Contemporary Mathematics

Notes on the Hopkins-Miller Theorem

Charles Rezk

Abstract. We give an exposition of the proof of a theorem of Hopkins and Miller, that the spectra E_{n} admit an action of the Morava stabilizer group.

1. Introduction

These expository notes were originally written as an aid to the author while he was giving a course on the proof of the Hopkins-Miller theorem (Theorem 2.1) at Northwestern in Spring 1997. The aim was to describe the theorem, and give a sketch of the proof.

The aim was achieved, though only by glossing over certain points. In particular, several places rely on the existence of closed model category structures on various categories of spectra, and on subtle properties of this structure; I have not tried to deal with these issues. A few other points are left as references. At some points, the exposition is rather terse. Otherwise, the proof is as complete as I could make it.

The matter of the final part of these notes, on the construction of A_{∞}-ring structures, has to some extent been superseded by recent work of Hopkins and Goerss. They have developed more powerful obstruction machinery for constructing rigid algebraic structures on spectra, and are able to prove results about the existence of E_{∞}-structures as well as of A_{∞}-structures.

The material is based primarily on lectures and courses given by Mike Hopkins which I have attended at one time or another, and on discussions with Mike Hopkins and Haynes Miller. Also of great use were notes prepared by Lee Nave at a course given by Mike Hopkins at the University of Washington in the Spring of 1996.

All errors are mine, most of the true bits belong to others.

1.1. Organization. After giving a statement of the main result in Section 2, we outline Lubin-Tate’s theory of universal deformations of formal groups over fields of positive characteristic, and its relation to homotopy theory, in Sections 3–7. In Sections 8–14 we summarize what we need about the theory of spectra and A_{∞}-ring spectra, and set up an obstruction theory and a spectral sequence for calculating maps between A_{∞}-ring spectra. In Sections 15–21 we give the calculation of the obstruction groups and the E_{2}-term of the spectral sequence,

\begin{matrix}c \\ ◯ \end{matrix}0000 (copyright holder)

1991 Mathematics Subject Classification. Primary 55N22; Secondary 55P42, 55S99.

assuming that we have an A_{∞}-ring spectrum structure on the spectra of interest; the obstruction groups vanish and the spectral sequence collapses for algebraic reasons. Finally, in Sections 22–23 we outline an obstruction theory for constructing an A_{∞}-ring spectrum structure, and compute the obstruction groups by reducing to the calculations already carried out.

2. The theorem

In this section we state the main theorem, modulo definitions of some of the key concepts. (Formal group laws are defined in Section 3, universal deformations of formal group laws are defined in Section 4, and A_{∞}-ring spectra are defined in Section 12.)

Let FG denote the category having as objects pairs (k, Γ), where k is a perfect field of finite characteristic p, and Γ is a formal group of height n over k, and with morphisms α: (k_{1}, Γ_{1}) → (k_{2}, Γ_{2}) consisting of a pair (i, f), where i is a map i: k_{2} → k_{1} of rings and f is an isomorphism f: Γ_{1} → i^{∗}Γ_{2} of formal group laws.

Theorem 2.1 (Hopkins-Miller). There exists a functor (k, Γ) ↦ E_{k,Γ} from FG^{op} to the category of A_{∞}-ring spectra, such that

1. E_{k,Γ} is a commutative ring spectrum,

2. there exists a unit in π_{2}E_{Γ},

3. π_{odd}E_{k,Γ} = 0, from which it follows that E_{k,Γ} is complex orientable,

4. and such that the corresponding formal group law over π_{0}E_{Γ} is the universal deformation of (k, Γ).

In particular, when (k, Γ) = (𝔽_{p^{n}}, F_{n}) where F_{n} is the unique p-typical formal group law of height n with [p]_{Γ}(x) = x^{p^{n}}, then E_{k,Γ} is called E_{n}, and

π_{0}E_{n} = 𝕎𝔽_{p^{n}}[[u_{1}, …, u_{n−1}]].

The Hopkins-Miller theorem shows that this spectrum admits an action by the group of G automorphisms of (𝔽_{p^{n}}, F_{n}) “on the nose”. This automorphism group is an extension of the Morava stabilizer group S_{n} by the Galois group of 𝔽_{p^{n}} over 𝔽_{p}.

One application of the Hopkins-Miller theorem is the construction of “higher real K-theories”, which are produced by taking the homotopy fixed points of E_{n} by the action of a finite subgroup of G. The fixed points of E_{n} by a maximal finite subgroup of G are usually denoted EO_{n}.

We will return to Theorem 2.1 in Section 7.

Part 1. Universal deformations

In the following sections, we define universal deformations and describe the Lubin-Tate moduli space. We use this to construct a functor from an appropriate category of formal group laws to the category of generalized homology theories (Section 6). It is this functor which the Hopkins-Miller theorem “lifts” to the category of A_{∞}-ring spectra.

3. Formal group laws

We define the category of formal group laws, and discuss some basic properties. More details on formal group laws may be found in [7], [1, part 2], and [13], among other places.

3.1. Definition of formal group laws. Let R be a commutative ring. A formal group law F over R is a formal power series F (x, y) ∈ R[[x, y]] such that

(a) F (x, 0) = x = F (0, x),

(b) F (x, y) = F (y, x),

(c) F (F (x, y), z) = F (x, F (y, z)). By (a) and (b) such a power series F (x, y) necessarily has the form

F (x, y) = x + y + ∑_{i,j≥1} a_{ij}x^{i}y^{j},

where a_{ij} = a_{ji} ∈ R. Sometimes I’ll write x +_{F} y for F (x, y).

Example 3.2. The additive formal group law is defined by x +_{G_{a}} y = x + y.

Example 3.3. The multiplicative formal group law is defined by 1+(x +_{G_{m}} y) =

(1 + x)(1 + y).

Example 3.4. Elliptic formal group laws. (See, e.g., Silverman [17].)

Let B is a complete local R-algebra with maximal ideal 𝔪; that is, B ≃ lim B/𝔪^{k} and ⋂_{k} 𝔪^{k} = 0. A formal group law F on R defines an abelian group structure on 𝔪. In particular, it defines an abelian group structure on the ideal of R[[x]] generated by x. We write a+_{F}b for the sum of a, b ∈ 𝔪 in this group structure.

A morphism f: F → G of formal group laws over R is a formal power series f(x) ∈ R[[x]] such that f(0) = 0 and f(x +_{F} y) = f(x) +_{G} f(y). The isomorphisms

are precisely those f(x) ∈ R[[x]] for which f(x) = ax + ⋯ with a an invertible element of R. Such an isomorphism is called a strict isomorphism if a = 1.

Given a formal group law F over R and a power series f(x) ∈ R[[x]] with f(0) = 0 and with f′(0) a unit in R, we can define a new formal group law G by

G(x, y) = f (F (f^{−1}(x), f^{−1}(y))) ,

which comes with an isomorphism f: F → G of formal group laws.

I’ll write fgl(R) for the groupoid consisting of all formal group laws on R and all isomorphisms between them. A map ϕ: R → S of rings induces a functor ϕ^{∗}: fgl(R) → fgl(S) by sending

F (x, y) = ∑_{i,j} a_{ij}x^{i}y^{j}  to  ϕ^{∗}F (x, y) = ∑_{i,j} ϕ(a_{ij})x^{i}y^{j}.

For any formal group law, one can write the n-series

[n]_{F} (x) = x +_{F} ⋯ +_{F} x,

by taking the n-fold sum of x in the group law F. The n-series is in fact an endomorphism of the formal group law F. Since [n]_{F} (x) = nx + ⋯, it is an automorphism if n is invertible in R.

3.5. Formal group laws over fields of characteristic p. We now consider formal group laws over a field k of characteristic p.

Proposition 3.6. If f: F → G is a homomorphism of formal group laws over k, and if f(x) ≠ 0, then

f(x) = g(x^{p^{n}})

for some n ≥ 0 and some g(x) ∈ k[[x]] with g(0) = 0 and g′(0) ≠ 0.

For such a homomorphism f: F → G the largest such integer n is called the height of f (or if f(x) = 0, we say f has infinite height.) If [p]_{F} (x) has height n then we say that the formal group law F has height n.

Proof of Proposition 3.6. Since f(F (x, y)) = G(f(x), f(y)), by applying \frac{∂}{∂y} and evaluating at y = 0 we see that

f′(x)F_{2}(x, 0) = G_{2}(f(x), 0)f′(0).

Since F_{2}(x, 0) = 1 + (higher terms), it has a multiplicative inverse, so we see that if f′(0) = 0 then f′(x) = 0, whence f(x) = g(x^{p}) for some power series g(x). The proof can be completed by induction if we can show that g(x) is itself a morphism of formal group laws.

Let σ: k → k denote the homomorphism x ↦ x^{p}. Then one can check that

F ⟶^{x^{p}} σ^{∗}F ⟶^{g} G

is a diagram of formal group laws and homomorphisms. In fact,

g(σ^{∗}F (x^{p}, y^{p})) = g(F (x, y)^{p}) = f(F (x, y)) = G(f(x), f(y)) = G(g(x^{p}), g(y^{p})),

whence

g(σ^{∗}F (x, y)) = G(g(x), g(y)),

as desired.

Example 3.7. The additive formal group law has infinite height, since the pseries is [p]_{G_{a}}(x) = px ≡ 0. The multiplicative formal group law has height 1, since [p]_{G_{m}}(x) = (1 + x)^{p} − 1 ≡ 1 + x^{p} − 1 = x^{p}. An elliptic formal group law has height 1 or height 2. (Silverman [17, Ch. 7].)

Example 3.8. Over k = 𝔽_{p} there is a (p-typical) formal group law F_{n} (the Honda formal group law) with [p]_{F_{n}}(x) = x^{p^{n}}. (Ravenel [13, App. 2].)

4. The universal deformations

We define deformations and the deformation category, and introduce the universal deformations.

4.1. Deformations of a formal group law. Let (k, Γ) be a pair consisting of a field k of characteristic p > 0 and a formal group Γ over k. A deformation of (k, Γ) to a complete local ring B (with maximal ideal 𝔪 = 𝔪_{B} and projection π: B → B/𝔪) is a pair (G, i) consisting of a formal group law G over B and a homomorphism i: k → B/𝔪_{B}, such that i^{∗}Γ = π^{∗}G.

A morphism of deformations (G_{1}, i_{1}) → (G_{2}, i_{2}) is defined only when i_{1} = i_{2}, in which case it consists of an isomorphism f: G_{1} → G_{2} of formal group laws over B such that π^{∗}f is the identity map of π^{∗}G_{1} = π^{∗}G_{2} = i_{1}^{∗}Γ = i_{2}^{∗}Γ. That is,

f(x) ≡ x mod 𝔪_{B}.

Such an isomorphism f is sometimes called a ⋆-isomorphism.

The category of deformations of Γ over B is denoted Def_{Γ}(B). It’s nice to think of this category using the following picture.

\xymatrix{Def_{Γ}(B) \ar[rr] \ar[d] &  & fgl(B) \ar[d] \\ ∐_{i: k→B/𝔪}\{Γ\} \ar[r] & ∐_{i: k→B/𝔪}fgl(k) \ar[r] & fgl(B/𝔪)}

This is a diagram of groupoids. The lower left-hand corner denotes the discrete groupoid, having as objects the set of maps i: k → B/𝔪, and as maps only identity maps. This groupoid sits inside a coproduct of copies of the groupoid fgl(k); the inclusion maps the “object” i to the pair (Γ, i), where G is a formal group law over k and i: k → B/𝔪 a homomorphism. The lower right-hand functor maps such objects to fgl(B/𝔪) by sending (G, i) to i^{∗}G. The diagram commutes, and is a pull-back square of groupoids.

Remark 4.2. Given a morphism f: C → D of groupoids, the simplicial nerve of f is a Kan fibration of simplicial sets if and only if f has the following property: for each object c ∈ ob C and morphism β: fc → d in D there exists an α: c → d′ in C such that fα = β.

Thus, if I apply the classifying space functor to the above diagram, the square remains a pull-back square; furthermore, the right-hand vertical arrow becomes a fibration. Thus, the diagram induces a homotopy pull-back square of classifying spaces.

4.3. The Lubin-Tate theorem. Given j: k → B/𝔪, let Def_{Γ}(B)_{j} denote the full subgroupoid of Def_{Γ}(B) consisting of those deformations (G, i) for which i = j.

I will always feel free to replace a groupoid by its classifying space, and thus will speak of π_{0} and π_{1} of a groupoid, meaning the corresponding homotopy groups of its classifying space. Of course, if C is a groupoid then π_{0}C is just the set of isomorphism classes in C, and π_{1}(C, X) (the fundamental group based at an object X) is the group of automorphisms of the object X in C.

Theorem 4.4 (Lubin-Tate). [10] If Γ is a formal group law of height n < ∞ over k, and if D = (G, i) is some deformation of (k, Γ), then

π_{1}(Def_{Γ}(B)_{i}, D) = {1},

and there is a bijection

π_{0}(Def_{Γ}(B)_{i}) ≃ 𝔪^{×(n−1)}.

Furthermore, there exists a complete local ring E(k, Γ), an isomorphism i: k ⟶^{∼} E(k, Γ)/𝔪, and a formal group law F on E(k, Γ) such that the pair (F, id) is a universal deformation, in the sense that the functor B ↦ π_{0} Def_{Γ}(B) is corepresented by the ring E(k, Γ), so that a map ϕ: E(k, Γ) → B corresponds to the isomorphism class of ϕ^{∗}F in Def_{Γ}(B).

An outline of the proof of Theorem 4.4 is given in Section 5.

Let me make clear what it means for (F, id) to be a universal deformation. Given a deformation (G, i) of (k, Γ) to B, we have the following diagrams of rings and formal group laws.

[figure: E(k, Γ); π; B; π; F; G; k; i; B/𝔪; Γ; i^{∗}Γ; π^{∗}G]

(The squiggly arrows are meant to represent the action of a functor on an object.) For (F, id) to be a universal deformation, there must exist a unique map ϕ: E(k, Γ) → B such that the object (ϕ^{∗}F, i) is isomorphic to (G, i) in the category Def_{Γ}(B). Then there exists a (necessarily unique) ⋆-isomorphism g: G → ϕ^{∗}F. In other words, you get a unique pair of diagrams

[figure: E(k, Γ); π; ϕ; B; π; F; ϕ^{∗}F; G; g; ∼; k; i; B/𝔪; Γ; i^{∗}Γ; π^{∗}G]

where g(x) ≡ x mod 𝔪_{B}.

The first part of Theorem 4.4 says that fgl(B) → fgl(B/𝔪) is a “covering space up to homotopy”. Thus, an element in π_{1} of the base “space” fgl(B/𝔪) induces an action on π_{0} of the “fiber” Def_{Γ}(B)_{i}. In particular, given γ ∈ π_{1}(fgl(k), Γ) ≃ Aut(Γ), we obtain an induced map γ̃: π_{0} Def_{Γ}(B)_{i} → π_{0} Def_{Γ}(B)_{i} for each i: k → B/𝔪. Putting these together over all the i’s we get a map γ̃: π_{0} Def_{Γ}(B) → π_{0} Def_{Γ}(B). It’s easy to see that this action is natural in B; in other words, there is an induced action of Aut(Γ) on the functor π_{0} Def_{Γ}(−), and hence on the ring E(k, Γ) which represents the functor.

If k = 𝔽_{p^{n}}, and Γ = F_{n} is the Honda formal group law of height n, then Aut(F_{n}) contains the n-th Morava stabilizer group S_{n} as the group of strict automorphisms of F_{n}.

We can sometimes get a bigger group action on E(k, Γ). Suppose k is a finite field, and fix a map i_{0} : k → B/𝔪. Then maps i: k → B/𝔪 can be factored i = i_{0}σ^{d}, where σ: k → k denotes σ(x) = x^{p}. If Γ is Galois invariant, i.e., if σ^{∗}Γ = Γ (for example, if Γ is the Honda formal group law), then there is an action of Gal(𝔽_{p^{n}}/𝔽_{p}) = C_{n} on Def_{Γ}(B), and hence on E(k, Γ).

4.5. The ring E(k, Γ). We can say something about the ring E(k, Γ), at least if k is a perfect field. Since it is a complete local ring with residue field k, it must contain the Witt vectors 𝕎k. For k = 𝔽_{p^{n}}, 𝕎k = ℤ_{p}[ζ], where ζ is a primitive (p^{n} − 1)st root of unity. For a perfect field k the Witt ring 𝕎k has the following universal property; given a diagram

\xymatrix{𝕎k \ar[d] \ar@{.>}[r] & B \ar[d] \\ k \ar[r] & B/𝔪}

where B is a complete local ring and all maps are continuous homomorphisms, there is a unique dotted arrow making the diagram commute. (This is essentially [16, II.5].) This is easy to see when k = 𝔽_{p^{n}}, since in this case the Teichmüller construction ensures that one can produce a canonical multiplicative section k → B.

Thus E(k, Γ) is a 𝕎k-algebra, and in fact contains 𝕎k (because there exists at least one deformation of Γ to 𝕎k, so that there exists a map from E(k, Γ) back down to 𝕎k.) Since a map i: k → B/𝔪 determines a unique map 𝕎k → B, making B also into a 𝕎k-algebra, then the isomorphism of Theorem 4.4

Hom_{𝕎k- alg}(E(k, Γ), B) ≃ π_{0} Def_{Γ}(B)_{i} ≃ 𝔪^{×(n−1)}

indicates that we should set

E(k, Γ) ≃ 𝕎𝔽_{p^{n}}[[u_{1}, …, u_{n−1}]].

4.6. Functoriality of the universal deformation. Consider a morphism α: (k_{1}, Γ_{1}) → (k_{2}, Γ_{2}) in the category FG; α is a pair (j, f) with j: k_{2} → k_{1} and f: Γ_{1} ⟶^{∼} j^{∗}Γ_{2}. Such a morphism induces a diagram

\xymatrix{∐_{k_{1}→B/𝔪}\{Γ_{1}\} \ar@{-->}[dr] \ar[dd] \\  & fgl(B/𝔪) \\ ∐_{k_{2}→B/𝔪}\{Γ_{2}\} \ar@{-->}[ur]}

which commutes up to a homotopy (determined by f). This induces a map

π_{0} Def_{Γ_{1}}(B) → π_{0} Def_{Γ_{2}}(B)

which is natural in B, and hence a map E(k_{2}, Γ_{2}) → E(k_{1}, Γ_{1}). Thus we obtain, according to Theorem 4.4, a functor

FG^{op} → {complete local rings}

sending

(k, Γ) ↦ E(k, Γ).

5. Construction of the universal deformation

I give a construction of the universal deformation, and at least indicate how the proof goes. I mainly follow Lubin-Tate’s original proof [10], though I also borrow an observation of Drinfel’d [4].

5.1. Deformations to a trivial square 0 extension over k. I want to give at least an idea of how the proof goes. So what I’ll do first is compute Def_{Γ}(k[ϵ]/ϵ^{2}).

Let R = k[ϵ]/ϵ^{2}. The inclusion k → R allows us to push a formal group law Γ on k to R; by abuse of notation, I will also refer to the pushed-forward-to-R formal group law as Γ. An arbitrary formal group law on R which lifts Γ must look something like

F (x, y) = x +_{Γ} y +_{Γ} ϵG(x, y)

where G(x, y) ∈ k[[x, y]]. For F to really be a formal group law, we must also have (a) G(x, 0) = 0 = G(0, y),

(b) G(x, y) = G(y, x),

(c) G(x +_{Γ} y, z) + G(x, y) = G(x, y +_{Γ} z) + G(y, z).

Such a G is called a Γ-symmetric 2-cocycle. I’ll write Z_{sym}^{2}(Γ) for the set of such cocycles.

Now suppose I have another formal group law F′(x, y) = x +_{Γ} y +_{Γ} ϵG′(x, y), and

a map f: F → F′ which reduces to the identity map mod ϵ. Such an f has the form,

f(x) = x +_{Γ} ϵg(x)

for some g(x) ∈ k[[x]] with g(0) = 0. We shall call such a g a Γ-symmetric 1-cocycle, and write Z_{sym}^{1}(Γ) for the set of all such cocycles.

Then f(x +_{F} y) = f(x) +_{F′} f(y) if and only if

G(x, y) + g(x +_{Γ} y) = G′(x, y) + g(x) + g(y).

This means that we can form a little chain complex:

\xymatrix{0 \ar[r] & \{ \begin{matrix}g(x) ∈ k[[x]], \\ g(0) = 0 \end{matrix} \} \ar[r]^{δ} & \{ \begin{matrix}G(x, y) ∈ k[[x, y]], \\ G a Γ-symmetric 2-cocycle \end{matrix} \} \ar[r] & 0}

where δg(x, y) = g(x) + g(y) − g(x +_{Γ} y). The homology of this chain complex is

precisely the homotopy of Def_{Γ}(R)_{id}. That is, the cokernel of δ may be identified with the set of isomorphism classes in Def_{Γ}(R)_{id}, and the kernel can be identified as the group of automorphisms of any chosen object of Def_{Γ}(R)_{id}.

5.2. Lemmas about formal group laws. To compute this complex, we need a couple of lemmas. First, note that an additive symmetric cocycle is a symmetric cocycle for the additive group.

Lemma 5.3. An additive symmetric cocycle G(x,y), homogeneous of degree d, has the form aC_{d}(x, y) for some a ∈ k, and where C_{d}(x, y) ∈ ℤ[[x, y]] is defined by

C_{d}(x, y) = \frac{1}{λ_{d}} (x^{d} + y^{d} − (x + y)^{d})

with

λ_{d} = \begin{cases}1 & if d ≠ p^{j} \\ p & if d = p^{j} \end{cases} .

Proof. See Fröhlich [7]. The proof is also given in [1, part 2].

Lemma 5.4. Let π: R → S be a surjective map of rings. If F (x, y) ∈ R[[x, y]] satisfies the axioms for a formal group law modulo terms of degree (d+1), and if π^{∗}F is a formal group law over S, then there exists a formal group law F′(x, y) ∈ R[[x, y]] such that F′ ≡ F mod degree (d + 1) terms and such that π^{∗}F′ = π^{∗}F.

Lemma 5.4 really follows from the theorem of Lazard.

Theorem 5.5 (Lazard). There exists a ring L ≃ ℤ[s_{1}, s_{2}, …] and a formal group law F over L such that the induced map

Hom_{Ring}(L, R) → ob fgl(R)

sending f ↦ f^{∗}F is a bijection. Furthermore, if we write

x +_{F} y = x + y + ∑_{i,j} a_{ij}x^{i}y^{j}

then a_{ij} ∈ L is a polynomial in the variables s_{1}, …, s_{i+j−1}.

Proof. See [7], [1, part 2] or [13, Appendix 2].

Lemma 5.3 shows that any Γ-symmetric 2-cocycle has the form

G_{d}(x, y) = aC_{d}(x, y) + higher degree terms,

since the lowest degree term of a Γ-symmetric 2-cocycle must necessarily be an additive cocycle. Lemma 5.4 shows that such a Γ-symmetric 2-cocycle exists, since the expression

F (x, y) = x +_{Γ} y +_{Γ} ϵC_{d}(x, y)

is a formal group law modulo degree (d + 1), and modulo 𝔪 is the formal group law Γ.

There is an obvious filtration ⋯ ⊃ F_{d}Z_{sym}^{2}(Γ) ⊃ F_{d+1}Z_{sym}^{2}(Γ) ⊃ ⋯ of the symmetric 2-cocycles in which F_{d}Z_{sym}^{2}(Γ) consists of all elements whose lowest degree term is at least d. Thus we have shown that the associated quotients of this complete filtration F_{d}Z_{sym}^{2}/F_{d+1}Z_{sym}^{2} are each isomorphic to k, generated by an element G_{d}.

Similarly, there is a filtration ⋯ ⊃ F_{d}Z_{sym}^{1}(Γ) ⊃ F_{d+1}Z_{sym}^{1}(Γ) ⊃ ⋯ where F_{d}Z_{sym}^{1}(Γ) consists of all elements whose lowest degree term is at least d. Thus each associated quotient F_{d}Z_{sym}^{1}/F_{d+1}Z_{sym}^{1} is isomorphic to k, generated by an element f_{d}(x) = x^{d}. Thus Z_{sym}^{•} is a filtered chain complex, and all we need to do is calculate the associated spectral sequence!

Let’s assume that

Γ(x, y) = x + y + bC_{p^{n}}(x, y) + · · · .

We can do this without loss of generality, by the following lemma, and by the fact that isomorphic formal group laws have isomorphic complexes Z_{sym}^{•} of symmetric cocycles.

Lemma 5.6. A height n formal group law Γ over k is isomorphic to a formal group law Γ′ having the form

Γ′(x, y) = x + y + bC_{p^{n}}(x, y) + ⋯,

with b ≠ 0.

Proof. This is because if Γ has the form

Γ(x, y) ≡ x + y + bC_{k}(x, y) + (x, y)^{k+1}

for k ≠ p^{j}, we can let f(x) = x + bx^{k} and Γ′(x, y) = f(Γ(f^{−1}x, f^{−1}y)), whence f: Γ → Γ′ is an isomorphism of formal group laws, and

Γ′(x, y) ≡ x + y + (x, y)^{k+1}.

If Γ has the form

Γ(x, y) = x + y + bC_{p^{j}}(x, y) + (x, y)^{p^{j}+1}

one can show by Lemma 5.7 that [p]_{Γ}(x) = bx^{p^{j}} + ⋯, hence b = 0 or j = n.

Lemma 5.7. If F is a formal group law with

x +_{F} y = x + y + bC_{k}(x, y) + ⋯

then

[n]_{F} (x) = nx + b\{\frac{1}{λ_{k}} (n − n^{k})\}x^{k} + · · · .

5.8. Computation of the chain complex. Let f_{k}(x) = x^{k}. Then

(δf_{k})(x, y) = x^{k} + y^{k} − (x +_{Γ} y)^{k}
= \begin{cases}C_{k}(x, y) + ⋯ & if k ≠ p^{j}, where j ≥ 0, \\ b^{p^{j}}C_{p^{n}}(x, y)^{p^{j}} + ⋯ & if k = p^{j}, where j ≥ 0. \end{cases}

Since C_{p^{n}}(x, y) = C_{p}(x, y)^{p^{n−1}}, we see that

δf_{k} = \begin{cases}G_{k} mod F_{k+1}Z_{sym}^{2} & if k ≠ p^{j}, \\ G_{p^{n+j}} mod F_{p^{n+j}+1}Z_{sym}^{2} & if k = p^{j}. \end{cases}

Thus the cokernel of δ has dimension (n − 1) over k, generated by elements of the form

G′_{p^{k}}(x, y) = C_{p^{k}}(x, y) + ⋯

for k = 1, …, n − 1. The kernel of δ is trivial. Hence

π_{0} Def_{Γ}(R)_{i} = H_{sym}^{2}(Γ) ≃ k^{×(n−1)},

and

π_{1} Def_{Γ}(R) ≃ 0.

5.9. The case of a general square-zero extension. The above argument proves the first part of the Lubin-Tate theorem for B = k[ϵ]/ϵ^{2}. The method of proof extends to prove a more general statement.

Let B be a complete local ring B with maximal ideal 𝔪 and residue field k = B/𝔪. Let F_{k} be a formal group law over B/𝔪^{k} and let Γ denote its pushforward to k. Suppose F_{k+1} is a formal group law over B/𝔪^{k+1} which lifts F_{k}. Then any other formal group law F′_{k+1} over B/𝔪^{k+1} which lifts F_{k} has the form

F′_{k+1}(x, y) = F_{k}(x, y) +_{F_{k+1}} G(x, y)

where G(x, y) ∈ Z_{sym}^{2}(Γ) ˆ⊗𝔪^{k}/𝔪^{k+1}. Furthermore, an isomorphism f: F_{k+1} → F′_{k+1} which lifts the identity map of F_{k} has the form

f(x) = x +_{F_{k+1}} g(x)

where g(x) ∈ Z_{sym}^{1}(Γ) ˆ⊗𝔪^{k}/𝔪^{k+1}.

Thus the “fiber” of fgl(B/𝔪^{k+1}) → fgl(B/𝔪^{k}) over F_{k} has H_{sym}^{2}(Γ)⊗𝔪^{k}/𝔪^{k+1} as its set of components, and H_{sym}^{1}(Γ) ⊗ 𝔪^{k}/𝔪^{k+1} as the set of isomorphisms of any object.

The above can be proved most easily by pulling back the map π: B/𝔪^{k+1} → B/𝔪^{k} along itself; the pulled-back map is a trivial square-0 extension, and the argument of the previous sections can be applied to it.

5.10. Construction of the universal deformation. To finish the proof, we need to construct the universal deformation. This will be a formal group law over the ring

E(k, Γ) = 𝕎k[[u_{1}, …, u_{n−1}]].

It turns out that it suffices to produce a formal group law over this ring which has the form

F (x, y) ≡ x +_{Γ} y +_{Γ} u_{1}C_{p}(x, y) +_{Γ} ⋯ +_{Γ} u_{n−1}C_{p^{n−1}}(x, y) +_{Γ} ⋯

modulo p and modulo (u_{1}, …, u_{n−1})^{2}. It is easy to construct such: we can define a formal group law over k[[u_{1}, …, u_{n−1}]]/(u_{1}, …, u_{n−1})^{2} by

x +_{F} y = x +_{Γ} y +_{Γ} u_{1}G_{p}(x, y) +_{Γ} ⋯ +_{Γ} u_{n−1}G_{p^{n−1}}(x, y),

where G_{p^{j}}(x, y) ∈ Z_{sym}^{2}(Γ) are the symmetric cocycles we constructed above. This group law then lifts up to 𝕎k[[u_{1}, …, u_{n−1}]], by Lazard’s Theorem 5.5.

The proof is completed by considering the map

Hom_{𝕎k- alg}(E(k, Γ), B) → π_{0} Def_{Γ}(B)_{i}

which sends f to the isomorphism class of f^{∗}F, where F is the above constructed formal group law. The left hand side is isomorphic as a set to 𝔪^{×(n−1)}, which has a natural filtration by powers of 𝔪. The right hand side also has a filtration via the kernels of the maps π_{0} Def_{Γ}(B)_{i} → π_{0} Def_{Γ}(B/𝔪^{k})_{i}, and one shows the map is an isomorphism by comparing filtration quotients; of course, if B = k[ϵ]/ϵ^{2}, then it is clearly an isomorphism by what we have shown.

Example 5.11. Let n = 1 and k = 𝔽_{p}, and consider the multiplicative formal group law G_{m}(x, y) = x + y + xy over 𝔽_{p}. Then

E(𝔽_{p}, G_{m}) ≃ ℤ_{p}

and we can take as the universal deformation F (x, y) = x + y + xy.

6. The homology theories E_{k,Γ}

In this section we use universal deformation theory to construct homology theory E_{k,Γ} for each formal group law (k, Γ). In fact, we construct a functor E from the category FG of formal group laws to the category of homology theories.

6.1. Complex orientations and formal group laws. A reference for material in this section is from [1, part 2].

A ring spectrum E is complex orientable if there exists a cohomology class x ∈ Ẽ^{2}ℂℙ^{∞} such that the restriction of this class to ℂℙ^{1} ≃ S^{2} is the generator of Ẽ^{2}S^{2} ≃ π_{0}E. Such a class determines a characteristic class c_{1}(L) ∈ E^{2}(X) for a complex line bundle L over X.

For a complex oriented cohomology theory, E^{∗}ℂℙ^{∞} ≃ E^{∗}[[x]]. Thus the orientation determines a formal group law F, induced by the map ℂℙ^{∞} × ℂℙ^{∞} → ℂℙ^{∞} which corresponds to tensor product of line bundles. Thus

c_{1}(L ⊗ L′) = c_{1}(L) +_{F} c_{1}(L′).

Note that this formal group law is of the form

F (x, y) = x + y + ∑_{ij} a_{ij}x^{i}y^{j}

for a_{ij} ∈ π_{2(i+j−1)}E. I’ll call this a degree −2 formal group law, since the equation defining F (x, y) is in degree −2. (That is, it’s an element of π_{−2}F(ℂℙ^{∞}, E), where F(X, Y) denotes the function spectrum from X to Y.)

For example, the complex cobordism spectrum MU is complex orientable. The two inclusions of η_{L}, η_{R}: MU ⇉ MU ∧ MU induce two formal group laws over MU_{∗}MU and a canonical strict isomorphism between them.

Theorem 6.2 (Quillen). The canonical formal group law F over MU_{∗} is the universal formal group law (of degree −2) in the sense that, given a formal group law G of degree −2 over a graded ring R_{∗}, there exists a unique map ϕ: MU_{∗} → R_{∗} such that ϕ^{∗}F = G.

Furthermore, the canonical isomorphism f: η_{R}^{∗}F → η_{L}^{∗}F over MU_{∗}MU is the universal strict isomorphism in the sense that, given a strict isomorphism g: G → G′ of formal group laws over R_{∗}, there exists a unique map ψ: MU_{∗}MU → R_{∗} such that ψ^{∗}f = g.

Sometimes we want to consider a cohomology theory E which is periodic; e.g., which has an element u ∈ π_{2}E which is invertible. Such an element will allow us to switch between degree −2 formal group laws and degree 0 ones (a degree 0 formal group law is one whose coefficients lie in degree 0).

Let R_{∗} be a graded ring, such that R_{2} contains an invertible element. Given a formal group law F defined over R_{0} and a chosen invertible element u_{F} ∈ R_{2}, we produce a degree −2 formal group law as follows: define

F̄(x, y) = u_{F}^{−1}F (u_{F} x, u_{F} y).

Thus if F (x, y) = ∑ a_{ij}x^{i}y^{j}, then F̄(x, y) = ∑ a_{ij}u_{F}^{i+j−1}x^{i}y^{j} is a degree −2 formal group law.

Given a morphism g: F → G of formal group laws over R_{0}, and chosen elements u_{F} , u_{G} ∈ R_{2}, we of course get corresponding formal group laws F̄ and Ḡ of degree −2; let

ḡ(x) = u_{G}^{−1}g(u_{F} x).

Then ḡ: F̄ → Ḡ is a morphism of degree −2 formal group laws. Note that ḡ′(0) = u_{G}^{−1}u_{F} g′(0).

6.3. The Landweber exact functor theorem. Given (k, Γ) ∈ FG we define a functor E_{k,Γ} on spaces by

(E_{k,Γ})_{∗}(X) ≃ E(k, Γ)[u^{±}] ⊗_{MU_{∗}}^{F̄} MU_{∗}(X).

The “F̄” sitting on top of the tensor product symbol is meant to symbolize the fact that the tensor product is induced by the map MU_{∗} → E(k, Γ)[u^{±}] which classifies F̄, which is the degree −2 formal group law induced from F using u_{F} = u. That this is a homology theory comes from the Landweber exact functor theorem, as we will show below.

Theorem 6.4 (Landweber). Let R be a MU_{∗} module. Suppose for each prime p and each n the map

R/p, …, v_{n−1} ⟶^{·v_{n}} R/p, …, v_{n−1}

is injective, where v_{n} is the coefficient of x^{p^{n}} in the p-series of the universal formal group law over MU_{∗}. (That is, p, v_{1}, v_{2}, … is a regular sequence for R.) Then the functor

X ↦ R_{∗} ⊗_{MU_{∗}} MU_{∗}(X)

is a homology theory.

Proof. Proved in [8].

Proposition 6.5. The map F̄: MU_{∗} → E(k, Γ)[u^{±}] which classifies a degree −2 universal deformation satisfies the hypotheses of the Landweber exact functor theorem.

We will return to the proof of Proposition 6.5 in section 6.7.

Remark 6.6.

1. If R_{∗} is a ring with a degree −2 formal group law F̄, we use the classifying map MU_{∗} → R_{∗} to produce the module structure on R_{∗}.

2. It turns out that if we have two degree −2 formal group laws F̄ and Ḡ over R_{∗} which satisfy the Landweber condition, and are strictly isomorphic by a strict isomorphism ḡ, there is an induced equivalence of homology theories, via a map

R_{∗} ⊗_{MU_{∗}}^{F̄} MU_{∗}(X) → R_{∗} ⊗_{MU_{∗}}^{F̄} MU_{∗}MU ⊗_{MU_{∗}} MU_{∗}(X) → R_{∗} ⊗_{MU_{∗}}^{Ḡ} MU_{∗}(X)

where the left hand map is induced by the comultiplication

MU_{∗} → MU_{∗}MU ⊗_{MU_{∗}} MU_{∗}X,

and the right hand map is induced by the map MU_{∗}MU → R_{∗} which classifies ḡ.

6.7. Construction of the functor. We define a functor

E: FG^{op} → {homology theories},

as follows. For each object (k, Γ) in FG, choose a universal deformation F of (k, Γ) to the ring E(k, Γ), and let E_{k,Γ} be the homology theory constructed above using the Landweber theorem. Let E(k, Γ)_{∗} = E(k, Γ)[u_{F}^{±}] denote the corresponding graded ring

Suppose we have a map α: (k_{1}, Γ_{1}) → (k_{2}, Γ_{2}) in FG; such an α consists of maps i: k_{2} → k_{1} and f: Γ_{1} → i^{∗}Γ_{2}. Given universal deformations F_{1} of (k_{1}, Γ_{1}) and F_{2} of (k_{2}, Γ_{2}), we get an induced map

ψ: E(k_{2}, Γ_{2}) → E(k_{1}, Γ_{1})

and an isomorphism

g: F_{2} → ψ^{∗}F_{1}

such that g ≡ f mod 𝔪. That is, (ψ, g) is the unique pair of maps fitting into the diagrams

[figure: E(k_{2}, Γ_{2}); π; ψ; E(k_{1}, Γ_{1}); π; F_{2}; ψ^{∗}F_{2}; F_{1}; g; ∼; k_{2}; i; k_{1}; Γ_{2}; i^{∗}Γ_{2}; Γ_{1}; f; ∼]

We extend ψ to a map E(k_{2}, Γ_{2})[u_{F_{2}}^{±}] → E(k_{1}, Γ_{1})[u_{F_{1}}^{±}] by sending

ψ(u_{F_{2}}) = g′(0)u_{F_{1}}.

The point here is that we have a diagram

\xymatrix{ & E(k_{2}, Γ_{2})_{∗} \ar[dd]^{ψ} \\ MU_{∗} \ar[ur]^{F̄_{2}} \ar[dr]_{F̄_{1}} \\  & E(k_{1}, Γ_{1})_{∗}}

which does not commute, since ψ^{∗}F̄_{2} ≠ F̄_{1}. Instead, we may define E_{α}: E_{k_{2},Γ_{2}} → E_{k_{1},Γ_{1}} to be the natural isomorphism of homology theories defined via the composite

E(k_{2}, Γ_{2})_{∗} ⊗_{MU_{∗}}^{F̄_{2}} MU_{∗}(X) → E(k_{1}, Γ_{1})_{∗} ⊗_{MU_{∗}}^{ψ^{∗}F̄_{2}} MU_{∗}(X) ⟵^{∼} E(k_{1}, Γ_{1})_{∗} ⊗_{MU_{∗}}^{F̄_{1}} MU_{∗}(X)

where the left hand arrow is induced by ψ, and the right hand arrow is induced by the strict isomorphism ḡ: F̄_{1} → ψ^{∗}F̄_{2} defined by

ḡ(x) = (ψu_{F_{2}})^{−1}g(u_{F_{1}}x) = x + · · · .

It is not too hard to show that E so defined is really a functor.

Example 6.8. Let (k, Γ) = (𝔽_{p}, G_{m}). Then Aut(k, Γ) ≃ ℤ_{p}^{×}, where n ∈ ℤ_{p} corresponds to the map

[n]_{Γ}(x) = (1 + x)^{n} − 1.

Since this has height one, E(k, Γ) ≃ ℤ_{p}. We can use the multiplicative formal group law as its own universal lift:

x +_{F} y = x + y + xy.

This induces a degree −2 formal group law on ℤ_{p}[u^{±}] given by

x +_{F̄} y = x + y + uxy.

The morphism [n]_{Γ}(x) therefore lifts to an automorphism [n]_{F} (x) = (1 + x)^{n} − 1 = nx+⋯ of F, and hence induces an automorphism [n]F̄(x) = u^{−1}((1+ux)^{n} −1) of F̄. We thus see that an automorphism [n]_{Γ} of Γ lifts to a map ψ_{n}: ℤ_{p}[u^{±}] → ℤ_{p}[u^{±}] which sends ψ_{n}(u) = nu.

The theory E_{𝔽_{p},G_{m}} is precisely complex K-theory completed at p, and the ψ_{n}’s are precisely the Adams operations.

6.9. Proof of Landweber exactness. It remains to show that the map MU_{∗} → E(k, Γ)[u^{±}] is Landweber exact. It clearly suffices to show that that the sequence p, w_{1}, w_{2}, … is regular in E(k, Γ), where

[p]_{F} (x) = px + ⋯ + w_{1}x^{p} + ⋯ + w_{2}x^{p^{2}} + · · · .

(In fact, w_{i} = v_{i}u^{1−p^{i}}.)

The cokernel of multiplication by p is k[[u_{1}, …, u_{n−1}]]. We need to compute the p-series of F over this ring. By Lemma 5.6 we can assume without loss of generality that x +_{Γ} y = x + y + bC_{p^{n}}(x, y) + ⋯, whence

x +_{F} y ≡ x + y + u_{1}G_{p}(x, y) + ⋯ + u_{n−1}G_{p^{n−1}}(x, y) + bC_{p^{n}}(x, y) + ⋯

modulo p and (u_{1}, …, u_{n−1})^{2} for certain Γ-symmetric 2-cocycles G_{i}. Recall that

G_{p^{i}}(x, y) = C_{p^{i}}(x, y) + higher terms.

Lemma 6.10. We have that

[p]_{F} (x) = u_{i}x^{p^{i}} + ⋯

modulo p, u_{1}, …, u_{i−1} and (u_{i}, …, u_{n−1})^{2}, and

[p]_{F} (x) = bx^{p^{n}} + ⋯

modulo p, u_{1}, …, u_{n−1}.

Proof. This is an immediate consequence of Lemma 5.7.

Lemma 6.10 implies that the inclusion

k[[w_{1}, …, w_{n−1}]] → k[[u_{1}, …, u_{n−1}]]

is in fact an isomorphism (since the Jacobian matrix (\frac{∂w_{i}}{∂u_{j}}) modulo (u_{1}, …, u_{n−1})^{2} is upper triangular with ones on the diagonal by Lemma 6.10.) Clearly, the sequence w_{1}, w_{2}, … is regular in this ring.

7. The construction

In the next few sections I will discuss the category of spectra (after Lewis-May-Steinberger), and categories of A_{∞}-ring spectra, denoted A_{∞}. The first thing to note is that A_{∞} is a topological category; i.e., for any two objects, there is a topological space of maps. (See Section 12.)

Let A_{∞}^{LT} ⊂ A_{∞} denote the full topological subcategory consisting of A_{∞}-ring spectra E such that

1. E is cofibrant as an A_{∞} ring, (see Section 12),

2. the underlying spectrum of E is a homotopy commutative ring spectrum, which

3. has π_{odd}E = 0,

4. has an invertible element in π_{2}E, and such that

5. π_{0}E is a complete local ring, with maximal ideal 𝔪, such that

6. π_{0}E/𝔪 is a perfect field of non-zero characteristic, for which

7. the degree 0 formal group law F over π_{0}E, (which exists because E is complex orientable by properties 2 and 3, see [1, part 2]), is a universal deformation of its reduction modulo 𝔪.

Thus there is a functor

A_{∞} ⊃ A_{∞}^{LT} ⟶^{π} FG^{op},

sending E ↦ (π_{0}E/𝔪, F).

Theorem 7.1. The functor π is a weak equivalence of topological categories. That is, π is surjective on isomorphism classes of FG, and for each E, F ∈ A_{∞}^{LT} induces a weak equivalence

map_{A_{∞}}(E, F) → FG(πF, πE)

where the set on the right hand side is viewed as a space with the discrete topology.

Proof. The theorem will follow from Corollary 14.6 and Corollary 22.7.

Here’s how you get Theorem 2.1 from Theorem 7.1. First, choose a small model for the category of formal group laws FG; for example, we could fix an algebraically closed field K of characteristic p and require that FG only contain those objects (k, Γ) for which k ⊂ K. Likewise, choose a small model for A_{∞}^{LT}. Now we can perform a “homotopy right Kan extension” of the topological functor i: A_{∞}^{LT} → A_{∞} along the functor π: A_{∞}^{LT} → FG^{op}, obtaining a functor ρ: FG^{op} → A_{∞}. (Homotopy right Kan extensions are a generalization of the notion of a homotopy inverse limit.) By some general nonsense about homotopy Kan extensions one discovers that ρ has the desired properties; in particular, if (k, Γ) = π(E), then ρ(k, Γ) is weakly equivalent to E.

Part 2. Spectra and A_{∞}-ring spectra

In the following sections I collect the results we need about spectra and A_{∞}- ring spectra. I only summarize the facts needed; one should consult the original sources for details and proofs of this material. The category of spectra I use is that of Lewis, May, and Steinberger [9]; in addition to their book, one should also consult the work of Elmendorf, Kriz, Mandell, and May [6], which refines and extends the constructions of [9]. (Note, however, that we make no use the theory of “S-modules” which is the main objective of [6].) In addition, I make use results of Hopkins, McClure, and Goerss on the closed model category structure for various categories of spectra; some of this work has not seen print.

I then (in Section 14) set up a spectral sequence and obstruction theory for computing maps between A_{∞}-ring spectra in the cases we need to consider.

8. Spectra

I give a sketch of the theory of spectra of Lewis, May, and Steinberger [9]; see also [6]. I will omit many details.

We work in the category of compactly generated weak Hausdorff spaces, denoted Top. The category of pointed compactly generated weak Hausdorff spaces is denoted 𝒯. This means that

map_{∗}(X ∧ Y, Z) ≃ map_{∗}(X, map_{∗}(Y, Z)),

where map_{∗} denotes the space of pointed maps with an appropriate topology.

A universe U is a countably dimensional real inner-product space. For example, ℝ^{∞} is a universe. An indexing set A in U is a collection of finite dimensional vector subspaces of U such that any finite dimensional subspace of U is contained in some element of A. Thus, A = {ℝ^{0}, ℝ^{1}, ℝ^{2}, … } is a indexing set in ℝ^{∞}. By abuse of notation, we let U denote the indexing set consisting of all finite dimensional subspaces of U.

Let S^{V} denote the one-point compactification of the vector space V, and let W − V denote the orthogonal complement of V in W. A prespectrum on A is a collection of spaces X_{V} for each V ∈ A and maps

i_{V,W} : X_{V} ∧ S^{W −V} → X_{W}

for each pair V, W ∈ A with V ⊂ W, such that for each V_{1} ⊂ V_{2} ⊂ V_{3} the diagram

[figure: S^{V_{3}−V_{2}} ∧ S^{V_{2}−V_{1}} ∧ X_{V_{1}}; ∼; S^{V_{3}−V_{1}} ∧ X_{V_{1}}; S^{V_{3}} ∧ X_{V_{2}}; X_{V_{3}}]

commutes.

A map of prespectra f: X → Y is a collection of maps f_{V} : X_{V} → Y_{V} commuting with all the structure. We denote the category of prespectra by PA.

Note that each structure map i_{V,W} is adjoint to a map

î_{V,W} : X_{V} → Ω^{W −V} X_{W} ,

where Ω^{V} X = map_{∗}(S^{V} , X). A spectrum on A is a prespectrum X on A for which each map î_{V,W} is a homeomorphism. The category of spectra SA is the full subcategory of PA consisting of the spectra.

Given two prespectra X and Y, we can topologize the pointed set of maps PA(X, Y) as a subspace of ∏_{V ∈A} map_{∗}(X_{V} , Y_{V}). It is not hard to see that the composition maps PA(X, Y) ∧ PA(Y, Z) → PA(X, Z) are continuous. Thus PA is a topological category. Likewise we set SA(X, Y) = PA(X, Y), making SA into a topological category.

First we note that these indexing sets are not too important.

Proposition 8.1. If A is an indexing set of U then the functor SU → SA induced by restriction is an equivalence of categories.

Proof. Given X ∈ SA, we can produce a spectrum X′ ∈ SU by setting X′_{V} = Ω^{W −V} X_{W} where W ∈ A is some subspace of U containing V. It straightforward to check that this is well-defined, and produces the equivalence of categories.

The following proposition says that any prespectrum can be turned into a spectrum. It is crucial and non-trivial.

Proposition 8.2. [9, Appendix] There is a pair of adjoint functors

ℓ: PA ⇆ SA : r

where r is inclusion of subcategories. Furthermore, ℓr ≃ 1, and the functors ℓ and r are continuous (i.e., induce continuous maps on function spaces).

As a consequence, it follows that spectra have all small limits and colimits. Prespectra have limits and colimits which are computed space-wise. Limits of spectra can also be computed space-wise, since Ω commutes with limits of spaces. Colimits of spectra are computed by colim_{I} X ≃ ℓX′ where X′ is the prespectrum defined by X′_{V} = colim_{I} X_{V}.

8.3. Constructions of spectra. Given a space X, define a prespectrum Σ_{pre}^{∞}X by

(Σ_{pre}^{∞}X)_{V} = S^{V} ∧ X,

with the obvious structure maps, and define the suspension spectrum Σ^{∞}X by Σ^{∞}X = ℓ(Σ_{pre}^{∞}X). This has the property that

SU(Σ^{∞}X, Y) ≃ map_{∗}(X, Y_{0}).

For each finite V ⊂ U we define a prespectrum S_{pre}^{−V} by

(S_{pre}^{−V})_{W} = \begin{cases}S^{W −V} & if V ⊂ W, \\ ∗ & otherwise, \end{cases}

with the obvious structure maps. Define a sphere spectrum S^{−V} by S^{−V} = ℓ(S_{pre}^{−V}). Such an object has the property that

SU(S^{−V} , Y) ≃ Y_{V} .

For an unpointed space K and a spectrum X over U we define spectra K ⊗ X and X^{K} as follows. Let X^{K} be defined by

X^{K} = map(K, X_{V}).

It is not hard to check that X^{K} is a spectrum. Let (K ⊗ X)^{pre} be the prespectrum defined by

(K ⊗ X)_{V}^{pre} = K_{+} ∧ X_{V} ,

and let K ⊗ X = ℓ(K ⊗ X)^{pre} denote the corresponding spectrum. We have the following fact.

Proposition 8.4. [9, I, Prop. 3.3] Given K ∈ Top and X, Y ∈ SU there are natural isomorphisms

SU(K ⊗ X, Y) ≃ map(K, SU(X, Y)) ≃ SU(X, Y^{K}).

This means we can define things like homotopies between maps. Given maps f, g: X → Y of spectra in SU a homotopy can be taken to be a map

H: I → SU(X, Y)

such that H(0) = f and H(0) = g, or equivalently as a map

H′: I ⊗ X → Y

or a map

H′′: X → Y^{I}.

We can think of the object I ⊗X as a cylinder on X, and Y^{I} as a spectrum of paths on Y. We can use these kinds of constructions to produce mapping-cylinders, path fibrations, and other nifty constructions.

There are constructions analogous to K ⊗ X and X^{K} which we can carry out when K is a pointed space. I only give one example: for a pointed space K and spectrum X let (K ∧ X)^{pre} be the prespectrum defined by

(K ∧ X)_{V}^{pre} = K ∧ X_{V} ,

and let K ∧ X = ℓ(K ∧ X)^{pre}. Then we have

Proposition 8.5. Given K ∈ 𝒯 and X, Y ∈ SU there is a natural isomorphism

SU(K ∧ X, Y) ≃ map_{∗}(K, SU(X, Y)).

8.6. Model category structure. We say a map f: X → Y in SU is a weak equivalence if for each V ⊂ U the induced map f_{V} : X_{V} → Y_{V} is a weak equivalence of pointed spaces. We can define homotopy groups of a spectrum X as follows: for each n ∈ ℤ, choose a vector space V ⊂ U such that n+dim(V) = m ≥ 0, and write

π_{n}(X) = π_{m}(X_{V}).

One can show that this doesn’t depend on the choice of V. Thus, a weak equivalence of spectra is just a map which induces isomorphisms on stable homotopy groups.

Proposition 8.7. [6, Thm. 4.4] The category of spectra SU admits the structure of a Quillen closed model category, such that a map f: X → Y is a

1. weak equivalence if each map f_{V} : X_{V} → Y_{V} is a weak equivalence of spaces,

2. fibration if each map f_{V} : X_{V} → Y_{V} is a Serre fibration,

3. cofibration if f is a retract of a map f′ which is obtained by successively gluing on “cells” of the form

∂Δ_{n} ⊗ S^{−V} → Δ_{n} ⊗ S^{−V}

where n ≥ 0, V ⊂ U, and Δ_{n} denotes the Euclidean n-simplex. Furthermore, this structure gives SU the structure of a topological model category.

I won’t say what all this means. Quillen closed model categories are defined in [11]. The existence of such a structure implies, among other things, that we can form a homotopy category Ho SU by formally inverting the weak equivalences in SU.

The statement that SU is a topological model category means that for a cofibration i: X → Y and a fibration p: E → F of spectra, the induced map of spaces

f: SU(Y, E) → SU(X, E) ×_{SU(X,F)}SU(Y, F)

is a Serre fibration, and that if either i or p is also a weak equivalence, then so is f. (This is analogous to Quillen’s notion of a simplicial model category [11].)

9. Smash products

In this section I develop the “external” smash product and related constructions.

9.1. Smash product. Given universes U and U′, then the product space U × U′ is another universe. Let

A = {V × W ⊂ U × U′ | V ⊂ U, V′ ⊂ U′}.

This is an indexing set of U × U′. Given X ∈ SU and Y ∈ U′ define

(X ∧ Y)_{V ×W}^{pre} = X_{V} ∧ Y_{W};

this is a prespectrum over A, where the structure maps are

S^{V′×W′−V ×W} ∧ X_{V} ∧ Y_{W} ≃ S^{V′−V} ∧ S^{W′−W} ∧ X_{V} ∧ Y_{W} → X_{V′} ∧ Y_{W′}.

Then let

X ∧ Y = ℓ(X ∧ Y)^{pre} ∈ SA ≃ S(U × U′).

Thus we obtain a smash product functor

− ∧ −: SU × SU′ → S(U × U′).

As an example, we have Proposition 9.2. For V ⊂ U and W ⊂ U′ there is a natural isomorphism

S^{−V} ∧ S^{−W} ≃ S^{−V ×W}

of spectra over U × U′.

Proof. We compute on the level of prespectra, and then spectrify. If V ⊂ V′ ⊂ U and W ⊂ W′ ⊂ U′ we get

(S^{−V} ∧ S^{−W})_{V′×W′}^{pre} ≃ S^{V′−V} ∧ S^{W′−W} ≃ S^{V′×W′−V ×W} ≃ (S^{−V ×W})_{V′×W′}^{pre}.

Hence the corresponding spectra are isomorphic.

9.3. Function spectra. Let’s fix a Y ∈ SU′. Then the functor − ∧ Y : SU → S(U × U′) has a right adjoint, namely the function spectrum

F (Y, −): S(U × U′) → SU.

By this adjoint functor characterization, we see that for V ⊂ U we must have

F (Y, Z)_{V} ≃ SU(S^{−V} , F (Y, Z)) ≃ S(U × U′)(S^{−V} ∧ Y, Z),

and since we have already defined the smash product, we can use this observation to define F (Y, Z) by the above formula. One has that for V ⊂ W ⊂ U,

S^{W −V} ∧ S^{−W} ≃ S^{−V}

as one can see by checking this formula on the level of prespectra:

(S^{W −V} ∧ S^{−W})_{W′}^{pre} ≃ S^{W −V} ∧ S^{W′−W} ≃ S^{W′−V} ≃ (S_{pre}^{−V})_{W′}.

Thus the structure maps of F (Y, Z) are defined to be the natural isomorphisms

F (Y, Z)_{V} = S(U × U′)(S^{−V} ∧ Y, Z)
≃ S(U × U′)(S^{W −V} ∧ S^{−W} ∧ Y, Z)
≃ Ω^{W −V} S(U × U′)(S^{−W} ∧ Y, Z)
= Ω^{W −V} F (Y, Z)_{W}

and we see that F (Y, Z) is indeed a spectrum.

One can show that if K is an unpointed space and X a spectrum, then X^{K} ≃ F (Σ^{∞}K_{+}, X).

9.4. Associativity. The significant property of this category of spectra which we need is associativity of the smash product. If X ∈ SU_{1}, Y ∈ SU_{2} and Z ∈ SU_{3}, then there is a natural isomorphism

(X ∧ Y) ∧ Z ≃ X ∧ (Y ∧ Z)

of spectra over U_{1} × U_{2} × U_{3}. In fact, both sides of the above equation are naturally isomorphic to the spectrification of the prespectrum defined by

V_{1} × V_{2} × V_{3} ↦ X_{V_{1}} ∧ Y_{V_{2}} ∧ Z_{V_{3}}.

10. Change of universe functors

I will talk about several different constructions which allow you to compare spectra which are defined over different universes. Some of the material of this section is in [9, Ch. II] and [6, App. A].

10.1. Push-forward and pull-back. Given a linear isometric embedding of universes f: U → U′, that is, a vector space map which preserves the inner product, we produce functors

f_{∗}: SU ⇄ SU′ : f^{∗}

called push-forward and pull-back. Pull-back is easy; for V ⊂ U and Y ∈ SU′ let

(f^{∗}Y)_{V} = Y_{fV} ,

where fV ⊂ U′ denotes the image of V under f. The natural homeomorphism S^{V} ≃ S^{fV} induced by f produces isomorphisms

Ω^{W −V} (f^{∗}Y)_{V} ≃ map_{∗}(S^{W −V} , Y_{fV}) ≃ map_{∗}(S^{fW −fV} , Y_{fV}) ≃ Y_{fW} = (f^{∗}Y)_{W} .

Thus f^{∗}Y is a spectrum over U.

Push-forward is a little more complicated. We define a pre-spectrum (f_{∗}X)^{pre} as follows. For W ⊂ U′ let

(f_{∗}X)_{W}^{pre} = S^{W −fV} ∧ X_{V}

where V = f^{∗}(fV ∩W). In other words, V is the largest subspace of U which maps into W. If W ⊂ W′ let V′ = f^{∗}(fV′ ∩ W′), whence we get structure maps

S^{W′−W} ∧ (f_{∗}X)_{W}^{pre} ≃ S^{W′−W} ∧ S^{W −fV} ∧ X_{V} ≃ S^{W′−fV′} ∧ S^{V′−V} ∧ X_{V}

⟶ S^{W′−V′} ∧ X_{V} = (f_{∗}X)_{W′}^{pre}.

Of course we let f_{∗}X = ℓ(f_{∗}X)^{pre} ∈ SU′.

We compute that f_{∗}S^{−V} ≃ S^{−fV}, by a sequence of adjunctions

SU′(f_{∗}S^{−V} , Y) ≃ SU(S^{−V} , f^{∗}Y) ≃ (f^{∗}Y)_{V} = Y_{fV} ≃ SU′ ∗ S^{−fV} , Y).

Note that if the linear isometry f: U → U′ is an isomorphism of universes, then the definition of f_{∗}X simplifies to (f_{∗}X)_{W} = X_{f^{∗}W}, and so f_{∗} and f^{∗} form an adjoint equivalence of categories.

10.2. Many smash products. We can now use the push-forward functors to define many smash product functors from the category of spectra over U to itself. Let U^{n} = U × ⋯ × U. Then for X_{1}, …, X_{n} ∈ SU and for some linear isometric map f: U^{n} → U we define a functor SU^{n} → SU by

X_{1}, …, X_{n} ↦ f_{∗}(X_{1} ∧ ⋯ ∧ X_{n}).

Note that if i: U_{1} → U_{2} and j: U_{2} → U_{3} then (j ∘ i)_{∗}X ≃ j_{∗}i_{∗}X. Thus, given f: U^{2} → U and g: U^{2} → U we can compose smash product functors, so that

f_{∗}(g_{∗}(X ∧ Y) ∧ Z) ≃ (f ∘ (g × 1))_{∗}(X ∧ Y ∧ Z)

where f ∘ (g × 1)): U^{3} → U denotes the composite map on universes.

10.3. “Twisted” constructions involving the linear isometries space. Let L(U, U′) denote the set of all linear isometries from U to U′. Topologize this space by the compact-open topology. Then one has the following result.

Proposition 10.4. [9, II, Lemma 1.5] The space L(U, U′) is contractible.

For us, this means that we have a space L(U^{n}, U) which parameterizes all possible ways of smashing n things together, and this parameter space is contractible. One would like to somehow glue “all” these smash-products together and get a super-universal smash product functor from SU to itself.

For the time being let’s write L for L(U, U′). For maps of topological spaces T → L and T′ → L let map_{L}(T, T′) ⊂ map(T, T′) denote the space of maps T → T′ which commute with the projections to L. Let Top denote the category of unpointed spaces, and let (Top ↓ L) denote the category of spaces over L.

Proposition 10.5. Given t: T → L ∈ (Top ↓ L), X ∈ SU and Y ∈ SU′ there exist functorial constructions

• T ⋊ X ∈ SU′,

• Y^{T} ∈ SU,
• S(X, Y) → L ∈ (Top ↓ L),

such that there are natural isomorphisms

SU′(T ⋊ X, Y) ≃ map_{L}(T, S(X, Y)) ≃ SU(X, Y^{T}),

and such that if T = {f} → L is a single point, then • {f} ⋊ X ≃ f_{∗}X,

• Y^{{f}} ≃ f^{∗}Y, and

• map_{L}({f}, S(X, Y)) ≃ SU′(f_{∗}X, Y) ≃ SU(X, f^{∗}Y), or in other words, the fiber of the map S(X, Y) → L(U, V) over the point f is homeomorphic to the above spaces.

The construction T ⋊ X is known as the twisted half-smash product. First I’ll outline a construction of T ⋊ X. First, suppose T is a compact space.

For V ⊂ U let T (V) ⊂ U′ denote the union of all fV ⊂ U′ as f ranges over all elements in the image of ξ: T → L. Since T is compact there exists a finite dimensional vector subspace W ⊂ U′ such that T (V) ⊂ W. Conversely, for each W ⊂ U′ there is a largest V ⊂ U such that T (V) ⊂ W.

Given such V and W with T (V) ⊂ W, let ξ(V) → T denote the vector subbundle of the trivial bundle W × T → T such that over t ∈ T the fiber is ξ(t)(V) ⊂ W. Let W − ξ(V) denote the orthogonal complement bundle of ξ(V) in W × T, and let T^{W −ξ(V)} denote the Thom space. Define

(T ⋊ X)_{W}^{pre} = T^{W −ξ(V)} ∧ X_{V} .

Suppose W ⊂ W′ ⊂ U′. Then there are corresponding spaces V ⊂ V′ ⊂ U and sub-bundles ξ(V) ⊂ ξ(V′) of W′ × T. Then

(W′ − W) × (W − ξ(V)) ≃ W′ − ξ(V) ≃ (W′ − ξ(V′)) × (V′ − V)

since ξ(V′) − ξ(V) can be identified with a trivial bundle with fiber V′ − V. Thus structure maps are defined by

S^{W′−W} ∧ (T ⋊ X)_{W}^{pre} ≃ S^{W′−W} ∧ T^{W −ξ(V)} ∧ X_{V}
≃ T^{W′−ξ(V′)} ∧ S^{V′−V} ∧ X_{V}
→ T^{W′−ξ(V′)} ∧ X_{V′} = (T ⋊ X)_{W′}^{pre}.

Of course, we now let T ⋊ X = ℓ(T ⋊ X)^{pre}. Note that if T = {f} ∈ L then we have just recovered the construction of f_{∗}X.

If T is not compact, then we let

T ⋊ X = colim_{K⊂T} K ⋊ X

as K ranges over all the compact subspaces of T.

For S(X, Y), we let this be equal as a set to

S(X, Y) = ∐_{f∈L} SU(X, f^{∗}Y).

This has a natural map π: S(X, Y) → L. We topologize it by giving it the largest topology such that a map f: T → S(X, Y) is continuous map if and only if

1. the composite πf: T → L is continuous, and

2. the corresponding map T ⋊ X → Y is a map of spectra. This last is a construction of Elmendorf [5].

10.6. Applications of the twisted constructions. We need one more set of facts about all these constructions; namely, that they preserve features of the model category structure as best as is possible. Since I haven’t been explicit about model category structures I won’t give the general statement, but here are some consequences.

1. If T → L with T a CW-complex, and X ∈ SU a cofibrant spectrum, then T ⋊ X ∈ SU′ is cofibrant.

2. If T ⟶^{f} T′ → L is such that f is a weak equivalence between CW-complexes, and X ∈ SU is a cofibrant spectrum, then T ⋊ X → T′ ⋊ X is a weak equivalence of spectra over U′.

3. If T → L with T a CW-complex, and X → Y a weak equivalence between cofibrant spectra over U, then T ⋊ X → T ⋊ Y is a weak equivalence of spectra over U′.

4. If X ∈ SU is a cofibrant spectrum, and Y → Y′ ∈ SU′ is a weak equivalence of cofibrant spectra, then X ∧ Y → X ∧ Y′ is a weak equivalence of spectra over U × U′.

5. If X ∈ SU is a cofibrant spectrum, and Y ∈ SU′, then S(X, Y) → L is a Serre fibration.

Corollary 10.7. If f, g: U → U′ are linear isometries, and X ∈ SU a cofibrant spectrum, then f_{∗}X and g_{∗}Y are weakly equivalent as spectra over U′, and furthermore this weak equivalence may be chosen canonically up to homotopy.

Proof. Chose a path γ: I → L between f and g (recall that L is contractible). Then we get a diagram

f_{∗}X ≃ {f} ⋊ X → I ⋊ X ← {g} ⋊ X

where each of the maps is a weak equivalence by (2) above, since the unit interval I is certainly a CW-complex. Similarly, given two homotopies γ and γ′, an argument involving homotopies between homotopies shows that the weak equivalence is canonical up to homotopy.

Corollary 10.8. For any two universes U and U′, the homotopy categories of spectra Ho SU and Ho SU′ are canonically equivalent.

Corollary 10.9. For any cofibrant X, Y ∈ SU and f, g ∈ L(U^{2}, U) the smash products f_{∗}(X ∧Y) and g_{∗}(X ∧Y) are canonically weakly equivalent, and thus there is a canonical smash product defined on the level of the homotopy category.

11. Operads

I give a couple of definitions of operads.

11.1. Definition of operads. An operad in Top is a collection of spaces {C[n]}_{n≥0} equipped with the following structure:

1. a right action of Σ_{n} on C[n],

2. a distinguished point 1 ∈ C[1],

3. and for each sequence I = (i_{1}, …, i_{k}) of non-negative integers (k ≥ 0), and i_{1} + ⋯ + i_{k} = n a map

C[k] × C[i_{1}] × ⋯ × C[i_{k}] → C[n].

Notationally, this map sends f ∈ C[k], g_{j} ∈ C[i_{j}] to f{g_{1}, …, g_{k}}. In addition, the following relations must hold.

1. 1{g} = g for g ∈ C[n],

2. f{1, …, 1} = f for f ∈ C[k],
3. for f ∈ C[k], g_{j} ∈ C[i_{j}], and h_{j,ℓ} ∈ C[m_{j,ℓ}],

f{g_{1}{h_{1,1}, …, h_{1,m_{1}}}, …, g_{k}{h_{k,1}, …, h_{k,m_{k}}} = (f{g_{1}, …, g_{k}}){h_{1,1}, …, h_{k,m_{k}}}},

4. (fσ){g_{1}, …, g_{k}} = f{g_{σ1}, …, g_{σk}} for f ∈ C[k], g_{j} ∈ C[i_{j}], and σ ∈ Σ_{k},

5. f{g_{1}σ_{1}, …, g_{k}σ_{k}} = f{g_{1}, …, g_{k}}σ_{1} × ⋯ × σ_{k} for f ∈ C[k], g_{j} ∈ C[i_{j}], and σ_{j} ∈ Σ_{i_{j}}.

A map of operads C → D is a collection of continuous maps C[n] → D[n] which commute with all the structure.

11.2. Examples of operads. Given a space X ∈ Top, define E_{X} by

E_{X}[n] = map(X^{n}, X).

It is not hard to see that E_{X} is an operad, under composition of functions. It is called the endomorphism operad of X. Likewise, the spaces L[n] = L(U^{n}, U) also form an operad under composition, called the linear isometries operad.

Another example of an operad is the associative operad. Let A[n] denote the set of monomials of length n in n distinct non-commuting formal variables t_{1}, …, t_{n}. E.g., A[2] = {t_{1}t_{2}, t_{2}t_{1}}. Write f(t_{1}, …, t_{n}) for a general element of A[n], and write 1 ∈ A[0] for the “empty” monomial, (not to be confused with the distinguished element of A[1], which is denoted t_{1}). Then A is an operad, under “composition” of formal monomials; that is,

f{g_{1}, …, g_{k}}(t_{1}, …, t_{n})

= f(g_{1}(t_{1}, …, t_{i_{1}}), g_{2}(t_{i_{1}+1}, …, t_{i_{1}+i_{2}}), …, g_{k}(t_{i_{1}+···+i_{k−1}+1}, …, t_{n})).

If X is a topological monoid, then you can produce a map A → E_{X} of operads by evaluation, i.e.,

f(t_{1}, …, t_{n}) ↦ ((x_{1}, …, x_{n}) ↦ f(x_{1}, …, x_{n})).

In fact, giving the structure of a topological monoid on X is precisely equivalent to giving a map of operads from A to E_{X}.

11.3. Σ-objects. I need a description of operads which shows off some of their other facets, and will be useful in Section 23. This definition works in any symmetric monoidal category. The primary example of interest here is the category of topological spaces, with cartesian product as monoidal product. To keep things simple, I state the definitions in terms of the category Top. But one should keep in mind that things work in a general symmetric monoidal category.

A Σ-object C is a collection of spaces {C[n]}_{n≥0} such that C[n] is equipped with a right action by Σ_{n}. Thus, any operad is a Σ-object, but not conversely.

Given a Σ-object C, define a functor F_{C}: Top → Top by

F_{C}(X) = ∐_{n≥0} C[n] ×_{Σ_{n}} X^{n}.

Thus each Σ-object produces a functor; likewise, each map of Σ-objects produces a natural transformation of functors. It is sometimes helpful to think of F_{C} as a “power series” with coefficients C[n].

Let I be the Σ-object defined by

I[n] = \begin{cases}pt & if n = 1, \\ ∅ & otherwise. \end{cases}

Then F_{I} is the identity functor.

Suppose D is another Σ-object. What is F_{C}(X) × F_{D}(X)? We compute

F_{C}(X) × F_{D}(X) ≃ ( ∐_{m≥0} C[m] ×_{Σ_{m}} X^{m}) × (∐_{n≥0} D[n] ×_{Σ_{n}} X^{n})

≃ ∐_{m,n≥0}(C[m] × D[n]) ×_{Σ_{m}×Σ_{n}} (X^{m} × X^{n})

≃ ∐_{d≥0} ( ∐_{m+n=d} C[m] × D[n] ×_{Σ_{m}×Σ_{n}} Σ_{d}) ×_{Σ_{d}} X^{d}.

Thus, if we define a new Σ-object C ⊗ D by

(C ⊗ D)[d] = ∐_{m+n=d} C[m] × D[n] ×_{Σ_{m}×Σ_{n}} Σ_{d},

then we see that F_{C}(X) × F_{D}(X) ≃ F_{C⊗D}(X). Note that this implies that F_{C}(X)^{×n} ≃ F_{C^{⊗n}}(X).

We can produce other formulas of the same type. Thus, if K is a space, define a Σ-object K × C by (K × C)[n] = K × C[n]. Then K × F_{C}(X) ≃ F_{K×C}(X). Also, if C_{α} is a diagram of Σ-objects, then colim (F_{C_{α}}(X)) ≃ F_{colim C_{α}}(X).

The formula we are really after is the one for the composition of functors F_{C}(F_{D}(X)). We compute

F_{C}(F_{D}(X)) ≃ ∐_{m≥0} C[m] ×_{Σ_{m}} F_{D}(X)^{m}

≃ ∐_{m≥0} C[m] ×_{Σ_{m}} F_{D^{⊗m}}(X)

≃ F_{(∐_{m≥0} C[m] ×_{Σm}D^{⊗m})}(X).

Thus, if we set

C ∘ D = ∐_{m≥0} C[m] ×_{Σ_{m}} D^{⊗m}

then we have F_{C} ∘ F_{D} ≃ F_{C∘D}. In full, this means

(C ∘ D)[d] = ∐_{m≥0} C[m] ×_{Σ_{m}} ( ∐_{i_{1}+···+i_{m}=d}D[i_{1}] × ⋯ × D[i_{m}] ×_{Σ_{i_{1}} ×···×Σ_{im}}Σ_{d}) .

This makes the category of Σ-objects into a monoidal category, where the monoidal structure corresponds to composition of functors.

11.4. Operads are monoids. Since the category of Σ-object is a monoidal category via the “∘” product (but not symmetric monoidal, since C ∘ D ≠ D ∘ C), we can think about monoids in this category. A monoid is just a Σ-object C equipped with maps η: I → C and μ: C ∘ C → C such that the following diagrams commute.

[figure: C ∘ C; μ; C ∘ C ∘ C; μ∘C; C∘μ; C ∘ C; μ; C ∘ I; C∘η; C; I ∘ C; η∘C; C ∘ C; μ; C]

Proposition 11.5. The category of operads is equivalent to the category of Σ-object-monoids.

Proof. Suppose C is a Σ-object-monoid. Then the map η: I → C is just the choice of a distinguished point in C[1], and the multiplication map μ: C ∘ C → C amounts to choosing a collection of maps

C[m] × C[i_{1}] × ⋯ × C[i_{m}] → C[d]

which satisfy axioms (4) and (5) for an operad. The three remaining axioms (1), (2), and (3) follow from the fact that C is a monoid. The converse argument is similar.

A Σ-object monoid C defines a functor F_{C}: Top → Top which is a triple. Thus, given an operad C, we can speak of algebras over the triple F_{C}.

Proposition 11.6. The category of C-algebras is equivalent to the category of F_{C}-algebras.

This implies in particular that for a space X, F_{C}(X) = ∐ C[m] ×_{Σ_{m}} X^{n} is a

C-algebra. In fact, it is the free C-algebra on X, with the property that

C- alg(F_{C}(X), Y) ≃ map(X, Y).

For example, if A is the associative operad,

F_{A}(X) ≃ ∐_{n≥0} X^{n}.

12. Algebra spectra

We discuss spectra which are also algebras over an operad.

12.1. Operads over the linear isometries operad. We want to have operads acting on spectra, rather than on spaces. First, note that there’s a natural inclusion Σ_{n} ⊂ L(U^{n}, U^{n}). Let X be a spectrum over U. For each σ ∈ Σ_{n} there’s a natural isomorphism X^{(n)} ≃ σ_{∗}X^{(n)} as follows. Recall that

(X_{pre}^{(n)})_{V_{1}×···V_{n}} = X_{V_{1}} ∧ ⋯ ∧ X_{V_{n}}.

This means that

(σ_{∗}X^{(n)})_{V_{1}×···V_{n}}^{pre} ≃ X_{V_{σ1}} ∧ ⋯ ∧ X_{V_{σn}}.

There’s a natural map (X^{(n)})_{pre} → (σ_{∗}X^{(n)})_{pre} induced by permuting the factors, which is an isomorphism. This specifies the desired isomorphism of spectra.

The space L[n] gets a right Σ_{n}-action by permuting the source universe U^{n}. Suppose T → L[n] is a map of Σ_{n}-spaces. Then T ⋊ X^{(n)} has a natural Σ_{n}-action, where σ ∈ Σ_{n} acts via the composite of isomorphisms

T ⋊ X^{(n)} ≃ T ⋊ σ_{∗}X^{(n)} ≃ T σ ⋊ X^{(n)} ≃ T ⋊ X^{(n)},

where T σ denotes the space over L[n] defined by the composite map T → L[n] ⟶^{σ} L[n], and the right-hand map in the sequence is induced by the isomorphism T σ → T of spaces over L[n].

By a Σ-object over the linear isometries operad, I mean a map C → L of Σ-objects. For X ∈ SU let F_{C}(X) denote the spectrum

F_{C}(X) = ⋁_{n≥0} C[n] ⋊_{Σ_{n}} X^{(n)}.

Thus F_{C}: SU → SU is a functor. (Note: C[0] ⋊ X^{(0)} is defined to be Σ^{∞}C[0]_{+}.)

Given an operad over the linear isometries operad, i.e., an operad map C → L, the functor F_{C} is a triple on SU, and we can speak of the category of algebras over C.

One example of an operad over L is the endomorphism operad of X. Define the endomorphism “operad” of a spectrum X to be an operad over L, by

E_{X}[n] = S(X^{(n)}, X) → L[n].

Then one can show that a C-algebra structure on X is precisely the same as a map of operads C → E_{X} making the diagram

\xymatrix{C \ar[dr] \ar[rr] &  & E_{X} \ar[dl] \\  & L}

commute.

The main examples of operads we’ll need are A_{∞}-operads. An A_{∞} operad A is one for which the operad π_{0}A of path components, defined by (π_{0}A)[n] = π_{0}A[n], is isomorphic to the associative operad, and each component of A[n] is contractible. We call a spectrum which is an algebra over such an operad an A_{∞}-ring spectrum.

For example, there is an operad with A[n] = L[n] × Σ_{n} which is an operad over the linear isometries operad; it induces a functor

F_{A}(X) ≃ ⋁_{n≥0} L[n] ⋊ X^{(n)}.

In [6] the authors use this operad A to define their notion of A_{∞}-ring spectrum. It will be necessary for us to use a different A_{∞}-operad which has technical properties we will need. As will be noted below, this is not a major problem.

Note that if X is a spectrum, then F_{C}(X) naturally is a C-algebra, and is in fact the free C-algebra on X, I’ll denote the free C-algebra by C(X). If Y is a C-algebra, I write UY for its underlying spectrum.

12.2. Properties of C-algebra spectra. The category of C-algebras has properties very much like those of the category of spectra SU. I’ll summarize them.

1. C- alg is a topological category. That is, for every two C-algebras X and Y the set of C-algebra maps C- alg(X, Y) ⊂ SU(X, Y) can be topologized as a subspace of the space of spectrum maps.

2. The category of C-algebras has all small limits and colimits. Colimits of C-algebras can be quite complicated. But note that limits are easy. For example, if X and Y are C-algebras, and UX and UY denote the underlying spectra of X and Y, then UX × UY admits a natural C-algebra structure given by

F_{C}(UX × UY) → F_{C}UX × F_{C}UY → UX × UY.

This is the product in the category of C-algebras. Thus, limits of C-algebras are computed on the underlying spectrum.

3. Given an unpointed space T and C-algebras X and Y, there are constructions T ⊗_{C} X and Y^{T}, with the property that

C- alg(T ⊗_{C} X, Y) ≃ map(T, C- alg(X, Y)) ≃ C- alg(X, Y^{T}).

Furthermore, U(Y^{T}) ≃ (UY)^{T}. (But note that U(T ⊗_{C} X) ≄ T ⊗ UX!) Let

E be a C-algebra. Then π_{∗}(UE)^{T} = π_{∗}U(E^{T}) ≃ E^{−∗}T. Thus, if E is an A_{∞}-ring spectrum, then E^{T} is an A_{∞}-ring spectrum whose homotopy is the cohomology of T.

4. The category of C-algebra spectra is a topological closed model category (this is non-trivial to prove). The weak equivalences and fibrations are taken to be those maps which are weak equivalences or fibrations on the underlying spectra.

Some of these results appear in EKMM [6]. The final result on the model category structure of C-algebras represents work of Hopkins. (EKMM prove the model category structure for certain special operads C, but not the general case.)

We note one more result, also due to Hopkins: the homotopy category of Calgebra spectra depends (up to equivalence) only on the weak homotopy type of the operad C. In particular, this means that all A_{∞}-operads give the same homotopy category of algebra spectra. Thus, we are always free to replace a given A_{∞} operad A with an A′ which is technically convenient for us.

13. Simplicial spectra

Let Δ denote the category having as objects the finite ordered sets [n] = {0, 1, …, n}, and as morphisms those set maps f: [m] → [n] for which i ≤ j implies f(i) ≤ f(j). This category has a representation in Top which is easier to picture: there’s a functor Δ: Δ → Top which sends [n] to the Euclidean n-simplex Δ_{n} which has vertices labelled v_{0}, v_{1}, …, v_{n}, and which sends f: [m] → [n] to the affine map which sends v_{i} to v_{f(i)}.

A simplicial object in a category C is a functor X_{•}: Δ^{op} → C, and we call the category of such sC. We define a realization functor |−|: sSU → SU as a quotient

∐_{n≥0} Δ_{n} ⊗ X_{n}/ ∼ .

More precisely, it is the coequalizer of

∐_{[m]→[n]∈Δ} Δ_{m} ⊗ X_{n} ⇉ ∐_{[d]∈Δ} Δ_{d} ⊗ X_{d},

in which the summand corresponding to δ: [m] → [n] ∈ Δ is mapped by the top arrow to δ ⊗ X_{n}, and by the bottom arrow to Δ_{m} ⊗ δ. Geometric realization is characterized by the fact that it is the left adjoint to a functor SU → sSU sending Y to [n] ↦ Y^{Δ_{n}}.

An augmented simplicial object is a simplicial object X_{•} together with a map ϵ: X_{0} → Y such that the two maps X_{1} ⇉ X_{0} → Y are the same. Thus, if X_{•} → Y is an augmented simplicial spectrum, then there’s an induced map |X_{•}| → Y.

13.1. Facts about geometric realizations of simplicial spectra. We need the following facts [6].

1. If X_{•} ∈ sSU and Y_{•} ∈ sSU′, then you can form a simplicial spectrum X_{•}∧Y_{•} in sS(U × U′), defined by [n] ↦ X_{n} ∧ Y_{n}. The crucial fact is that there’s a natural isomorphism of spectra

|X_{•} ∧ Y_{•}| ≃ |X_{•}| ∧ |Y_{•}|.

2. If T → L(U, U′) and X_{•} ∈ sSU, then there’s a natural isomorphism

|T ⋊ X_{•}| ≃ T ⋊ |X_{•}|.

3. Geometric realization commutes with colimits.

So if C → L is a Σ-object over L, then there’s a natural isomorphism

F_{C}|X_{•}| ≃ |F_{C}(X_{•})|.

Now suppose X_{•} is a simplicial C-algebra for some operad C, i.e., a functor X_{•}: Δ^{op} → C- alg. Let UX_{•} denote the underlying simplicial spectrum. Then we can make |UX_{•}| into a C-algebra, by a map

F_{C}|UX_{•}| ≃ |F_{C}UX_{•}| → |UX_{•}|.

So the geometric realization of a simplicial C-algebra is again a simplicial C-algebra.

13.2. Resolutions of a C-algebra. Given a C-algebra X, let CX denote the free C-algebra on the underlying spectrum of X, so UCX ≃ ⋁_{n≥0} C[n] ⋊_{Σ_{n}} X^{(n)}.

Then we get an augmented simplicial object

X ← CX ⇇ CCX ⋯

which I’ll denote by C^{•+1}X. (By abuse of notation I omit the forget functor U.)

We need a model category fact: for good (i.e., cofibrant) choices of X and C, the natural map |C^{•+1}X| → X is a weak equivalence of C-algebras, and |C^{•+1}X| is itself a cofibrant C-algebra, and that C^{•+1}X is “Reedy cofibrant” in the category of simplicial C-algebras. This means in particular that

C- alg(X, E) → C- alg(|C^{•+1}X|, E)

is a weak equivalence of spaces. We will compute A_{∞} maps between A_{∞}-ring spectra using a spectral sequence based on this resolution, using an A_{∞}-operad C.

Remark 13.3. It is important that we be able to produce a Reedy cofibrant resolution, as we will see below. In order to produce such it is convenient to have an operad C for which a cofibrant C-algebra X has an underlying spectrum UX which is cofibrant as a spectrum. It turns out that this will be the case when C is a cofibrant operad; these will be discussed in Section 22.2. Thus, we will assume henceforth that C is a cofibrant A_{∞}-operad.

The proof that a cofibrant operad suffices is quite simple, so we sketch it here (using notions from Section 22.2). Let Y be a cofibrant C-algebra over a cofibrant operad C. It will suffice to can produce a C-algebra X for which UX is cofibrant as a spectrum together with a trivial fibration p: X → Y of C-algebras, for then the theory of model categories tells us that p admits a section, and so UY is a retract of the cofibrant spectrum UX and thus cofibrant.

We produce such an X by means of a dirty trick. Start with a trivial fibration of spectra p: X → UY in which X is a cofibrant spectrum. We need to put a C-algebra structure on X so that p becomes a map of C-algebras. Consider the diagram

\xymatrix{ & \{S(X^{(n)}, X) ×_{S(X^{(n)},Y)} S(Y^{(n)}, Y)\} \ar[r] \ar[d]_{f} & \{S(X^{(n)}, X)\} \ar[d]_{g} \\ C \ar[r] \ar@{.>}[ur] & \{S(Y^{(n)}, Y)\} \ar[r] & \{S(X^{(n)}, Y)\}}

The map g is bunch of trivial fibrations of spaces (one for each n ≥ 0), because X^{(n)} is a cofibrant spectrum and p is a trivial fibration of spectra. One can show that the pull-back f is actually a map between operads; since f is a trivial fibration space-wise, it is a trivial fibration of operads, and the cofibrancy property of C implies that the dotted arrow exists as a map of operads. You can check that such a dotted arrow gives precisely a C-algebra structure on X making p into a map of C-algebras, as desired!

14. Homotopy spectral sequence

We describe the Bousfield-Kan homotopy spectral sequence of a pointed cosimplicial space, and the Bousfield obstruction theory for an unpointed cosimplicial space.

14.1. Spectral sequence. A functor Δ → C is called a cosimplicial object in C. Thus

[n] ↦ Y^{n} = C- alg(C^{n+1}X, E)

is a cosimplicial space Y^{•}.

If X_{•} is a simplicial object in some category, there is a “skeletal” filtration

X_{0} ≃ |X_{•}|_{0} → |X_{•}|_{1} → |X_{•}|_{2} → ⋯ → |X_{•}|

where |X_{•}|_{k} is a quotient of ∐_{0≤n≤k} Δ_{n} ⊗ X_{n}. This gives a tower

C- alg(|X_{•}|_{0}, E) ← C- alg(|X_{•}|_{1}, E) ← ⋯ ← C- alg(|X_{•}|, E)

of spaces. If X_{•} is “Reedy cofibrant”, then the induced tower is a tower of fibrations.

The spaces in the tower, called the Tot-tower, can be derived directly from the cosimplicial space Y^{•}. The total space Tot(Y^{•}) is a subspace of ∏_{n≥0} map(Δ_{n}, Y^{n}). The kth approximation to the total space Tot^{k}(Y^{•}) is the corresponding subspace of ∏_{0≤n≤k} map(Δ_{n}, Y^{n}). One has

C- alg(|C^{•+1}|_{k}, E) ≃ Tot^{k} Y^{•}.

(The Reedy cofibrancy condition on X_{•} is the exact condition needed for the cosimplicial space Y^{•} to be fibrant in the sense of Bousfield and Kan [3].)

Suppose we have chosen a C-algebra map f: X → E. This gives us base-points C^{n+1}X → X → E in Y^{n}, and a base-point in Tot Y^{•}. The Bousfield-Kan spectral sequence of a pointed cosimplicial space [3] is a second quadrant spectral sequence “converging” to π_{∗}(Tot Y^{•}, f), where

E_{2}^{s,t} = π^{s}π_{t}(Y^{•}, f) ⟹ π_{t−s}(Tot Y^{•}, f).

There are differentials

d_{r}: E_{r}^{s,t} → E_{r}^{s+r,t+r−1}.

Some comment on the E_{2}-term is appropriate. E_{2}^{∗,t} is the “cohomotopy” of the cosimplicial object

\xymatrix{π_{t}Y^{0} \ar[r] \ar[r] & π_{t}Y^{1} \ar[r] \ar[r] \ar[r] \ar[l] & π_{t}Y^{2} \ar[r] \ar[r] \ar[r] \ar[r] \ar[l] \ar[l] & ⋯ \ar[l] \ar[l] \ar[l]}

where homotopy groups are taken at the base-point determined by f. If t ≥ 2, then the cohomotopy of the cosimplicial object is just the cohomology of the corresponding co-chain complex. The same also holds for t = 1 if all Y^{s} are look spaces. The “corner” E_{2}^{0,0} is only a pointed set, defined as the equalizer

E_{2}^{0,0} = Ker(π_{0}Y^{0} ⇉ π_{0}Y^{1}).

14.2. Obstruction theory. The above spectral sequence can be defined once we have chosen a point in Tot Y^{•}. But what if we don’t know whether such a point exists (this is one of the things we want to prove). There is an obstruction theory, due to Bousfield [2], which lets us solve this problem.

First, suppose we choose a point y_{0} ∈ Tot^{0} Y^{•} ≃ Y^{0}. This defines a class [y_{0}] ∈ π_{0}Y^{0}, which we can think of as E_{1}^{0,0}. We can lift y_{0} to Tot^{1} Y^{•} if and only if [y_{0}] is in the equalizer of π_{0}Y^{0} → π_{0}Y^{0}. That is, y_{0} lifts only if the class [y_{0}] “survives” to E_{2}^{0,0}.

The next step is to lift to Tot^{2} Y^{•}. The problem here involves extending a map from ∂Δ_{2} → Y^{2} to all of Δ_{2}; it turns out that if a point y_{0} ∈ Tot^{0} Y^{•} can be lifted to Tot^{1} Y^{•}, then the obstruction to lifting y_{0} to Tot^{2} Y^{•} lies in E_{2}^{2,1} (at least if all the π_{1}Y^{s} act trivially on π_{t}Y^{s}, so that this group is well defined; there is a description of this obstruction even when this is not true, but then the obstruction is just a relation, rather than a map).

This process continues; to lift to Tot^{3} Y^{•} we have to show that an obstruction in E_{3}^{3,2} vanishes. The result we need is

Theorem 14.3 (Bousfield [2, 6.1]). If Y^{•} is a cosimplicial space, and given a vertex y_{0} ∈ Y^{0} such that [y_{0}] ∈ Ker(π_{0}Y^{0} ⇉ π_{0}Y^{1}), and if

E_{2}^{s,s−1} = 0

for s ≥ 2, then y_{0} lifts to Tot Y^{•}.

14.4. Computing spaces of maps of A_{∞} algebras. Let E and F be A_{∞}- ring spectra (over our chosen A_{∞}-operad C), such that the underlying spectra of E and F are equivalent to E_{k_{1},Γ_{1}} and E_{k_{2},Γ_{2}} respectively. Suppose furthermore that F is cofibrant as a C-algebra spectrum. Then we can construct a simplicial resolution of F, and thereby get an obstruction theory and potentially a spectral sequence computing the homotopy of C- alg(F, E).

Theorem 14.5. In the above situation (1a)

E_{2}^{0,0} ≃ Hom_{E_{∗}- alg}(E_{∗}F, E_{∗}),

(1b)

E_{2}^{s,t} ≃ Der_{E_{∗}- alg}^{s}(E_{∗}F, E_{∗+t}), for t − s ≥ −1, t > 0.

(2a)

Hom_{E_{∗}- alg}(E_{∗}F, E_{∗}) ≃ FG((k_{1}, Γ_{1}), (k_{2}, Γ_{2})),

(2b)

Der_{E_{∗}- alg}^{s}(E_{∗}F, E_{∗+t}) ≃ 0, for all s, t ≥ 0.

Proof. Part (1a) is proved in Section 16. Part (1b) is proved in Section 18. Part (2a) is proved in Section 17. Part (2b) is proved in Sections 19, 20, and 21.

Corollary 14.6. The topological functor

A_{∞}^{LT} → FG^{op}

is fully faithful (i.e., induces weak equivalence on mapping spaces for all objects of A_{∞}^{LT}.)

Proof. Given objects E, F ∈ A_{∞}^{LT} which map to formal group laws (k_{1}, Γ_{1}) and (k_{2}, Γ_{2}) respectively, we see by Theorem 14.3 and Theorem 14.5 that A_{∞}(F, E) is a homotopy discrete space with π_{0}A_{∞}(F, E) ≃ FG((k_{1}, Γ_{1}), (k_{2}, Γ_{2})).

Part 3. Calculations

In the following sections we compute the obstruction groups and E_{2}-term of the spectral sequence described in Section 14. These calculations will also be used in Section 23 when we construct A_{∞}-structures.

15. Preliminary calculations

Let E and F be Landweber exact spectra. Then

E_{∗}F ≃ E_{∗} ⊗_{MU_{∗}} MU_{∗}MU ⊗_{MU_{∗}} F_{∗}.

Proposition 15.1. For Landweber exact theories E_{∗} and F_{∗}, the module E_{∗}F is flat over E_{∗}.

Proof. It suffices to show that E_{∗}MU is flat as an MU_{∗} module. That is, we must show that the functor

{MU_{∗}-modules} ⟶ {E_{∗}-modules}

defined by N ↦ E_{∗}MU ⊗_{MU_{∗}} N is exact. But this functor factors

{MU_{∗}-modules} ⟶ {MU_{∗}MU-comodules} ⟶ {E_{∗}-modules},

sending

N ↦ MU_{∗}MU ⊗_{MU_{∗}} N ↦ E_{∗} ⊗_{MU_{∗}} MU_{∗}MU ⊗_{MU_{∗}} N ≃ E_{∗}MU ⊗_{MU_{∗}} N,

where the left-hand functor is exact since MU_{∗}MU is a flat module over MU_{∗}, and the right-hand functor is exact by Landweber’s theorem.

15.2. Universal coefficient and Künneth theorems. Let E be a ringspectrum, and X and Y arbitrary spectra. One would like to have a Künneth spectral sequence

Tor_{s}^{E_{∗}}(E_{∗}X, E_{∗+t}Y) ⟹ E_{s+t}(X ∧ Y)

and a coefficient spectral sequence

Ext_{E_{∗}}^{s}(E_{∗}X, E_{∗+t}) ⟹ E^{t−s}X.

This doesn’t happen in general, but it does happen in many cases for good spectra E. Adams [1, part 3] gave a general criterion for the existence of such spectral sequences. We need Adams’ criterion in one case; it is a nice observation of Hopkins.

Proposition 15.3. If E is a Landweber exact theory, then there are Künneth and universal coefficient spectral sequences as described above.

Proof. Adams constructs the spectral sequences for E using the fact that for any arbitrary spectrum X there exists a map P → X of spectra such that

(a) E_{∗}P → E_{∗}X is surjective,

(b) E_{∗}P is projective over E_{∗},

(c) E_{∗}(P ∧ Y) ≃ E_{∗}P ⊗_{E_{∗}} E_{∗}Y (for the Künneth theorem), and

(d) E^{∗}P ≃ Hom_{E_{∗}}(E_{∗}P, E_{∗}) (for the coefficient theorem). In the case when E = MU, Adams shows that it suffices to take P to be an appropriate wedge of Spanier-Whitehead duals of Thom spectra of finite complex Grassmanians.

If P → X is the map Adams constructed for MU, and E is a Landweber exact theory, then properties (a), (b), (c), and (d) hold for E: (a) and (b) follow because tensoring preserves surjections and projectives, (c) is trivial, and (d) follows because P is a wedge of finite spectra T, and because E^{∗}T ≃ E_{∗}DT is a free module over E_{∗} by computation, since DT is a Thom spectrum of a finite complex Grassmanian.

Corollary 15.4. If E and F_{i} (i = 1, …, n) are Landweber exact spectra, and F ≃ F_{1} ∧ ⋯ ∧ F_{n}, then E_{∗}F is flat over E_{∗}, and

E_{∗}F ≃ E_{∗}F_{1} ⊗_{E_{∗}} ⋯ ⊗_{E_{∗}} E_{∗}F_{n}.

Proposition 15.5. If E = E_{k_{1},Γ_{1}} and F = E_{k_{2},Γ_{2}}, then the evaluation map

E^{∗}F → Hom_{E_{∗}}(E_{∗}F, E_{∗})

is an isomorphism.

Proof. By the universal coefficient theorem, we only need to prove that the Ext^{s} groups vanish for s > 0. This follows from Lemma 15.6.

Lemma 15.6. Let M be a flat E_{∗}-module. Then

Ext_{E_{∗}}^{s}(M, E_{∗}) ≃ 0

for s > 0.

Proof. Let 𝔪 = (p, u_{1}, …, u_{n−1}) denote the maximal ideal of E_{∗}. We first prove that Ext_{E_{∗}}^{s}(M, E_{∗}/𝔪^{d}) ≃ 0 for each d.

For d = 1 we have

Ext_{E_{∗}}^{s}(M, E_{∗}/𝔪) ≃ Ext_{E_{∗}/𝔪}^{s}(M ⊗_{E_{∗}} E_{∗}/𝔪, E_{∗}/𝔪)

by flat base change. Since E_{∗}/𝔪 ≃ k[u^{±}] is injective as a graded module over itself, these Ext groups are 0 for s > 0.

Note that 𝔪^{d}/𝔪^{d+1} is a finite k[u^{±}]-vector space, so Ext_{E_{∗}}^{s}(M, 𝔪^{d}/𝔪^{d+1}) ≃ 0 for s > 0. Thus for d > 1, the exact sequence

0 → 𝔪^{d}/𝔪^{d+1} → E_{∗}/𝔪^{d+1} → E_{∗}/𝔪^{d} → 0

shows that Ext_{E_{∗}}^{s}(M, E_{∗}/𝔪^{d+1}) = 0 for s > 0, as well as showing that the map Hom_{E_{∗}}(M, E_{∗}/𝔪^{d+1}) → Hom_{E_{∗}}(M, E_{∗}/𝔪^{d}) is surjective.

Finally, use the exact sequence

0 → E_{∗} → ∏ E_{∗}/𝔪^{d} ⟶^{1−S} ∏ E_{∗}/𝔪^{d} → 0

in Ext_{E_{∗}}^{s}(M, −) to prove the lemma. (Here S denotes the “shift” map.)

Remark 15.7. Note that Lemma 15.6 and hence Proposition 15.5 (and, as we shall see, Lemma 18.4 and Corollary 19.3) can fail for an arbitrary Landweber exact theory E_{∗}. For example, if E = E(n), the Johnson-Wilson theory with E(n)_{∗} ≃ ℤ_{(p)}[v_{1}, …, v_{n−1}, v_{n}^{±}], then it is not the case that Ext_{E_{∗}}^{1}(E_{∗}E, E_{∗}) vanishes, even though E_{∗}E is flat over E_{∗}.

16. Identification of E_{2}^{0,0}

In this section we show that

E_{2}^{0,0} ≃ Hom_{E_{∗}- alg}(E_{∗}F, E_{∗}),

proving part (1a) of Theorem 14.5 We need to compute the equalizer of the pair of maps

π_{0}C- alg(CF, E) ⇉ π_{0}C- alg(C^{2}F, E).

First note that for any C-algebra Y there is a natural map

π_{0}C- alg(Y, E) ⟶ Hom_{E_{∗}- alg}(E_{∗}Y, E_{∗}).

This map sends a C-algebra map f: Y → E to the map

E_{∗}Y ⟶^{E_{∗}f} E_{∗}E → E_{∗}.

Note that E_{∗}Y is in fact an associative E_{∗} algebra, since the A_{∞}-structure on Y

induces maps E_{∗}Y ⊗_{E_{∗}} E_{∗}Y → E_{∗}(Y ∧ Y) → E_{∗}Y.

Now suppose Y = CX for some spectrum X. Given an E_{∗} module M, let T M denote the E_{∗} tensor algebra on M; i.e.,

T M = ⨁_{n≥0} M^{⊗n} ≃ E_{∗} ⊕ M ⊕ (M ⊗_{E_{∗}} M) ⊕ · · · .

At the level of the homotopy category of spectra,

Y = CX ≃ ⋁_{n≥0} X^{(n)} ≃ S^{0} ∨ X ∨ X^{(2)} ∨ · · · .

The inclusion X → CX induces a map E_{∗}X → E_{∗}CX of E_{∗}-modules, and hence a map α_{X}: T (E_{∗}X) → E_{∗}CX of algebras.

Lemma 16.1. If E_{∗}X is flat over E_{∗}, then α_{X} is an isomorphism.

Proof. This is clear from Corollary 15.4.

In particular Lemma 16.1 holds for X = C^{n}F.

Now consider the commutative diagram

\xymatrix{π_{0}C- alg(CX, E) \ar[r] \ar[d] & Hom_{E_{∗}- alg}(E_{∗}CX, E_{∗}) \ar[d] \\ [X, E] \ar[r] & Hom_{E_{∗}}(E_{∗}X, E_{∗})}

The left-hand vertical arrow is an isomorphism, since CX is the free C-algebra on X. If E_{∗}X is flat over E_{∗}, the bottom and right-hand vertical arrows are isomorphisms by Proposition 15.5 and Lemma 16.1; hence the top arrow is also an isomorphism. Thus, we have reduced our problem to that of computing the equalizer of the pair of maps

Hom_{E_{∗}- alg} (T (E_{∗}F), E_{∗}) ⇉ Hom_{E_{∗}- alg} (T^{2}(E_{∗}F), E_{∗}) .

But this is precisely just the equalizer of

Hom_{E_{∗}}(E_{∗}F, E_{∗}) ⇉ Hom_{E_{∗}} (T (E_{∗}F), E_{∗})

where the top arrow sends f: E_{∗}F → E_{∗} to the composite

T (E_{∗}F) → E_{∗}F ⟶^{f} E_{∗}

and the bottom arrow sends it to the composite

T (E_{∗}F) ⟶^{T (f)} T (E_{∗}) → E_{∗}.

Thus

E_{2}^{0,0} = Hom_{E_{∗}- alg}(E_{∗}F, E_{∗}).
17. Computation of Hom_{E_{∗}- alg}(E_{∗}F, E_{∗})

In this section we show that

Hom_{E_{∗}- alg}(E_{∗}F, E_{∗}) ≃ FG((k_{1}, Γ_{1}), (k_{2}, Γ_{2})),

which proves part (2a) of Theorem 14.5.

Let E = E_{k_{1},Γ_{1}}, F = E_{k_{2},Γ_{2}}. There are universal deformations (G_{e}, i_{e}) of the formal groups (k_{e}, Γ_{e}), where G_{e} is a degree 0 formal group law over the ring

E(k_{e}, Γ_{e}) = π_{0}E_{k_{e},Γ_{e}} ≃ 𝕎k[[u_{1}, …, u_{n−1}]].

Recall from Section 6 that chosen invertible elements u_{G_{e}} ∈ π_{2}E_{k_{e},Γ_{e}} allow us to produce degree −2-formal group laws Ḡ_{e}.

We need to understand the set

Hom_{E_{∗}- alg}(E_{∗}F, B_{∗}).

Recall from Section 15 that

E_{∗}F ≃ E_{∗} ⊗_{MU_{∗}} MU_{∗}MU ⊗_{MU_{∗}} F_{∗}.

Thus by Theorem 6.2 a map E_{∗}F → B_{∗} of commutative graded rings corresponds precisely to the following data: a map ϕ: E_{∗} → B_{∗}, a map ψ: F_{∗} → B_{∗}, and a strict isomorphism ḡ: ψ^{∗}Ḡ_{2} → ϕ^{∗}Ḡ_{1} of formal group laws. Represent such data by the triple (ϕ, ḡ, ψ).

Suppose we fix once and for all an E_{∗}-algebra structure on B_{∗}, in other words, a map ϕ. Then elements of the set Hom_{E_{∗}- alg}(E_{∗}F, B_{∗}) correspond to data (ψ, ḡ). That is,

Hom_{E_{∗}- alg}(E_{∗}F, B_{∗}) ≃ {(ψ: F_{∗} → B_{∗}, ḡ: ψ^{∗}Ḡ_{2} ⟶^{∼} ϕ^{∗}Ḡ_{1})}.

Write ψ_{0} : F_{0} → B_{0} for the restriction of ψ to degree 0. Since F_{∗} ≃ F_{0}[u_{G_{2}}^{±}], the set of homomorphisms ψ: F_{∗} → B_{∗} corresponds bijectively to the set of pairs (ψ_{0}, ψ(u_{G_{2}})) where ψ_{0} is a map F_{0} → B_{0} and ψ(u_{G_{2}}) is an invertible element in B_{2}.

Recall from Section 6 that the strict isomorphism ḡ: ψ^{∗}Ḡ_{2} → ϕ^{∗}Ḡ_{1} of degree −2 formal group laws and the elements ϕ(u_{G_{1}}) and ψ(u_{G_{2}}) induce an isomorphism g: ψ_{0}^{∗}G_{2} → ϕ_{0}^{∗}G_{1} of degree 0 formal group laws, not necessarily strict, by g(x) = ϕ(u_{G_{1}})ḡ(ψ(u_{G_{2}})^{−1}x). Note that if ḡ(x) = x + a_{2}x^{2} + a_{3}x^{3} + ⋯, then

g(x) = (ϕ(u_{G_{1}})ψ(u_{G_{2}})^{−1})x + a_{2}(ϕ(u_{G_{1}})ψ(u_{G_{2}})^{−2})x^{2} + · · · .

Thus, given g(x) and ϕ(u_{G_{1}}), we can recover ψ(u_{G_{2}}) (since ϕ(u_{G_{1}}) = g′(0)ψ(u_{G_{2}})), and hence recover ḡ(x).

Putting this together, we see that the data (ψ, ḡ) corresponds bijectively to data (ψ_{0}, g). Thus, we have proved

Proposition 17.1. There is an isomorphism of sets

Hom_{E_{∗}- alg}(E_{∗}F, B_{∗}) ≃ {(ψ_{0} : F_{0} → B_{0}, g: ψ^{∗}G_{2} ⟶^{∼} ϕ^{∗}G_{1})}.

Remark 17.2. There is another way to describe this set. It is easy to see that E_{∗}F ≃ E_{0}F [u_{G_{1}}], so that

Hom_{E_{∗}- alg}(E_{∗}F, B_{∗}) ≃ Hom_{E_{0}- alg}(E_{0}F, B_{0}).

Thus, a map from E_{0}F to a ring B corresponds precisely to: a map ϕ: E_{0} → B, a map ψ: F_{0} → B, and an isomorphism g: ψ^{∗}G_{2} ⟶^{∼} ϕ^{∗}G_{1} (not necessarily strict).

Equivalently, there is a square diagram of groupoids

\xymatrix{Hom_{Ring}(E_{0}F, B) \ar[r] \ar[d] & Hom_{Ring}(E_{0}, B) \ar[d] \\ Hom_{Ring}(F_{0}, B) \ar[r] & fgl(B)}

commuting up to a natural isomorphism, which is a homotopy pull-back square. (A set S is regarded as a groupoid having as objects the set S, and as maps only identity maps.) This way of viewing things will come in handy in Section 21.

Suppose now that B = B_{0} is a complete local ring, and that B is a continuous E_{0}-algebra, i.e., the map ϕ_{0} : E_{0} → B is a continuous map between complete local rings. The continuity hypothesis is equivalent to requiring that ϕ_{0}(𝔪_{E}) ⊂ 𝔪_{B}, where 𝔪_{E} = (p, u_{1}, …, u_{n−1}) ⊂ E_{0} is the maximal ideal. Given some E_{0}-algebra map E_{0}F → B, I would like to know that the induced map F_{0} → E_{0}F → B between complete rings is continuous. This is ensured by the following well-known lemma.

Lemma 17.3. Let 𝔪_{E} ⊂ E_{0} and 𝔪_{F} ⊂ F_{0} be the maximal ideals of these rings. Then 𝔪_{E}E_{0}F = 𝔪_{F} E_{0}F as ideals of E_{0}F.

Proof. The ideal 𝔪_{E} is generated by the coefficients of the terms of the pseries [p]_{G_{1}}(x) which are of degree less than p^{n}. Likewise, the ideal 𝔪_{F} is generated by the coefficients of the terms of the p-series [p]_{G_{2}}(x) which are of degree less than p^{n}. There is an isomorphism of formal group laws g: G_{1} → G_{2} defined over the ring E_{0}F, and hence an equation g ∘[p]_{G_{1}} = [p]_{G_{2}} ∘g. The lemma follows easily.

We have fixed a map ϕ_{0} : E_{0} → B and hence a formal group law ϕ^{∗}G_{1} over B which is a deformation of (k_{1}, Γ_{1}) to B. Maps ψ: F_{0} → B classify ⋆-isomorphism classes of deformations of (k_{2}, Γ_{2}) to B. Consider the diagram of groupoids

\xymatrix{P \ar[r] \ar[d] & \{ϕ^{∗}G_{1}\} \ar[d] \\ Def_{Γ_{2}}(B) \ar[r] \ar[d] & fgl(B) \ar[d] \\ ∐_{j: k_{2}→B/𝔪_{B}}\{i^{∗}Γ_{2}\} \ar[r] & fgl(B/𝔪_{B})}

in which both squares are homotopy-pullback squares of groupoids. We get two descriptions of P from this diagram. The first comes from the top square: since Def_{Γ_{2}}(B) is homotopy discrete and π_{0} Def_{Γ_{2}}(B) ≃ Hom_{Ring}^{cts}(F_{0}, B), we see that P is equivalent to a disjoint union of a bunch of loop-spaces of fgl(B), and in fact P is homotopy discrete with

π_{0}P ≃ ∐_{ψ: F_{0}⟶^{cts}B} Hom_{fgl(B)}(ϕ^{∗}G_{1}, ψ^{∗}G_{2}).

Thus, P ≃ Hom_{E_{∗}- alg}(E_{∗}F, B_{∗}) by Proposition 17.1 and Lemma 17.3.

The second description comes from the big rectangle. This also describes P as a union of loop-spaces, and shows that P is homotopy discrete with

π_{0}P ≃ ∐_{j: k_{2}→B/𝔪_{B}} Hom_{fgl(B/𝔪_{B})}(i^{∗}Γ_{1}, j^{∗}Γ_{2}),

where i: k_{1} → B/𝔪_{B} is the map induced by ϕ_{0} : E_{0} → B modulo maximal ideals. Thus we conclude

Hom_{E_{∗}- alg}(E_{∗}F, B_{∗}) ≃ P ≃ FG((B/𝔪_{B}, i^{∗}Γ_{1}), (k_{2}, Γ_{2})).

If we let B_{∗} = E_{∗} with ϕ = id, then we get an isomorphism

Hom_{E_{∗}- alg}(E_{∗}F, E_{∗}) ≃ FG((k_{1}, Γ), (k_{2}, Γ))

as desired.

Remark 17.4. Note that if we let B_{∗} = E_{∗}/𝔪^{d} for any d ≥ 1 we still get an isomorphism

Hom_{E_{∗}- alg}(E_{∗}F, E_{∗}/𝔪^{d}) ≃ FG((k_{1}, Γ_{1}), (k_{2}, Γ_{2})).

In particular, this holds for d = 1. If we restrict to degree 0 and then tensor down via the map E_{0} → E_{0}/𝔪 = k_{1}, we see that

Hom_{k_{1}- alg}(E_{0}F/𝔪, k_{1}) ≃ FG((k_{1}, Γ_{1}), (k_{2}, Γ_{2}))

where 𝔪 denotes the ideal generated by p, u_{1}, …, u_{n−1} ∈ E_{0} ⊂ E_{0}F. This means one can give a description of E_{0}F/𝔪 in many cases using formal group law theory.

For example, if Γ_{1} = Γ_{2} = Γ is the Honda formal group law, so that E = F = E_{n}, then FG((𝔽_{p^{n}}, Γ), (𝔽_{p^{n}}, Γ)) is isomorphic to the set of pairs (σ, f) where σ: 𝔽_{p^{n}} → 𝔽_{p^{n}} is a Galois automorphism, and f(x) = e_{0}x +_{Γ} e_{1}x^{p} +_{Γ} e_{2}x^{p^{2}} +_{Γ} ⋯ is a

power series with coefficients e_{i} ∈ 𝔽_{p^{n}}, with e_{0} ≠ 0 (see [13, App. 2]). This shows that

E_{0}F/𝔪 ≃ (𝔽_{p^{n}} ⊗_{𝔽_{p}} 𝔽_{p^{n}})[e_{0}, e_{1}, e_{2}, …]/(e_{0}^{p^{n}−1}− 1, e_{1}^{p^{n}}− e_{1}, e_{2}^{p^{n}}− e_{2}, …).

(Note that a map of 𝔽_{p^{n}}-algebras 𝔽_{p^{n}} ⊗_{𝔽_{p}} 𝔽_{p^{n}} → 𝔽_{p^{n}} corresponds precisely to a map

of rings σ: 𝔽_{p^{n}} → 𝔽_{p^{n}}.)

18. Derivations of associative algebras

In this section we define the derived functors of associative algebra derivations and show part (1b) of Theorem 14.5.

18.1. A complex for the E_{2}-term. Suppose Y is a C-algebra, and choose a map f: Y → E. The set π_{t}(C- alg(Y, E), f) is the same as homotopy classes of maps S^{t} → C- alg(Y, E) which send the base-point of the sphere to f, and where homotopies are base-point preserving.

\xymatrix{pt \ar[r]^{f} \ar[d] & C- alg(Y, E) \\ S^{t} \ar@{-->}[ur]^{g}}

By adjunction, these are the same as homotopy classes of lifts in the diagram

\xymatrix{ & E^{S^{t}} \ar[d] \\ Y \ar[r]^{f} \ar@{-->}[ur]^{ĝ} & E}

of C-algebras, where homotopies are ones which are fiberwise over f. Recall from Section 12.2 that E^{S^{t}} is a C-algebra whose underlying spectrum is isomorphic to F (Σ^{∞}S_{+}^{t}, E) ≃ E ∨ Σ^{−t}E. Thus π_{∗}E^{S^{t}} ≃ E^{−∗}S^{t} ≃ E_{∗}[ϵ]/ϵ^{2}, with |ϵ| = −t. The spectrum E^{S^{t}} is naturally a ring spectrum over E. Applying E_{∗}(−) everywhere and using the ring structure of E_{∗} gives us a diagram of algebras

\xymatrix{ & E_{∗}E^{S^{t}} \ar[d] \ar[r] & E_{∗}[ϵ]/ϵ^{2} \ar[d] \\ E_{∗}Y \ar[r]^{E_{∗}f} \ar@{-->}[ur]^{E_{∗}ĝ} & E_{∗}E \ar[r] & E_{∗}}

and thus we get a map

π_{t} (C- alg(Y, E), f) → Hom_{(E_{∗}- alg↓E_{∗})}(E_{∗}Y, E_{∗}[ϵ_{t}]/ϵ_{t}^{2}),

where the right hand side denotes maps in the category of E_{∗}-algebras augmented over E_{∗}. Note that E_{∗}Y is augmented over E_{∗}, via the composite E_{∗}Y → E_{∗}E → E_{∗}; this augmentation itself depends on the choice of f: Y → E.

Now suppose Y = CX where E_{∗}X is a flat E_{∗}-module. Then using the commutative diagram

\xymatrix{π_{0}C- alg(CX, E^{S^{t}}) \ar[r] \ar[d] & Hom_{E_{∗}- alg}(T (E_{∗}X), E_{∗}[ϵ_{t}]/ϵ_{t}^{2}) \ar[d] \\ [X, E ∨ Σ^{−t}E] \ar[r] & Hom_{E_{∗}}(E_{∗}X, E_{∗} ⊕ E_{∗+t})}

and an argument similar to that of Section 16 one shows that the top horizontal map is a bijection. Thus, we have

E_{2}^{s,t} ≃ H^{s} [Hom_{(E_{∗}- alg↓E_{∗})}(T^{•+1}(E_{∗}F), E_{∗}[ϵ_{t}]/ϵ_{t}^{2})] ,

for t > 0 and t − s ≥ −1.

18.2. Associative algebra derivations. Let A be a commutative ring, and suppose R is an associative A-algebra. Let M be an R-bimodule relative to A; that is, M is a module over the ring R^{e} = R ⊗_{A} R^{op}. Then form a ring R ⊕ M with

multiplication given by

(r, m) · (r′, m′) = (rr′, rm′ + mr′)

for r, r′ ∈ R, m, m′ ∈ M. The inclusion map η: R → R ⊕ M and the projection map π: R ⊕ M → R are algebra maps, and M ⊂ R ⊕ M is an ideal such that M^{2} = 0.

Consider sections s: R → R ⊕ M of the projection π. Such an s = (id, D) where D: R → M is an A-module map such that

D(rr′) = r(Dr′) + (Dr)r′

for r, r′ ∈ R. Such a map D: R → M is called an A-derivation of R into M. The set of A-derivations is written Der_{A}(R, M), whence

Hom_{(A- alg↓R)}(R, R ⊕ M) ≃ Der_{A}(R, M).

More generally, let ϕ: R → S be a map of associative A-algebras, and let M be an S-bimodule. The map ϕ makes M into an R-bimodule by restriction. Hence

Hom_{(A- alg↓S)}(R, S ⊕ M) ≃ Der_{A}(R, M).

Note that Der_{A}(R, M) is naturally an A-module. If A = S = E_{∗}, R = E_{∗}Y, and M = E_{∗+t}, then this module is

Der_{E_{∗}}(E_{∗}Y, E_{∗+t}) ≃ Hom_{(E_{∗}- alg↓E_{∗})}(E_{∗}Y, E_{∗}[ϵ_{t}]/ϵ_{t}^{2}).

Thus,

E_{2}^{s,t} ≃ H^{s} [Der_{E_{∗}}(T^{•+1}(E_{∗}F), E_{∗+t})] .

18.3. Derived functors of associative algebra derivations. Let sA- alg denote the category of simplicial associative A-algebras. This has a model category structure as follows. The weak equivalences f: R_{•} → S_{•} are maps which induce isomorphisms on homology of the associated chain-complexes. A cofibration f is a retract of a free map. A free map f: R_{•} → S_{•} is a map which in each simplicial degree is an inclusion

R_{n} → S_{n} ≃ R_{n}⟨X_{n}⟩,

where R⟨X⟩ denotes the algebra obtained from R by adjoining the elements of the set X as free generators, and such that the the sets X_{•} are closed under the degeneracy operators of S_{•}. (See Quillen [11].)

A cofibrant resolution of a simplicial algebra R_{•} is a diagram A ⟶^{i} P_{•} ⟶^{f} R_{•}, where i is a cofibration, and f is a weak equivalence. There is a functorial cofibrant resolution P_{•} of R, given by P_{n} = F^{n+1}(R), where F B denotes the free associative A-algebra on the underlying set of R. Another resolution is the complex Q_{n} = T^{n+1}(R). It is a cofibrant resolution if R is a projective A-module; this is easy to see if R is a free A-module, since if X is a set of module generators of R, then the set of monomials in X is a set of module generators of T R.

Given an associative A-algebra R over S, and an S-bimodule M, define

Der_{A}^{s}(R, M) = H^{s}[Der_{A}(P_{•}, M)],

the s-th derived functor of derivations from B to M, where P_{•} → R is a cofibrant resolution of R. The derived functors do not depend on the choice of cofibrant resolution.

Our complex T^{n+1}(E_{∗}F) of E_{∗}-algebras is not cofibrant, since E_{∗}F is not projective over E_{∗}; nonetheless, we will show below that this complex computes derived functors of derivations.

Suppose R = T N, where N is an A-module, and let M be an S-bimodule. Then if P_{•} → N is a free resolution of N by a simplicial A-module, then T P_{•} is a cofibrant resolution of R = T N. Then, since

Der_{A}(T N, M) ≃ Hom_{(A- alg↓S)}(T N, S ⊕ M) ≃ Hom_{A}(N, M)

it is not hard to see that

Der_{A}^{s}(R, M) ≃ Ext_{A}^{s}(N, M).

Thus, Lemma 15.6 gives

Lemma 18.4. If we take A = S = E_{∗}, M = E_{∗+t}, and R = T N where N is a flat E_{∗}-module, then

Der_{E_{∗}}^{s}(T N, E_{∗+t}) = 0, s > 0.

18.5. The E_{2}-term is a derived functor. Suppose R_{•} is a simplicial Aalgebra, augmented over S, and let M be an S-module Then there is a spectral sequence of A-modules

E_{1}^{p,q} = Der_{A}^{p}(R_{q}, M) ⟹ Der_{A}^{p+q}(R_{•}, M).

This spectral sequence is constructed by producing a double complex P_{p,q} of Aalgebras, in which each row P_{•,q} → R_{q} is a cofibrant resolution of R_{q}, and so that the diagonal is a cofibrant resolution of R_{•}. Such a resolution can be constructed by the functorial cofibrant resolution P_{p,q} = F^{p+1}R_{q}.

Now letting A = S = E_{∗}, M = E_{∗+t} and R_{q} = T^{q+1}(E_{∗}F) we see from Lemma 18.4 applied to N = E_{∗}F that the spectral sequence at the E_{1}-term is

E_{1}^{p,q} = Der_{E_{∗}}^{p}(T^{q+1}(E_{∗}F), E_{∗+t}) = \begin{cases}0 & if p > 0, \\ Der_{E_{∗}}(T^{q+1}(E_{∗}F), E_{∗+t}) & if p = 0. \end{cases}

Thus the spectral sequence is concentrated in a single column of the E_{1}-term, and collapses at E_{2}. In fact, E_{1}^{0,∗} is precisely the cochain complex whose cohomology we want to compute, and by the convergence of the spectral sequence, what it computes is Der_{E_{∗}}^{s}(E_{∗}F, E_{∗+t}).

19. Change of ground ring

In this section we show that in order to prove that Der_{E_{∗}}^{s}(E_{∗}F, E_{∗+t}) vanishes, it suffices to show that Der_{E_{0}/𝔪}^{s}(E_{0}F/𝔪, E_{t}/𝔪) vanishes.

19.1. Associative algebra differentials. Given an A-algebra R, define I_{R/A} to be an R-bimodule equipped with a A-derivation d: R → I_{R/A} which induces an isomorphism

Der_{A}(R, M) ≃ Hom_{R^{e}}(I_{R/A}, M).

Such an object always exists: I_{R/A} can be constructed as the cokernel of δ in

R ⊗_{A} R ⊗_{A} R ⊗_{A} R ⟶^{δ} R ⊗_{A} R ⊗_{A} R ⟶^{p} I_{R/A} → 0

where

δ(a ⊗ b ⊗ c ⊗ d) = ab ⊗ c ⊗ d − a ⊗ bc ⊗ d + a ⊗ b ⊗ cd.

The universal derivation d: R → I_{R/A} is given by d(r) = p(1 ⊗ r ⊗ 1). By the bar complex, we see that I_{R/A} is also the kernel of μ in

0 → I_{R/A} → R ⊗_{A} R ⟶^{μ} R.

We call I_{R/A} the R-bimodule of associative algebra differentials of R.

Let R = T N be the tensor algebra on an A-module N. Then derivations D: T N → M to some R-bimodule M correspond bijectively to maps

Hom_{(A- alg↓R)}(T N, R ⊕ M) ≃ Hom_{A}(N, M),

and hence

Der_{A}(T N, M) ≃ Hom_{A}(N, M).

Since I_{T N/A} should be an R-bimodule, we see that

I_{T N/A} ≃ T N ⊗_{A} N ⊗_{A} T N.

Note that this means that I_{R/A} is a free R-bimodule when R is a free algebra.

Suppose R is an associative A-algebra. Define

D_{R/A} = I_{P_{•}/A} ⊗_{P_{•}^{e}} R^{e}

for some choice P_{•} → R of cofibrant resolution of R. Thus D_{R/A} is a complex of projective R-bimodules, and

Ext_{R^{e}}^{s}(D_{R/A}, M) ≃ H^{s}[Hom_{R^{e}}(I_{P_{•}/A} ⊗_{P_{•}^{e}} R^{e}, M)]

≃ H^{s}[Der_{A}(P_{•}, M)] ≃ Der_{A}^{s}(R, M).

Here “Ext” really means some kind of “hyper-Ext”.

We therefore get a coefficient long exact sequence for derived functors of derivations.

Proposition 19.2. Let 0 → M′ → M → M′′ → 0 be a short exact sequence of R-bimodules. Then there is a long exact sequence

0 → Der_{A}(R, M′) → Der_{A}(R, M) → Der_{A}(R, M′′) → Der_{A}^{1}(R, M) → · · · .

Corollary 19.3. If Der_{E_{∗}}^{s}(E_{∗}F, E_{∗+t}/𝔪) = 0 for all s, then

Der_{E_{∗}}^{s}(E_{∗}F, E_{∗+t}) = 0

for all s.

Proof. The proof is similar to that of Lemma 15.6.

19.4. Change of ground ring. Suppose A → A′ is a map of commutative rings. Then given an A-algebra R we get an induced A′-algebra R′ = R ⊗_{A} A′.

Proposition 19.5. If R is flat as an A-module, and M is an R′-bimodule, then

Der_{A}^{s}(R, M) ≃ Der_{A′}^{s}(R′, M).

Proof. Let P_{•} → R be a cofibrant A-algebra resolution of R. Then P′_{•} = P_{•} ⊗_{A} A′ is a cofibrant A′-algebra, since A⟨X⟩ ⊗_{A} A′ ≃ A′⟨X⟩ for a set X. Since P_{•} is

also a complex of projective A-modules, it forms a projective A-module resolution of R. Thus, if R is flat, P′_{•} is a cofibrant resolution of R′.

Using E_{∗} → E_{∗}/𝔪 and the flatness of E_{∗}F over E_{∗}, we get

Corollary 19.6. There is an isomorphism

Der_{E_{∗}}^{s}(E_{∗}F, E_{∗+t}/𝔪) ≃ Der_{E_{∗}}^{s}(E_{∗}F/𝔪, E_{∗+t}/𝔪).

Using E_{0}/𝔪 → E_{∗}/𝔪 and the fact that E_{0}/𝔪 is a field, we get

Corollary 19.7. There is an isomorphism

Der_{E_{∗}}^{s}(E_{∗}F/𝔪, E_{∗+t}/𝔪) ≃ Der_{E_{0}/𝔪}^{s}(E_{0}F/𝔪, E_{t}/𝔪).

20. Reduction to commutative algebras

20.1. Commutative algebra derivations. We now let B be a commutative A-algebra, and let M be a B-module. Then B ⊕ M can be given the structure of an A-algebra augmented over B. The sections of this augmentation correspond to (commutative) A-algebra derivations of B into M. A commutative algebra derivation is the same thing as an associative algebra derivation for which M is viewed as a B-bimodule with the same B-module structure on both sides.

There are natural isomorphisms

Der_{A}(B, M) ≃ Hom_{B^{e}}(I_{B/A}, M) ≃ Hom_{B}(I_{B/A} ⊗_{B^{e}} B, M)

using the map B^{e} = B ⊗_{A} B → B. We call Ω_{B/A} = I_{B/A} ⊗_{B^{e}} B the B-module of

commutative algebra differentials on B; there is a universal derivation d: B → Ω_{B/A}. Because of the universal property of Ω_{B/A}, we see that Ω_{B/A} may also be described as the free B-module on elements dx for x ∈ B, modulo the relations da = 0 for a ∈ A and d(xy) = xdy + ydx. In particular, if B = A[X] is a polynomial algebra generated by a set X, then Ω_{B/A} is a free B-module generated by the set of symbols {dx | x ∈ X}.

The category of simplicial commutative algebras has a model category structure, in which a weak equivalence B_{•} → B′_{•} is a map which induces an isomorphism on the homology of the associated chain complex, and a cofibration is a retract of a free map. A free map f: B_{•} → B′_{•} is a map which in each simplicial degree is an inclusion

B_{n} → B′_{n} ≃ B_{n}[X_{n}],

where B[X] denotes the polynomial algebra over B generated by elements of X, and such that the sets X_{•} are closed under degeneracy operators of B′_{•}. (See Quillen [11].)

Thus, we can define derived functors of commutative algebra derivations, called André-Quillen cohomology, by

H_{A}^{s}(B, M) = H^{s}[Der_{A}(Q_{•}, M)]

where Q_{•} → B is a cofibrant resolution of B as a commutative algebra. (This is Quillen’s construction of André-Quillen cohomology; see Quillen [12].)

If Q_{•} → B is a cofibrant commutative algebra resolution of B, write L_{B/A} =

Ω_{Q_{•}/A} ⊗_{Q_{•}} B. Then

H_{A}^{s}(B, M) ≃ H^{s}[Hom_{B}(L_{B/A}, M)].

20.2. Comparing associative and commutative derivations. Let’s fix a commutative A-algebra B and a B-module M. Then let

Λ_{B/A} ≃ D_{B/A} ⊗_{B^{e}} B.

Thus, Λ_{−/A} is a construction which produces a simplicial B-module from the commutative A-algebra B. It has the property that

Der_{A}^{s}(B, M) ≃ H^{s}[Hom_{B}(Λ_{B/A}, M)].

On the other hand, we have the gadget L_{B/A} which is another simplicial B-module. We want to show that if L_{B/A} is acyclic (i.e., H_{s}L_{B/A} ≡ 0 for all s), then Λ_{B/A} is acyclic. In fact

Proposition 20.3. If B is a commutative ring which is flat over A, then there is a first quadrant spectral sequence

H_{s} (Λ_{B}^{t+1}(L_{B/A})) ⟹ H_{s+t}Λ_{B/A}.

(Here Λ_{B}^{d} denotes the dth exterior power functor; Λ_{B}^{t+1}L_{B/A} is the simplicial B-module obtained by applying the (t + 1)st exterior power functor degree-wise to L_{B/A}.)

Corollary 20.4. Let A = E_{0}/𝔪 = k_{1} and B = E_{0}F/𝔪. If L_{B/A} is acyclic, then Λ_{B/A} is acyclic, whence

Der_{E_{0}/𝔪}^{s}(E_{0}F/𝔪, E_{t}) = 0.

Proof. Since L_{B/A} is an acyclic simplicial module which is degree-wise projective, it has a simplicial contracting homotopy. Hence so does Λ_{B}^{t+1}L_{B/A}, whence the result follows by Proposition 20.3.

20.5. Construction of the spectral sequence. First, let Q_{•} → B be a vertical cofibrant resolution of B as a commutative algebra; without loss of generality we can assume that each Q_{p} is a polynomial algebra. Now, let P_{p,•} → Q_{p} be a horizontal functorial cofibrant resolution of Q_{p} as an associative algebra. Hence, we get a double complex P_{•,•} of associative algebras; the diagonal of this double complex is a cofibrant resolution of B as an associative algebra.

We apply the functor P ↦ I_{P/A} ⊗_{P^{e}} B to each spot in the double complex.

The total homology of the resulting complex of B modules is precisely H_{∗}(Λ_{B/A}). Taking horizontal homology in the complex we get

E_{1}^{p,q} = H_{q}(I_{P_{p,•}} ⊗_{P_{p,•}^{e}} B) ≃ H_{q}(Λ_{Q_{p}/A} ⊗_{Q_{p}} B).

Lemma 20.6. Given a flat commutative A-algebra Q there is a natural isomorphism

Tor_{q+1}^{Q^{e}}(Q, Q) ⟶^{∼} H_{q}(Λ_{Q/A}),

and a natural map

i: Ω_{Q/A} ⟶^{∼} H_{0}Λ_{Q/A}.

Lemma 20.6 is proved below.

Note that Tor_{∗}^{Q^{e}}(Q, Q) has the structure of a graded commutative algebra. This algebra structure arises from the fact that there are algebra maps Q → Q^{e} = Q ⊗_{A} Q → Q. Furthermore, if Q is a polynomial algebra, then Tor_{∗}^{Q^{e}}(Q, Q) is an

exterior algebra generated by Tor_{1}. Thus, in the case that Q is a polynomial algebra over A, (whence Q^{e} is a polynomial algebra over Q,) the natural map i extends in a natural way to an isomorphism of algebras

Λ_{B}^{∗+1}(Ω_{Q/A}) ⟶^{∼} Tor_{∗+1}^{Q^{e}}(Q, Q) ⟶^{∼} H_{∗}(Λ_{Q/A}).

In particular,

H_{q}(Λ_{Q_{p}/A}) ≃ Λ_{Q_{p}}^{q+1}(Ω_{Q_{p}/A})

whence

E_{1}^{p,q} ≃ Λ_{Q_{p}}^{q+1}(Ω_{Q_{p}/A}) ⊗_{Q_{p}} B ≃ Λ_{B}^{q+1}(Ω_{Q_{p}/A} ⊗_{Q_{p}} B).

(This last follows because Λ_{Q_{p}}^{∗}(Ω_{Q_{p}/A}) is a projective Q_{p}-module.) The right-hand side of this equation is the pth degree of Λ_{B}^{q+1}L_{B/A}, and thus Proposition 20.3 holds.

Proof of Lemma 20.6. Recall that Λ_{Q/A} = I_{P_{•}/A} ⊗_{P_{•}^{e}} Q where P_{•} → Q is a

cofibrant resolution of Q by associative algebras, and that there are exact sequences

\xymatrix{0 \ar[r] & I_{P_{•}/A} \ar[r] \ar[d]_{f} & P_{•} ⊗_{A} P_{•} \ar[r] \ar[d] & P_{•} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & I_{Q/A} \ar[r] & Q ⊗_{A} Q \ar[r] & Q \ar[r] & 0}

The right-hand vertical arrow induces an isomorphism on homology by construction. Since Q is flat over A the middle vertical arrow also induces an equivalence on homology. Hence by the 5-lemma the map f is an equivalence.

This map f factors into a sequence of maps of simplicial modules

I_{P_{•}/A} ≃ I_{P_{•}/A} ⊗_{P_{•}^{e}} P_{•}^{e} → I_{P_{•}/A} ⊗_{P_{•}^{e}} Q^{e} → I_{Q/A}.

The left-hand arrow in this sequence is an equivalence by a straightforward spectral sequence argument (since I_{P_{•}/A} is a projective P_{•}^{e}-module). Since the composite arrow is an equivalence, it follows that I_{P_{•}/A} ⊗_{P_{•}^{e}} Q^{e} is a projective Q-bimodule

resolution of I_{Q/A}.

Hence the complex I_{P_{•}/A} ⊗_{P_{•}^{e}} Q^{e} ⊗_{Q^{e}} Q ≃ I_{P_{•}/A} ⊗_{P_{•}^{e}} Q (which is precisely Λ_{Q/A})

computes Tor_{∗}^{Q^{e}}(I_{Q/A}, Q). Using the long exact sequence of Tor applied to the sequence 0 → I_{Q/A} → Q^{e} → Q → 0, and the fact that Tor_{∗}^{Q^{e}}(Q^{e}, Q) vanishes for ∗ > 0, we see that

Tor_{q+1}^{Q^{e}}(Q, Q) ⟶^{∼} H_{q}(Λ_{Q/A})

as desired.

21. End of the calculation

In this section we show that the André-Quillen cohomology groups, denoted H_{E_{0}/𝔪}^{s}(E_{0}F/𝔪, E_{t}), vanish, which by Corollary 20.4, Corollary 19.4, Corollary 19.6, and Corollary 19.3 proves part (2b) of Theorem 14.5.

21.1. Differentials and the Frobenius map. Suppose that A is an 𝔽_{p}- algebra, and B an A-algebra. Then the map σ_{B}: B → B defined by σ_{B}(x) = x^{p} is a homomorphism. For an algebra map A → B we get a commutative diagram

\xymatrix{B \ar[r]^{σ_{B}} & B \\ A \ar[r]^{σ_{A}} \ar[u] & A \ar[u]}

Of course, σ_{B} is not in general a map of A-algebras. Let σ^{∗}B denote the pushout of commutative rings in

\xymatrix{B \ar[r] & σ^{∗}B \\ A \ar[u] \ar[r]^{σ_{A}} & A \ar[u]}

Then there is an A-algebra map ϕ: σ^{∗}B → B (the Frobenius map). If B = A[X] is a polynomial algebra over A generated by the set X, then σ^{∗}B ≃ A[X], and ϕ sends x ∈ X to x^{p}.

Proposition 21.2. For any A-algebra B as above, the map

L_{σ^{∗}B/A} → L_{B/A}

induced by ϕ: σ^{∗}B → B is the 0 map.

Proof. Let Q_{•} → B be a cofibrant commutative A-algebra resolution of B. We may suppose that Q_{n} = A[X_{n}] for some set X_{n}. Then σ^{∗}Q_{•} → σ^{∗}B is a commutative A-algebra resolution of σ^{∗}B. If we apply the differentials functor Ω_{−/A} we get induced maps which send

dx ↦ dϕ(x) = d(x^{p}) = px^{p−1}dx = 0

for each x ∈ X_{n} ⊂ σ^{∗}Q_{n}. Thus, on the level of complexes, we get the zero map L_{σ^{∗}B/A} → L_{B/A}.

Corollary 21.3. If B is an A-algebra and if the maps σ_{A} and σ_{B} are isomorphisms, then H_{∗}(L_{B/A}) ≡ 0.

Proof. If σ_{A} and σ_{B} are isomorphisms, then ϕ: σ^{∗}B → B is an isomorphism of A-algebras, whence Lϕ is an equivalence. But Lϕ ∼ 0 by Proposition 21.2.

21.4. The Frobenius map on E_{0}F/𝔪. Recall from Remark 17.4 that if E = F = E_{n}, then one can compute that

B = E_{0}F/𝔪 ≃ (𝔽_{p^{n}} ⊗_{𝔽_{p}} 𝔽_{p^{n}})[e_{0}, e_{1}, e_{2}, …]/(e_{0}^{p^{n}−1}− 1, e_{1}^{p^{n}}− e_{1}, e_{2}^{p^{n}}− e_{2}, …).

Clearly the Frobenius maps on A = 𝔽_{p^{n}} and on B are isomorphisms, so in this case we are done.

More generally, let E = E_{k_{1},Γ_{1}} and F = E_{k_{2},Γ_{2}}, and let A = E_{0}/𝔪 = k_{1}, B = E_{0}F/𝔪 and let Q be a commutative ring of characteristic p. Then given a map of rings f: B → Q there is a commutative diagram

\xymatrix{B \ar[d]^{f} \ar[r]^{σ_{B}} & B \ar[d]^{f} \\ Q \ar[r]^{σ_{Q}} & Q}

In other words, there are two maps

(σ_{B})_{∗}, (σ_{Q})^{∗}: Hom_{Ring}(B, Q) ⇉ Hom_{Ring}(B, Q);

the top map takes f to σ_{Q}f, the bottom map to fσ_{B}, and (σ_{B})_{∗} = (σ_{Q})^{∗}. Thus to show that σ_{B} is an isomorphism, it suffices to show that (σ_{Q})^{∗} is an isomorphism for all Q.

Recall (from Remark 17.2) our homotopy pull-back square

\xymatrix{Hom_{Ring}(E_{0}F/𝔪, Q) \ar[r] \ar[d] & Hom_{Ring}(k_{2}, Q) \ar[d] \\ Hom_{Ring}(k_{1}, Q) \ar[r] & fgl(Q)}

which is functorial in Q. The automorphism σ_{Q} induces a map between such squares. Since k_{1} and k_{2} are perfect fields, the induced pth-power maps (σ_{Q})^{∗} on Hom_{Ring}(k_{1}, Q) and Hom_{Ring}(k_{2}, Q) are isomorphisms.

Let fgl_{n}(Q) ⊂ fgl(Q) denote the full subgroupoid of formal group laws of height n. (Although Q is only a ring of characteristic p, and not in general a field, we define height as before: a formal group law F has height n if [p]_{F} (x) = ax^{p^{n}} + ⋯ where a ∈ Q is an invertible element. Under this definition it is possible for a formal group law over Q to not have a well-defined height. Note however that if f: F → G is an isomorphism of formal group laws over Q, and if F is of height n, then G is also of height n, since [p]_{G} = f ∘ [p]_{F} ∘ f^{−1}.)

Proposition 21.5. There is a natural isomorphism of the functor

(σ_{Q}^{n})^{∗}: fgl_{n}(Q) → fgl_{n}(Q)

with the identity functor, whence σ_{Q}^{∗} is an equivalence of categories.

Corollary 21.6. The Frobenius map on E_{0}F/𝔪 is an isomorphism.

Recall that there is a homomorphism α_{F} : F → σ^{∗}F of formal group laws given by α_{F} (x) = x^{p}.

Lemma 21.7. Given a map f: F → G of formal group laws over a ring of characteristic p, such that f′(0) = 0, there exists a unique map g: σ^{∗}F → G of formal group laws such that g ∘ α_{F} = f.

Proof. Because f(F (x, y)) = G(fx, fy), applying \frac{∂}{∂y} and setting y = 0 gives f′(x)F_{2}(x, 0) = G_{2}(fx, 0)f′(0), whence f′(x) = 0. Thus f(x) = g(x^{p}) for a (unique) power series g. Since

g(σ^{∗}F (x^{p}, y^{p})) = g(F (x, y)^{p}) = f(F (x, y)) = G(fx, fy) = G(g(x^{p}), G(y^{p}))

we see that g(x +_{F} y) = g(x) +_{G} g(y).

Proof of Proposition 21.5. If F is a formal group law of height n, then the p-series [p]_{F} (x): F → F has the form [p]_{F} (x) = ax^{p^{n}} + ⋯, with a invertible. Thus by Lemma 21.7 there exists a unique isomorphism β_{F} : (σ^{n})^{∗}F → F such that β_{F} ∘ α_{F}^{n} = [p]_{F}.

If h: F → G is a map between height n formal group laws, we see that

h(β_{F} (x^{p^{n}})) = h([p]_{F} (x)) = [p]_{G}(h(x)) = β_{G}(h(x)^{p^{n}}) = β_{G}(h(x^{p^{n}})),

whence h ∘ β_{F} = β_{G} ∘ h, whence β: (σ^{n})^{∗} → id is an natural isomorphism of functors. Thus (σ^{n})^{∗} is an equivalence of categories.

Part 4. Construction of the A_{∞}-ring structures

In the following sections we show that the homology theories introduced in Section 6 admit A_{∞}-ring structures, by showing that certain obstructions to the existence of such structures vanish. This technique originated in work of Robinson [15] on the existence of A_{∞}-structures for Morava K-theories (see also Remark 23.8).

22. Producing an algebra structure

We still need to show that for each formal group law (k, Γ) the ring spectrum E = E_{k,Γ} admits an A_{∞}-structure. That is, for an appropriate A_{∞}-operad C, we want a map of operads making the diagram

\xymatrix{C \ar@{.>}[rr] \ar[dr] &  & E_{E} \ar[dl] \\  & L}

commute. Recall from Section 12.1 that E_{E} is an operad with E_{E}[n] = S(E^{(n)}, E). We will choose our operad C so that it is a cofibrant A_{∞} operad.

22.1. Free operads. If A is a Σ-object in some symmetric monoidal category, the free operad FA on A is an operad equipped with a map of Σ-objects A → FA inducing an isomorphism

Hom_{oper}(FA, C) ≃ Hom_{Σ}(A, C),

for each operad C. There is a free operad functor which is left adjoint to the forgetful functor from operads to Σ-objects.

Since operads are monoids in the category of Σ-objects, the free operad should really be a kind of “free monoid”. This is true, but the formula for such a free monoid is somewhat strange. Morally speaking,

FA = I ⨿ A ∘ (I ⨿ A ∘ (I ⨿ A ∘ (⋯))).

More precisely, FA can be written as a countable colimit of expressions of the above form which terminate at some finite depth.

The reason for this strange formula is that the monoidal structure ∘ on Σobjects is not “bi-distributive”; rather, it distributes over coproducts on the right, but not on the left. I.e., in general (A⨿B)∘C ≃ (A∘C)⨿(B ∘C), but A∘(B ⨿C) ≄ (A ∘ B) ⨿ (A ∘ C).

22.2. Model category for operads. We give a model category structure for the category s oper of simplicial operads.

Say a Σ-object A is Σ-free if each A[n] is free as a Σ_{n}-simplicial set. We define a closed model category for simplicial operads by letting f: A → B be

1. a weak equivalence if each map f[n]: A[n] → B[n] is a weak equivalence on the underlying simplicial set, and

2. a cofibration if it is a retract of a free map. A free map is a map f of the

form

f_{n}: A_{n} → B_{n} ≃ A_{n} ⨿_{oper} FX_{n}

where X_{n} is a discrete, Σ-free Σ-object, and such that the subobjects X_{n} ⊂ B_{n} are closed under degeneracy operators.

That this indeed gives a model category structure follows from general results of Quillen [11].

To produce our desired operad C, first let A → L be an arbitrary A_{∞}-operad over the linear isometries operad. Let Sing A be the simplicial operad obtained by taking singular simplices, and let C̃ → Sing A be a cofibrant resolution of Sing A; we can choose C̃ so that C̃_{n} = FK_{n} for some Σ-free Σ-object K_{n}. Then by adjunction we get a map C = |C̃| → A which is a weak equivalence of topological operads.

22.3. The spectral sequence. We want to show that map_{(oper↓L)}(C, E_{E}) is non-empty. The simplicial object C̃_{n} = FK_{n} leads to a cosimplicial space Y^{•}, with

Y^{n} = map_{(oper↓L)}(FK_{n}, E_{E}),

with Tot Y^{•} = map_{(oper↓L)}(|C̃|, E_{E}). Thus, if we choose an element σ ∈ π_{0}Y^{0} ⇉ π_{0}Y^{1}, there is an obstruction theory / spectral sequence of the cosimplicial space (see Section 14) with

E_{2}^{s,t} = π^{s}π_{t}(map_{(oper↓L)}(FK_{n}, E_{E}), σ)).

22.4. Identification of E_{2}^{0,0}. We need to compute the equalizer of the pair of arrows

π_{0} map_{(oper↓L)}(FK_{0}, E_{E}) ⇉ π_{0} map_{(oper↓L)}(FK_{1}, E_{E}).

Note that for any operad B over L there is a natural map

π_{0} map_{(oper↓L)}(B, E_{E}) → Hom_{oper}(π_{0}B, π_{0}E_{E})

which sends an operad map f: B → E_{E} over L to the map π_{0}f: π_{0}B → E_{E}, where π_{0}B denotes the discrete operad with (π_{0}B)[n] = π_{0}(B[n]).

Since E_{E}[n] ≃ S(E^{(n)}, E), we see that π_{0}E_{E}[n] ≃ [E^{(n)}, E]. In fact, by Proposition 15.5,

π_{0}E_{E}[n] ≃ Hom_{E_{∗}}((E_{∗}E)^{⊗n}, E_{∗}).

Now suppose B = FK for some discrete Σ-object K equipped with a map K → L. Such a K is necessarily Σ-free, since L is. There is a commutative diagram

\xymatrix{π_{0} map_{(oper↓L)}(FK, E_{E}) \ar[r] \ar[d] & Hom_{oper}(FK, π_{0}E_{E}) \ar[d] \\ π_{0} map_{(Σ↓L)}(K, E_{E}) \ar[r] & Hom_{Σ}(K, π_{0}E_{E})}

The vertical arrows are isomorphisms by the characteristic property of a free operad. The lower horizontal arrow is also an isomorphism, since K is discrete and Σ-free. Thus the upper horizontal arrow is an isomorphism. Thus we see that E_{2}^{0,0} is isomorphic to the equalizer of the pair of arrows

E_{2}^{0,0} → Hom_{oper}(FK_{0}, π_{0}E_{E}) ⇉ Hom_{oper}(FK_{1}, π_{0}E_{E}).

Since π_{0}C, the coequalizer of FK_{1} ⇉ FK_{0}, is the associative monoid operad, it follows that E_{2}^{0,0} ≃ Hom_{oper}(π_{0}C, π_{0}E_{E}). This is precisely the set of ring spectrum structures on E; that is, an associative ring spectrum structure on E (in the “classical” sense, rather than in the A_{∞} sense) is precisely equivalent to a map from the associative operad to π_{0}E_{E}.

22.5. Identification of E_{2}-term. Let us now choose the element σ ∈ E_{2}^{0,0} corresponding to the “usual” ring spectrum structure on E. We obtain as before an obstruction theory / spectral sequence with

E_{1}^{s,t} ≃ π_{t}(map_{(oper↓L)}(FK_{s}, E_{E}), σ)

and “converging” to π_{t−s}(map_{(oper↓L)}(C, E_{E}), σ).

Theorem 22.6. The E_{2}-term of this spectral sequence is given by

E_{2}^{s,t} = \begin{cases}0 & if s > 0, \\ [Σ^{t}E, E] & if s = 0, t > 0. \end{cases}

Proof. This follows from the discussion of Section 23.1, Theorem 23.5, Remark 23.6, and Proposition 23.7.

In particular, the obstruction groups E_{2}^{s,s−1} vanish identically, and so

Corollary 22.7. The spectrum E admits an A_{∞}-ring structure whose underlying ring structure is equivalent to the usual one.

23. Computation of algebra-structure obstruction groups

If B is an operad and K is an unpointed space, let map(K, B) denote the operad with map(K, B)[n] = map(K, B[n]), and with operad structure given by “point-wise composition”. Now if given a map B → L of operads, define an operad B^{K} over the linear isometries operad via the pullback square

\xymatrix{B^{K} \ar[r] \ar[d] & map(K, B) \ar[d] \\ L = map(pt, L) \ar[r] & map(K, L)}

To prove Theorem 22.6, we must first compute the E_{1}-term. Suppose that B → L is a map of operads, and σ: B → E_{E} is some fixed map of operads over L. Then the set

π_{t}(map_{(oper↓L)}(B, E_{E}), σ)

is the same as the set of homotopy classes of maps S^{t} → map_{(oper↓L)}(B, E_{E}) which send the base-point of the sphere to σ, and where homotopies are base-point preserving.

\xymatrix{pt \ar[r]^{σ} \ar[d] & map_{(oper↓L)}(B, E_{E}) \\ S^{t} \ar@{-->}[ur]}

By adjunction, these are the same as homotopy classes of lifts in the diagram

\xymatrix{ & E_{E}^{S^{t}} \ar[d] \ar[dr] \\ B \ar@{-->}[ur] \ar[r] & E_{E} \ar[r] & L}

That is, there is a natural isomorphism

π_{t}(map_{(oper↓L)}(B, E_{E}), σ) → π_{0}(map_{(oper↓E_{E})}(B, E_{E}^{S^{t}})).

There is a natural map

π_{0} map_{(oper↓L)}(B, E_{E}^{S^{t}}) → Hom_{oper}(π_{0}B, π_{0}E_{E}).

If B = FK and K is discrete and Σ-free there is a square

\xymatrix{π_{0} map_{(oper↓L)}(FK, E_{E}^{S^{t}}) \ar[r] \ar[d] & Hom_{oper}(FK, π_{0}E_{E}^{S^{t}}) \ar[d] \\ π_{0} map_{(Σ↓L)}(K, E_{E}^{S^{t}}) \ar[r] & Hom_{oper}(K, π_{0}E_{E}^{S^{t}})}

The vertical arrows are isomorphisms by the universal property of free operads. The lower horizontal arrow is an isomorphism since K is discrete and Σ-free. Thus the top horizontal arrow is an isomorphism. This isomorphism, and the fact that the map E_{E}^{S^{t}} → E_{E} is a split fibration (using the base-point of S^{t}), implies that the map

π_{0}(map_{(oper↓E_{E})}(FK, E_{E}^{S^{t}})) → map_{(oper↓π_{0}E_{E})}(FK, π_{0}E_{E}^{S^{t}})

is an isomorphism.

Since E_{E}^{S^{t}}[n] is weakly equivalent to map(S^{t}, E_{E}[n]), it follows that

π_{0}E_{E}^{S^{t}}[n] = Hom_{E_{∗}}((E_{∗}E)^{⊗n}, E_{∗}[ϵ]/ϵ^{2}),

where |ϵ| = −t; furthermore, the projection map π_{0}E_{E}^{S^{t}} → π_{0}E_{E} sends ϵ to 0.

Since the pinch map S^{t} → S^{t} ∨ S^{t} makes the sphere into a cogroup up to homotopy, the induced map of operads

π_{0}E_{E}^{S^{t}} ×_{π_{0}E_{E}} π_{0}E_{E}^{S^{t}} ≃ π_{0}E_{E}^{S^{t}∨S^{t}} → π_{0}E_{E}^{S^{t}}

makes π_{0}E_{E}^{S^{t}} into an abelian group object in the category of operads over π_{0}E_{E}.

23.1. Quillen cohomology of an operad. Suppose A → B is a map of discrete operads. Let C_{•} be a cofibrant resolution of a discrete operad A. Let N → B be an abelian group operad over an operad B. Then we define the Quillen cohomology of an operad by

H^{s}(A, N) = H^{s}[Hom_{(oper↓B)}(C_{•}, N)].

In our case we take A to be the associative operad, B = π_{0}E_{E}, and we take N = π_{0}E_{E}^{S^{t}} for t > 0. Thus we have that

E_{2}^{s,t} = H^{s}(A, π_{0}E_{E}^{S^{t}}).

23.2. Bimodules. Let A be an operad. An A-bimodule is a Σ-object M equipped with maps ℓ: A ∘ M → M and r: M ∘ A → M such that the diagrams

\xymatrix{A ∘ A ∘ M \ar[r]^{μ∘M} \ar[d]^{A∘ℓ} & A ∘ M \ar[d]^{ℓ} & M ∘ A ∘ A \ar[r]^{M∘μ} \ar[d]^{r∘A} & M ∘ A \ar[d]^{r} \\ A ∘ M \ar[r]^{ℓ} & M & M ∘ A \ar[r]^{r} & M}

and

[figure: A ∘ M; ℓ; M; η∘M; M∘η; M ∘ A; r; M]

commute.

There are several examples of bimodules.

1. A is itself an A-bimodule.

2. If K is a Σ-object, then A∘K∘A is an A-bimodule, called the free A-bimodule on K

3. If X and Y are objects, define a Σ-object 𝔥𝔬𝔪(X, Y) by

𝔥𝔬𝔪(X, Y)[n] = Hom(X^{⊗n}, Y).

If X = Y, then 𝔥𝔬𝔪(X, X) is the endomorphism operad of X. If X and Y are A-algebras, then 𝔥𝔬𝔪(X, Y) is an A-bimodule.

An A-bimodule M induces a functor from A-algebras to A-algebras as follows. Define M ∘_{A} (X) by the coequalizer

M ∘_{A} (X) = Cok ((M ∘ A)(X) ⇉ M(X)) .

Note that if M = A, then A ∘_{A} (X) ≃ X; i.e., the bimodule A induces the identity

functor on A-algebras. Likewise, if M = A^{∘d}, then A^{∘d} ∘_{A} (X) ≃ A^{d−1}(X). There is an adjunction

(23.3) Hom_{A- bimod} (M, 𝔥𝔬𝔪(X, Y)) ≃ Hom_{A- alg} (M ∘_{A} (X), Y ) .

23.4. Bar complex. If M and N are A-bimodules, let B(M, A, N) denote the simplicial object

\xymatrix{M ∘ N & M ∘ A ∘ N \ar[l] \ar[l] & M ∘ A ∘ A ∘ N \ar[l] \ar[l] \ar[l] & ⋯ \ar[l] \ar[l] \ar[l] \ar[l]}

In particular, B(A, A, A) is a resolution of A by free A-bimodules.

If N → B is a coefficient system (i.e., an abelian group object over B), then we define the Hochschild cohomology of A by

HH^{s}(A, N) = H^{s}[Hom_{(A- bimod↓B)}(B(A, A, A), N)].

The fact that relates the Quillen cohomology and Hochschild cohomology of an operad is the following.

Theorem 23.5. [14] If A is a discrete, Σ-free operad, and N is a coefficient system for A, then there is an exact sequence

0 → HH^{0}(A, N) → HH^{0}(I, N) → H^{0}(A, N) → HH^{1}(A, N) → 0

and isomorphisms

H^{s}(A, N) ≃ HH^{s+1}(A, N)

for s > 0.

Remark 23.6. Recall that I denotes the “trivial operad”. Since I ∘ I ≃ I, the bar complex B(I, I, I) is constant, so the Hochschild cohomology of I is concentrated in degree 0. In the case of interest to us, in which the coefficient system N is π_{0}E_{E}^{S^{t}} → π_{0}E_{E} pulled back along I → π_{0}E_{E}, it is trivial to show that

HH^{0}(I, N) ≃ Hom_{(I- bimod↓π_{0}E_{E})}(I, π_{0}E_{E}^{S^{t}})
≃ Hom_{E_{∗}}(E_{∗}E, E_{∗+t})
≃ [Σ^{t}E, E],

and that

HH^{s}(I, N) = 0 for s > 0.

Proposition 23.7. If A is the associative operad, then HH^{s}(A, π_{0}E_{E}^{S^{t}}) ≃ 0 for all s ≥ 0 and t > 0.

Proof. These cohomology groups are computed using the co-chain complex

C^{n} = Hom_{(A- bimod↓π_{0}E_{E})}(A^{∘n+2}, π_{0}E_{E}^{S^{t}}).

Note that

Hom_{A- bimod}(A^{∘n+2}, π_{0}E_{E}^{S^{t}}) ≃ Hom_{A- bimod}(A^{∘n+2}, 𝔥𝔬𝔪_{E_{∗}}(E_{∗}E, E_{∗}[ϵ]/ϵ^{2}))

≃ Hom_{A- alg}(A^{∘n+2} ∘_{A} (E_{∗}E), E_{∗}[ϵ]/ϵ^{2})

≃ Hom_{E_{∗}- alg}(T^{n+1}(E_{∗}E), E_{∗}[ϵ]/ϵ^{2})),

using the adjunction of (23.3). Likewise,

Hom_{A- bimod}(A^{∘n+2}, π_{0}E_{E}) ≃ Hom_{E_{∗}- alg}(T^{n+1}(E_{∗}E), E_{∗}).

Thus,

C^{n} ≃ Hom_{(E_{∗}- alg↓E_{∗})}(T^{n+1}(E_{∗}E), E_{∗}[ϵ]/ϵ^{2}),

and as we know from Sections 20 and 21, this complex computes Der_{E_{∗}}^{s}(E_{∗}E, E_{∗+t}), which vanishes.

Remark 23.8. Compare this to Robinson’s calculation [15]. If we had taken E ≃ K(n), the nth Morava K-theory, and if p is odd (so that K(n) is a commutative ring spectrum), then the proof of Proposition 23.7 shows that HH^{s}(A, π_{0}E_{E}^{S^{t}}) is the Quillen cohomology of the K(n)-algebra K(n)_{∗}K(n) with coefficients in the module K(n)_{∗+t}. By a result of Quillen [12], these groups are the same up to a shift in dimension as the Hochschild cohomology of the algebra K(n)_{∗}K(n); this recovers Robinson’s calculations of the obstruction groups for this spectrum.

The Hochschild cohomolgy of K(n)_{∗}K(n) is in general non 0. However, the algebra K(n)_{∗}K(n) ≃ Σ(n)⊗Λ(n), where Λ(n) is an exterior algebra on n-generators, and the Hochschild cohomology of K(n)_{∗}K(n) originates almost entirely from this exterior factor. By replacing K(n) with E_{k,Γ} we get rid of this contribution.

Proof of Theorem 23.5. Let C_{•} → A be a cofibrant resolution of the operad A by free operads; thus, C_{n} = FK_{n}. We obtain a double complex D_{m,n} = B(C_{n}, C_{n}, C_{n})_{m} of bimodules using the bar construction. The diagonal of D_{•,•} is a C_{•}-bimodule resolving C_{•}. If N is a coefficient system for A, then by applying Hochschild cohomology we obtain a spectral sequence

HH^{s}(C_{n}, N) ⟹ HH^{s+n}(A, N).

Thus the theorem easily follows from the following lemma.

Lemma 23.9. Let C be a discrete operad. Then there is a sequence

0 → HH^{0}(C, N) → HH^{0}(I, N) → H^{0}(C, N) → HH^{1}(C, N) → 0

natural in C; furthermore, if C = FK for some Σ-object K, the sequence is exact, and HH^{s}(C, N) ≃ 0 for s > 1.

Proof. See [14].

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Department of Mathematics, Northwestern University, Evanston, IL 60208

 E-mail address: rezk@math.nwu.edu