GAMMA HOMOLOGY, LIE REPRESENTATIONS

AND E_{∞} MULTIPLICATIONS

Alan Robinson

Abstract. We prove that the stable homotopy of any Γ-module F is the homology of a bicomplex Ξ(F), in which the (q − 1)st row is the two-sided bar construction B(Lie_{q}^{∗}, Σ_{q}, F [q]). This gives a natural homotopical cotangent bicomplex for graded commutative algebras, in a form suitable for use in a new obstruction theory for classifying E_{∞} ring structures on spectra. The E_{∞} structure on certain Lubin-Tate spectra is a corollary.

Introduction

Let F be any functor from finite based sets to spaces. The stable homotopy groups of F, which we denote by π_{∗}(F), are defined as the homotopy groups of the spectrum ∥F ∥ associated with F by the well known construction introduced by G. Segal.

We consider the discrete abelian case, where F takes values in k-modules for some commutative ring k. In §3 we give an explicit functorial bicomplex Ξ(F) the homology of which is π_{∗}(F). The proof is an application of recent theory due to Pirashvili and Richter [10],[11].

In §4 we apply the Ξ-construction to the functor which takes a based set S_{+} to the module B^{⊗S} ⊗ M, where B is a given graded commutative algebra and M a B-module. This gives a canonical, functorial cotangent complex for graded commutative algebras. Its homology is the Γ-homology HΓ_{∗}(B|A; M), where A is the ground ring. Theorem 4.2 reconciles this construction with other definitions of Γ-homology, E_{∞}-homology and topological André-Quillen homology for commutative rings as defined by Basterra, Kriz and others (see [2],[17]). The description of Γ-homology as stable homotopy yields some calculations [14],[15].

Our interest in the Ξ-construction was inspired by a question in homotopy theory: when does a ring spectrum admit a multiplication satisfying the E_{∞} homotopy associativity and commutativity conditions? In the language of S-modules, what conditions are sufficient for enhancing a multiplication in the homotopy category of S-modules to a commutative S-algebra structure? In §5 we answer this question by setting up an obstruction theory based upon Γ-homology. The Ξ-complex arises there in a natural way. Other very different approaches to this problem have been given by Basterra [1] and Kriz, and in unpublished work by Goerss and Hopkins.

The author expresses his thanks to Birgit Richter and to the referee. Their valuable comments led to substantial improvements in this paper.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 55P43 (13D03, 18G60).

1. Γ-modules

Let Γ denote the category of finite based sets and basepoint-preserving maps. A construction of G. Segal [18], refined by Bousfield and Friedlander [3], assigns a spectrum ∥F ∥ to each functor F from Γ to based simplicial sets. By applying F to the set of simplices at each level in a simplicial model of S^{n}, Bousfield and Friedlander obtain a bisimplicial set. The diagonal simplicial set of this is the nth space in the spectrum ∥F ∥. The homotopy groups of the spectrum ∥F ∥ are called the (stable) homotopy groups of F. We denote them by π_{∗}F.

We are concerned with the discrete abelian case. Let us fix a commutative ring k with unit, to be our ground ring. A (left) Γ-module is a functor Γ → k-mod. Since a k-module can be regarded as a discrete simplicial set, this fits into the above context. If F is a Γ-module, then the spaces in the spectrum ∥F ∥ are simplicial k-modules. The theory of Γ-modules and their homotopy has been substantially developed in recent work of Teimuraz Pirashvili and Birgit Richter [10] [13].

The most fundamental Γ-module is the functor L which assigns to each based set X the free k-module k·X/k·0 generated by X, modulo the submodule generated by the basepoint. On simplicial sets, L is the reduced chain functor. Let us consider the case of a Γ-module which has the particular form F = Ψ ∘L, where Ψ : k-mod → k-mod is any functor. From the fact that L(S^{n}) is a projective resolution of (k, n) in the sense of [4] it follows that the stable homotopy groups π_{∗}(Ψ ∘ L) are the stable derived functors of Ψ in the sense of Dold and Puppe, evaluated on k.

The main result of [11] gives an explicit chain complex for calculating the stable homotopy of any Γ-module. This complex is based upon the nerve of the subcategory of all surjections in Γ. Consequently it is quite large, even when Γ is replaced (as it usually is) by a minimal skeleton with one set of each cardinality. In §2 we give a much smaller complex Ξ(F), based upon the Lie operad, which has the same property.

2. The modules Lie_{n} and the standard bicomplex

As we turn to the combinatorial details, we replace the category Γ by the equivalent full subcategory in which the objects are the finite based sets [n] = {0, 1, 2, …, n} for all n ≥ 0, where 0 ∈ [n] is the basepoint. We shall denote this subcategory henceforth by Γ.

The group of automorphisms in Γ of the object [n] is the permutation group Σ_{n}. The important subcategory of all surjections in Γ is generated by these automorphisms together with the special surjections c_{ij} : [n] → [n − 1] which are defined for 0 ≤ i < j ≤ n by the formulae

(2.0)  c_{ij}(t) = \{ \begin{matrix}t & for 0 ≤ t < j \\ i & for t = j \\ t − 1 & for j < t ≤ n. \end{matrix}

Thus c_{ij} maps j to i and is strictly order-preserving on [n] \ {j}.

Γ-HOMOLOGY, LIE REPRESENTATIONS AND E_{∞} MULTIPLICATIONS 3

2.1 The Lie representations. Let L_{n} be the free Lie algebra over k on the set of generators {x_{i}}_{1≤i≤n}. We denote by Lie_{n} the so-called multilinear part of L_{n}. This can be described in many different ways. First, it is defined as the direct summand of L_{n} spanned by all Lie monomials containing each of the n generators exactly once. Second, it is the nth module in the Lie operad. Third, it is isomorphic to the module of all natural transformations Φ^{⊗n} → Φ, where Φ be the forgetful functor from Lie algebras (over k) to k-modules.

The symmetric group Σ_{n} acts upon Lie_{n} by permuting the n generators. The kΣ_{n}-module thus obtained is known as the Lie representation. It has many applications in combinatorics, geometry and homotopy theory. Very often it occurs twisted by the sign character, and this is the version we require. We define the left action of Σ_{n} on the k-module Lie_{n} by setting

σ · f(x_{1}, …, x_{n}) = ε(σ) f(x_{σ(1)}, …, x_{σ(n)})

for every multilinear Lie monomial f and every σ ∈ Σ_{n}, where ε(σ) is the sign of σ. Let Lie_{n}^{∗} be the dual k-module Hom(Lie_{n}, k), which is thus a right Σ_{n}-module. We shall require the following properties of Lie_{n} (see [20], 2.3):

(1) the left regulated Lie brackets

σ · [x_{1}, [x_{2}, [x_{3}, …, [x_{n−1}, x_{n}]..]]] for σ ∈ Σ_{n−1}

form a k-basis of Lie_{n}. Therefore

(2) the k-modules Lie_{n} and Lie_{n}^{∗} are free of rank (n − 1)!, and

(3) the restricted Σ_{n−1}-modules Res_{Σ_{n−1}}^{Σ_{n}}Lie_{n} and Res_{Σ_{n−1}}^{Σ_{n}}Lie_{n}^{∗} are respectively isomorphic to the left and right regular representations.

2.2 The homomorphisms γ_{ij} : Lie_{n}^{∗} → Lie_{n−1}^{∗}.

We now introduce certain linear maps γ_{ij} : Lie_{n}^{∗} → Lie_{n−1}^{∗}. These are induced by the standard surjections in Γ, but they do not make the duals of the Lie representations into a Γ-module, as 2.4 below makes clear.

Since Lie_{n}^{∗} is for each n a finitely-generated projective k-module, we can define the γ_{ij} completely by giving their duals γ_{ij}^{∗} : Lie_{n−1} → Lie_{n}. For 0 ≤ i < j ≤ n we set

(γ_{ij}^{∗}f)(x_{1}, …, x_{n}) = \{ \begin{matrix}(−1)^{j+1}[x_{j}, f(x_{1}, .., x̂_{j}, .., x_{n})] & if i = 0 \\ (−1)^{j+1}f(x_{1}, .., x_{i−1}, [x_{i}, x_{j}], x_{i+1}, .., x̂_{j}, .., x_{n}) & if i > 0 \end{matrix}

for each multilinear Lie monomial f(x_{1}, …, x_{n−1}), where the circumflex accent ˆ means that the accented term is omitted.

2.3 The standard relations.

It follows from the multilinearity of f and from the Jacobi identity that for i < j < k

((γ_{0j}^{∗}γ_{0,k−1}^{∗} + γ_{0k}^{∗}γ_{0}^{∗}j)f)(x_{1}, …, x_{n}) = (−1)^{j+k+1}[x_{j}, [x_{k}, f(x_{1}, .., x̂_{j}, .., x̂_{k}, .., x_{n})]]
+ (−1)^{j+k}[x_{k}, [x_{j}, f(x_{1}, .., x̂_{j}, .., x̂_{k}, .., x_{n})]]
= (−1)^{j+k+1}[[x_{j}, x_{k}], f(x_{1}, .., x̂_{j}, .., x̂_{k}, .., x_{n})]
= − (γ_{jk}^{∗}γ_{0j}^{∗}f)(x_{1}, …, x_{n})

and that

(γ_{ij}^{∗}γ_{i,k−1}^{∗} + γ_{ik}^{∗}γ_{ij}^{∗} + γ_{jk}^{∗}γ_{ij}^{∗})f(x_{1}, …, x_{n})

= (−1)^{j+k+1}f(x_{1}, …, [[x_{i}, x_{j}], x_{k}] − [[x_{i}, x_{k}], x_{j}] − [x_{i}, [x_{j}, x_{k}]], .., x̂_{j}, .., x̂_{k}, .., x_{n}) = 0 .

Passing to the dual in the above gives all cases (i = 0 and i > 0) of the first relation in the following lemma. The other relations are even more straightforward to verify.

2.4 Lemma. When i < j < k < l

γ_{ij}γ_{jk} + γ_{ij}γ_{ik} + γ_{i,k−1}γ_{ij} = 0
γ_{ij}γ_{kl} + γ_{k−1,l−1}γ_{ij} = 0
γ_{ik}γ_{jl} + γ_{j,l−1}γ_{ik} = 0
γ_{i,l−1}γ_{jk} + γ_{jk}γ_{il} = 0 .  □

2.5 The bicomplex Ξ(F).

Let F be any Γ-module. Then F [n] is a left Σ_{n}-module by restriction of structure, which we can combine with the right Σ_{n}-module Lie_{n}^{∗} in the two-sided bar construction B(Lie_{n}^{∗}, Σ_{n}, F [n]). We take this as the (n − 1)st row in our standard bicomplex, so that we have

Ξ_{p,q}(F) = Lie_{q+1}^{∗} ⊗ k[Σ_{q+1}^{p}] ⊗ F [q + 1] .

the horizontal differential ∂′ being the alternating sum ∑_{i=0}^{p}(−1)^{i}∂′_{i} of face operators:

∂′(z ⊗ [σ_{1}|σ_{2}| … |σ_{p}] ⊗ y) = zσ_{1} ⊗ [σ_{2}| … |σ_{p}] ⊗ y +

+ ∑_{i=1}^{p−1}(−1)^{i}(z ⊗ [σ_{1}|σ_{2}| … |σ_{i}σ_{i+1}| … |σ_{p}] ⊗ y) +

+ (−1)^{p}(z ⊗ [σ_{1}|σ_{2}| … |σ_{p−1}] ⊗ σ_{p}y) .

We define a vertical differential ∂′′ : Ξ_{p,q} ⟶ Ξ_{p,q−1}, that is

Lie_{q+1}^{∗} ⊗ k[Σ_{q+1}^{p}] ⊗ F [q + 1] ⟶ Lie_{q}^{∗} ⊗ k[Σ_{q}^{p}] ⊗ F [q]

in terms of the operators c_{ij} (see (2.0)) in the Γ-structure on F, and the homomorphisms γ_{ij} of 2.2, by setting

(−1)^{p} ∂′′(z⊗[σ_{1}|σ_{2}| … |σ_{p}]⊗y) = ∑_{0≤s<t≤q+1} γ_{st}z⊗c_{st}[σ_{1}|σ_{2}| … |σ_{p}]⊗c_{(σ_{1}...σ_{p})^{−1}{s,t}} y

where c_{st}[σ_{1}|σ_{2}| … |σ_{p}] is the p-simplex of the nerve of Σ_{q} which forms the bottom row of the commutative diagram

[q + 1] ⟶^{σ_{p}} [q + 1] ⟶^{σ_{p−1}} ⋯ ⟶^{σ_{2}} [q + 1] ⟶^{σ_{1}} [q + 1]
c_{(σ_{1}...σ_{p})^{−1}{st}} ↓  ↓ c_{(σ_{1}...σ_{p−1})^{−1}{st}}  ↓ c_{σ_{1}^{−1}{st}}  ↓ c_{st}
[q] ⟶ [q] ⟶ ⋯ ⟶ [q] ⟶ [q]

arising as follows. Given an isomorphism σ_{1} in the category Γ and a surjection c_{st} as here, the right-hand square above can be uniquely completed with a surjection c_{σ_{1}^{−1}{st}} and an isomorphism c_{st}[σ_{1}] : [q] → [q]. Repeating this procedure p times constructs the whole diagram, and the simplex c_{st}[σ_{1}|σ_{2}| … |σ_{p}]. (The notation c_{st} when s > t means c_{ts}, and c_{σ{st}} is an abbreviation for c_{σ(s),σ(t)}.)

2.6 Proposition. Ξ(F) is a bicomplex.

Proof. The relation ∂′∂′ = 0 holds because the two-sided bar resolution (see for instance [19], §2.3), is a chain complex. On the other hand ∂′′∂′′ = 0 because ∂′′ is on the first tensor factor the Leibniz differential ([8], 10.6.2) on a generic Lie algebra. The vanishing of ∂′′∂′′ can also be explicitly verified by a calculation using Lemma 2.4.

Finally, we must verify that ∂′∂′′ + ∂′′∂′ = 0. In fact, ∂′′ anticommutes with each of the horizontal face operators ∂′_{i}. For all faces except ∂′_{0} this is an immediate consequence of the commutative diagram used above to define c_{st}[σ_{1}|σ_{2}| … |σ_{p}]. The face ∂′_{0} however involves the action of σ_{1} on Lie_{q+1}. We first note that the signs of σ_{1} and of c_{σ_{1}{st}}(σ_{1}) are related by ε(c_{σ_{1}{st}}(σ_{1})) = (−1)^{t−σ_{1}(t)}ε(σ_{1}). For any f ∈ Lie_{q} we therefore have

σ_{1}(γ_{0t}^{∗}f)(x_{1}, …, x_{q+1})
= (−1)^{t}ε(σ_{1})[x_{σ_{1}(t)}, f(x_{σ_{1}(1)}, .., x̂_{σ_{1}(t)}, .., x_{σ_{1}(q+1)})]

= (−1)^{t}ε(σ_{1})ε(c_{σ_{1}{0t}}σ_{1})[x_{σ_{1}(t)}, (c_{σ_{1}{0t}}σ_{1})·f(x_{1}, .., x̂_{t}, .., x_{q+1})]

= γ_{0,σ_{1}(t)}^{∗} (c_{σ_{1}{0t}}σ_{1}) · f(x_{1}, .., x_{q+1})

which proves that σ_{1}γ_{0t}^{∗} = γ_{0,σ_{1}(t)}^{∗} (c_{σ_{1}{0t}}σ_{1}). Since γ_{0t} is defined as the adjoint of γ_{0t}^{∗}, we have γ_{0t}(zσ_{1}) = γ_{0,σ_{1}(t)}(z) c_{σ_{1}{0t}}(σ_{1}) for all z ∈ Lie_{q}^{∗}. We show in the same way that γ_{st}(zσ_{1}) = γ_{σ_{1}(s,t)}(z) c_{σ_{1}{st}}(σ_{1}) when s > 0, though here we have to separate the cases σ_{1}(s) < σ_{1}(t) and σ_{1}(s) > σ_{1}(t). The relations just established are precisely those needed to prove that ∂′′ anticommutes with the remaining face operator ∂′_{0}. Thus ∂′∂′′ + ∂′′∂′ = 0, and the lemma is proved. □

3. The complex Tot Ξ and stable homotopy

In this section we use the theory of [10] and [11] to prove that the homology of the Ξ-complex is stable homotopy.

3.1 Definition. We denote by HΞ_{∗}(F) the homology of the total complex Tot Ξ(F).

Let L be the Γ-module (denoted t^{∗} in [10]) which assigns to [n] the free k-module generated by [n] modulo the submodule generated by the basepoint:

L[n] = k[n]/k[0] .

We recall that tensor products over k of Γ-modules are defined objectwise:

(F ⊗ G)[n] = F [n] ⊗ G[n] .

3.2 Lemma. For any Γ -module F

HΞ_{i}(F ⊗ L) ≈ \{ \begin{matrix}F [0] & for i = 0 \\ 0 & for i ≠ 0. \end{matrix}

Proof. We calculate the homology of the bicomplex Ξ(F ⊗ L) using the spectral sequence which first calculates the homology of the rows. The (n − 1)st row is the bar construction B(Lie_{n}^{∗}, Σ_{n}, F [n] ⊗ L[n]), so the horizontal homology E_{n−1,∗}^{1} is Tor_{∗}^{Σ_{n}}(Lie_{n}^{∗}, F [n] ⊗ L[n]). Let us denote the trivial representation of any group by η. From the observation that L[n] is the induced Σ_{n}-module Ind_{Σ_{n−1}}^{Σ_{n}} η, we have

F [n]⊗L[n] ≈ F [n]⊗Ind_{Σ_{n−1}}^{Σ_{n}}η ≈ Ind_{Σ_{n−1}}^{Σ_{n}}(Res_{Σ_{n−1}}^{Σ_{n}}F [n] ⊗ η) ≈ Ind_{Σ_{n−1}}^{Σ_{n}}Res_{Σ_{n−1}}^{Σ_{n}}F [n] .

Using this and the result §2.1(3) that Lie_{n}^{∗} restricts to the regular representation Ind_{Σ_{1}}^{Σ_{n−1}} η of Σ_{n−1}, we can reduce the groups E_{n−1,∗}^{1} in two stages: first to the homology of Σ_{n−1}, and then to that of the trivial group Σ_{1}:

E_{n−1,∗}^{1} ≈ Tor_{∗}^{Σ_{n}}(Lie_{n}^{∗}, F [n] ⊗ L[n]) ≈ Tor_{∗}^{Σ_{n}}(Lie_{n}^{∗}, Ind_{Σ_{n−1}}^{Σ_{n}}Res_{Σ_{n−1}}^{Σ_{n}}F [n])

≈ Tor_{∗}^{Σ_{n−1}}(Res_{Σ_{n−1}}^{Σ_{n}}Lie_{n}^{∗}, Res_{Σ_{n−1}}^{Σ_{n}}F [n])

≈ Tor_{∗}^{Σ_{n−1}}(Ind_{Σ_{1}}^{Σ_{n−1}}η, Res_{Σ_{n−1}}^{Σ_{n}}F [n])
≈ Tor_{∗}^{k}(k, F [n]) ,

which is F [n] in degree zero, and zero elsewhere. Therefore the E^{1}-term reduces to the vertical edge, with E_{0,n−1}^{1} ≈ F [n].

We must calculate d_{n,0}^{1} : F [n + 1] → F [n]. Rewritten as

d_{n,0}^{1} : Lie_{n+1}^{∗} ⊗_{Σ_{n+1}}(F [n + 1] ⊗ L[n + 1]) → Lie_{n}^{∗} ⊗_{Σ_{n}}(F [n] ⊗ L[n])

this is induced by the differential ∂′′ = ∑_{0≤s<t≤n+1}(γ_{st} ⊗ c_{st}). We calculate the operators γ_{st} ⊗ c_{st} on Lie_{n+1}^{∗} ⊗L[n + 1] from their adjoints

c_{st}^{∗} ⊗ γ_{st}^{∗} : L[n]^{∗} ⊗_{Σ_{n}} Lie_{n} ⟶ L[n + 1]^{∗} ⊗_{Σ_{n+1}} Lie_{n+1} .

Let U_{n} denote the left regulated Lie bracket [x_{1}, [x_{2}, [x_{3}, …, [x_{n−1}, x_{n}]..]]], which we shall denote more briefly by [1, [2, [3, …, [n − 1, n]..]]]. The elements σU_{n}, for σ ∈ Σ_{n−1}, form a k-basis for Lie_{n}. The cosets e_{j} = (j n)Σ_{n−1}, for 1 ≤ j ≤ n, form a k-basis of L[n]^{∗} ≈ Σ_{n}/Σ_{n−1}, and the single element e_{n} ⊗ U_{n} is a basis for the rank one k-module L[n]^{∗} ⊗_{Σ_{n}} Lie_{n}.

We take first the case s = 0. The definitions of c_{0t} and of γ_{0t}^{∗} in §§1-2 show that for t < n + 1, the homomorphism c_{0t}^{∗} ⊗ γ_{0t}^{∗} maps e_{n} ⊗ U_{n} into

(−1)^{t+1}e_{n+1} ⊗ [t, [1, [2, …, [t − 1, [t + 1, …, [n, n + 1]..]]]]]

which equals e_{n+1} ⊗ U_{n+1} in L[n + 1]^{∗} ⊗_{Σ_{n+1}} Lie_{n+1} because the permutation (t 1 2 … t − 1) needed to standardize the order of the basis elements in the Lie bracket has sign (−1)^{t+1}. For t = n + 1, from c_{0,n+1}(n) = n we obtain

(c_{0,n+1}^{∗} ⊗ γ_{0,n+1}^{∗})(e_{n} ⊗ U_{n}) = (−1)^{n}e_{n} ⊗ [n + 1, [1, [2, …, [n − 1, n]..]]]

= (−1)^{n}e_{n+1} ⊗ (n n + 1)[n + 1, [1, [2, …, [n − 1, n]..]]]

= (−1)^{n+1}e_{n+1} ⊗ [n, [1, [2, …, [n − 1, n + 1]..]]]
= e_{n+1} ⊗ U_{n+1}

since (n n+1) has sign −1 and the n-cycle (1 2 3 … n) has sign (−1)^{n+1}. Therefore in all cases (c_{0t}^{∗} ⊗ γ_{0t}^{∗})(e_{n} ⊗ U_{n}) = e_{n+1} ⊗ U_{n+1}. Similar calculations show that (c_{st}^{∗} ⊗ γ_{st}^{∗})(e_{n} ⊗ U_{n}) = (−)^{s}2e_{n+1} ⊗ U_{n+1} for s > 0. These calculations of adjoint homomorphisms establish that the differential d_{n,0}^{1} : F [n + 1] → F [n] has the following expression in terms of the Γ-operators on F:

d_{n,0}^{1} = ∑_{t=1}^{n+1} c_{0t} + 2 ∑_{1≤s<t≤n+1}(−1)^{s}c_{st} .

We define an augmentation E_{0,0}^{1} ≈ F [1] → F [0] by the same formula, which since n = 1 reduces at this level to the single term c_{01}.

Let κ_{n} : [n] → [n + 1] be the morphism of Γ defined by κ_{n}(i) = i, for 1 ≤ i ≤ n. The κ_{n} induce a sequence of morphisms in the Γ-module F which we also denote by κ. In Γ there are identities as follows: c_{st}κ_{n} = κ_{n}c_{st} for 0 ≤ s < t ≤ n, but c_{s,n+1}κ_{n} = 1 for 0 ≤ s ≤ n. From this and the above formula for d_{n,0}^{1} it follows that κ_{n−1}d_{n−1,0}^{1} + d_{n,0}^{1}κ_{n} = (−1)^{n−1}. This means that the morphisms (−1)^{n}κ_{n} give a chain contraction of the augmented vertical edge E_{∗,0}^{1} → F [0]. Therefore E_{p,q}^{2} is F [0] when (p, q) = (0, 0), and is zero when (p, q) ≠ (0, 0). The lemma is proved. □

3.3 Lemma. If F is a constant Γ -module, then Ξ(F) is acyclic. In fact, all the columns of Ξ(F) are acyclic.

Proof. It is clearly enough to show that all the columns are acyclic when F is the Γ-module with constant value k. Since all the modules Ξ_{pq}(k) = Lie_{q+1}^{∗} ⊗ k[Σ_{q+1}^{p}] are finite and free, it suffices to show that the complexes dual to the columns are acyclic.

The dual of the column Ξ_{0,∗}(k) at the edge has modules Ξ_{0,q}^{∗}(k) = Lie_{q+1} and differential ∂′′^{∗} = ∑_{0≤s<t≤q+1} γ_{st}^{∗}. By 2.1(3) the set of left regulated Lie brackets [α_{1}, [α_{2}, .., [α_{q}, q + 1]..]] where α ∈ Σ_{q} forms a k-basis of Ξ_{0,q}^{∗}. We define linear maps Δ^{∗} : Ξ_{0,q}^{∗} → Ξ_{0,q−1}^{∗} by the formula

Δ^{∗}[α_{1}, [α_{2}, .., [α_{q}, q + 1]..]] = \{ \begin{matrix}0 & if α_{1} ≠ 1 \\ [β_{2}, [β_{3}, .., [β_{q}, q]..]] & if α_{1} = 1, \end{matrix}

where β_{j} = α_{j} − 1. From this definition it follows that Δ[1, g(2, …, q + 1)] = g(1, …, q) for every Lie polynomial g. (We are still using the abbreviated notation introduced in 3.2, whereby the basis element x_{i} is denoted by i.) Let us verify that Δ^{∗} is a contraction of the cochain complex Ξ_{0,∗}^{∗}. Suppose that ω = [α_{1}, [α_{2}, .., [α_{q}, q + 1]..]] is a basis element with α_{1} ≠ 1. Then Δ^{∗}ω = 0. On the other hand Δ^{∗}γ_{st}^{∗}ω = 0 for all (s, t) except for

Δ^{∗}γ_{01}^{∗}ω = Δ^{∗}[1, [α_{1} + 1, [α_{2} + 1, .., [α_{q} + 1, q + 2]..]]] = ω ,

and therefore (Δ^{∗}∂′′^{∗} + ∂′′^{∗}Δ^{∗}) ω = ω. Next we must consider the second type of basis element, namely those ω of the form [1, [α_{2}, .., [α_{q}, q + 1]..]]. For these we have still Δ^{∗}γ_{01}^{∗}ω = ω and Δ^{∗}γ_{0t}^{∗}ω = 0 for t > 0; but now

Δ^{∗}γ_{st}^{∗} ω = − γ_{s−1,t−1}^{∗}Δ^{∗}ω  for 1 < s < t ≤ q + 2
Δ^{∗}γ_{1t}^{∗} ω = Δ^{∗}[[1, t], [ε_{2}, […, [ε_{q}, ε_{q+1}]..]]]

where ε_{j} = α_{j} if α_{j} < t, and ε_{j} = α_{j} + 1 if α_{j} ≥ t; here α_{q+1} means q + 1;

= Δ^{∗}[1, [t, [ε_{2}, […, [ε_{q}, ε_{q+1}]..]]]
= [t − 1, [ε_{2} − 1, […, [ε_{q} − 1, ε_{q+1} − 1]..]]]
= − γ_{0,t−1}^{∗}Δ^{∗}ω for all 1 < t ≤ q + 2 .

Therefore (Δ^{∗}∂′′^{∗} + ∂′′^{∗}Δ^{∗}) ω = ω for ω of the second type. This proves that Δ^{∗} contracts Ξ_{0,∗}^{∗}(k). Hence its dual Δ is a chain contraction of the column Ξ_{0,∗}(k).

The p th column Ξ_{p,∗}(k) has Ξ_{p,q} = Lie_{q+1}^{∗} ⊗ k[Σ_{q+1}^{p}]. To contract this column we define θ : k[Σ_{q}^{p}] → k[Σ_{q+1}^{p}] by

θ [σ_{1}|σ_{2}| … |σ_{p}] = [σ̄_{1}|σ̄_{2}| … |σ̄_{p}]

where σ̄_{j} is the shift of σ_{j} having σ̄_{j}(1) = 1 and σ̄_{j}(t) = σ_{j}(t−1)+1 for t > 1. This θ has the following commutation relations with the Γ-operators defined in 2.5: first, c_{01}θ = 1 and c_{0t}θ = θc_{0,t−1} for t > 1; and second, c_{st}θ = θ c_{s−1,t−1} for 0 < s < t. Combining these with the duals of the above formulae for Δ^{∗}c_{st}^{∗}, we find that the homomorphisms

(Δ ⊗ θ) : Lie_{q}^{∗} ⊗ k[Σ_{q}^{p}] ⟶ Lie_{q+1}^{∗} ⊗ k[Σ_{q+1}^{p}]

give a chain contraction of the p th column. We have now shown that Ξ(k) is an acyclic bicomplex, because the homology of every column vanishes. □

3.4 Proposition. The homology groups HΞ_{i}(L^{⊗r}) of the Γ -modules L^{⊗r} are as follows.

HΞ_{i}(L) ≈ \{ \begin{matrix}k & for i = 0 \\ 0 & for i > 0, \end{matrix}

and HΞ_{i}(L^{⊗r}) ≈ 0 for all i when r = 0 and when r > 1.

Proof. In the case r ≥ 1, the result follows when we apply Lemma 3.2 to calculate HΞ_{i}(L^{⊗(r−1)} ⊗ L). The single non-zero value occurs when r = 1, when L^{⊗(r−1)} is the Γ-module with constant value k.

The case r = 0 follows from Lemma 3.3, because L^{⊗0} is the constant Γ-module with value k. □

3.5 Theorem. There is a isomorphism HΞ_{∗}(F) ≈ Tor_{∗}^{Γ}(L^{∗}, F), functorial in the left Γ -module F.

Proof. It is sufficient to establish a natural isomorphism HΞ_{0}(F) ≈ L^{∗} ⊗_{Γ} F, and to prove that HΞ_{i}(P) = 0 for i > 0 when P is projective.

The right Γ-module L^{∗} is identified with the functor t of ([10], 1.4) in which t[n] is the module of basepoint-preserving maps from [n] to the ring k (where the basepoint of k is 0).

Let Γ_{n} be the free right Γ-module on one element situated at the object [n]. Explicitly, Γ_{n}[m] is the free k-module on Hom_{Γ}([m], [n]), and the Γ-module structure arises from composition. We compute L^{∗} ⊗_{Γ} F from the exact sequence of ([10], 1.4.1)

Γ_{2} ⟶^{α} Γ_{1} ⟶^{β} L^{∗} ⟶ 0

in which β maps the generator of Γ_{1} to the based map [1] → k taking 1 to 1, and α maps the generator of Γ_{2} to p_{{12}} − p_{{1}} − p_{{2}}, where p_{X} : [2] → [1] is defined by p_{X}^{−1}{1} = X. Tensoring this sequence with F and using the isomorphisms Γ_{n} ⊗_{Γ} F ≈ F [n], we deduce that L^{∗} ⊗_{Γ} F is the cokernel of

(3.5.1)  c_{12} − c_{01} − c_{02} : F [2] ⟶ F [1] .

On the other hand, HΞ_{0}(F) is by definition the cokernel of

∂′ + ∂′′ : (Lie_{1}^{∗} ⊗ k[Σ_{1}] ⊗ F [1]) ⊕ (Lie_{2}^{∗} ⊗ F [2]) ⟶ Lie_{1}^{∗} ⊗ F [1] .

Since Lie_{1}^{∗} and Lie_{2}^{∗} are isomorphic to k, and ∂′ is zero in this bidegree (from the bar construction on the trivial group Σ_{1}), this formula for HΞ_{0}(F) reduces to (3.5.1). The isomorphism HΞ_{0}(F) ≈ L^{∗} ⊗_{Γ} F therefore holds.

We now need to show that HΞ_{i}(P) = 0 for i > 0 when P is projective. Since the tensor powers L^{⊗r} are projective generators of Γ-mod (by [10] or [11]), and HΞ_{∗} commutes with sums, it is enough to prove this for the cases P = L^{⊗r} for r ≥ 0. This was done in 3.4, so the theorem is completely proved. □

3.6 Corollary. The complex Ξ(kΓ) is a projective resolution of the right Γ - module L^{∗}.

Proof. The Γ-bimodule kΓ is projective both as a left Γ-module and as a right Γ-module. We have

Ξ_{p,q}(kΓ) ≈ Lie_{q+1}^{∗} ⊗ k[Σ_{q+1}^{p}] ⊗ Γ_{q+1} .

where Γ_{q+1} is the free right Γ-module as in 3.5. Therefore each Ξ_{p,q}(kΓ) is projective in mod-Γ, as both k[Σ_{q+1}^{p}] and Lie_{q+1} (by 2.1) are k-free. Thus Ξ(kΓ) is a projective complex of right Γ-modules; it is augmented over L^{∗} by the following composite involving the map β of 3.5:

Ξ_{0,0}(kΓ) = Lie_{1}^{∗} ⊗ Γ_{1} ≈ Γ_{1} ⟶^{β} L^{∗} .

By Theorem 3.5, the homology of Ξ(kΓ) is Tor_{∗}^{Γ}(L^{∗}, kΓ). This vanishes in positive degrees, since kΓ is a free left module; and β induces an isomorphism of L^{∗} ⊗_{Γ} kΓ with L^{∗} by the calculation in the proof of 3.5. Hence the augmented total complex is exact, and Ξ(kΓ) is a projective resolution of L^{∗}. □

Using the cyclic property of the Lie operad one can construct analogues of 3.5 and 3.6 for F-modules, where F is the category of non-empty finite sets (without basepoint). This variant will be treated elsewhere.

3.7 Corollary. There is a functorial isomorphism π_{∗}(F) ≈ HΞ_{∗}(F) for Γ -modules F.

Proof. This follows immediately from Theorem 3.5 and the natural isomorphism Tor_{∗}^{Γ}(L^{∗}, F) ≈ π_{∗}(F) of ([10], 2.2). □

The bicomplex Ξ(F) is much smaller, and more transparent in its structure, than the Robinson-Whitehouse complex which served the same purpose in [11]. The application which we give in §5 makes clear that Ξ(F) has precise geometrical origins in the theory of E_{∞}-structures and infinite delooping.

4. The Γ-homology and Γ-cohomology of graded commutative algebras

Let A = {A_{n}}_{n∈ℤ} be an associative graded ring with unit which is commutative in the usual graded sense: that is, yx = (−1)^{mn}xy when x ∈ A_{m} and y ∈ A_{n}. Let B be an A-algebra, and G a B-module. (Here and henceforth the qualification “graded” is usually suppressed, but is to be understood; and unmarked tensor products are over the ground ring A.)

We denote by (B|A)^{⊗} the tensor algebra of B over A. Then (B|A)^{⊗} ⊗ G has a natural Γ-module structure over the ring B: if φ : [n] → [m] is any morphism in Γ, we set

φ_{∗}(b_{1} ⊗ ⋯ ⊗ b_{n} ⊗ g) = ε c_{1} ⊗ ⋯ ⊗ c_{m} ⊗ h

in which

c_{i} = b_{i_{1}} … b_{i_{r}}  if φ^{−1}(i) = {i_{1}, …, i_{r}} where i_{1} < i_{2} < ⋯ < i_{r}
h = b_{j_{1}} … b_{j_{s}}g  if φ^{−1}(0) = {0, j_{1}, …, j_{s}} where j_{1} < j_{2} < ⋯ < j_{s}

and in which ε is the sign of the permutation that rearranges {1, 2, …, n} in the order in which b_{1},...,b_{n} appear in the expansion of the product c_{1} … c_{m}. When φ happens to be a permutation σ (regarded as an isomorphism in Γ) this means that φ_{∗} rearranges the factors and multiplies by the sign (compare [10], p.158):

σ_{∗}(b_{1} ⊗ ⋯ ⊗ b_{n} ⊗ g) = ε(σ) b_{σ^{−1}1} ⊗ ⋯ ⊗ b_{σ^{−1}n} ⊗ g .

4.1 Definition. The Γ -cotangent complex of B over A is the bicomplex of Bmodules resulting from applying the Ξ-construction of §2 to the above Γ-module:

K(B|A) = Ξ((B|A)^{⊗} ⊗ B) .

The Γ -homology and Γ -cohomology of B relative to A, with coefficients in the B-module G, are defined as the homology and cohomology of K(B|A):

HΓ_{∗}(B|A; G) = HΞ_{∗}(K(B|A) ⊗_{B} G) ≈ HΞ_{∗}((B|A)^{⊗} ⊗_{A} G)

HΓ^{∗}(B|A; G) = HΞ^{∗} Hom_{B}(K(B|A), G) ≈ HΞ^{∗} Hom_{A}((B|A)^{⊗}, G)

where the bicomplexes on the right are given the differentials induced from those in the centre column. Since the Γ-modules here are graded, all these constructs have a further internal grading.

Conditions for the homotopy invariance of these definitions are given in 4.5 below.

4.2 Theorem. For any graded A-algebra B and B-module G

HΓ_{∗}(B|A; G) ≈ π_{∗}((B|A)^{⊗} ⊗ G) .

Proof. This is a special case of 3.7. □

4.3 Corollary. The above definition of Γ -homology is consistent with the definitions given for ungraded commutative algebras in [11] and in [17].

Proof. In the ungraded case, the Γ-module K(B|A) ⊗ G is the Loday functor of ([9], 3.2). In [11] and [17] it is proved that the original definitions of Γ-homology give groups isomorphic to the stable homotopy of this functor. □

4.4 Homotopy invariance of the cotangent complex.

In applications we often need a homotopy invariance property for the cotangent complex and for Γ-cohomology. By this we mean that the levelwise extensions of these functors to the category of simplicial graded commutative algebras should respect weak equivalences. This is true for the cotangent complex and for Γ-homology provided that B is flat over A. It is true for Γ-cohomology if B is a projective Amodule, or if B is flat and Ext_{A}^{s}(B, G) = 0 for all s > 0, where G is the coefficient module. When these conditions do not hold, a homotopy invariant cotangent complex can be obtained by replacing B by a simplicial projective resolution, then proceeding as above.

General properties of Γ-homology, such as the transitivity exact sequence, are proved in [17]. Some calculations are given in [14] and [15]. In particular, the Γ-homology of a polynomial algebra in one variable is isomorphic to the topological homology of the Eilenberg-Mac Lane spectrum Hℤ.

5. Application: obstruction theory for E_{∞} multiplications on ring spectra

5.1 Multiplicative actions of E_{∞} operads on spectra.

Let V be a commutative ring spectrum in the sense of classical homotopy theory. Thus there are a given unit map η : S → V and a multiplication μ : V ∧ V → V satisfying the homotopy associativity, homotopy commutativity and homotopy unit axioms. We work in a category of spectra which is symmetric monoidal with respect to smash product, such as the category of S-modules [5].

We investigate E_{∞} structures on V. An E_{∞} structure means an action of an E_{∞} operad C on V. This is prescribed by maps

μ_{m} : C_{m} ⋉_{Σ_{m}} V^{(m)} ⟶ V

where V^{(m)} is the mth smash power. The μ_{m} are required to be compatible with composition (that is, to form a morphism of non-unital operads from C to the endomorphism operad End(V)); and μ_{2} must be the given multiplication μ. In the category of S-modules, such an E_{∞} structure guarantees that V is equivalent to a commutative S-algebra: there are no obstructions to higher coherence of the unit η.

We shall take C to be the E_{∞} operad T introduced in ([17], 1.5). This operad of cell complexes is the product of the Barratt-Eccles operad with the tree operad. The Barratt-Eccles operad is E_{∞}: it has mth space EΣ_{m}, and composition is induced by wreath product of permutations. The mth space in the tree operad is the space T̃_{m} of trees with root and leaves labelled by the elements of [m] = {0, 1, …, m}, and composition is by grafting trees. As we explained in [17], the reason for our choice is that the product operad T , unlike the Barratt-Eccles operad, is a cofibrant E_{∞} operad in the model category structure defined by Hinich [7].

It is useful to note that the operad T can also be written T_{m} ≈ ẼΣ_{m} ×_{Σ_{m}} T̃_{m} where ẼΣ_{m} is the two-sided bar construction on the symmetric group Σ_{m}, and the action of Σ_{m} in the operad structure is the left action on ẼΣ_{m}.

We shall now explain how we propose to classify E_{∞} ring structures by an inductive procedure.

5.2 The boundary and the diagonal filtration in T .

The bar filtration on EΣ_{m} induces a filtration on T_{m}. Let us denote the rth space in the filtration of T_{m} by T_{m}^{r}. Thus T_{m}^{0} = Σ_{m} × T̃_{m}, and T_{m}^{r} = ∅ when r < 0.

We shall denote by ∂T̃_{m} the subspace of decomposable or fully-grown trees in T̃_{m}. These are the trees in which at least one internal edge has length 1. (This is the space denoted by T_{m} in [17].)

For each j-element subset of {1, 2, …, m}, the tree space T̃_{m} has a face homeomorphic to T̃_{i} × T̃_{j}, when i, j ≥ 2 and i + j = m + 1. It contains just those trees in which an edge of length 1 separates the j leaves labelled by the subset from the i−1 other leaves and the root. The stabilizer of this face is a conjugate of Σ_{i−1}×Σ_{j} in Σ_{m}. The union of all the faces is ∂T̃_{m}, and the intersection of two faces is either empty or is a common subface.

In the operad T , we define a face of T_{m} to be a subspace EΣ_{m} × F, where F is a face of the tree space T̃_{m}. The boundary ∂T_{m} of T_{m} is EΣ_{m} × ∂T̃_{m}, which is the same as the union of all the faces. In any action

μ_{m} : T_{m} ⋉_{Σ_{m}} V^{(m)} ⟶ V

of T on a spectrum V, the restriction of μ_{m} to any face (EΣ_{m} × F) ⋉ V^{(m)} is determined, up to coherent homotopy, by the corresponding composite

μ_{i} × μ_{j} : (T_{i} × T_{j}) ⋉_{Σ_{i−1}×Σ_{j}} V^{(m)} ⟶ V .

This is true because EΣ_{m} has EΣ_{i−1} × EΣ_{j} as a (filtered) (Σ_{i−1}×Σ_{j})-equivariant deformation retract. There is an explicit standard retraction EΣ_{m} ⟶ EΣ_{i−1} × EΣ_{j}, defined using shuffles, which we always use to extend the above composite μ_{i} × μ_{j} over the whole face. These retractions preserve filtration, and are coherent with respect to iterated face inclusions. Hence the restriction of μ_{m} to T_{m}^{r} ∩ ∂T_{m} is determined by the restrictions of the μ_{i}×μ_{j} to (T_{i}^{r}×T_{j}^{r})⋉V^{(m)}, where i+j = m+1. We now define the diagonal filtration ∇ on T by setting ∇^{r}T_{m} = T_{m}^{r−m}; and we define ∂∇^{r}T_{m} to be ∇^{r}T_{m} ∩ ∂T_{m}. It follows that μ_{m} restricted to ∂∇^{r}T is determined by the μ_{i} × μ_{j} restricted to ∇^{r−1}T × ∇^{r−1}T , because face inclusions strictly increase the lower index. This permits us to make inductive constructions.

5.3 Definition. An n-stage for an E_{∞} ring structure on V is a sequence of maps

μ_{m} : ∇^{n}T_{m} ⋉_{Σ_{m}} V^{(m)} ⟶ V

which on their restricted domain of definition satisfy the requirements for a morphism of operads from T to End(V).

A 2-stage for an E_{∞} ring structure is nothing but a map μ : V ∧ V → V. A 3-stage incorporates an associativity homotopy μ(1 ∧ μ) ≃ μ(μ ∧ 1) and a commutativity homotopy μ ≃ τμ, where τ interchanges factors. This 3-stage extends to a 4-stage if and only if these homotopies satisfy the well-known pentagonal and hexagonal homotopy conditions and the condition that the commutativity homotopy be homotopy commutative.

We denote by V_{∗}(X) and V^{∗}(X) the V -homology and V -cohomology of the spectrum or space X. The multiplication on V gives all the usual products in these theories. We denote by R the graded coefficient ring V_{∗}(S^{0}), and by Λ the dual Steenrod algebra V_{∗}(V), which is a graded Hopf R-algebroid. We make the standing assumption that Λ is R-flat, and that there is a perfect universal coefficient isomorphism for all n

V^{∗}(V^{(n)}) ≈ Hom_{R}(Λ^{⊗n}, R) .

The augmentation Λ → R makes R into a Λ-module, allowing us to define the complex Hom_{Λ}(K(Λ|R), R) and its cohomology HΓ^{∗}(Λ|R; R). The grading on the rings gives a third grading on the bicomplex K(Λ|R). The associated single complex Tot K(Λ|R) and the cohomology HΓ^{∗}(Λ|R; R) are therefore bigraded.

5.4 Proposition. Let V be a ring spectrum satisfying the above universal coefficient condition. Then the obstruction to extending an n-stage on V to an (n + 1)- stage is an (n, 2 − n)-cocycle of the complex Tot Hom_{Λ}(K(Λ|R); R). Given a fixed (n − 1)-stage, the obstruction cocycles to extending different n-stages form a single cohomology class in this complex.

Proof. In order to extend an n-stage {μ_{m}} to an (n + 1)-stage, we require for each integer m in the range 2 ≤ m ≤ n + 1 an extension of μ_{m} : ∇^{n}T_{m} ⋉_{Σ_{m}} V^{(m)} ⟶ V over ∇^{n+1}T_{m} ⋉_{Σ_{m}} V^{(m)}. The extension is determined on ∂∇^{n+1}T_{m} ⋉_{Σ_{m}} V^{(m)} already by the condition that {μ_{m}} should respect compositions, as explained in 5.2. The obstruction to extending μ_{m} therefore lies in the group

V^{1}((∇^{n+1}T_{m}/(∇^{n}T_{m} ∪ ∂∇^{n+1}T_{m})) ∧_{Σ_{m}} V^{(m)}) .

But we have

∇^{n+1}T_{m}/(∇^{n}T_{m} ∪ ∂∇^{n+1}T_{m}) ≈ (EΣ_{m}^{n−m+1}/EΣ_{m}^{n−m}) ∧ (T̃_{m}/∂T̃_{m})

where ∂T̃_{m} is the space of fully-grown (or decomposable) trees, and Σ_{m} acts freely upon the factor EΣ_{m}^{n−m+1}/EΣ_{m}^{n−m}, which is a wedge of spheres of dimension n − m + 1, indexed by Σ_{m}^{n−m+2}. It is known that T̃_{m}/∂T̃_{m} has the homotopy type of a wedge of (m−1)! spheres of dimension m−2, and that its homology is isomorphic as a Σ_{m}-module to Lie_{m}^{∗} with the twisted sign as in 2.1 (see for instance [20]). Thus the quotient ∇^{n+1}T_{m}/(∇^{n}T_{m}∪ ∂∇^{n+1}T_{m}) is a wedge of (n−1)-dimensional spheres. From the perfect universal coefficient formula V^{∗}(V^{(m)}) ≈ Hom_{R}(Λ^{⊗m}, R), we therefore have obstructions (to the existence of an n-stage) in all the groups

V^{1}((∇^{n+1}T_{m}/(∇^{n}T_{m} ∪ ∂∇^{n+1}T_{m})) ∧_{Σ_{m}} V^{(m)})

≈ Hom_{R}^{2−n}(Lie_{m}^{∗} ⊗R[Σ_{m}^{n−m+1}] ⊗ Λ^{⊗m}, R)

≈ Hom_{Λ}^{n−m+1, m−1, 2−n}(K(Λ|R), R)

from the definition of K(Λ|R) in 4.1. This sequence of elements, for m in the range 2 ≤ m ≤ n+1, forms an (n, 2−n)-cochain θ of Tot Hom_{Λ}(K(Λ|R); R). (The second grading 2 − n is the internal grading of the cohomology theory V^{∗}.)

A completely analogous argument shows that the various n-stages which extend a given (n − 1)-stage are enumerated by difference (n − 1, 2 − n)-cochains of Tot Hom_{Λ}(K(Λ|R); R).

We must now investigate how the obstruction cochain θ changes when the n-stage is varied. This corresponds to analysing the attaching maps connecting the quotients in the diagonal filtration: we show that these correspond to the differentials in the Ξ-complex. Suppose the n-stage is altered by a difference (n − 1, 2 − n)- cochain ρ. The obstruction component θ^{n−m+1, m−1, 2−n} is altered by

δ′ρ^{n−m, m−1, 2−n} + δ′′ρ^{n−m+1, m−2, 2−n}

where the two terms correspond to the two factors in the smash product decomposition of ∇^{n+1}T_{m}/(∇^{n}T_{m} ∪ ∂∇^{n+1}T_{m}) displayed above. In the first of these two terms, δ′ is the cohomology differential in the bar construction, arising from the smash factor EΣ_{m}^{n−m+1}/EΣ_{m}^{n−m}. In the second term, δ′′ is the cohomology dual of the vertical differential in the Ξ-construction (2.5), arising from the formula for the contribution to the boundary from the smash factor T̃_{m}/∂T̃_{m}. Here we use an explicit isomorphism between the homology of the space T̃_{m}/∂T̃_{m} and the representation Lie_{m}^{∗}, which was established in [20]. The calculation of the boundary is given in the proof of 6.4 of [17], and the evaluation of δ′′ then proceeds as in [16].

The above analysis of the coboundary shows that the indeterminacy in θ obtained by altering the n-stage therefore equals the group of (n, 2 − n)-coboundaries in Tot Hom_{Λ}(K(Λ|R); R). The same analysis shows in the usual way that θ is a cocycle. This completes the proof of the proposition. □

5.5 Theorem. Let V be a ring spectrum satisfying the universal coefficient condition as in 5.4. Suppose given an (n − 1)-stage μ for V which can be extended to an n-stage. Then there is a natural Γ-cohomology class [θ] ∈ HΓ^{n,2−n}(Λ|R; R), the vanishing of which is necessary and sufficient for μ to be extendable to an (n + 1)- stage. □

5.6 Theorem. Let V be as in 5.5 above. Suppose V is homotopy commutative and homotopy associative, and that HΓ^{n,2−n}(Λ|R; R) = 0 for all n ≥ 4. Then V has an E_{∞}-structure. If further HΓ^{n,1−n}(Λ|R; R) = 0 for all n ≥ 3, then this E_{∞}-structure is unique up to homotopy. □

Goerss and Hopkins have proved a very similar theorem to 5.6 in the course of extensive unpublished work based upon [6]. An important application shows that the Lubin-Tate spectra of [12] have E_{∞} structures.

5.7 Corollary (Goerss-Hopkins). The Lubin-Tate spectrum E_{n} of a Honda formal group law of height n has an E_{∞} multiplicative structure, and this structure is unique up to homotopy.

Proof. We need the results of the elegant homological treatment of Λ = V_{∗}V, due to Hopkins and Miller, which is presented in Part 3 of [12]. These arguments show that the perfect universal coefficient theorem holds for V = E_{n}, and that the André-Quillen cohomology D^{∗}(Λ|R; R) is all zero. Further, they show that Ext_{R}^{s}(Λ, R) = 0 for s > 0, so that the conditions in 4.4 for the homotopy invariance of Γ-cohomology are met. Therefore the dual in cohomology of Richter’s Atiyah-Hirzebruch spectral sequence [14] can be applied to show that HΓ^{∗}(Λ|R; R) is zero. □

An alternative proof that the Γ-cohomology of the dual Steenrod algebra for these spectra is zero is given in [15].

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Mathematics Institute, University of Warwick, Coventry CV4 7AL, England

E-mail address: car@maths.warwick.ac.uk