Homotopy theory of A_{∞} ring spectra and applications to MU-modules

A.Lazarev

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

Abstract

We give a definition of a derivation of an A_{∞} ring spectrum and relate this notion to topological Hochschild cohomology. Strict multiplicative structure is introduced into Postnikov towers and generalized Adams towers of A_{∞}-ring spectra. An obstruction theory for lifting multiplicative maps is constructed. The developed techniques are then applied to show that a broad class of complex-oriented spectra admit structures of MU-algebras where MU is the complex cobordism spectrum. Various computations of topological derivations and topological Hochschild cohomology are made.

Key words: Topological Hochschild cohomology, R-algebras, Morava K-theories.

1 Introduction

It has long been proved useful in topology to exploit multiplicative structures. The notion of an A_{∞} ring spectrum or, in more modern terminology, an Salgebra is an analogue in stable homotopy theory of the algebraic notion of an associative ring. The analogue of a commutative ring is the so-called E_{∞} ring spectrum or a commutative S-algebra. There has been much interest recently to these objects, especially after the appearance of the seminal work [7]. For instance, M.Mandell used them in [14] to give an algebraic characterization of the category of p-complete nilpotent spaces. Another prominent example is the Hopkins-Miller theorem (cf. [16]) asserting the existence of the action of the Morava stabilizer group on the Johnson- Wilson spectrum Ê_{n}.

Until recently only a handful of spectra were known to possess an A_{∞} or E_{∞} structure. Those are K-theoretic spectra arising from permutative categories, Eilenberg-MacLane spectra, various bordism spectra and their Bousfield localizations. The main purpose of the present paper is to devise a general method of proving the existence of A_{∞}-structures.

We give a definition of the topological derivation spectrum of an S-algebra in the sense of [7] and, more generally, of and R-algebra for an S-algebra R. This definition is based on the notion of a derivation of an R-algebra. A derivation of an R-algebra A with coefficients in an A-bimodule M is a map of R-algebras A → A ∨ M commuting with the canonical projection onto A. An analogous theory for commutative R-algebras was developed in by M.Basterra in [3]. The topological derivation spectrum and the topological Hochschild (co)homology of an R-algebra are related via cofibre sequences (2) and in good many cases they can be recovered from each other. We use these tools to develop an obstruction theory for lifting R-algebra maps and to introduce (strict) multiplicative structures into Postnikov towers of R-algebras and generalized Adams towers. We compute the topological derivations of the Eilenberg-MacLane spectrum HZ/p as an MU-algebra and compare the result with the classical computation of [8]. A curious consequence is that the mod p Moore spectrum does not admit a structure of an A_{∞} ring spectrum for any p. Further we prove that Morava K-theories (as well as a much broader class of complex-oriented spectra) admit structures of MU-algebras. By neglect of structure they also admit an A_{∞}- structure. The problem of introducing multiplicative structures into MU-modules was considered by Elmendorf, Kriz, Mandell and May in [7] and by Strickland in [18]. However these authors only prove the existence of such structures up to homotopy whereas the present paper deals with strict products. Such strict products (for Morava K-theories only) were first studied in [17]. 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). Our method involves building up a complex-oriented spectrum starting from the Eilenberg-MacLane spectrum (which reduces to the Postnikov tower in the case of Morava K-theories) in the spirit of algebraic deformation theory.

Finally we compute the topological derivations and topological Hochschild cohomology of K(n) (modulo additive extensions). We show how to deal with these extensions in a forthcoming paper. Our interest in topological Hochschild (co)homology of general spectra was inspired by the work of Mc-Clure and Staffeldt [15] where computations were made for the connective K-theory spectrum.

This paper is based on two preprints of the author [11] and [12], though the content of these has been considerably revised. Further extensive revisions were made following suggestions of the referees.

2 Derived module of differentials

Let R be a commutative S-algebra (in older terminology, an E_{∞} ring spectrum), A a (not necessarily commutative) R-algebra. Without loss of generality we assume that R is q-cofibrant as a commutative S-algebra and A is q-cofibrant as an R-algebra in the sense of [7]. Throughout the paper the notation ∧ will mean ∧_{R}.

Denote by Ω̃_{A/R} the (functorial) homotopy fibre of the multiplication map taken in the category of A-bimodules:

Ω̃_{A/R} → A ∧ A → A.

That is, Ω̃_{A/R} is the pullback A ∧ A ×_{A} F (I, A) where I is the unit interval R-module. The pullbacks of A-bimodules are created in the category of Rmodules and thus Ω_{A/R} is an A-bimodule.

Definition 2.1 The (derived) module of differentials Ω_{A/R} or just Ω_{A} if the “ground ring” R is clear from the context is the canonical cell approximation of Ω̃_{A/R} in the category of A-bimodules.

So we still have the homotopy fibre sequence of A-bimodules

Ω_{A/R} → A ∧ A → A  (1)

The exact triangle (1) is split by the map

A ≅ A ∧ R →^{id∧1} A ∧ A

in the homotopy category of left A-modules. Here (and later on in the paper) 1 : R → A stands for the unit map of the R-algebra A. Therefore Ω_{A} is weakly equivalent as a left A-module to A ∧ A/R. Symmetrically, Ω_{A} is weakly equivalent to A/R ∧ A as a right A-module.

Now let M be an A-bimodule. Recall from [7], X.1 that M is then a left A ∧ A^{op} module where A^{op} is an R-algebra which as an R-module coincides with A but with opposite multiplication. Define the topological derivation homology spectrum Der^{R}(A, M) as

Der^{R}(A, M) = Ω_{A} ∧_{A∧A^{op}} M

and the topological derivation cohomology spectrum Der_{R}(A, M) as

Der_{R}(A, M) = F_{A∧A^{op}}(Ω_{A}, M).

Here Der stands for “derivations”, and the meaning of this notation will become clear shortly.

Recall that the standard definition of Hochschild homology and cohomology (cf. [7]) is respectively

THH^{R}(A, M) = A ∧_{A∧A^{op}} M;
THH_{R}(A, M) = F_{A∧A^{op}}(A, M)

Now from (1) we see that the two definitions are related via cofibre sequences of R-modules:

Der^{R}(A, M) → M → THH^{R}(A, M);
THH_{R}(A, M) → M → Der_{R}(A, M)  (2)

Now define the derived module of derivations of A with values in M as

Der(A, M) = [A, A ∨ M]_{R−alg/A}  (3)

The right hand side of the last formula means homotopy classes of maps in the category of R-algebras over A, that is, R-algebras B supplied with a map B → A. This category inherits the structure of a topological model category from the category of R-algebras and therefore it is legitimate to consider homotopy classes of maps in R − alg/A. Further, A is an R-algebra over A in the obvious way and A ∨ M is given the structure of an R-algebra as follows:

(A ∨ M) ∧ (A ∨ M) = A ∧ A ∨ A ∧ M ∨ M ∧ A ∨ M ∧ M → A ∨ M,

where the last map is induced by the multiplication on the first summand, by the A-bimodule structure on M on the second and third summands and zero on the last summand. The structure map A ∨ M → A is the usual projection.

We will see shortly that Der(A, M) is an abelian group and

Der(A, M) = [Ω_{A}, M]_{A−bimod}

The right hand side of the last equality is the homotopy classes of maps from Ω_{A} to M in the category of A-bimodules (or, equivalently, in the category of left A ∧ A^{op}-modules). This gives, therefore, another definition of topological derivation groups, namely

Der_{R}^{n}(A, M) = Der(A, Σ^{n}M)

(and explains the notation Der_{R}(A, M)).

Denote the category of R-algebras over A by C_{R/A} and its homotopy category by hoC_{R/A}. Let B ∈ C_{R/A} and denote the A-bimodule

A ∧_{B} Ω_{B} ∧_{B} A ≅ A ∧ A^{op} ∧_{B∧B^{op}} Ω_{B}  (4)

by Ω_{A}^{B}. Since Ω_{B} is by definition a cell B ∧ B^{op}-module the smash products in (4) represent the derived smash product in the category of B-bimodules. Therefore the functor Ω_{A}^{B} could be interpreted as a point-set functor from the category of R-algebras over A to the category of A-bimodules or as a homotopy functor between the corresponding homotopy categories.

Theorem 2.2 There is a natural equivalence

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

In other words the functors

B → Ω_{A}^{B} : hoC_{R/A} → ho(A − bimod) and
M → A ∨ M : ho(A − bimod) − hoC_{R/A}

are adjoint.

Proof. As a first step we construct the “universal map” of A∨M-bimodules

Ω_{A}^{A∨M} → M.

Since Ω_{A}^{A∨M} = A ∧_{A∨M} Ω_{A∨M} ∧_{A∨M} A it is enough to construct a map of A ∨ M-bimodules Ω_{A∨M} → M. Consider the following strictly commutative diagram of A ∨ M-modules

A ∧ A ∨ A ∧ M ∨ M ∧ A ∨ M ∧ M → A ∨ M
↓  ↓
A ∨ M ∨ M  → A ∨ M
↓  ↓
M  → pt

Here the vertical arrow going out of the left upper corner is determined by the algebra structure on A and an A-bimodule structure on M on the first three summands, and is zero on the last summand. The arrow going to the lower left corner is zero on A, id on the second summand and −id on the last summand. (Of course, the category of A ∨ M-modules is not additive, but we can still choose a functorial representative of the negative of a map). Denote the homotopy fibre of the middle row by M̃. The homotopy fibre of the last row is the same as the actual fibre and is equal to M. Then take as our universal map the composite

Ω_{A∨M} → Ω̃_{A∨M} → M̃ → M.

So we constructed a map Ω_{A}^{A∨M} → M. Now for f : B → A ∨ M we have a map

Ω_{A}^{B} → Ω_{A}^{A∨M} → M

and therefore a correspondence

D : [B, A ∨ M]_{R−alg/A} → [Ω_{A}^{B}, M]_{A−bimod}.

Notice that we actually proved the existence of D on the point-set level. This more precise result will be used below. We need a special case of the Theorem 2.2 to prove the general case:

Lemma 2.3 For

B = T V = {R ∨ V ∨ V^{∧2} ∨ … →^{μ} A} ∈ C_{R/A},

the free associative algebra on an R-module V over A this correspondence establishes an isomorphism

[T V, A ∨ M]_{R−alg/A} → [Ω_{A}^{T V} , M]_{A−bimod}  (5)

Proof. First compute both sides of (5) separately. Obviously, the left hand side is equal to

[V, A ∨ M]_{R−mod/A} = [V, M]_{R−mod}.

We have the following homotopy fibre sequence of T V -bimodules:

(T V) ∧ V ∧ T V →^{f} T V ∧ T V →^{m} T V  (6)

where f = f_{1} − f_{2} and

f_{1} : T V ∧ V ∧ T V →^{m∧id}T V ∧ T V

f_{2} : T V ∧ V ∧ T V →^{id∧m}T V ∧ T V;

m is the multiplication map. Smashing (6) with A on both sides over T V we get

A ∧ V ∧ A → A ∧ A → A ∧_{T V} A.

Therefore Ω_{A}^{T V} = A ∧ V ∧ A and

[Ω_{A}^{T (V)}, M]_{A−bimod} = [V, M]_{R−mod} = [T (V), A ∨ M]_{hoC_{R/A}}.

The second equality uses the (obvious) fact that the functor

M → A ∨ M : ho(R − mod) → ho(R − mod/A)

is right adjoint to the forgetful functor

ho(R − mod/A) → ho(R − mod).

So we identified each side of (5) with [V, M]_{R−mod}. To prove our lemma we have to show that D respects this identification. In other words for an R-module map f : V → M the composite

Ω_{T V} → Ω_{A∨M} → M

should correspond under this identification to the map f.

Consider now the diagram

T V ∧ V ∧ T V ⟶  Ω_{A∨M}  ⟶  M
↓  ↓  ↓
T V ∧ T V  ⟶ A ∧ A ∨ A ∧ M ∨ M ∧ A ∨ M ∧ M ⟶ A ∨ M ∨ M
↓  ↓  ↓
T V  ⟶  A ∧ M  =  A ∧ M

Here all three columns are homotopy cofibre sequences. Also recall that the upper left vertical map is m ∧ id − id ∧ m. Clearly the composite

V → T V ∧ V ∧ T V → T V ∧ T V →
A ∧ A ∨ A ∧ M ∨ M ∧ A ∨ M∧ → A ∨ M ∨ M → M ∨ M

coincides with the composite

V → V ∨ V → A ∧ M ∨ M ∧ A → M ∨ M

where the first map is id ∨ 0 − 0 ∨ id, the second map is μ ∧ f − f ∧ μ and the last map is induced by the A-bimodule structure on M. Therefore the composite map V → M ∨ M is f ∨ 0 − 0 ∨ f. Taking into account that the exact triangle

M → M ∨ M → M

splits we see that there is a unique up to homotopy lifting

V → T V ∧ V ∧ T V → M

and it coincides with f. The lemma is proved.

Return now to the general case. Without loss of generality we assume that B is a q-cofibrant R-algebra and resolve it by the monadic bar-construction B_{⋆}. Recall from [7], Chapter 12 that B_{⋆} = {B_{i}} is a simplicial R-algebra with B_{i} = T^{i}B and there is a weak equivalence |B_{⋆}| → B where |B_{⋆}| is the geometric realization of B_{⋆}. According to [7], Proposition 3.3 Chapter VII this geometric realization is the same in the category of R-algebras as in the underlying category of R-modules).

The problem with B_{⋆} is that its ith simplices need not be q-cofibrant Ralgebras which means that its realization |B_{⋆}| is not necessarily q-cofibrant. Therefore we can’t use it to compute the homotopy type of the function space out of B. To make the cofibrant resolution of B we use the argument due to M.Basterra, cf. [3]. Namely, replace B_{⋆} by the simplicial R-algebra ΓB_{⋆} that is obtained from B_{⋆} by the application of the functorial cell R-algebra approximation functor Γ component-wise. Then each component ΓB_{i} is a cell R-algebra, all degeneracy operators are cell inclusions and the face maps are sequentially cellular. Therefore we conclude by [7], Chapter 10, Theorem 2.7 that the realization |ΓB_{⋆}| is a cell R-algebra. Moreover, the simplicial components of ΓB_{⋆} are weakly equivalent to free R-algebras. Also |ΓB_{⋆}| is weakly equivalent to B. That means that we have a weak equivalence of topological spaces

[B, A ∨ M]_{R−alg/A} ≃ [|ΓB_{⋆}|, A ∨ M]_{R−alg/A}

Denote F_{C_{A/R}}(−, −) the topological space of maps between two objects in C_{A/R} (not homotopy classes of maps). We have:

F_{C_{A/R}}(|ΓB_{⋆}|, A ∨ M) ≅ F_{C_{A/R}}(ΓB_{⋆}, A ∨ M)

where F_{C_{A/R}}(ΓB_{⋆}, A ∨ M) is the topological space of maps between two simplicial R-algebras ΓB_{⋆} and A∨M, the latter regarded as a constant simplicial algebra.

Next, F_{C_{A/R}}(ΓB_{⋆}, A ∨ M) is the total space of the cosimplicial topological space

T^{⋆} = {F_{C_{A/R}}(ΓB_{i}, A ∨ M)}.  (7)

The monadic bar-construction B_{⋆} is a proper simplicial spectrum and so is ΓB_{⋆}. That ensures that the cosimplicial space (7) is fibrant in the sense that there is a tower of fibrations

T ot^{0}(T) ← T ot^{1}(T) ← …

where T ot^{i}(T) is the ith cosimplicial approximation of T^{⋆} and whose inverse limit is T ot(T^{⋆}).

Likewise

F_{A−bimod}(Ω_{A}^{ΓB_{⋆}}, M) = F_{ΓB_{⋆}−bimod}(Ω_{ΓB_{⋆}}, M)

is the total space of the cosimplicial space

{T^{⋆}′} = F_{ΓB_{i}−bimod}(Ω_{ΓB_{i}}, M)}.  (8)

The cosimplicial space T^{⋆′} does not necesssarily give rise to a tower of fibrations since the simplicial spectrum Ω_{ΓB_{i}} need not be proper. (Recall that the definition of the module of differentials involves taking the homotopy fibre which could destroy properness). Replace {Ω_{ΓB_{i}}} functorially by a proper simplicial spectrum and denote the corresponding cosimplicial space by T̃^{⋆}. Then T̃^{⋆} is fibrant and we still have a map T^{⋆} → T̃^{⋆}. According to the previous lemma this map induces induces an equivalence on the ith cosimplices for all i. It follows that this map induces an isomorphism of the corresponding spectral sequences. Therefore it is a weak equivalence and Theorem 2.2 is proved.

Remark 2.4 It will be useful to generalize the notion of the module of differentials a little. Let R′ be a (not necessarily commutative ) R-algebra and consider the category of R′-algebras which means just R-algebras A equipped with a map R′ → A, not necessarily central as usually required. For an R′- algebra A we can define Ω_{A/R′} just as we did before (replacing ∧ by ∧_{R′}). The analogue of Theorem (2.2) reads as follows: for an R′-algebra B over an R′-algebra A one has a natural equivalence

[B, A ∨ M]_{hoC_{R′/A}} ≅ [Ω_{A/R′}^{B}, M]_{A−bimod}

where Ω_{A/R′}^{B} := A ∧_{B} Ω_{A/R′} ∧_{B} A.

The proof is the same, one only has to notice that there still are adjunctions

R′ − algebras →^{←} R′ − bimodules;

A − bimodules →^{←} R′ − bimodules

where the “free” R′-algebra functor is given for an R′-bimodule V as

T V = R′ ∨ V ∨ V ∧_{R′} V …

and the “free” A-bimodule on a R′-bimodule V is A ∧_{R′} V ∧_{R′} A. The R′- bimodule F_{A−bimod}(Ω_{A/R′}, M) will be called R′-relative topological derivation spectrum of A with coefficients in M and denoted as Der(A; R′, M). Notice, that even if R′ happens to be commutative Der(A; R′, M) is not the same as Der_{R′}(A, M) since the latter is relevant to the category of R′-relative Abimodules, that is, those A-bimodules whose induced R′-bimodule structure is symmetric.

Corollary 2.5 The functor

B → Ω_{A}^{B} : C_{R/A} → A − bimod

respects homotopy colimits.

Proposition 2.6 (Transitivity exact sequence) Let A → B → C be morphisms of R-algebras. Then one has the following homotopy cofibre sequence of C-bimodules:

C ∧_{B} Ω_{B/A} ∧_{B} C → Ω_{C/A} → Ω_{C/B}

Proof. Without loss of generality we assume that both maps A → B and B → C are q-cofibration of R-algebras. We have the following homotopy cofibre sequence of B-bimodules:

Ω_{B/A} → B ∧_{A} B → B.

Smashing it with C over B on both sides we get the homotopy cofibre sequence of C-bimodules:

C ∧_{B} Ω_{B/A} ∧_{B} C → C ∧_{A} C → C ∧_{B} C.

Consider the following homotopy commutative diagram:

C ∧_{B} Ω_{B/A} ∧_{B} C ⟶ C ∧_{A} C ⟶ C ∧_{B} C
↓  ↓  ↓
0  ⟶  C  ⟶  C

Here the right and the middle vertical arrows are induced by multiplication on C. Taking the homotopy fibres of the vertical arrows we get the desired cofibre sequence.

Corollary 2.7 For a C-bimodule M we have the following homotopy cofibre sequence

Der(C; B, M) → Der(C; A, M) → Der(B; A, M)

Proof. Apply the functor F_{C−bimod}(?, M) to the transitivity exact sequence.

Remark 2.8 Notice that the transitivity exact sequence is functorial with respect to the strict morphisms of triples {A → B → C} (essentially because the construction of the modules of differentials is functorial).

3 Extensions of R-algebras

In this section we define and study extensions of R-algebras. Our results turn out to be in some ways analogous to the classical theory for associative algebras (cf., for example, [13], Chapter X). A more modern treatment suitable for operadic algebras is found in the recent work of Hopkins and Goerss [9].

Definition 3.1 An extension of an R-algebra B by an R-algebra I without unit is the following homotopy fibre sequence of R-modules:

I → A → B.  (9)

where A is an algebra, and both arrows are algebra maps.

Next denote I/I^{2} the homotopy cofibre

I ∧_{A} I → I → I/I^{2}.

(notice that I is naturally an A-bimodule.) Then we have the following

Lemma 3.2 (Conormal exact sequence) The R-module I/I^{2} is weakly equivalent as an R-module to a B-bimodule (which we will still denote as I/I^{2}) and there is a homotopy cofibre sequence of B-bimodules

I/I^{2} → B ∧_{A} Ω_{A} ∧_{A} B → Ω_{B}  (10)

Proof. We have the following transitivity exact sequence (corresponding to algebra maps R → A → B):

B ∧_{A} Ω_{A} ∧_{A} B → Ω_{B} → Ω_{B/A}.

So it suffices to show that Σ(I/I^{2}) = Ω_{B/A}. Consider the following diagram:

I ∧_{A} I ⟶ I ⟶ I/I^{2}
↓  ↓  ↓
I  ⟶ A ⟶  B
↓  ↓  ↓
I ∧_{A} B ⟶ B ⟶ B ∧_{A} B

Here the arrows going to the right lower corner are both equal to 1 ∧ id, the arrows going out of the left upper corner are multiplication maps on I and the map from I to I ∧_{A} B is id ∧ 1. This diagram is commutative in the homotopy category of R-modules and all rows and columns are homotopy cofibre sequences of R-modules. Indeed, the middle row and column are cofibre sequences by hypothesis, the first column and last row are likewise, since smash product preserves such sequences, the first row is a cofibre sequence by definition of I/I^{2}, and the last column of induced maps is also a cofibre sequence. To be precise, the map from the first column to the middle column can be rigidified in the category of R-modules, hence the last column can be considered as the functorial cofibre of the relevant maps and is itself therefore a cofibre sequence. Next we have the following commutative diagram in the homotopy category of R- modules:

I/I^{2} ⟶ 0 ⟶ Ω_{B/A}
↓ ↓ ↓
I/I^{2} ⟶ B ⟶ B ∧_{A} B
↓ ↓ ↓
0 ⟶ B ⟶  B

Here the rightmost column is a cofibre sequence by definition of Ω_{B/A}, the middle row is also such by the previous discussion. It follows that the upper row is a cofibre sequence. Therefore Ω_{B/A} ≅ ΣI/I^{2} and the lemma is proved.

A derivation d : B → B ∨ ΣI gives rise to an extension as follows.

Consider the diagram

\begin{matrix}I & ≅ & I \\ ↓ & & ↓ \\ X & ⟶ & B \\ ↓ & & ↓ \\ B & ⟶^{d} & B ∨ ΣI \end{matrix}  (11)

Here the lower vertical map is the canonical inclusion of a retract; the lower square is a homotopy pullback in the category of R-algebras (which is also the homotopy pullback of underlying R-modules), therefore the left column is a homotopy cofibre sequence of R-modules and an extension of an R-algebra B by I. We will call I → X → B the extension associated with the derivation d.

Definition 3.3 An extension I′ → X′ → B′ is called singular if there exists an extension I → X → B associated with a derivation d : B → B ∨ ΣI and a strictly commutative diagram of algebras

I → X → B
↓ ↓ ↓
I′ → X′ → B

where the vertical arrows are weak equivalences.

The next proposition explains the analogy with usual singular extensions of algebras (hence the name); the multiplication map on the “ideal” I (at least up to homotopy)is zero.

Proposition 3.4 Let I → X → B be a singular extension. Then the multiplication map I ∧_{X} I → I is homotopic to zero as a map of R-modules, so the homotopy cofibre sequence

I ∧_{X} I → I → I/I^{2}

splits and I/I^{2} is weakly equivalent as an R-module to I ∨ Σ(I ∧_{X} I).

Proof. Let us consider first the “universal” singular extension

I → B → B ∨ ΣI.

Then the multiplication map I ∧_{B} I → I can be factored up to homotopy as follows

I ∧_{B} I →  I
↘  ∥
B ∧_{B} I

Since the map I → B is homotopically zero we conclude that the map I ∧_{B} I → I is also so.

Next if we have a singular extension I → X → B associated with a derivation B → B ∨ ΣI it is clear that the multiplication map I ∧_{X} I → I factors through I ∧_{B} I → I and is therefore homotopic to zero.

The next proposition describes more concretely the boundary map

Ω_{B} → ΣI/I^{2}

in the case of a singular extension.

Proposition 3.5 Let (9) be a singular extension, so ΣI/I^{2} ≃ ΣI ∨ΣI ∧_{A}ΣI. Denote by

d : B → ΣI

the boundary map in (9). Recall that there is an R-module weak equivalence Ω_{B} ≅ B ∧ B/R. Then for the boundary map in (10)

∂ : B ∧ B/R → ΣI ∨ ΣI ∧_{A} ΣI

we have ∂ = ∂_{1} + ∂_{2} where

∂_{1} : B ∧ B/R →^{id∧d} B ∧ ΣI → B ∧_{B} ΣI ≃ ΣI;

and ∂_{2} is the composite map

B ∧ B/R →^{d∧d} ΣI ∧ ΣI → ΣI ∧_{A} ΣI

Proof. Consider the following homotopy commutative diagram of R-modules:

B = B ∧_{A} A → B ∧_{A} B →^{id∧d} B ∧_{A} ΣI
∥ ↑ ↑id ∧ d
B = B ∧_{A} A → B ∧ B → B ∧ B/R

Here both rows are split homotopy cofibre sequences , the R-module B ∧_{A} ΣI is homotopy equivalent to ΣI/I^{2} as we saw in the proof of Lemma 3.2 and the right vertical map is the map ∂ that we are interested in. Now the proposition follows from the following diagram , where again both rows are split homotopy cofibre sequences of R-modules:

ΣI  = A ∧_{A} ΣI → B ∧_{A} ΣI → ΣI ∧_{A} ΣI
↑  ↑id ∧ d  ↑id ∧ d
A ∧ B/R  →  B ∧ B/R → ΣI ∧ B/R

Remark 3.6 There is a slight catch in the above proposition. We considered the map

id ∧ d : B ∧ B/R → B ∧ ΣI

denoting the extension of the map B → ΣI to B/R by the same letter d. This extension is not unique, but the indeterminacy vanishes upon smashing with B.

4 The universal derivation

In this section we will compute the homotopy type of the “universal derivation”, that is, of the composite map

A → A ∨ Ω_{A} → Ω_{A}

where the first map is the algebra map adjoint to the identity map Ω_{A} → Ω_{A} and the second map is the projection.

Let A be an R-algebra. Recall that there is an equivalence of R-modules

Ω_{A} ≅ A ∧ A/R  (12)

Theorem 4.1 Upon the identification (12) the universal derivation A → Ω_{A} coincides with the map

A →^{id∧d} A ∧ A/R.

where d is defined by the homotopy cofibre sequence

R → A →^{d} A/R.

More symmetrically: the universal derivation is the unique (in the homotopy category of R-modules) lifting A → Ω_{A} in the diagram

Ω_{A}
↓
A ≅ A →^{f} A ∧ A

where f = 1 ∧ id − id ∧ 1

Proof. Consider the following extension

I → R → A

where the second map is just the unit map and I is the kernel of this map. From the conormal sequence of this extension we see that there is a map

Ω_{A} → ΣI/I^{2}  (13)

and, consequently, a singular extension with the kernel I/I^{2}:

I/I^{2} → X → I.  (14)

We have the following homotopy commutative diagram of R-modules:

Ω_{A} ⟶^{g} ΣI/I^{2} ∨ Σ^{2}I/I^{2} ∧_{X} I/I^{2}
↑ ↗
A

Here the horizontal row is a fragment of the conormal sequence corresponding to (13), the vertical arrow is the universal derivation and the slanted arrow is the boundary map in (13).

We claim that the map g is the canonical inclusion of a retract:

f : Ω_{A} ≃ ΣI/I^{2} → ΣI/I^{2} ∨ Σ^{2}I/I^{2} ∧_{X} I/I^{2}.

To see that observe that there is a map of extensions

I → R → A
↓ ↓ ↓
I/I^{2} → X → A

and therefore the following homotopy commutative diagram of R-modules:

Ω_{A} ≃ I/I^{2}  → A →^{1∧id} A ∧ A
↓  ↓
Ω_{A/X} ≃ I/I^{2} ∨ Σ^{2}I/I^{2} ∧_{X} I/I^{2} → A →^{1∧id} A ∧_{X} A

Notice that since both rows split there exists only one up to homotopy map

Ω_{A} → Ω_{A/X}

making the diagram commute and this map is obviously the inclusion of a retract. Our claim is proved. Therefore, our universal derivation is just the boundary map (14):

A → ΣI/I^{2}.

Next from the diagram

I → R → A → ΣI = A/R
↓ ↓ ∥ ↓
I/I^{2} → R/I^{2} → A → ΣI/I^{2} = A ∧ A/R

we see that the boundary map A → ΣI/I^{2} is the composition

A → A/R ≃ ΣI → ΣI/I^{2}

and the diagram

I → R →  A  →  A/R
↓  ↓  1 ∧ id ↓  1 ∧ id ↓
I/I^{2} → A →^{1∧id}  A ∧ A  → A ∧ A/R

tells us that the map ΣI → ΣI/I^{2} is the same as

A/R →^{1∧id} A ∧ A/R

and the first part of our theorem is proved.

Consider now the following homotopy commutative diagram

Ω_{A} → A ∧ A → A ∧ A/R
f ↑ ↗
A

Here the slanted arrow is id ∧ d and the composite of the two horizontal arrows is an isomorphism identifying Ω_{A} with A ∧ A/R. Clearly the map A → Ω_{A} fitting in the above diagram exists and is unique up to homotopy. With this our theorem is proved.

The universal derivation allows us to define a useful forgetful map

l : Der_{R}(A, M) → F_{R}(A, M)

as the composite

Der_{R}(A, M) = F_{A−bimod}(Ω_{A}, M) → F (Ω_{A}, M) → F (A, M)

where the last map is induced by d.

We will also denote by l the corresponding map on homotopy groups

Der_{R}^{•}(A, M) → [A, M]_{R}^{•}.

At the level of homotopy groups the map l has the following simple description: let d : A → A ∨ M be a derivation of A with coefficients in M. Then l(d) is the homotopy class of the following composite map

A →^{d} A ∨ M → M

where the second map is just the projection onto the wedge summand.

Lemma 4.2 Let d ∈ Der_{R}^{•}(A, A). Then l(d) ∈ [A, A]_{R}^{•} is contained in the subspace of primitive cohomology operations from A into itself.

Proof. Let d have degree i, so it determines a map of algebras A → A∨Σ^{i}A. Then the following diagram is homotopy commutative:

A ∧ A  ⟶ A
↓  ↓
Σ^{i}A ∧ A ∨ A ∧ Σ^{i}A → A

Here the upper arrow is the multiplication map m, the lower arrow is m ∨ m, the right vertical arrow is d and the left vertical arrow is d ∧ 1 ∨ 1 ∧ d. This is the definition of a primitive operation and the lemma is proved.

5 Pullbacks and pushouts of singular extensions

Here we will describe some basic constructions with singular extensions. Let

E : I → A → B  (15)

be a singular extension and

f : C → B

a map of algebras. Consider the strictly commutative diagram

\begin{aligned}I & → & X & → & C \\ ↓ & & ↓ & & ↓ \\ I & → & A & → & B \end{aligned}  (16)

where the right vertical map is f and the right square is a homotopy pullback of R-algebras. It is easy to see that the upper row in (16) is a singular extension (the associated derivation is induced by the map C → B). We will denote it f^{⋆}(E).

We have the following simple interpretation of f^{⋆}(E) in terms of the topological module of differentials. Let

Ω_{B} → ΣI

be the derivation corresponding to the singular extension E. Then it is easy to see that the composite map

Ω_{C} → Ω_{B} → ΣI

represents the element in Der_{R}^{1}(C, I) that corresponds to f^{⋆}(E). Next consider again an extension (15) and a map of B-bimodules

g : I → M

The map

d : Ω_{B} → ΣI → ΣM

gives rise to an element in Der_{R}^{1}(B, M) and, therefore, to a singular extension of B by M. We will call this extension the pushout of E by g and denote it g_{⋆}(E).

Proposition 5.1 There is a map from E to g_{⋆}(E) which is the identity on B.

Proof. Consider the diagram

I
↓
A → B  ≅  B
↓  ↓  ↓
B →^{d} B ∨ ΣI → B ∨ ΣM

Here the left square is a homotopy pullback, and from the commutativity of the outer square it follows that there is a map from it to the homotopy pullback diagram

A → B
↓  ↓
B → B ∨ ΣI

and therefore from E to g_{⋆}(E) and the proposition is proved.

Remark 5.2 Consider the diagram

I → A → B
↓ ↓ ↓
M → Y → B

where the upper row is E and the lower row is g_{⋆}(E). Since both rows are homotopy cofibre sequences of R-modules, Y is homotopically the same as A ∨_{I} M but it is not clear apriori how to introduce strict multiplication on A ∨_{I} M

6 Obstruction theory.

In this section we consider the problem of lifting an algebra map. We restrict ourselves to the case of a singular extension only. However in the next section we will see that a fairly broad class of algebra maps can be decomposed as a sequence of singular extensions, which we call a generalized Adams resolution. Together these results provide a sequence of obstructions for lifting an arbitrary map of R-algebras.

Theorem 6.1 Let I → B → A be a singular extension of R-algebras associated with a derivation d : B → B ∨ ΣI and f : X → A a map of R-algebras where X is q-cofibrant. Then:

(i) f lifts to an R-algebra map X → B iff the induced derivation d ∘ f ∈ Der^{1}(X, I) is homotopic to zero. (ii) Assuming that a lifting exists in the fibration

F_{R−alg}(X, B) → F_{R−alg}(X, A)  (17)

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

Proof. (i). If a lifting of f exists, then d ∘ f factors as

X →^{f} B →^{ϵ} A ∨ ΣI

which means that the derivation d ∘ f is trivial, Conversely, if d ∘ f is trivial then the diagram of R-algebras

X →^{f} A
f ↓  ↓ ϵ
A →^{d} A ∨ ΣI

commutes up to homotopy and by the universal property of the homotopy pullback there is a map f̃: X → B lifting f.

(ii). Suppose that the lifting of f exists, so the (homotopy) fibre of the map (17) is nonempty. We have the following diagram of R-algebras.

B  ⟶  A
| ↘  ↙  |
|  A  |
↓ ↗  ↖  ↓
A  ⟶^{d}  A ∨ ΣI

Here the northwest arrow is the canonical projection onto the wedge summand. Changing B in its homotopy class if necessary we can arrange that this diagram be strictly commutative. Notice that the outer square is a homotopy pullback of R-algebras. Applying the functor F_{R−alg}(X, −) to this diagram we get the diagram of spaces

F_{R−alg}(X, B)  ⟶  F_{R−alg}(X, A)
|  ↘  ↙  |
|  F_{R−alg}(X, A)  |
↓  ↗  ↖  ↓
F_{R−alg}(X, A)  ⟶  F_{R−alg}(X, A ∨ ΣI)

Again the outer square is a homotopy pullback (of spaces). Taking the homotopy fibres of the maps from the outer square to the center (over f ∈ F_{R−alg}(X, A)) we get the following homotopy pullback of spaces:

hofibF_{R−alg}(X, B) → F_{R−alg}(X, A) ⟶  pt
↓  ↓
pt  ⟶ hofibF_{R−alg}(X, A ∨ ΣI) → F_{R−alg}(X, A)

(18)

Notice that the space in the right lower corner of (18) is canonically weakly equivalent to Ω^{∞}Der_{R}(X, ΣI). Since according to our assumption the space

hofib F_{R−alg}(X, B) → F_{R−alg}(X, A)

is nonempty the images of the lower and right arrows in (18) coincide and therefore

hofib F_{R−alg}(X, B) → F_{R−alg}(X, A) ≅ Ω(Ω^{∞}Der_{R}(X, ΣI)) ≅ Ω^{∞}Der_{R}(X, I).

With this our theorem is proved.

7 Ideals of R-algebras and generalized Adams resolutions

Let

I → A → B  (19)

be an extension of R-algebras (not necessarily singular.) In such a situation we will call I an ideal of A. To stress the analogy with algebra we will use the notation A/I instead of B in the sequel so (19) becomes

I → A → A/I  (20)

One essential feature of ideals is that one can define R-algebras of the form A/I^{n} for n > 1. More precisely:

Theorem 7.1 Associated to the extension (20) is a tower of R- algebras (a generalized Adams resolution) of the form

… → A/I^{n} → A/I^{n−1} → … → A/I  (21)

such that

(i) there are projections p_{n} : A → A/I^{n} compatible with the maps in the tower and the corresponding extensions

I^{n} → A → A/I^{n}

where

I^{n} := I ∧_{A} … ∧_{A} I;

(ii) each successive stage A/I^{n} is a singular extension of the previous stage A/I^{n−1} by a certain A/I^{n−1}-bimodule which we denote I^{n}/I^{n−1}

Proof. We know that there is there is a weak equivalence

Ω_{(A/I)/A} ≃ ΣI/I^{2}.

where I/I^{2} is determined as the homotopy cofibre of the multiplication map I ∧_{A} I → I. Associated to the (universal) derivation

d_{A/I} : A/I → Ω_{(A/I)/A} ≃ ΣI/I^{2}

is the following singular extension

I/I^{2} → X → A/I.

We claim that there exists a map A → X making the diagram

\begin{aligned}& & X \\ & ↗ & ↓ \\ A & →^{f} & A/I \end{aligned}

commute. Indeed, the obstruction to the existence of such a map is an element

f^{⋆}(d_{A/I}) ∈ Der(A, ΣI/I^{2})

which is zero by virtue of the transitivity exact sequence corresponding to the morphisms of algebras

R → A → A/I.

Therefore we have a morphism of extensions

\begin{aligned}I & → & A & → & A/I \\ ↓ & & ↓ & & ∥ \\ I/I^{2} & → & X & → & A/I \end{aligned}  (22)

Now set A/I^{2} := X. From (22) it is easy to see that there is an extension

I ∧_{A} I → A → A/I^{2}

(which justifies the notation A/I^{2}).

Proceeding by induction assume that we have constructed the tower {A/I^{k}} for k ≤ n together with algebra maps p_{n} : A → A/I^{n}. Consider the following homotopy commutative diagram of A/I^{n}-bimodules where both rows are homotopy fibre sequences:

?  → A/I^{n} ∧_{A} A/I → A/I
g ↑ ↑ ↑
Ω_{(A/I^{n})/A} → A/I^{n} ∧_{A} A/I^{n} → A/I^{n}

Here the lower row is the definition of Ω_{(A/I^{n})/A} and the upper row represents the (left) action of A/I^{n} on A/I. The right and middle vertical arrows are obvious maps of A/I^{n}-bimodules. Therefore we can choose the left vertical arrow (denoted by g) to be also a map of A/I^{n}-bimodules. Notice, that despite the choices made in the underlying category of R-modules this map still remains unique (up to homotopy).

Further notice that the A/I^{n}-bimodule ? is weakly equivalent as an Rmodule to ΣI^{n} ∧_{A} A/I. Denoting by I^{n}/I^{n+1} the homotopy cofibre of the multiplication map I^{n} ∧_{A} I → I^{n} we see that I^{n}/I^{n+1} is weakly equivalent to I^{n} ∧_{A} A/I. So the A/I^{n}-bimodule ? is weakly equivalent as an R-module to ΣI^{n}/I^{n+1}. From now on we will (slightly abusing notations) use the symbol ΣI^{n}/I^{n+1} to denote the corresponding A/I^{n}-bimodule.

Now we can construct the singular extension

I^{n}/I^{n+1} → A/I^{n+1} → A/I^{n}

associated with the derivation

g : Ω_{(A/I^{n})/A} →? = ΣI^{n}/I^{n+1}.

The induction step is concluded and our theorem is proved.

Remarks 1. In the special case when A = S, the sphere spectrum, the tower (21) is just the canonical Adams resolution of S, cf. [1] and its homotopy inverse limit is the nilpotent completion of S at the ideal I (cf. [5]), which under favorable circumstances coincides with the localization of S with respect to the spectrum S/I. This is why we called the tower (21) the generalized Adams resolution.

2. It is not hard to show that the tower (21) is determined in the homotopy category of R-algebras up to a (noncanonical) isomorphism. We leave the details to the interested reader.

Examples 1. Take A = k(n), the nth connective Morava K-theory, A/I := HZ/p, A → A/I the canonical map inducing an isomorphism on π_{0}. Then the generalized Adams resolution of k(n) is exactly its Postnikov tower.

2. Truncated polynomial algebra. Denote R[t] the free associative algebra on the R-module Σ^{d}R, i.e.

R[t] ≃ R ∨ Σ^{d}R ∨ Σ^{2d}R ∨ …

The parameter t has formal degree d. The coefficient ring of R[t] is then R_{•}[t] - the polynomial algebra over R_{•} on one generator of degree d. Notice, that the S-algebra R[t] need not be commutative unless d = 0.

Define the ideal tR[t] from the following homotopy fibre sequence:

tR[t] → R[t] → R.

Then it is easy to see that the nth Adams stage

R^{n−1} := R[t]/(tR[t])^{n}

splits as a spectrum as

R^{n−1} ≃ R ∨ Σ^{d}R ∨ Σ^{2d}R ∨ … ∨ Σ^{d(n−1)}R

and R_{•}^{n−1} = R_{•}[t]/t^{n}. Moreover, the homotopy fibre sequence

Σ^{d(n−1)}R → R^{n−1} → R^{n−2}

is a singular extension.

3. Let E be a complex-oriented cohomology theory obtained by killing a regular ideal I_{•} = (x_{i_{1}}, x_{i_{2}}, …) in MU_{•} = Z[x_{1}, x_{2}, …]. By [7], chapter V and [18] we know that E is an MU-module with homotopy associative multiplication (or MU-ring spectrum). We will see that E possesses the structure of an MU-algebra (i.e. with strict multiplication). Therefore we have the generalized Adams resolution

→ MU/I^{n} → … → MU/I^{2} → MU/I = E

which is related to the algebraic tower {MU_{•}/I_{•}^{n}} and whose homotopy inverse limit is the spectrum MU itself.

4. Let E_{n} be the Johnson-Wilson theory with coefficient ring E_{n} = Z_{(p)}[v_{1}, v_{2}, …, v_{n}, v_{n}^{−1}]. It can be constructed from MU by killing a collection of polynomial generators and is, therefore, an MU-algebra. We have a canonical MU-algebra map E_{n} → K(n) with the kernel I_{n}. Associated with this map is a generalized Adams resolution

K(n) = E_{n}/I_{n} ← E_{n}/I_{n}^{2} ← …

whose homotopy inverse limit is equivalent to the K(n)-localization of E_{n}. This tower is closely related to the so-called I_{n}-adic tower studied in, for example, [2].

8 Postnikov towers of R-algebras

In this section we prove, as an application of the techniques developed before, that a Postnikov tower of an R-algebra is a tower of R-algebras. The analogous statement for commutative R-algebras was proved in [10] and [3]. The proof presented here is somewhat sketchy since it differs little from the one found in the above mentioned references.

Theorem 8.1 Assume that R is a connective commutative S-algebra and A is a connective R-algebra. Then there exists a tower of R-algebras

… ← A_{i} ← A_{i+1} ← …

and maps of R-algebras

f_{i} : A → A_{i}

compatible with the maps in the tower such that 1)π_{k}(A_{i}) = 0 for k > i and 2)f_{i} induces an isomorphism π_{k}(A) → π_{k}(A_{i}) for k ≤ i. Moreover, the

homotopy fibre sequence

K(π_{i+1}(A), i + 1) → A_{i+1} → A_{i}

is a singular extension and the whole construction is functorial in the homotopy category of R-algebras

Remark 8.2 The connectivity assumptions on R and A are needed in order to be able to kill higher homotopy groups.

Proof. Take A_{0} to be equal to K(π_{0}(A,)0). This is an R-algebra and according to [7], Prop. 4.3 there exists an R-algebra map

A → A_{0}

realizing the isomorphism on π_{0}.

Suppose by induction that the map A → A_{i} has already been constructed. Consider the following diagram:

A → A_{i}  →^{k} ΣK(π_{i+1}(A), i + 1)
↓ ↗
Ω_{A_{i}/A}

Here the vertical map is the canonical derivation, k is the ith k - invariant. The existence of the diagonal arrow can be seen as follows. The first nontrivial homotopy group of Ω_{A_{i}/A} is the same as the single nontrivial homotopy group of ΣK(π_{i+1}(A), i + 1), so we could construct a map

Ω_{A_{i}/A} → ΣK(π_{i+1}(A), i + 1)

by attaching cells in the category of A_{i}-bimodules. This gives a derivation

A_{i} → A_{i} ∨ ΣK(π_{i+1}(A), i + 1)

and the corresponding singular extension

K(π_{i+1}(A), i + 1) → A_{i+1} → A_{i}

which is the (i+1)th stage of the Postnikov tower and the theorem is proved.

9 The spectral sequence for topological Hochschild cohomology

In this section we discuss the spectral sequence that relates topological Hochschild cohomology and ordinary Hochschild cohomology of graded algebras. Since we know of no published source which describes algebraic Hochschild cohomology the way we need it we begin with an outline of the classical Hochschild cohomology.

Definition 9.1 . Let R_{•} be a commutative graded ring, A_{•} is an algebra over R_{•}, M_{•} is a graded module over A_{•}. Then Hochschild cohomology of A_{•} with coefficients in M_{•} is the module

HH_{R_{•}}^{•}(A_{•}, M_{•}) = Ext_{A_{•}⊗_{R•}^{L}A_{•op}}(A_{•}, M_{•})

where ⊗_{R_{•}}^{L} denotes derived tensor product over R_{•}

From now on we will write ⊗ instead of ⊗_{R_{•}} Notice that if A_{•} is flat over R_{•} then this definition coincides with the standard one (cf.[6]). There is also a generalization of the standard complex which computes Hochschild cohomology; let Ã_{•} be a differential graded R_{•}-algebra which is quasiisomorphic to A_{•} and R_{•}-flat. Then HH_{R_{•}}^{•}(A_{•}, M_{•}) is the cohomology of the bicomplex

C^{ij}(A_{•}, M_{•}) = Hom_{R_{•}}^{i}(Ã_{•}^{⊗j}, M_{•}).

The differentials in C^{••} are the internal one induced by the differential in Ã_{•} and the standard bar-differential.

We will be interested mostly in the truncated version of Hochschild cohomology which will be denoted as Der_{R_{•}}^{•}(A_{•}, M_{•}) (in keeping with the topological notations)

To define it let us introduce the module of differentials Ω_{A_{•}/R_{•}}^{•} from the following short exact sequence of complexes:

Ω_{A_{•}/R_{•}}^{•} → Ã_{•} ⊗ Ã_{•} →^{m} Ã_{•},

the second arrow being the multiplication map.

Now define the truncated Hochschild cohomology of A_{•} with coefficients in M_{•} as

Der_{R_{•}}^{•}(A_{•}, M_{•}) = Ext_{Ã_{•}⊗Ã_{•}}^{•}(Ω_{A_{•}/R_{•}}^{•}, M_{•}).

As in the topological case there is a “universal derivation”

d : Ã_{•} → Ω_{A_{•}/R_{•}}^{•}

whose composition with the inclusion

Ω_{A_{•}/R_{•}}^{•} ↪ Ã_{•} ⊗ Ã_{•}

is the map 1 ∧ id − id ∧ 1. It gives, therefore, the forgetful map

l : Der_{R_{•}}^{•}(A_{•}, M_{•}) → Ext_{R_{•}}^{•}(A_{•}, M_{•})

Lemma 9.2 Consider the truncated Hochschild complex

\overline{C}^{ij}(A_{•}, M_{•}) = Hom_{R_{•}}^{i}(Ã_{•}^{⊗j−1}, M_{•})

with the same differentials as in C^{••}(A_{•}, M_{•}). Then the cohomology of \overline{C}^{••}(A_{•}, M_{•}) coincides with Der_{R_{•}}(A_{•}, M_{•}) and the forgetful map l is induced by the projection

\overline{C}^{••}(A_{•}, M_{•}) → \overline{C}^{•0}(A_{•}, M_{•}) = Hom_{R_{•}}^{•}(Ã_{•}, M_{•})

times (−1).

Proof. Considering the standard two-sided bar-resolution of Ã_{•}:

Ã_{•} ←^{m} Ã_{•} ⊗ Ã_{•} ← Ã_{•}^{⊗3} ← …

we see that the complex Ã_{•}^{⊗3} ← Ã_{•}^{⊗4} ← … maps quasiisomorphically onto the complex Ω_{A_{•}/R_{•}}^{•} = Ker(m). The universal derivation

d : Ã_{•} → Ω_{A_{•}/R_{•}}^{•}

lifts to the map

d̃: Ã_{•} → Ã_{•}^{⊗3};
d̃(a) = −1 ⊗ a ⊗ 1.

Now the statement of the lemma becomes clear after we apply the functor Hom_{Ã_{•}⊗Ã_{•}^{op}}(−, M_{•}) to the complex Ã_{•}^{⊗3} ← Ã_{•}^{⊗4} ← . . ..

We now return to the topological situation; as before A is an R-algebra, M is an A-bimodule; denote π_{•}(A) as A_{•}, π_{•}(R) as R_{•} and π_{•}(M) as M_{•}.

Proposition 9.3 Suppose that 1)the Kunneth spectral sequence for π_{•}(A ∧ A^{op}) collapses so we have

π_{•}(A ∧ A^{op}) = T or_{•}^{R_{•}}(A_{•}, A_{•});

2)the algebras A_{•} ⊗_{R_{•}}^{L} A_{•} and T or_{•}^{R_{•}}(A_{•}, A_{•}) are quasiisomorphic. Then there are the following spectral sequences:

1E_{2}^{ij} = Der_{R_{•}}^{ij}(A_{•}, M_{•}) ⟹ Der_{R}^{i+j}(A, M);
2E_{2}^{ij} = Ext_{R_{•}}^{ij}(A_{•}, M_{•}) ⟹ π_{i+j}F_{R}(A, M)

and a map

1E_{2}^{ij} → ^{2}E_{2}^{ij}

which at the level of E_{∞} terms gives the forgetful map

l : grDer_{R_{•}}(A, M) → gr[A, M]_{•}

Proof. There is a spectral sequence (cf. [7], Thm 4.1)

E_{2}^{•,•} = Ext_{π_{•}(A∧A^{op})}(π_{•}Ω_{A}, M_{•}) ⟹ π_{•}F_{A∧A^{op}}(Ω_{A}, M) = Der_{R}^{•}(A, M)

Our assumptions 1) and 2) ensure that

π_{•}(A ∧ A^{op}) = A_{•} ⊗_{R_{•}}^{L} A_{•}

and

π_{•}(Ω_{A}) = Ω_{A_{•}/R_{•}},

so we identify the E_{2}-term of our spectral sequence as Der_{R_{•}}^{••}(A_{•}, M_{•}).

The second spectral sequence is just the hypercohomology spectral sequence for calculating π_{•}F_{R}(A, M). Next from the naturality of the hypercohomology spectral sequences it follows that we have the following commutative diagram of spectral sequences:

Ext_{π_{•}(A∧A^{op})}^{•}(π_{•}Ω_{A}, M_{•}) → Ext_{R)_{•}}^{•}(π_{•}Ω_{A}, M_{•}) → Ext_{R_{•}}^{•}(A_{•}, M_{•})
∥ ∥ ∥
Der_{R_{•}}^{•}(A_{•}, M_{•})  → Ext_{R_{•}}^{•}(Ω_{A_{•}/R_{•}}, M_{•}) → Ext_{R_{•}}^{•}(A_{•}, M_{•})

(23)

and since the composite map

A_{•} → π_{•}Ω_{A} ≃ Ω_{A_{•}/R_{•}}

is just the universal derivation of the algebra A_{•}, we conclude that the lower row in (23) gives us the forgetful map

l : Der_{R_{•}}^{••}(A_{•}, M_{•}) → Ext_{R_{•}}^{•}(A_{•}, M_{•}).

Our proposition is proved.

10 Calculations with HZ/p

In this section we calculate the topological Hochschild cohomology of the Eilenberg-MacLane spectrum HZ/p which will be denoted (in this section only) by H considered as an algebra over the complex cobordism spectrum MU. Before starting any calculations we record the following result of a general nature. It really should have belonged to [7], but unfortunately, is not found there. The argument below was communicated to the author by M.Mandell.

Lemma 10.1 Let M and N are R-ring spectra. Then the spectral sequence

T or_{R_{•}}(M_{•}, N_{•}) ⟹ π_{•}(M ∧_{R} N)  (24)

is one of differential graded R_{•}-algebras.

Sketch proof. To deal with multiplicative structures we need a slightly different construction then the one given in [EKMM], IV, section 5. Let

… → F_{p} →^{d_{p}} F_{p−1} → … →^{ϵ} M_{•}

be an R_{•}-free resolution of M_{•}. Let Q_{0} = Ker ϵ and Q_{p} = ker d_{p}. For p ≥ 0 denote by F_{p} the wedge of sphere R-modules with π_{•}(F_{p}) = F_{p}. Now let M′_{0} = F_{0} and choose a map ϕ_{0} : M′_{0} → M that represents ϵ on the level of homotopy groups. Let Q_{0} be the homotopy fibre of ϕ_{0}. Then π_{•}(Q_{0}) = Q_{0}, so we can choose a map F_{1} → Q_{0} the given map on the homotopy groups. Take M′_{1} to be the cofibre of the composite map

F_{1} → Q_{0} → F_{0}.

Further the map ϕ_{0} canonically extends to a map

ϕ_{1} : M′_{1} → M.

Denote the homotopy fibre of this map by Q_{1}. Then

π_{•}(Σ^{−1}Q_{1}) = Q_{1}

so we can find a map Σ^{−1}F_{2} → Q_{1} realizing the given map on the homotopy groups etc.

Continuing in this way we construct a direct system

M′_{0} → M′_{1} → …  (25)

(Notice that we used the notation M′_{p} to distinguish it from M_{p} of [7].) The telescope M′ of this direct system is weakly equivalent to M and we can assume that all consecutive maps are inclusions of cell subcomplexes, i.e. we have a filtration on M′. Smashing this with N we get a filtration on M′ ∧_{R} N and the associated spectral sequence (24) converging to π_{•}(M ∧_{R} N).

Now recall that M (and therefore M′) is an R-ring spectrum. Smashing (25) with itself we get a filtration on M′ ∧_{R} M′:

… → ∨_{i+j=k}M′_{i} ∧ M′_{j} → ∨_{i+j=k+1}M′_{i} ∧ M′_{j} → .  (26)

Proceeding by induction we realize the multiplication map M′ ∧_{R} M′ → M′ as a map of filtered R-modules so that on the cofibres of (26) it agrees with the pairing of algebraic resolutions F_{•} ⊗_{R_{•}} F_{•} → F_{•}.

So, we constructed a collection of maps M′_{i} ∧_{R} M′_{j} → M′_{i+j}. Using these maps and the multiplication on N we construct the maps

M′_{i} ∧_{R} N ∧_{R} M′_{j} ∧_{R} N → M′_{i+j} ∧_{R} N.

This induces the required pairing of spectral sequences and we are done.

Theorem 10.2 (i)

THH_{MU}^{•}(H, H) = Z/pZ[y_{2}, y_{4}, …, y_{2n}, …];
Der_{MU}(H, H)^{•−1} = Z/pZ[y_{2}, y_{4}, …]/(Z/pZ).

(ii)

[H, H]_{MU}^{•} = Λ_{Z/pZ}(z_{1}, z_{3}, …, z_{2n+1}, …)

and the map

Der_{MU}(H, H)^{•} → [H, H]_{MU}^{•}

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

[H, H]_{MU}^{•} → [H, H]_{S}^{•} = A_{p}  (27)

the elements z_{2p^{i}−1} get mapped to the elements Q_{i} which form the set of primitive elements in the Steenrod algebra, and all other z’s get mapped to zero.

Lemma 10.3 Let HZ denote the integral Eilenberg-MacLane spectrum. Then

π_{•}(H ∧_{MU} HZ) = Λ(x_{3}, x_{5}, …, x_{2n+1}, …).

Proof. Consider first the spectrum HZ ∧_{MU} HZ.We have a strongly converging spectral sequence

T or^{MU^{•}}(Z, Z) = Λ(x_{3}, … x_{2n+1}, …) → π_{•}(H ∧_{MU} H).  (28)

To prove that it collapses we compare it to the spectral sequence

T or^{MU^{•}}(Z, HQ) = T or^{MU_{Q}^{•}}(HQ, HQ) = Λ_{Q}(x_{3}, … x_{2n+1}, …) → π_{•}(H∧_{MU}H)

(29)

Here HQ is the rational Eilenberg-MacLane spectrum and MU_{Q} is the rationalization of MU. We see that the integral spectral sequence injects into the the rational one and therefore it is enough to prove the collapse of the latter. This is an essentially trivial fact and can be seen as follows. The rational spectrum MU_{Q} is equivalent as a commutative S-algebra to the infinite smash product of S-algebras of the form HQ[x_{n}] where

HQ[x_{n}] = HQ ∨ Σ^{2n}HQ ∨ ….

Here x_{n} is a formal symbol of degree 2n which corresponds to the polynomial generator of MU in degree 2n. Then we have π_{•}HQ[x_{n}] = Q[x_{n}]. Further the spectral sequence

T or_{HQ[x_{n}]_{•}}(HQ, HQ) = Λ_{Q}(y_{n+1}) ⟹ π_{•}(H ∧_{HQ[x_{n}]} H)

collapses for dimensional reasons and it follows that (29) splits as an infinite product of these little spectral sequence and therefore itself collapses. So we showed that (28) collapses. The spectral sequence for computing π_{•}(H ∧_{MU} HZ) is just the reduction of (28) modulo p and the statement of our lemma follows.

Proof of Theorem 10.2. First compute π_{•}(H ∧_{MU} H). We have the following spectral sequence

T or^{MU^{•}}(Z/pZ, Z/pZ) = Λ_{Z/pZ}(x_{1}, x_{3}, … x_{2n+1}, …) ⟹ π_{•}(H ∧_{MU} H) (30)

We want to show that (30) collapses. To this end consider the following homotopy cofibre sequence of MU-modules:

H ∧_{MU} HZ →^{id∧p} H ∧_{MU} HZ → H ∧_{MU} H.

From this cofibre sequence and taking into account that π_{•}(H ∧_{MU} HZ) is p-torsion we see that

π_{•}(H ∧_{MU} H) = π_{•}(H ∧_{MU} HZ) ⊕ Σπ_{•}(H ∧_{MU} HZ).

Here Σ means shifting the grading by 1. It follows that (30) collapses (the permanent cycle x_{1} in (30) is accountable for the second direct summand in the last equality). So we conclude that

π_{•}(H ∧_{MU} H) = Λ_{Z/pZ}(x_{1}, x_{3}, …, x_{2n+1}, …).

Next consider the spectral sequence

Ext_{π_{•}(H∧_{MU} H)}^{••}(Z/pZ, Z/pZ) ⟹ THH_{MU}^{•}(H, H).

Its E_{2}-term is easily computed and equal to

Ext_{Λ_{Z/pZ}(x_{1},x_{3},...x_{2n+1},...)}(Z/pZ, Z/pZ) = Z/pZ[y_{2}, y_{4}, …, y_{2n} …].

Again, all higher differentials are zero this time because the spectral sequence is even. Now we know that our spectral sequence lies in the right half plane and converges conditionally. Since all higher differentials are zero Boardman’s theorem (cf. [4], Theorem 7.1) tells us that it converges strongly. All extension problems are clearly trivial and we conclude that

THH_{MU}^{•}(H, H) = Z/pZ[y_{2}, y_{4}, …, y_{2n} …].

The formula for Der_{MU}^{•}(H, H) follows from the homotopy cofibre sequence

THH_{MU}(H, H) → H → Der_{MU}(H, H)

and part (i) of our theorem is proved.

For part (ii) notice that H ∧_{MU} H is a left H-module and we have an equality

F_{H}(H ∧_{MU} H, H) = F_{MU}(H, H)

in the homotopy category of H-modules. Therefore [H, H]_{MU}^{•} is the Z/pZ dual of the cooperation algebra π_{•}(H ∧_{MU} H) and (ii) follows.

To compute the image of the map

Der_{MU}^{•}(H, H) → [H, H]_{MU}^{•}  (31)

we first look at the map

Der_{MU_{•}}(Z/pZ, Z/pZ) → Ext_{MU_{•}}(Z/pZ, Z/pZ)

and then apply Proposition 9.3.

It is an easy exercise in homological algebra to show the elements

y_{2i} ∈ HH_{MU_{•}}^{1,2i−1}(Z/pZ, Z/pZ) = Z/pZ[y_{2}, y_{4} …]

correspond to the elements

z_{2i+1} ∈ Ext_{Z[x_{1},x_{2}...]}^{1,2i}(Z/pZ, Z/pZ),

so Proposition 9.3 tells us that the map (31) is as stated at the level of E_{2}- terms. However, the image of (31) should be contained in the subgroup of primitive cohomology operations by Lemma 4.2, which is spanned by the elements z_{2i+1} ∈ [H, H]_{MU} and therefore no elements of higher bar filtration might appear in this image. Part (ii) is proved.

To prove part (iii) notice that the MU-Steenrod algebra as well as the usual Steenrod algebra are Hopf algebras and the map (27) respects this structure. Therefore the primitive elements z_{2i+1} should map to the primitive elements in the Steenrod algebra. Hence only the elements z_{2p^{i}−1} could have a nonzero image (namely, Q_{i}). We will show now that this is indeed the case. Consider the spectrum k(n), the nth connective Morava K-theory. According to [7], Chapter V, Section 4 k(n) is an MU-module. Therefore the Postnikov tower of k(n) can be constructed within the category of MU-modules, and in particular the first nontrivial k-invariant of k(n) (which is just Q_{n}) is given as a map of MU-modules H → H. In other words the cohomological operation Q_{n} can be lifted as a map of MU-modules and part (iii) is proved.

Remark 10.4 Recall that for the ordinary topological Hochschild cohomology of K we have (cf.[8],Theorem 7.3)

THH^{•}(H, H) = Sym(e_{0}, e_{1}, … e_{h} …)/ < (e_{h})^{p}, h ∈ N >

where e_{h} ∈ THH^{2p^{h}}(H, H). It follows that

Der_{MU}^{•−1}(H, H) = THH^{•}(H, H)/(Z/pZ).

Then under the map

l : Der_{MU}^{•}(H, H) → [H, H]_{S}^{•} = A_{p}  (32)

the elements e_{h} correspond to Q_{h} ∈ A_{p} and the rest get mapped to zero. Indeed, the Hochschild cohomology classes should give rise to primitive cohomology operations, therefore only elements e_{h} may have a nonzero image. But we know that the elements Q_{i} do lift to Hochschild cohomology classes (even in the category of MU-modules). Therefore the map (32) is as described.

Corollary 10.5 The mod p Moore spectrum M_{p} cannot be given a structure of an A_{∞} ring spectrum for any p.

Indeed, for p = 2 M_{p} is not even a ring spectrum up to homotopy. For an odd p the second nontrivial homotopy groupg of M_{p} is Z/pZ = π_{2p−3}(M_{p}) and therefore the first k-invariant of M_{p} is a cohomology operation of degree 2p − 2 and cannot be one of Q_{i}’s which is a cohomology operation of odd degree. (In fact, the first Postnikov k-invariant of M_{p} is P^{∞}). We see that already the second Postnikov stage of M_{p} cannot be A_{∞}.

11 Morava K-theories as MU-algebras

Theorem 11.1 For an odd prime p let L_{n} be the MU-module obtained by killing the regular ideal generated by (p, x_{1}, x_{2}, …, x̂_{n}, …) in the ring MU_{•} = Z[x_{1}, x_{2}, …]. Then L_{n} admits a MU-algebra structure (which extends the MU_{•}-algebra structure on L_{n•} = Z/pZ[x_{n}]).

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_{n}^{i} plus all the remaining polynomial generators and the prime p in the ring MU_{•}, so that

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

Lemma 11.2

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 spectral sequence

T or_{st}^{Z[x_{1},x_{2},...]}(Z/pZ[x_{n}]/x_{n}^{i}, Z/pZ[x_{n}]/x_{n}^{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_{1}, x_{2}, …] ⊗ Λ(z_{1}, z_{3}, …),

where

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

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

T or^{Z[x_{1},x_{2},...]}(Z/pZ[x_{n}]/x_{n}^{i}, Z/pZ[x_{n}]/x_{n}^{i} = A_{∗} ⊗_{Z[x_{1},x_{2},...]} Z/pZ[x_{n}]/x_{n}^{i+1}

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

To show that all higher differentials are zero we introduce MU-modules L̃_{n}^{(i)} with

π_{•}(L̃_{n}^{(i)}) = Z[x_{n}]/x_{n}^{i}.

The MU-modules L̃_{n}^{(i)} are obtained from MU similar to L_{n}^{(i)} but without killing the prime p. Then the rationalization arguments as in the proof of Lemma (10.3) with L̃_{n}^{(i)} in place of the integral Eilenberg-MacLane spectrum show that our spectral sequence collapses.

Clearly there is no room for no multiplicative extensions in the spectral sequence and we have

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

Next we have the following spectral sequence:

E_{2}^{st} = Ext_{Λ_{Z/pZ}(z_{1},z_{3},...)⊗Z/pZ[x_{n}]/x_{n}^{i}}^{st}(Z/pZ[x_{n}]/x_{n}^{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 and our lemma is proved.

Remark 11.3 We don’t claim that the isomorphism of Lemma 11.2 is actually multiplicative.It is likely to be the case, however the proof of this seems to require the fact that the Yoneda product in the Ext-spectral sequence is associated with the composition product. This is claimed in [7], IV, Proposition 4.4 . Unfortunately the proof of this fact given in the cited reference is incorrect.

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

Lemma 11.4

[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. Notice that both HZ/p and L_{n}^{(i)} ∧_{MU} L_{n}^{(i)} are naturally left L_{n}^{(i)}- modules. We have the following weak equivalence of MU-modules:

F_{L_{n}^{i}}(L_{n}^{(i)} ∧_{MU} L_{n}^{(i)}, HZ/p) = F_{MU}(L_{n}^{(i)}, HZ/p).

Now the statement follows from the computation of L_{n}^{(i)} ∧_{MU} L_{n}^{(i)} in Lemma 11.2.

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

Lemma 11.5

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) → ΣDer_{MU}(L_{n}^{(i)}, HZ/p)

from which the result follows.

Remark 11.6 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 11.7

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

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

Proof. This is analogous to the statement (ii) of Theorem 10.2 of 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)}  (33)

We want to prove that (33) 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_{1}, x_{2}, …, x_{n}^{2}, x_{n+1}, …)

Notice that (33) gives rise to the following (algebraic) singular extension of MU_{•} - algebras:

Σ^{2n}Z/pZ → X_{•} → Z/pZ  (34)

If (34) 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_{n}]/x_{n}^{2} → Z/pZ

which is what we need. Therefore, we have to eliminate the possibility for (34) to be trivial, i.e. split. Suppose, on the contrary, that this is the case. We will deduce from it that (33) is split in the homotopy category of MUmodules, which would be a contradiction by Lemma (11.7)

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 (33) 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_{1}, …, x_{n−1}, x_{n}^{i+1}, x_{n+1}, …).

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)}  (35)

Just as before, we show that (35) is nontrivial. But now the space of extensions like (35) 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_{1}, …, x_{n−1}, x_{n}^{i+1}, x_{n+1}, …)

and the other as

Y_{•} = MU_{•}/(p, x_{1}, …, x_{n−1}, x_{n}^{i}, x_{n+1}, …, 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 so on the level of E_{2}-terms of the corresponding spectral sequences. Furthermore, since f(\overline{y}_{2ni+2}) is 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  (36)

where

\overline{Y}_{•} = MU_{•}/(p, x_{1}, …, x_{ni}, x_{ni+1}^{2}, …).

Therefore the extension corresponding to \overline{y}_{2ni+2} is the extension induced from (36) by the augmentation map Z/pZ[x_{n}]/x_{n}^{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}. With this our theorem is proved.

Corollary 11.8 The spectrum k(n), the nth connective Morava K-theory admits a structure of an MU-algebra.

Remark 11.9 P. Goerss pointed out to the author that there are in fact uncountably many different MU-algebra structures on L_{n}. For example, when considering case 1) above we could have altered the element y_{2n+2} by adding to it any decomposable element of degree 2n + 2 in the graded algebra Z/pZ[y_{2}, y_{4}, …] = T HH_{MU}^{•}(L_{n}^{i}, HZ/p). In fact, to have a unique MUalgebra structure on L_{n} we should fix an A_{∞}-MU_{•}-algebra structure on L_{n•} which amounts to infinitely many conditions.

Remark 11.10 Let L be an MU-module obtained by killing an arbitrary subset of polynomial generators of MU_{•} and/or p. Then the exact same method as above and obvious induction show that L admits a structure of an MU-algebra.

By comparison the results of [7] and [18] give only structures of MUring spectra on various MU-modules, which is an up to homotopy notion. It should be noted, however, that in contrast with the cited papers we never address the question of commutativity of our products, up to homotopy or otherwise.

12 Topological Hochschild cohomology of Morava K-theories

Now we know that the spectra k(n) are MU-algebras (hence so are the nonconnective Morava K-theories K(n) as being Bousfield localizations of k(n)’s) 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. [15]). In this section we compute T HH_{MU}(K(n), K(n)) and T HH_{S}(K(n), K(n)) for odd primes modulo additive extensions. The possibly nontrivial additive extensions might arise because K(n) is not a commutative MU-algebra. We hope to return to this problem in a future paper. Apart from this complication the results are similar to the calculation of topological Hochschild cohomology of the Eilenberg-MacLane spectrum HZ/p in section 9.

In the next theorem gr(?) stands for the associated graded of a filtered module.

Proposition 12.1 (i)

grT HH_{MU}^{•}(K(n), K(n)) = K(n)_{•}[[y_{2}, y_{4}, … ŷ_{2p^{n}}, …]];
grDer_{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 (representatives of) elements y_{2i} ∈ grDer_{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 [2]) and all other z’s go to zero.

Proof. The proof is along the same lines as the proof of Theorem 10.2, and we will be somewhat sketchy therefore. First compute π_{•}(K(n) ∧_{MU} K(n)). We have the our usual T or- spectral sequence

T or^{Z[x_{1},x_{2},...]}(K(n)_{•}, K(n)_{•}) = Λ_{K(n)_{•}}(z_{1}, z_{3}, …, ẑ_{2p^{n}−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}, …, ẑ_{2p^{n}−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}^{••} = Ext_{Λ_{K(n)•}(z_{1},z_{3},...)}^{••}(K(n)_{•}, K(n)_{•}) = K(n)_{•}[[y_{2}, y_{4}, …]].

This spectral sequence collapses since it is even, and so part (i) of Proposition 12.1 is proved. Notice however that there could possibly 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.

Part (ii) is proved similarly to Theorem 10.2. An easy calculation shows that

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

where Λ̂(?) denotes the completed exterior algebra. 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 similarly to the corresponding statement in Theorem 10.2. The only nontrivial thing is to prove that the element z_{2p^{i}−1} ∈ [K(n), K(n)]_{MU}^{2p^{i}−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_{p^{i}−1} and x_{p^{i}}. Consider the following homotopy cofibre sequence of MU-modules:

Σ^{2p^{i}−2}MU(i, n) ⟶^{x_{p^{i}−1}} 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 proposition describes topological Hochschild cohomology of K(n) as an S-algebra.

Proposition 12.2 Let p be an odd prime. Then

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

where α_{k} ∈ T HH^{2p^{k}}(K(n), K(n)). Moreover under the forgetful map

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

the image of y_{2p^{i}} for i = 0, 1 …, n − 1 is α_{i} and the rest of y_{2i} get mapped to 0 (on the level of associated graded modules)

Proof. We have a spectral sequence

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

and the calculation of Robinson in [17] shows that

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

This spectral sequence converges strongly to its target and collapses at the E_{2}-term.

To see that the image of y_{2i} is as stated consider the forgetful map

π_{•}(K(n) ∧_{S} K(n) → π_{•}(K(n) ∧_{MU} K(n).

We know (from, e.g. [19]) that

π_{•}(K(n) ∧_{S} K(n) ≅ K(n)_{•}[t_{1}, t_{2}, …,]/(v_{n}t_{i}^{p^{n}} − v_{n}^{p^{i}}t_{i}) ⊗ Λ(a_{0}, a_{1}, … a_{n−1}).

where the degree of a_{i} is 2p^{i} − 1 and that of t_{i} is 2(p^{i} − 1). It follows that the exterior generators a_{i} correspond to z_{2p^{i}−1} ∈ π_{•}(K(n) ∧_{MU} K(n)). After that the statement about the image of y_{2i} follows from comparing spectral sequences for T HH_{S}^{•}(K(n), K(n)) and T HH_{MU}^{•}(K(n), K(n)).

Acknowledgement. The author would like to express his gratitude to Max-Planck-Institut fur Mathematik where large part of the work was completed. Thanks are also due to Nick Kuhn and P.Goerss for fruitful conversations and comments on the subject of the paper, to M.Mandell who provided the proof of Lemma 10.1 and the referees for a host of useful suggestions..

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