ R/I_{∗}.

Proof. We will only give a sketch since the arguments of Strickland ([13], Proposition 4.17) carry over almost verbatim. Let η : R → R/I[u_{k}]/u_{k}^{i} be the unit map and J_{∗} = (u_{1}, u_{2} …, u_{k−1}, u_{k}^{i}, u_{k+1,...}) be the kernel of the map η_{∗} : R_{∗} → R/I[u_{k}]/u_{k}^{i}_{∗}. Given a primitive operation Q : R/I[u_{k}]/u_{k}^{i} → Σ^{s}R/I define the function d(Q) : J_{∗} → R/I[u_{k}]/u_{k}^{i}_{∗} as follows. For any x ∈ J_{∗} we have a cofire sequence

Σ^{|x|}R →^{x} R →^{ρ_{x}} R/x →^{β_{x}} Σ^{|x|+1}R.

Then there is a unique map f_{x} : R/x → R/I[u_{k}]/u_{k}^{i} such that f_{x} ∘ ρ_{x} = η. Further there is a unique map y : Σ^{|x|+1}R → Σ^{s}R/I such that Q∘f_{x} = y∘β_{x}. We define d(Q)(x) := y ∈ π_{|x|+1−s}R/I_{∗}.

It follows that the function d actually embeds the set of primitive operations from R/I[u_{k}]/u_{k}^{i} into R/I into Hom_{R_{∗}}(J/J^{2}, R/I_{∗}). Further it is straightforward to check that

d(Q_{s})(u_{t}) = δ_{st} for s, t ≠ k, d(Q_{s})(u_{k}^{i}) = 0;
d(Q_{i,k})(u_{j}) = 0 for j ≠ k, d(Q_{i,k})(u_{k}^{i}) = 1.

Therefore the elements Q_{i,k}, Q_{s}, s ≠ k form a basis in Hom(J_{∗}/J_{∗}^{2}, R/I_{∗}) dual to the basis (u_{1}, u_{2}, …, u_{k−1}, u_{k}^{i}, u_{k+1}, …) in J_{∗}/J_{∗}^{2} and our lemma is proved.

Now we come to the crucial part of the proof. We are going to show that the Bokstein operation Q_{i,k} can be improved to a topological derivation from which it would follow that R/I[u_{k}]/u_{k}^{i+1} is an R-algebra and R_{i,k} : R/I[u_{k}]/u_{k}^{i+1} → R/I[u_{k}]/u_{k}^{i} lifts to an R-algebra map.

The following lemma (interesting in its own right) is preparatory for computing the topological Hochschild cohomology of R/I[u_{k}]/u_{k}^{l} with coefficients in R/I.

Lemma 2.6 For any regular ideal J_{∗} = (x_{1}, x_{2}, …) in R_{∗} and any product on R/J there is a multiplicative isomorphism

π_{∗}(R/I ∧_{R} R/J^{op}) ≅ Λ_{R_{∗}/J_{∗}}(τ_{1}, τ_{2}, …)

where |τ_{i}| = |x_{i}| + 1.

Proof. The standard spectral sequence arguments show that π_{∗}(R/J ∧_{R} R/J^{op}) is a filtered R_{∗}/J_{∗}-algebra whose associated graded is isomorphic to Λ_{R_{∗}/J_{∗}}(τ_{1}, τ_{2}, …) so the problem is to show that the possible multiplicative extensions are trivial. More precisely one needs to show that all skew-commutators of the elements τ_{i} in π_{∗}(R/J ∧_{R} R/J^{op}) as well as their squares are zero. Consider the map of R-ring spectra f : R/J ∧_{R} R/J^{op} → F_{R}(R/J, R/J) which is induced by the structure of a left R/J ∧_{R} R/J^{op}- module spectrum on R/J. Then f induces a map of R_{∗}/J_{∗}-algebras f_{∗} : π_{∗}R/J ∧_{R} R/J^{op} → π_{∗}F_{R}(R/J, R/J). By [13], Proposition 4.15 the ring π_{∗}F_{R}(R/J, R/J) is a completed exterior algebra over R_{∗}/J_{∗}, in particular it is (graded) commutative. Therefore all possible nontrivial relations between the elements τ_{i} map to zero in the R_{∗}/J_{∗}-algebra π_{∗}F_{R}(R/J, R/J). Since these relations belong to R_{∗}/I_{∗} and f_{∗} is R_{∗}/I_{∗}-linear we conclude that they are actually zero and π_{∗}(R/J ∧_{R} R/J^{op}) is in fact a (graded) commutative R_{∗}/J_{∗}-algebra. This finishes the proof of lemma 2.6.

Remark 2.7 The R_{∗}/J_{∗}-algebra π_{∗}(R/J ∧_{R} R/J) need not be commutative in general. For example take R/J = MU/2, the reduction of MU modulo 2. Then it can be shown using the methods of [8] that MU/2 ∧_{MU} MU/2 is an MU/2_{∗}-algebra on one generator in degree 1 whose square is equal to x_{1} ∈ MU/2_{∗}. We will discuss this and related phenomena elsewhere.

Lemma 2.8 There is the following isomorphism of graded R_{∗}-modules:

grT HH^{∗}(R/I[u_{k}]/u_{k}^{i}, R/I) = R/I_{∗}[ũ_{1}, ũ_{2}, …, ũ_{k}, …];
grDer^{∗−1}(R/I[u_{k}]/u_{k}^{i}, R/I) = R/I_{∗}[ũ_{1}, ũ_{2}, …, ũ_{k}, …]/(R/I_{∗}).

Here gr(?) denotes the associated graded module. Moreover the image of the forgetful map

Der^{∗}(R/I[u_{k}]/u_{k}^{i}, R/I) → [R/I[u_{k}]/u_{k}^{i}, R/I]^{∗}

is the set of all primitive operations from R/I[u_{k}]/u_{k}^{i} to Σ^{∗}R/I.

Proof. Denote by J_{∗} the ideal in R generated by (u_{1}, u_{2}, …, u_{k−1}, u_{k}^{i}, u_{k+1}, …). Then we have the following spectral sequence

Ext_{π_{∗}R/I[u_{k}]/u_{k}^{i}∧R/I[u_{k}]/u_{k}^{i}^{op}}^{∗∗}(R/I[u_{k}]/u_{k}^{i}_{∗}, R/I_{∗})
= Ext_{π_{∗}R/J∧R/J^{op}}^{∗∗}(R/J_{∗}, R/I_{∗}) ⇒ T HH^{∗}(R/I[u_{k}]/u_{k}^{i}, R/I).

By Lemma 2.6 π_{∗}R/J ∧ R/J^{op} = Λ_{R/J_{∗}}(τ_{1}, τ_{2} …). Therefore

Ext_{π_{∗}R/J∧R/J^{op}}^{∗∗}(R/J_{∗}, R/I) = R/I_{∗}[ũ_{1}, ũ_{2}, …, ũ_{i}, …]

where |ũ_{l}| = −|u_{l}|+2 for l ≠ k and |ũ_{k}| = −|u_{k}|i+2. Our spectral sequence collapses and we obtain the desired isomorphism

grT HH^{∗}(R/I[u_{k}]/u_{k}^{i}, R/I) = R/I_{∗}[ũ_{1}, ũ_{2}, …, ũ_{i}, …].

The isomorphism involving Der^{∗}(R/I[u_{k}]/u_{k}^{i}, R/I) is obtained similarly. Now consider the spectral sequence

Ext_{R∗}^{∗∗}(R/I[u_{k}]/u_{k}^{i}_{∗}, R/I_{∗}) = Hom(Λ(z_{1}, z_{2}, …), R/I_{∗}) ⇒ [R/I[u_{k}]/u_{k}^{i}, R/I]^{∗}.

Here |z_{l}| = |u_{l}| + 1 for l ≠ k and |z_{k}| = |u_{k}|i + 1. This spectral sequence is easily seen to collapse and denoting by z̃_{i} the elements in the dual basis in Λ(z_{1}, z_{2}, …) we identify its E_{2} = E_{∞}-term with Λ_{R/I_{∗}}(z̃_{1}, z̃_{2}, …). (Of course this is an identification only as R/I_{∗}-modules, not as rings.) It is clear that the elements z_{l} correspond to the primitive operations Q_{l} : R/I[u_{k}]/u_{k}^{i} → Σ^{|u_{l}|+1}R/I for l ≠ k whilst z_{k} corresponds to Q_{i,k} : R/I[u_{k}]/u_{k}^{i} → Σ^{|u_{l}|i+1}R/I.

Furthermore the forgetful map l (operating on the level of E_{2}-terms) sends ũ_{l} ∈ grDer^{∗}(R/I[u_{k}]/u_{k}^{i}, R/I) to z̃_{l}. In other words all primitive operations R/I[u_{k}]/u_{k}^{i} → Σ^{∗}R/I are covered by l up to higher filtration terms. Since the image of l : Der^{∗}(R/I[u_{k}]/u_{k}^{i}, R/I) → [R/I[u_{k}]/u_{k}^{i}, R/I]^{∗} is contained in the subspace of the primitive operations no higher filtration terms are present and we conclude that all Bokstein operations Q_{l} and Q_{i,k} are in the image of l. With this Lemma 2.8 is proved.

So we proved that there exists a topological derivation Q̃_{i,k} : R/I[u_{k}]/u_{k}^{i} → R/I[u_{k}]/u_{k}^{i} ∨Σ^{|u_{k}|i+1}R/I such that its composition with the projection onto the wedge summand R/I[u_{k}]/u_{k}^{i} ∨ Σ^{|u_{k}|i+1}R/I → Σ^{|u_{k}|i+1}R/I is the Bokstein operation Q_{i,k}. Associated with Q̃_{i,k} is a topological singular extension

Σ^{|u_{k}|i}R/I → X → R/I[u_{k}]/u_{k}^{i}

such that X is weakly equivalent to R/I[u_{k}]/u_{k}^{i+1}. In other words the Rmodule R/I[u_{k}]/u_{k}^{i+1} admits a structure of an R-algebra so that the reduction map R/I[u_{k}]/u_{k}^{i+1} → R/I[u_{k}]/u_{k}^{i} is an R-algebra map. The inductive step is completed and Theorem 2.3 is proved.

The singular extension of R-algebras

R/I[u_{k}]/u_{k}^{i+1} ⟶^{R_{i,k}} R/I[u_{k}]/u_{k}^{i} ⟶^{Q_{i,k}} Σ^{|u_{k}|i+1}R/I

is of fundamental importance to us. We will call the R-algebra R/I[u_{k}]/u_{k}^{i+1} an elementary extension of R/I[u_{k}]/u_{k}^{i}. Sometimes we will use the notation R/I[u_{k}] for holim_{i→∞}R/I[u_{k}]/u_{k}^{i}. Of course the coefficient ring of R/I[u_{k}] is not always a polynomial algebra on R/I_{∗}.

Now let R= MU, the complex cobordism spectrum. It is well-known that MU is a commutative S-algebra and MU_{∗} = Z[x_{1}, x_{2}, …], the polynomial algebra on infinitely many generators in even degrees.

Corollary 2.9 Let J_{∗} be a regular ideal in MU_{∗} = Z[x_{1}, x_{2}, …] generated by any subsequence of the regular sequence p^{ν_{0}}, x_{1}^{ν_{1}}, x_{2}^{ν_{2}}, … where ν_{l} ≥ 1 for all l. Then MU/J admits a structure of an MU-algebra.

Proof. First assume that the element pν 0 does not belong to J_{∗}. Denote by I_{∗} be the ideal in MU_{∗} generated by all polynomial generators (x_{1}, x_{2}, …). Then MU/I is the integral Eilenberg-MacLane spectrum HZ which possesses a canonical structure of an MU-algebra (even a commutative MUalgebra).

Taking successive elementary extensions corresponding to elements x_{i_{k}}^{ν_{i_{k}}} with ν_{i_{k}} > 1 we construct an MU-algebra

M̃U/J = ((HZ[x_{i_{1}}]/x_{i_{1}}^{ν_{i_{1}}} )[x_{i_{2}}]/x^{ν_{i_{k}}}) ….

Similarly taking elementary extensions corresponding to those polynomial generators {u_{1}, u_{2}, …} whose powers are not in J_{∗} and passing to the (inverse) limit we construct the MU-algebra M̃U/J[u_{1}, u_{2}, …] whose underlying MU-module is MU/J.

If the element p^{ν_{0}} does belong to J_{∗} the only difference is that we start our induction with the Eilenberg-MacLane MU-algebra HZ/p^{ν_{0}} instead of HZ. With this Corollary 2.9 is proved.

Remark 2.10 Notice that our method also shows that the MU-algebra structures on MU/J for different J are compatible in the sense that various reduction maps given by killing elements x_{i}^{ν_{i}} are actually MU-algebra maps.

Recall that the Brown-Peterson spectrum BP is obtained from MU by killing all polynomial generators in MU_{∗} except for v_{i} = x_{2(p^{i}−1)} and localizing at the prime p. Therefore BP_{∗} = Z_{(p)}[v_{1}, v_{2}, …]. It follows that BP possesses an MU-algebra structure. We define

BP < n >= BP/(v_{i}, i > n) = HZ_{(p)}[v_{1}, v_{2}, …, v_{n}]
P (n) = BP/(v_{i}, i < n) = HF_{p}[v_{n}, v_{n+1}, …]
B(n) = v_{n}^{−1}P (n) = v_{n}^{−1}HF_{p}[v_{n}, v_{n+1}, …]
k(n) = BP/(v_{i}, i ≠ n) = HF_{p}[v_{n}]
K(n) = v_{n}^{−1}BP/(v_{i}, i ≠ n) = HF_{p}[v_{n}, v_{n}^{−1}]
E(n) = v_{n}^{−1}BP/(v_{i}, i > n) = HZ_{(p)}[v_{1}, v_{2}, …, v_{n}, v_{n}^{−1}]

Since inverting an element in the coefficient ring is an instance of Bousfield localization and Bousfield localization preserves algebra structures we conclude that all spectra listed above admit structures of MU-algebras.

We conclude this section with a few remarks about the commutativity of the products on the MU-algebras considered. First in the case of the odd prime p BP and other spectra derived from it have coefficient rings concentrated in degrees congruent to 0 mod 4 and therefore all products are unique up to homotopy and automatically commutative. If p=2 then Strickland shows that BP still admits a structure of a commutative MUring spectrum, but this does not follow directly from our construction. We conjecture that in the context of Corollary 2.9 one can choose an MU-algebra structure on MU/J compatible with any given structure of an MU-ring spectrum on MU/J.

The situation with E(n) is different. The spectrum E(n) is Landweber exact regardless of the prime p. Therefore it possesses a unique homotopy associative and commutative multiplication compatible with its structure (in the classical sense) of an MU-algebra spectrum. Our results show that this multiplication can be improved to an MU-algebra structure which in turn gives an A_{∞}-structure. On the other hand Strickland shows in [13] that at the prime 2 there is no homotopy commutative multiplication on E(n) in the category of MU-modules (to get a homotopy commutative product one has to choose a different set of generators). Therefore, quite curiously, E(n) is a homotopy commutative A_{∞}-ring spectrum and simaltaneously a MU-algebra that is not commutative even up to homotopy.

3 Separable extensions of R-algebras

We keep our convention that R is a commutative S-algebra with coefficient ring R_{∗} concentrated in even degrees. Let A be an R-algebra with coefficient ring A_{∗} and A_{∗} ⊂ \overline{A}_{∗} a ring extension in the usual algebraic sense. In this section we consider the problem of finding an R-algebra \overline{A} together with an R-algebra map A → \overline{A} wich realizes the given extension A_{∗} ⊂ \overline{A}_{∗} in homotopy. If such an algebra \overline{A} exists we say that the algebraic extension A_{∗} ⊂ \overline{A}_{∗} admits a topological lifting.

We now describe a general framework in which it is possible to prove that topological liftings exist. Suppose that R_{∗} → \overline{R}_{∗} is a separable extension of R. That means that R_{∗} is a subring of a graded commutative ring \overline{R}_{∗} and \overline{R}_{∗} is a separable algebra over R_{∗}, i.e. \overline{R}_{∗} is a projective module over \overline{R}_{∗} ⊗_{R_{∗}} \overline{R}_{∗}. In addition we assume that \overline{R}_{∗} is a projective R_{∗}-module. Then for any ideal I_{∗} in R_{∗} the R_{∗}-algebra \overline{R_{∗}/I}_{∗} := R_{∗}/I_{∗} ⊗_{R_{∗}} \overline{R}_{∗} is a separable R_{∗}/I_{∗}-algebra so R_{∗}/I_{∗} → \overline{R_{∗}/I}_{∗} is also a separable extension.

Further assume that the ideal I_{∗} is generated by a regular sequence (u_{1}, u_{2}, …) (possibly infinite), the R-module R/I is supplied with a structure of an R-algebra and the algebraic extension R_{∗}/I_{∗} → \overline{R_{∗}/I}_{∗} admits a topological lifting. In other words there exists an R-algebra \overline{R/I} and an R-algebra map R/I → \overline{R/I} realizing the given separable extension on the level of coefficient rings.

Recall that in the previous section we constructed an R-algebra structure on the R-module

R/I[u_{k}]/u_{k}^{l} := R/u_{1} ∧ R/u_{2} … ∧ R/u_{k−1} ∧ R/u_{k}^{l} ∧ R/u_{k+1} ∧ ….

Then we have the following

Theorem 3.1 There exist R-algebras \overline{R/I[u_{k}]/u_{k}^{l}} realizing in homotopy the R_{∗}-module \overline{R/I[u_{k}]/u_{k}^{l}}_{∗} := R/I[u_{k}]/u_{k}^{l}_{∗} ⊗_{R_{∗}} \overline{R} and reduction maps

\overline{R_{i,k}} : \overline{R/I[u_{k}]/u_{k}^{l+1}} → \overline{R/I[u_{k}]/u_{k}^{l}}.

Moreover the algebraic extension of rings R/I[u_{k}]/u_{k}^{l}_{∗} → \overline{R/I[u_{k}]/u_{k}^{l}}_{∗} admits a topological lifting so that the following diagram of R-algebras is homotopy commutative:

R/I[u_{k}]/u_{k}^{l+1} ⟶^{R_{i,k}} R/I[u_{k}]/u_{k}^{l}
↓  ↓
\overline{R/I[u_{k}]/u_{k}^{l+1}} ⟶^{\overline{R_{i,k}}} \overline{R/I[u_{k}]/u_{k}^{l}}

Proof. Proceeding by induction suppose that the R-algebras \overline{R/I[u_{k}]/u_{k}^{l}} with required properties were constructed for l ≤ i Consider the spectral sequence

T or_{∗∗}^{R_{∗}}(\overline{R/I[u_{k}]/u_{k}^{i}}_{∗}, \overline{R/I[u_{k}]/u_{k}^{i}}_{∗}^{op})
= \overline{R}_{∗} ⊗_{R_{∗}} \overline{R}_{∗} ⊗_{R_{∗}} T or_{∗∗}^{R_{∗}}(R/I[u_{k}]/u_{k}^{i}_{∗}, R/I[u_{k}]/u_{k}^{i}_{∗}^{op})
= \overline{R}_{∗} ⊗_{R_{∗}} \overline{R}_{∗} ⊗_{R_{∗}} Λ_{R/I[u_{k}]/u_{k}^{i}}_{∗}(τ_{1}, τ_{2}, …)
= \overline{R/I[u_{k}]/u_{k}^{i}}_{∗} ⊗_{R/I[u_{k}]/u_{k}^{i}}_{∗} \overline{R/I[u_{k}]/u_{k}^{i}}_{∗} ⊗ Λ(τ_{1}, τ_{2}, …)
⇒ π_{∗}\overline{R/I[u_{k}]/u_{k}^{i}} ∧_{R} \overline{R/I[u_{k}]/u_{k}^{i}}^{op}.

Since by Lemma 2.6

π_{∗}R/I[u_{k}]/u_{k}^{i} ∧_{R} R/I[u_{k}]/u_{k}^{i}^{op} = Λ_{R/I[u_{k}]/u_{k}^{i}}_{∗}(τ_{1}, τ_{2}, …)

we see that the exterior generators τ_{i} are permanent cycles and it follows that our spectral sequence collapses multiplicatively so the ring π_{∗}\overline{R/I[u_{k}]/u_{k}^{i}} ∧_{R} \overline{R/I[u_{k}]/u_{k}^{i}}^{op} is isomorphic to the exterior algebra on τ_{1}, τ_{2}, … with coefficients in \overline{R/I[u_{k}]/u_{k}^{i}}_{∗} ⊗_{R/I[u_{k}]/u_{k}^{i}}_{∗} \overline{R/I[u_{k}]/u_{k}^{i}}_{∗}.

Further since \overline{R/I[u_{k}]/u_{k}^{i}}_{∗} is a separable R/I[u_{k}]/u_{k}^{i}_{∗}-algebra we see that

grDer^{∗−1}(\overline{R/I[u_{k}]/u_{k}^{i}}, \overline{R/I})
= \overline{R/I[u_{k}]/u_{k}^{i}}_{∗} ⊗_{R/I[u_{k}]/u_{k}^{i}}_{∗} ⊗\overline{R/I}_{∗}[ũ_{1}, ũ_{2}, …]/(\overline{R/I}_{∗}).

Similarly

grDer^{∗−1}(R/I[u_{k}]/u_{k}^{i}, \overline{R/I})
= \overline{R/I[u_{k}]/u_{k}^{i}}_{∗} ⊗_{R/I[u_{k}]/u_{k}^{i}}_{∗} ⊗\overline{R/I}_{∗}[ũ_{1}, ũ_{2}, …]/(\overline{R/I}_{∗}).

Therefore the map Der(\overline{R/I[u_{k}]/u_{k}^{i}}, \overline{R/I}) → Der(R/I[u_{k}]/u_{k}^{i}, \overline{R/I}) induced by the R-algebra map R/I[u_{k}]/u_{k}^{i} → \overline{R/I[u_{k}]/u_{k}^{i}} is an isomorphism in homotopy.

Consider the following homotopy commutative diagram of R-algebras:

\begin{matrix}\overline{R/I[u_{k}]/u_{k}^{i}} & ⟶^{d} & \overline{R/I} ∨ Σ^{|u_{k}|i+1}\overline{R/I} \\ ↑ & & ↑ \\ R/I[u_{k}]/u_{k}^{i} & ⟶ & R/I ∨ Σ^{|u_{k}|i+1}\overline{R/I} \\ || & & ↑ \\ R/I[u_{k}]/u_{k}^{i} & ⟶^{Q_{i,k}} & R/I ∨ Σ^{|u_{k}|i+1}R/I \end{matrix}  (4)

Here the horizontal arrows are topological derivation. The lower arrow is the Bokstein operation Q_{i,k} constructed in the previous section,the middle arrow is the only one that makes the lower square commute. Since derivations of \overline{R/I[u_{k}]/u_{k}^{i}} with values in \overline{R/I} are in one-to-one correspondence with derivations of R/I[u_{k}]/u_{k}^{i} with values in \overline{R/I} the upper horizontal arrow d exists making the upper square commute.

Now the derivation d gives rise to a topological singular extension

Σ^{|u_{k}|i}\overline{R/I} →? → \overline{R/I[u_{k}]/u_{k}^{i}}

which on the level of coefficient rings reduces to the algebraic singular extension

Σ^{|u_{k}|i}\overline{R/I}_{∗} → \overline{R/I[u_{k}]/u_{k}^{i+1}}_{∗} → \overline{R/I[u_{k}]/u_{k}^{i}}_{∗}

Therefore we can denote ? by \overline{R/I[u_{k}]/u_{k}^{i+1}} and because of the diagram (4) we have a map of topological singular extensions

Σ^{|u_{k}|i}R/I → R/I[u_{k}]/u_{k}^{i+1} → R/I[u_{k}]/u_{k}^{i}
↓ ↓ ↓
Σ^{|u_{k}|i}\overline{R/I} → \overline{R/I[u_{k}]/u_{k}^{i+1}} → \overline{R/I[u_{k}]/u_{k}^{i}}

With this the inductive step is completed and Theorem 3.1 is proved.

We now show that Theorem 3.1 gives a way of adjoining roots of unity to various MU-algebras. Let F_{p} ↪ L be a finite separable extension of the field F_{p} (typically obtained by adjoining roots of an irreducible factor of the cyclotomic polynomial x^{p^{n}−1} − 1 for some n). Let R = MU_{p}, the pcompletion of the complex cobordism spectrum MU and \overline{R}_{∗} = W (L)⊗MU_{∗} where W (L) is the ring of Witt vectors of L. Since L is a separable extension of F_{p} the algebra W (L) is separable over the p-adic integers Ẑ_{p} and it follows that W (L) ⊗ MU_{∗} is a separable algebra over MU_{p∗} = Ẑ_{p} ⊗ MU_{∗}.

Corollary 3.2 Let J_{∗} be the ideal in MU_{∗} = Z[x_{1}, x_{2}, …] generated by (p, x_{i_{1}}, x_{i_{2}}, …) where x_{i_{k}} are polynomial generators in MU_{∗}. Then for any finite separable extension F_{p} ↪ L there exist MU_{p}-algebras MU/J, MU/J_{L}, MU/J_{W (F_{p})} and MU/J_{W (L)} together with the commutative diagram in the homotopy category of MU_{p}-algebras

\begin{aligned}MU/J_{W (F_{p})} & → & MU/J_{W (L)} \\ ↓ & & ↓ \\ MU/J & → & MU/J_{L} \end{aligned}  (5)

which reduces in homotopy to the following diagram of MU_{∗}-algebras:

Ẑ_{p} ⊗ MU_{∗}/(x_{i_{1}}, x_{i_{2}} …) → W (L) ⊗ MU_{∗}/(x_{i_{1}}, x_{i_{2}} …)
modp ↓  ↓ modp
MU_{∗}/J_{∗}  →  L ⊗ MU_{∗}/J_{∗}

Proof. Denote by u_{1}, u_{2}, … the collection of polynomial generators of MU_{∗} which are not in J_{∗}. There is an isomorphism of rings MU_{∗}/J_{∗} ≅ F_{p}[u_{1}, u_{2}, …]. Let I be the maximal ideal (p, x_{1}, x_{2}, …) in MU_{∗} = Z[x_{1}, x_{2}, …]. Then MU/I is the Eilenberg-MacLane spectrum HF_{p} and the canonical map MU_{p} → MU/I is a commutative S-algebra map. Therefore MU/I is an MU_{p}-algebra (even a commutative MU_{p}-algebra). Further the extension F_{p} ↪ L determines a map of commutative S-algebras HF_{p} → HL which is also a map of commutative MU_{p}-algebras. Using Theorem 3.1 we can adjoin the variable u_{i} to the topological extension HF_{p} → HL and obtain the tower of topological extensions

{HF_{p}[u_{i}]/[u_{i}^{k}] → HL[u_{i}]/[u_{i}^{k}]}.

Passing to the limit we get the topological extension HF_{p}[u_{i}] → HL[u_{i}].

This procedure could be repeated so we can adjoin another polynomial generator to the extension HF_{p}[u_{i}] → HL[u_{i}] (or any number of polynomial generators). In this way we construct the extension

HF_{p}[u_{1}, u_{2}, …] → HL[u_{1}, u_{2}, …]

which is the same as the extension MU/J → MU/J_{L} for the ideal J_{∗} = (p, x_{i_{1}}, x_{i_{2}}, …) in MU_{∗}. At this point we adjoin the indeterminate corresponding to the prime p. We then get the following diagram of MU-algebras:

MU/J ← MU/J_{W_{2}(F_{p})} ← … ← MU/J_{W_{n}(F_{p})} ←
↓ ↓ ↓
MU/J_{L} ← MU/J_{W_{2}(L)} ← … ← MU/J_{W_{n}(L)} ←

where the MU_{p}-algebras MU/J_{W_{n}(F_{p})} and MU/J_{W_{n}(L)} realize the MU_{p∗}- algebras W_{n}(F_{p}) ⊗ MU_{∗}/(x_{i_{1}}, x_{i_{2}}, …) and W_{n}(L) ⊗ MU_{∗}/(x_{i_{1}}, x_{i_{2}}, …) respectively.

Finally taking the (homotopy) inverse limit we get an MU_{p}-algebra map MU/J_{W (F_{p})} → MU/J_{W (L)} which realizes in homotopy the extension of rings Ẑ_{p} ⊗ MU/J_{∗} → W (L) ⊗ MU/J_{∗}. Corollary 3.2 is then proved.

Remark 3.3 It is not hard to prove that the diagram (5) is equivariant with respect to Gal(L/F_{p}) in a suitable sense but we save this observation for future work.

4 Localized towers of MU-algebras

In this section we investigate the behaviour of the towers of MU-algebras constructed in the previous section under Bousfield localization. We start with an almost trivial

Theorem 4.1 Let

I → A → B  (6)

be a singular extension of R-algebras for a commutative S-algebra R. Then for any R-module M the cofibre sequence

I_{M} → A_{M} → B_{M}  (7)

is also a singular extension of R-algebras (here ?_{M} denotes Bousfield localization with respect to M)

Proof. The singular extension (6) is associated with a certain topological derivation d : B → B ∨ ΣM so that there is a homotopy pullback square of R-algebras

A →  B
↓  ↓
B →^{d} B ∨ ΣI

where the vertical map B → B ∨ΣM is the canonical inclusion of the wedge summand. Localizing this diagram with respect to M we get the homotopy pullback diagram

A_{M} →  B_{M}
↓  ↓
B_{M} →^{d_{M}} B_{M} ∨ ΣI_{M}

Therefore (7) is a singular extension associated with the topological derivation B_{M} →^{d_{M}} B_{M} ∨ ΣI_{M} and the theorem is proved.

We now specialize to the case R = MU. Let I_{∗} be a regular ideal (u_{i_{1}}, u_{i_{1}}, …) in MU_{∗} such that the elements u_{i_{k}} have positive degrees and MU/I is an MU-algebra. Consider the MU-algebra MU/I[u_{i_{1}}, u_{i_{1}}, …, u_{i_{k}}] obtained by adjoining the indeterminates u_{i_{1}}, u_{i_{1}}, …, u_{i_{k}} to MU/I as explained in Section 2. Set

MU/I[[u_{i_{2}}, u_{i_{3}}, …, u_{i_{k}}]][u_{i_{1}}, u_{i_{1}}^{−1}] = MU/I[u_{i_{2}}, u_{i_{3}}, …, u_{i_{k}}]_{MU/I[u_{i_{1}},u_{i_{1}}^{−1}]},

the Bousfield localization of the MU-algebra MU/I[u_{i_{2}}, u_{i_{3}}, …, u_{i_{k}}] with respect to MU/I[u_{i_{1}}, u_{i_{1}}^{−1}] := u_{i_{1}}^{−1}MU/I[u_{i_{1}}]. The reason for this notation is the following Proposition 4.2 The coefficient ring of MU/I[[u_{i_{2}}, u_{i_{3}}, …, u_{i_{k}}]][u_{i_{1}}, u_{i_{1}}^{−1}] as an MU_{∗}-algebra is MU/I_{∗}[[u_{i_{2}}, u_{i_{3}}, …, u_{i_{k}}]][u_{i_{1}}, u_{i_{1}}^{−1}].

Proof. We have the following tower of MU-algebras:

MU/I[u_{i_{1}}] ← MU/I[u_{i_{1}}][u_{i_{2}}]/u_{i_{2}}^{2} ← … ← MU/I[u_{i_{1}}][u_{i_{2}}]/u_{i_{2}}^{l} ← …

Bousfield-localizing it with respect to MU/I[u_{i_{1}}, u_{i_{1}}^{−1}] we get the tower

MU/I[u_{i_{1}}, u_{i_{1}}^{−1}] ← MU/I[u_{i_{1}}, u_{i_{1}}^{−1}][u_{i_{2}}]/u_{i_{2}}^{2} ← …

Notice that the first term of the localized tower as well as its successive subquotients (isomorphic to suspensions of MU/I[u_{i_{1}}, u_{i_{1}}^{−1}]) are MU/I[u_{i_{1}}, u_{i_{1}}^{−1}]- local and therefore so is its homotopy inverse limit. Denote this limit by MU/I[u_{i_{1}}, u_{i_{1}}^{−1}][[u_{i_{2}}]], clearly

MU/I[u_{i_{1}}, u_{i_{1}}^{−1}][[u_{i_{2}}]]_{∗} = MU/I_{∗}[u_{i_{1}}, u_{i_{1}}^{−1}][[u_{i_{2}}]]

Further the canonical map

holim(MU/I[u_{i_{1}}][u_{i_{2}}]/u_{i_{2}}^{l}) ≃ MU/I[u_{i_{1}}, u_{i_{2}}] → MU/I[u_{i_{1}}, u_{i_{1}}^{−1}][[u_{i_{2}}]]

is MU/I[u_{i_{1}}, u_{i_{1}}^{−1}]-equivalence and we conclude that

MU/I[u_{i_{1}}, u_{i_{1}}^{−1}][[u_{i_{2}}]] ≅ MU/I[u_{i_{1}}, u_{i_{2}}]_{MU/I[u_{i_{1}},u_{i_{1}}^{−1}]}  (8)

We then proceed by adjoining u_{i_{3}}. There is a tower of MU-algebras

MU/I[u_{i_{1}}, u_{i_{2}}] ← MU/I[u_{i_{1}}, u_{i_{2}}][u_{i_{3}}]/u_{i_{3}}^{2} ← …

where the successive stages

Σ^{|u_{i_{3}}|s}MU/I[u_{i_{1}}, u_{i_{2}}][u_{i_{3}}] → MU/I[u_{i_{1}}, u_{i_{2}}][u_{i_{3}}]/u_{i_{3}}^{s+1} → MU/I[u_{i_{1}}, u_{i_{2}}][u_{i_{3}}]/u_{i_{3}}^{s}

are singular extensions of MU-algebras. Localizing this tower with respect to MU/I[u_{i_{1}}, u_{i_{1}}^{−1}] and using (8) and Theorem 4.1 we get the tower

MU/I[[u_{i_{2}}]][u_{i_{1}}, u_{i_{1}}^{−1}] ← MU/I[[u_{i_{2}}]][u_{i_{1}}, u_{i_{1}}^{−1}][u_{i_{3}}]/u_{i_{3}}^{2} ← ….

whose homotopy limit

holim(MU/I[[u_{i_{2}}]][u_{i_{1}}, u_{i_{1}}^{−1}][u_{i_{3}}]/u_{i_{3}}^{k}) = MU/I[[u_{i_{2}}, u_{i_{3}}]][u_{i_{1}}, u_{i_{1}}^{−1}]

is clearly weakly equivalent to the localization of MU/I[u_{i_{1}}, u_{i_{2}}, u_{i_{3}}] with respect to MU/I[u_{i_{1}}, u_{i_{1}}^{−1}. Moreover

MU/I[[u_{i_{2}}, u_{i_{3}}]][u_{i_{1}}, u_{i_{1}}^{−1}]_{∗} = MU/I_{∗}[[u_{i_{2}}, u_{i_{3}}]][u_{i_{1}}, u_{i_{1}}^{−1}]

Repeating this process for u_{i_{4}}, u_{i_{4}}, …, u_{i_{k}} we obtain the desired isomorphism

MU/I[[u_{i_{2}}, u_{i_{3}}, …, u_{i_{k}}]][u_{i_{1}}, u_{i_{1}}^{−1}]_{∗} = MU/I_{∗}[[u_{i_{2}}, u_{i_{3}} …, u_{i_{k}}]][u_{i_{1}}, u_{i_{1}}^{−1}].

Our main application of the developed localization techniques is to Johnson-Wilson theories E(n).

Recall that the MU-algebra spectrum E(n) has the coefficient ring E(n)_{∗} = Z_{(p)}[v_{1}, v_{2}, …, v_{n−1}][v_{n}, v_{n}^{−1}]. The MU_{∗}-algebra structure on E(n)_{∗} is defined by the correspondence x_{i} → 0 for i ≠ 2(p^{n} − 1) and x_{2(p^{n}−1)} → v_{n} where x_{i} are polynomial generators of MU_{∗}. We have the following

Corollary 4.3 There exist MU-algebras Ê(n), Ê_{W (L)}(n), K(n), K(n)_{L}, and the homotopy commutative diagram of MU-algebras

\begin{aligned}Ê(n) & → & Ê_{W (L)}(n) \\ ↓ & & ↓ \\ K(n) & → & K(n)_{L} \end{aligned}  (9)

which realizes in homotopy the diagram of MU_{∗}-algebras

Ẑ_{p}[[v_{1}, v_{2}, …, v_{n−1}]][v_{n}, v_{n}^{−1}] → W (L)[v_{1}, v_{2}, …, v_{n−1}][[v_{n}, v_{n}^{−1}]]
↓  ↓
Z/pZ[v_{n}, v_{n}^{−1}]  →  L[v_{n}, v_{n}^{−1}]

Proof. We first construct the map of MU-algebras k(n) → k(n)_{L} by adjoining v_{n} to the extension HF_{p} → HL. Here k(n) and k(n)_{L} are the connective Morava K-theories with coefficient rings k(n)_{∗} = F_{p}[v_{n}] and k(n)_{∗} = L[v_{n}]. Adjoining indeterminates p, v_{1}, v_{2}, …, v_{n−1} we construct the homotopy commutative diagram of MU-algebras

\begin{aligned}B̂P < n > & → & B̂P < n >_{W (L)} \\ ↓ & & ↓ \\ k(n) & → & k(n)_{L} \end{aligned}  (10)

which realizes in homotopy the diagram

Ẑ_{p}[v_{1}, v_{2}, …, v_{n−1}][v_{n}] → W (L)[v_{1}, v_{2}, …, v_{n−1}][v_{n}]
↓  ↓
Z/pZ[v_{n}]  →  L[v_{n}]

According to Proposition 4.2 Bousfield localization of B̂P < n > with respect to K(n) on the level of coefficient rings amounts to inverting v_{n} and completing at the ideal (v_{1}, v_{2}, …, v_{n−1}). Further notice that since π_{∗}B̂P < n >_{W (L)} is a free π_{∗}B̂P < n >-module of finite rank (which is equal to the degree of the field extension L/F_{p}) the MU-algebra B̂P < n >_{W (L)} is a finite cell B̂P < n >-module and therefore its K(n)-localization is equivalent to

B̂P < n >_{K(n)} ∧_{_{BP} <n>} B̂P < n >_{W (L)}

and its coefficient ring is

W (L) ⊗ π_{∗}B̂P < n >_{K(n)} = W (L)[[v_{1}, v_{2}, …, v_{n−1}]][v_{n}, v_{n}^{−1}].

Therefore we obtain the diagram (9) by localizing (10) with respect to K(n) in the category of MU-algebras. Corollary 4.3 is proved.

5 Computing spaces of S-algebra maps

In this section we investigate spaces of strictly multiplicative maps from Ê(n)_{W (L)} to certain MU-algebras which we call called “strongly K(n)_{L}- complete.”. As a consequence we obtain versions of the Hopkins-Miller theorem as well as splitting theorems for such spectra. Such theorems (in a weaker, up to homotopy form) were previously obtained by methods of formal group theory, cf. [2].

Let A, B be S-algebras, which we will assume without loss of generality to be q-cofibrant in the sense of [4]. That means in particular, that the topological space of S-algebra maps B → A has the “correct” homotopy type (i.e. the one that depends only on the homotopy type of A and B as S-algebras). Denote this topological space by F_{S−alg}(B, A). Also denote the set of multiplicative up to homotopy maps from A to B by Mult(A, B).

We recall one result from [6] which will be needed later on.

Theorem 5.1 Let Σ^{−1}M → X → A be a singular extension of S-algebras associated with a derivation d : A → A ∨ M and f : B → A a map of R-algebras. Then f lifts to an R-algebra map B → X iff a certain element in Der^{0}(B, M) is zero. Assuming that a lifting exists the map

F_{R−alg}(B, X) → F_{R−alg}(B, A)

has homotopy fibre over the point f ∈ F_{R−alg}(B, A) weakly equivalent to Ω^{∞}Der(B, Σ^{−1}M) (the 0th space of the spectrum Der(X, Σ^{−1}M)).

Recall from the previous section that the spectrum Ê(n)_{W (L)} with coefficient rings W (L)[[v_{1}, v_{2}, …, v_{n−1}]][v_{n}, v_{n}^{−1}] has a structure of an S-algebra.

We say that the generalized Hopkins-Miller theorem holds for an Salgebra E if π_{0}F_{S−alg}(Ê(n)_{W (L)}, E) = Mult(Ê(n)_{W (L)}, E) whilst π_{i}F_{S−alg}(Ê(n)_{W (L)}, E) = 0 for i > 0.

Proposition 5.2 The generalized Hopkins-Miller theorem holds for the Salgebra K(n)_{L}.

Proof. Consider the Bousfield-Kan spectral sequence (cf.[3]) for the mapping space F_{S−alg}(A, B). The identification of the E_{2}-term is standard and we refer the reader to [10] for necessary details. Here we only mention that the key ingredient in this identification is the existence of the Kunneth formula

K(n)_{L∗}(Ê(n)_{W (L)} ∧ Ê(n)_{W (L)})
= K(n)_{L∗}(Ê(n)_{W (L)}) ⊗_{K(n)_{L∗}} K(n)_{L∗}(Ê(n)_{W (L)}).

This is the result we need:

E_{2}^{st} = Der_{K(n)_{L∗}}^{st}(Ê(n)_{W (L)∗}K(n)_{L}, K(n)_{L∗}) for s, t ≠ 0;
E_{2}^{00} = Mult(Ê(n)_{W (L)}, K(n)_{L}).

Here Der_{k}^{∗∗}(?,?) is defined for a graded k-algebra R_{∗} and a graded R_{∗}- bimodule M_{∗} as the shifted Hochschild cohomology:

Der_{k}^{∗∗}(A_{∗}, M_{∗}) := HH_{k}^{∗+1∗}(A_{∗}, M_{∗})

(the second grading reflects the fact that A_{∗} and M_{∗} are graded objects). Further

Ê(n)_{W (L)∗}K(n)_{L} = Ê(n)_{∗}K(n) ⊗ W (L) ⊗ L = E(n)_{∗}K(n) ⊗ L ⊗ L

and computations in [9], Chapter VI show that

E(n)_{∗}K(n) = Σ_{∗} = F_{p}[v_{n}, v_{n}^{−1}][t_{k}|k > 0]/(t_{k}^{p^{n}} − v_{n}^{p^{k}−1}t_{k})

where the degree of t_{k} is 2(p^{k} − 1). We have:

HH_{K(n)_{L∗}}^{∗∗}(Σ_{∗} ⊗ L ⊗ L, K(n)_{L∗}) = HH_{K(n)_{∗}}^{∗∗}(Σ_{∗} ⊗ L, K(n)_{∗} ⊗ L)
= HH_{K(n)_{∗}}^{∗∗}(Σ_{∗}, K(n)_{∗}) ⊗ HH_{F_{p}}^{∗}(L, L) = HH_{K(n)_{∗}}^{∗∗}(Σ_{∗}, K(n)_{∗}) ⊗ L.

An easy cohomological calculation (due to A.Robinson, cf. [11]) shows that HH_{K(n)_{∗}}^{∗∗}(Σ_{∗}, K(n)_{∗}) = K(n)_{∗} and therefore

HH_{K(n)_{L∗}}^{∗∗}(Ê(n)_{W (L)_{∗}}K(n)_{L}) = K(n) ⊗ L = K(n)_{L∗}  (11)

Thus the Bousfield-Kan spectral sequence reduces to its corner term and Proposition 5.2 is proved.

Remark 5.3 Notice that even though in Proposition 5.2 and we used the results in the previous section that spectra K(n)_{L}, and Ê(n)_{W (L)} admit Salgebra structures we did not specify which structures are used. In other words Proposition 5.2 is valid with any choice of S-algebra structures on the spectra in question.

We can now regard K(n)_{L} as a bimodule over Ê(n)_{W (L)} by choosing an S-algebra map Ê(n)_{W (L)} → K(n)_{L} (which is supplied by Propositions 5.2). Therefore it makes sense to consider spectra of topological Hochschild cohomology and topological derivations of Ê(n)_{W (L)} with values in K(n)_{L}. It turns out that these spectra do not depend up to weak equivalence which bimodule structure we choose.

Proposition 5.4 The canonical map of S-modules

THH(Ê(n)_{W (L)}, K(n)_{L}) → K(n)_{L};

is a weak equivalences. Furthermore the S-module Der(Ê(n)_{W (L)}, K(n)_{L}) is contractible.

Proof. Since there is a cofibre sequences of S-modules

Σ^{−1}Der(Ê(n)_{W (L)}, K(n)_{L}) → THH(Ê(n)_{W (L)}, K(n)_{L}) → K(n)_{L}

it suffices to prove the statement about THH. We have the following spectral sequence

Ext_{Ê(n)_{W (L)∗} Ê(n)_{W (L)}}^{∗∗}(Ê(n)_{W (L)_{∗}}, K(n)_{L∗}) ⇒ T HH^{∗}(Ê(n)_{W (L)}, K(n)_{L})

(12)

Since Ê(n)_{W (L)} is a Landweber exact theory the π_{∗}Ê(n)_{W (L)} - module Ê(n)_{W (L)_{∗}} Ê(n)_{W (L)} is flat and by flat base change we obtain

Ext_{Ê(n)_{W (L)∗} Ê(n)_{W (L)}}^{∗∗}(Ê(n)_{W (L)_{∗}}, K(n)_{L∗})

= Ext_{Ê(n)_{W (L)∗}K(n)_{L}}^{∗∗}(K(n)_{L∗}, K(n)_{L∗})
= HH_{K(n)_{L∗}}^{∗}(Ê(n)_{W (L)∗}K(n)_{L}, K(n)_{L∗}).

The isomorphism (11) shows that the spectral sequence (12) collapses giving the weak equivalence

THH(Ê(n)_{W (L)}, K(n)_{L}) ≃ K(n)_{L}

and our theorem is proved.

We now describe a particular class of MU-algebras E which will be called strongly K(n)_{L} - complete and for which the set of multiplicative maps Ê(n)_{W (L)} → E has an especially simple form.

Let ν = {ν_{0}, ν_{1}, …, ν̂_{2(p^{n}−1)}, …} (i.e.the 2(p^{n} − 1)th spot is missed) be a sequence of positive integers. Associated to ν is the regular sequence p^{ν_{0}}, x_{1}^{ν_{1}}, x_{2}^{ν_{2}}, … in MU_{∗}. Consider the MU-module

MU(ν) := x_{2(p^{n}−1)}^{−1}MU_{W (L)}/(p^{ν_{0}}, x_{1}^{ν_{1}}, x_{2}^{ν_{2}}, …)
= v_{n}^{−1}MU_{W (L)}/(p^{ν_{0}}, x_{1}^{ν_{1}}, x_{2}^{ν_{2}}, …).

Since MU(ν) can be constructed from K(n)_{L} by a sequence (possibly infinite) of elementary extensions we see that there is a structure of an MU-algebra on MU(ν) compatible with the MU_{∗}-algebra structure on v_{n}^{−1}MU(ν)_{∗} = W (L) ⊗ v_{n}^{−1}MU_{∗}/(p^{ν_{0}}, x_{1}^{ν_{1}}, x_{2}^{ν_{2}}, …). Notice that there is a reduction MU-algebra map MU(ν) → K(n)_{L}. Furthermore the filtration on MU(ν) by powers of the maximal ideal has subquotients isomorphic to products of copies of K(n)_{L∗} = L[v_{n}, v_{n}^{−1}].

Let E and F be two MU-algebras supplied with an MU-algebra map E → K(n)_{L} and F → K(n)_{L}. We say that an MU-algebra map E → F is compatible if it is a map over K(n)_{L}.

Definition 5.5 The class SC(K(n)_{L}) of strongly K(n)_{L}-complete MUalgebras is the smallest class of MU-algebras over K(n)_{L} that satisfies the following two axioms:

(i) SC(K(n)_{L}) contains MU(ν) for any ν = {ν_{1}, ν_{2}, …, ν̂_{2(p^{n}−1)}, …} as above; (ii) For any sequential inverse system of compatible maps in SC(K(n)_{L}) its homotopy inverse limit is also in SC(K(n)_{L}).

In other words an MU-algebra is K(n)_{L}-complete if it can be built from K(n)_{L} by taking elementary extensions and (homotopy) inverse limits. If E_{1} ← E_{2} ← … is an inverse system in SC(K(n)_{L}) then π_{∗}holim(E_{n}) = lim_{n→∞}π_{∗}E_{n} since coefficients rings of E_{n} are all in even degrees. Therefore we never have to worry about lim^{1}-problems.

Examples of strongly K(n)_{L} - complete MU-algebras include K(n)_{L}, Ê(n)_{W (L)} and also v_{n}^{−1}B̂P_{W (L)} and v_{n}^{−1}P̂(n)_{W (L)}, the Artinian completions of v_{n}^{−1}BP_{W (L)} and v_{n}^{−1}P (n)_{W (L)}

Let E be a strongly K(n)_{L}-complete MU-algebra. The canonical Salgebra map (which is also an MU-algebra map) E → K(n)_{L} determines via composition the morphism of topological spaces (unbased)

i : F_{S−alg}(E(n)_{W (L)}, E) → F_{S−alg}(E(n)_{W (L)}, K(n)_{L})  (13)

Proposition 5.6 The map (13) is a weak equivalence. In particular F_{S−alg}(E(n)_{W (L)}, E) is a homotopically discrete space. Moreover the MUmodule Der(E(n)_{W (L)}, E) is contractible.

Proof. First assume that E = MU(ν) where all ν_{l} except for ν_{i} are equal to one, i.e. that E = K(n)_{L}[x_{k}]/x_{k}^{ν_{i}}. The we have the tower

K(n)_{L} ← K(n)_{L}[x_{k}]/x_{k}^{2} ← … ← K(n)_{L}[x_{k}]/x_{k}^{ν_{i}} = E.

Applying to this tower the functor F_{S−alg}(E_{W (L)},?) we get the tower of topological spaces

F_{S−alg}(E_{W (L)}, K(n)_{L}) ← F_{S−alg}(E_{W (L)}, K(n)_{L}[x_{k}]/x_{k}^{2}) ← …
← F_{S−alg}(E_{W (L)}, K(n)_{L}[x_{k}]/x_{k}^{ν_{i}}) = F_{S−alg}(E_{W (L)}, E)  (14)

Using Theorem 5.1 and Proposition 5.4 we see that the homotopy fibre of each map

F_{S−alg}(E_{W (L)}, K(n)_{L}[x_{k}]/x_{k}^{s}) ← F_{S−alg}(E_{W (L)}, K(n)_{L}[x_{k}]/x_{k}^{s+1})

is weakly equivalent to the space Ω^{∞}Der(E_{W (L)}, Σ^{|x_{k}|s}K(n)_{L}) which is contractible. Hence all maps in (14) are weak equivalences of spaces and in particular F_{S−alg}(E_{W (L)}, K(n)_{L}[x_{k}]/x_{k}^{ν_{i}}) is weakly equivalent to F_{S−alg}(E_{W (L)}, K(n)_{L}. Similarly Der(E_{W (L)}, K(n)_{L}[x_{k}]/x_{k}^{ν_{i}}) is weakly equivalent to Der(E_{W (L)}, K(n)_{L} and is contractible.

Further obvious induction shows that the statement of Proposition 5.6 holds for an MU-algebra of the form MU(ν) where all ν_{l} except for for a finite number are equal to one. Passing to the limit we obtain the result for an arbitrary MU-algebra MU(ν).

Now using transfinite induction suppose that the conclusion of Proposition 5.6 holds for an inverse system {E_{l}} of strongly K(n)_{L}-complete MUalgebras consisting of compatible maps and let E = holim E_{l}. Applying the functor Map_{S−alg}(E_{W (L)},?) to {E_{l}} we get an inverse system of topological spaces {F_{S−alg}(E_{W (L)}, E_{l})}. By inductive assumption all spaces F_{S−alg}(E_{W (L)}, E_{l}) are weakly equivalent to F_{S−alg}(E_{W (L)}, K(n)_{L}) and the maps in the inverse system {F_{S−alg}(E_{W (L)}, E_{l})} respect this equivalence. We conclude that the space F_{S−alg}(E_{W (L)}, E) is also weakly equivalent to F_{S−alg}(E_{W (L)}, K(n)_{L}). Similar arguments show that Der(E_{W (L)}, K(n)_{L}) is contractible and Proposition 5.6 is proved.

Remark 5.7 We see that for any strongly K(n)_{L}-complete MU-algebra E the canonical S-algebra map Ê(n)_{W (L)} → K(n)_{L} lifts uniquely to a map Ê(n)_{W (L)} → E. Therefore E is naturally a (bi)module over Ê(n)_{W (L)}.

The following lemma descibes the structure of cohomology operations from Ê(n)_{W (L)} to a strongly K(n)_{L}-complete MU-algebra E.

Lemma 5.8 Let E be a strongly K(n)_{L}-complete MU-algebra. Then the evaluation map

E^{∗}Ê(n)_{W (L)} → Hom_{Ê(n)_{W (L)∗}}(Ê(n)_{W (L)_{∗}} Ê(n)_{W (L)}, E_{∗})

is an isomorphism.

Proof. We have the following natural isomorphism of S-modules

F_{S}(Ê(n)_{W (L)}, E) ≅ F_{Ê(n)_{W (L)}}(Ê(n)_{W (L)} ∧ Ê(n)_{W (L)}, E)

Consider further the spectral sequence

Ext_{Ê(n)_{W (L)∗}}^{∗∗}(Ê(n)_{W (L)∗}Ê(n)_{W (L)}, E_{∗})
⇒ [Ê(n)_{W (L)} ∧ Ê(n)_{W (L)}, E]_{(Ê(n)_{W (L)})}^{∗} = [Ê(n)_{W (L)}, E]^{∗}  (15)

We claim that all higher Ext groups vanish so that our spectral sequence reduces to Ext_{Ê(n)_{W (L)∗}}^{0∗}(Ê(n)_{W (L)_{∗}} Ê(n)_{W (L)}, E_{∗}). (This would clearly give

us the statement of the lemma). To see this first assume that E is of the form MU(ν) for some sequence ν.

Then since for the maximal ideal M of E_{∗} the E_{∗}-module M^{n}/M^{n+1} is isomorphic to direct product of copies of K(n)_{L∗} it is enough to prove that

Ext_{Ê(n)_{W (L)∗}}^{i}(Ê(n)_{W (L)∗}Ê(n)_{W (L)}, K(n)_{L}) = 0

for i > 0 (here we suppressed the second, internal grading on Ext from the notations). Since the Ê(n)_{W (L)∗}-module Ê(n)_{W (L)∗}Ê(n)_{W (L)} is flat we have by flat base change:

Ext_{Ê(n)_{W (L)∗}}^{i}(Ê(n)_{W (L)∗}Ê(n)_{W (L)}, K(n)_{L∗})
= Ext_{K(n)_{L∗}}^{i}(Ê(n)_{W (L)∗}Ê(n)_{W (L)} ⊗_{Ê(n)_{W (L)∗}} K(n)_{L∗}, K(n)_{L∗})

and the last Ext-group is zero for i > 0 since K(n)_{L∗} is injective as a graded module over itself.

Therefore our claim about the vanishing of higher Ext-groups in (15) is proved for E = MU(ν). To get the general case suppose that E is the homotopy inverse limit of strongly K(n)_{L}-complete MU-algebras E_{l} for which higher Ext’s do vanish. Then

Ext_{Ê(n)_{W (L)∗}}^{i∗}(Ê(n)_{W (L)∗}Ê(n)_{W (L)}, E_{∗})
= Ext_{Ê(n)_{W (L)∗}}^{i∗}(Ê(n)_{W (L)∗}Ê(n)_{W (L)}, lim_{l→∞}E_{l∗})
lim_{l→∞}Ext_{Ê(n)_{W (L)∗}}^{i∗}(Ê(n)_{W (L)∗}Ê(n)_{W (L)}, E_{l∗}) = 0

for i > 0. With this Lemma 5.8 is proved.

We can now formulate our main

Theorem 5.9 The generalized Hopkins-Miller theorem holds for any strongly K(n)_{L}-complete MU-algebra E.

Proof. By Proposition 5.6 we know that the space Map_{S−alg}(Ê(n)_{W (L)}, E) is homotopically discrete with

π_{0}(Map_{S−alg}(Ê(n)_{W (L)}, E)) = Hom_{K(n)_{L∗}−alg}(Σ_{∗} ⊗ L, K(n)_{L_{∗}}).

So it remains to show that an arbitrary multiplicative operation Ê(n)_{W (L)} → E lifts to an S-algebra map. Since any such operation determines an Ê(n)_{W (L)∗}-algebra map Ê(n)_{W (L)∗}E(n)_{W (L)} → E_{∗} we conclude by Lemma 5.8 that there is an injective map

Mult(Ê(n)_{W (L)}, E) → Hom_{Ê(n)_{W (L)∗}−alg}(Ê(n)_{W (L)∗}Ê(n)_{W (L)}, E_{∗}) (16)

We claim that any Ê(n)_{W (L)∗}-algebra map Ê(n)_{W (L)∗}Ê(n)_{W (L)} → K(n)_{L∗} lifts uniquely to a map Ê(n)_{W (L)∗}Ê(n)_{W (L)} → E_{∗} so that there is an isomorphism

Hom_{Ê(n)_{W (L)∗}−alg}(Ê(n)_{W (L)∗}Ê(n)_{W (L)}, E_{∗}
≅ Hom_{Ê(n)_{W (L)∗}−alg}(Ê(n)_{W (L)∗}Ê(n)_{W (L)}, K(n)_{L∗})  (17)

To see this assume first that E is of the form MU(ν). Induction up the filtration of the Ê(n)_{W (L)∗}-module E_{∗} by the powers of the maximal ideal shows that our claim is equivalent to the vanishing of certain Hochschild cohomology classes in HH_{Ê(n)_{W (L)∗}}^{∗}(Ê(n)_{W (L)∗}Ê(n)_{W (L)}, K(n)_{L∗}). Flat base change gives an isomorphism

HH_{Ê(n)_{W (L)∗}}^{∗}(Ê(n)_{W (L)∗}Ê(n)_{W (L)}, K(n)_{L∗})
= HH_{K(n)_{L∗}}^{∗}(Ê(n)_{W (L)∗}Ê(n)_{W (L)} ⊗_{Ê(n)_{W (L)∗}} K(n)_{L∗}, K(n)_{L∗})
= HH_{K(n)_{L∗}}^{∗}(K(n)_{L∗}Ê(n)_{W (L)}, K(n)_{L∗}) = K(n)_{L∗}.

We see that all higher Hochschild cohomology groups are zero and our claim is proved (in the special case E = MU(ν)). Then the standard by now arguments (which we omit) show that (17) holds for any E.

Further we have an obvious change of rings isomorphism

Hom_{Ê(n)_{W (L)∗}−alg}(Ê(n)_{W (L)∗}Ê(n)_{W (L)}, K(n)_{L∗})
= Hom_{K(n)_{L∗}−alg}(K(n)_{L∗}Ê(n)_{W (L)}, K(n)_{L∗})

The last term corresponds bijectively by Proposition 5.6 to homotopy classes of S-algebra maps from Ê(n)_{W (L)} to E. This shows that any multiplicative operation Ê(n)_{W (L)} → E lifts to an S-algebra map and Theorem 5.9 is proved.

Remark 5.10 The class of strongly K(n)_{L}-complete MU-algebras is probably not the most general one for which the conclusion of Theorem 5.9 holds. It is plausible that one has the generalized Hopkins-Miller theorem for any K(n)_{L}-local MU-algebra but such a result seems out of reach at present.

Tracing the proofs of Proposition 5.2 and Theorem 5.9 we see that they don’t depend on the choice of the S-algebra structure on Ê(n)_{W (L)} as long as the ring spectrum structure is fixed. (But they do depend on the choice of the S-algebra structure on E.) This gives the following

Corollary 5.11 The spectrum Ê(n)_{W (L)} has a unique (up to a noncanonical isomorphism) structure of an S-algebra compatible with its structure of a ring spectrum.

Proof. Let Ê′(n)_{W (L)} denote the S-algebra whose underlying ring spectrum is equivalent to Ê(n)_{W (L)} but the S-algebra structure is possibly different. Then the given multiplicative up to homotopy weak equivalence Ê′(n)_{W (L)} → Ê(n)_{W (L)} can be lifted to an S-algebra map which shows that the would-be exotic S-algebra structure on Ê′(n)_{W (L)} is actually isomorphic to the standard one.

Remark 5.12 A similar statement about the uniqueness of the A_{∞} structure on Ê(n) was proved by A.Baker in [1]. However the proof in the cited reference was based on the obstruction theory of A.Robinson and it is unclear at this time to what extent this theory carries over in the present context.

Theorem 5.9 allows one to obtain splittings of various strongly K(n)_{L}- complete MU-algebras. For example consider v_{n}^{−1}B̂P_{W (L)}, the Artinian completion of the v_{n}-localization of the Brown-Peterson spectrum BP_{W (L)} (cf.[2] concerning Artinian completions). Then v_{n}^{−1}B̂P_{W (L)} is a strongly K(n)_{L}-complete MU-algebra obtained from K(n)_{W (L)} by adjoining the collection of indeterminates {p, v_{1}, v_{2}, …}:

v_{n}^{−1}B̂P_{W (L)} = K(n)_{W (L)}[[v_{1}, v_{2}, …]].

There is a canonical map v_{n}^{−1}B̂P_{W (L)} → Ê(n)_{W (L)} which on the level of coefficient ring reduces to killing the ideal (v_{n+1}, v_{n+2}, …) in v_{n}^{−1}B̂P_{W (L)_{∗}}. From Theorem 5.9 we conclude that any S-algebra map Ê(n)_{W (L)} → Ê(n)_{W (L)} lifts uniquely to an S-algebra map Ê(n)_{W (L)} → v_{n}^{−1}B̂P_{W (L)}. In particular the identity map id : Ê(n)_{W (L)} → Ê(n)_{W (L)} can be so lifted and we obtain

Theorem 5.13 The S-algebra map v_{n}^{−1}B̂P_{W (L)} → Ê(n)_{W (L)} admits a unique S-algebra splitting Ê(n)_{W (L)} → v_{n}^{−1}B̂P_{W (L)}

An analogous theorem was proved (in a weaker, up to homotopy form) in [2].

Acknowledgement. The author is indebted to A. Baker for many useful discussions on the subject of the paper.

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