Γ-HOMOLOGY OF COMMUTATIVE

RINGS AND OF E_{∞} RING SPECTRA

Alan Robinson and Sarah Whitehouse

Introduction

In stable homotopy theory, the notion of E_{∞} ring spectrum is a substantial generalization of the algebraic notion of commutative ring. In this paper we introduce the natural homology and cohomology theories for E_{∞} ring spectra, which arise in problems of classification and extension of these generalized rings. The theory can be specialized to commutative rings in the ordinary algebraic sense by applying it to the corresponding Eilenberg-Mac Lane spectra. This gives a completely new homology theory for commutative rings, which has applications in homotopy theory and probably elsewhere. It is analogous to, but distinct from, the homology theory for commutative rings constructed by André [1] and by Quillen [10] (after work by Grothendieck, Harrison, Lichtenbaum and Schlessinger). One would not expect our theory to specialize to André’s, because polynomial algebras (which are acyclic in André/Quillen homology) do not give free E_{∞} ring spectra, and there is no reason why they should be acyclic in the homology which is appropriate there.

The first author’s work in this field began in the late 1980’s, and was announced at the Adams Memorial Conference in 1990. The second author has contributed in her Ph.D. thesis, written during 1992-94 [14], and since. There is also unpublished work of Waldhausen which, starting from different motivation and using an entirely different construction, obtains a homology theory for E_{∞} algebras by using methods of stable homotopy theory. This theory has been developed for discrete commutative algebras by Schwede [13]. Their work and ours are far apart in concept and in detail, but it is likely that they will eventually be proved equivalent where they overlap. We have furthermore very recently received a draft paper by I. Kriz which discusses E_{∞} cohomology.

1. The Γ-cotangent spectrum

Recent progress by Elmendorf, Kriz, Mandell and May [4] in the foundations of stable homotopy theory has lightened our task of exposition. There is now a straightforward definition of commutative ring spectrum, equivalent to the standard notion of E_{∞} ring spectrum, over which tensor products (among other concepts) can be handled as easily as in ordinary commutative algebra. Henceforth we adopt the standpoint of [4] in all foundational matters concerning spectra, though we shall often use traditional E_{∞} terminology.

Typeset by AMS-TEX

If P is a commutative ring spectrum and X is a P -module, then the free commutative ring spectrum generated by X is

ℙ_{P} X = ⋁_{j≥0} X^{(j)}/Σ_{j} ,

the wedge of the symmetric powers, where X^{(j)} is the jth power of X with respect to the smash product over the ring P. The functor ℙ_{P} is a monad on the category of P -modules. Let R be a commutative P -algebra: then R is an algebra for this monad, so there is a natural ring map ℙ_{P} R → R. Suppose given any R-module K. We define a functor Ω_{K} on all intermediate ring spectra (i.e. all P -algebras over R) by setting

Ω_{K}(Q) = Q ∧_{P} K ,

and we define a right action Ω_{K}ℙ_{P} → Ω_{K} of the monad ℙ_{P} on that functor as follows: on the summand Q^{(j)}/Σ_{j} of ℙ_{P} Q we take the track sum of the maps

Q^{(j)} ∧_{P} K ⟶^{t^{i}∧1} Q^{(j)} ∧_{P} K = Q ∧_{P} Q^{(j−1)} ∧_{P} K ⟶^{1∧μ} Q ∧_{P} K

for 1 ≤ i ≤ j, where t generates the cyclic permutations of the smash product Q^{(j)} and μ is the action of Q^{(j−1)} on K via Q → R; then using the freeness of the action of Σ_{j} on the smash power, together with the coherence of track addition, we obtain a map

(Q^{(j)}/Σ_{j}) ∧_{P} K → Q ∧_{P} K .

These maps give an action of the monad ℙ_{P} upon Ω_{K}, up to coherent homotopy, which is quite sufficient for the coend below to be constructed.

We regard Ω_{K}, with this right ℙ_{P} -action, as the module of K-valued Kähler differentials in the homotopical situation. (The action of ℙ_{P} incorporates the P - linear Leibnitz rule.) We form the Γ-cotangent spectrum by using the monadic bar construction B(−, −, −) of May [8]:

C(R/P; K) = B(Ω_{K}, ℙ_{P} , R) = ∫ ^{j} Δ^{j} ⋉ Ω_{K}ℙ_{P}^{j} R .

This is a R-module. In the absolute case, where P is the sphere spectrum S, we denote it by C(R; K), and we write ℙ for ℙ_{S}. On the other hand, when the module K is the ring R itself, we just write C(R/P).

The Γ-homology of R relative to P, with coefficients in the module K, is defined to be the homotopy of the cotangent spectrum:

HΓ_{∗}(R/P; K) = π_{∗}C(R/P; K) ;

and the Γ-cohomology is the homotopy of the dual R-module

HΓ^{∗}(R/P; K) = π_{−∗}F_{R}(C(R/P), K) .

Evidently these are homological functors of the R-module K. The following result is not so trivial.

1.1 Theorem (Transitivity). Let P → Q → R be maps of commutative ring spectra. Then there is a natural cofibration sequence of R-module spectra

C(Q/P; K) ⟶ C(R/P; K) ⟶ C(R/Q; K) ⟶ ΣC(Q/P; K)

for any R-module K. Consequently there is a long exact sequence in Γ-homology

… → HΓ_{n+1}(R/Q; K) → HΓ_{n}(Q/P; K) → HΓ_{n}(R/P; K) → HΓ_{n}(R/Q; K) → …

and a corresponding sequence in cohomology, which are analogues of André’s Jacobi-Zariski sequences. □

Of special interest is the discrete case, in which B is an A-algebra and M a Bmodule in the ordinary sense of commutative algebra. Then we define the cotangent spectrum C(B/A; M), and hence Γ-(co)homology, by applying the above constructions to the Eilenberg-Mac Lane spectra P = HA, R = HB and K = HM. Being a connective HB-module spectrum, C(B/A; M) corresponds to an object in the derived category D_{B}^{+} of B-modules. We realize this object as a chain bicomplex in §3. When B contains a field of characteristic zero, it is equivalent to the complexes defined by Harrison, André and Quillen. In other cases Γ-homology is completely different, though some formal properties carry over from the André/Quillen theory.

2. A spectral sequence

The cotangent spectrum C(R/P; K) is filtered, because it arises from a bar construction. It also has a filtration by degree, which we now explain. Since ℙ_{P} R = ⋁_{j≥0} R^{(j)}/Σ_{j} is graded by j, the iterate ℙ_{P}^{i} R inherits a gradation by the total number of smash factors R involved; and so does Ω_{K}ℙ_{P}^{i} R. Each of the face operators in this simplicial space preserves or decreases the degree of this gradation. Therefore the realization C(R/P; K) receives a filtration. Combinatorial analysis of the iterates of the functor ℙ_{P} shows that the kth quotient in this filtration has the homotopy type of the double suspension of

T_{k} ⋊_{Σ_{k}} EΣ_{k} ⋉_{Σ_{k}} R^{(k)} ∧_{P} K ,

where T_{k} is the space of fully-grown k-trees [12], with its natural Σ_{k}-action, and R^{(k)} is the kth smash power of the algebra R, taken over the ring spectrum P. There results a spectral sequence

E_{p,q−1}^{1} ≈ π_{p−2}(T_{q} ⋊_{Σ_{q}} EΣ_{q} ⋉_{Σ_{q}} R^{(q)} ∧_{P} K) ⟹ HΓ_{p+q−1}(R/P; K) .

The E^{1}-term is therefore the homotopy of an extended power of R over P, twisted by T_{q} (which is a wedge of (q − 3)-spheres) and by K. In general one can hope to calculate this for each q by the standard spectral sequence

H_{s}(Σ_{q}; V_{q} ⊗ π_{t}(R^{(q)} ∧_{P} K)) ⟹ E_{s+t+2,q−1}^{1}

where V_{q} is the Σ_{q}-module H̃_{q−3}(T_{q}).

Let us now specialize to the discrete case, where P = HA, R = HB, K = HM and the commutative algebra B is flat over A. Then, by ([11],Thm.12.2), R^{(q)} ∧_{P} K ≃ H(B_{A}^{⊗q} ⊗_{A} M). So the second spectral sequence collapses to the edge, yielding the following.

2.1 Theorem. Let B be a flat commutative algebra over the discrete ring A, and let M be a B-module. Then there is a first-quadrant spectral sequence

E_{p,q}^{1} ≈ H_{p}(Σ_{q+1}; V_{q+1} ⊗ B^{⊗q+1} ⊗ M) ⟹ HΓ_{p+q}(B/A; M)

where V_{q+1} is the Σ_{q+1}-module given by the homology of the tree-space T_{q+1}, and d^{r} has bidegree (r − 1, −r).

When B is projective, there is a dual spectral sequence in cohomology

E_{1}^{p,q} ≈ H^{p}(Σ_{q+1}; V^{q+1} ⊗ Hom(B^{⊗q+1}, M)) ⟹ HΓ^{p+q}(B/A; M) . □

3. Explicit complexes in the discrete case

The cotangent spectrum C(B/A; M), where B still denotes a flat commutative A-algebra, is constructed as described in §1 by using iterates of the free E_{∞} monad ℙ_{HA}. From general considerations, we know that the HB-module C(B/A; M) corresponds to some object in the derived category D_{B}^{+}. We may ask how to construct this chain complex. Under the flatness hypothesis, smash products over HA correspond precisely to tensor products over A, as noted above; so the monad ℙ_{HA} corresponds to the monad which takes an E_{∞} differential A-algebra X to

⨁_{j≥0} C_{∗}(Σ_{j}) ⊗_{Σ_{j}} X^{⊗j} ,

the tensor products being over A. The Kähler differentials functor of §1 translates directly. So one can translate C(B/A; M) into the realization of a bar construction in the derived category. This yields a bicomplex which, by the same combinatorial arguments as referred to in §2, is equivalent to a bicomplex of the form

(3.1)  CΓ_{p,q}(B/A; M) = ⨁_{k≥1} B^{⊗k} ⊗ C_{p}(Σ_{k}) ⊗ C̃_{q−2}(T_{k}) ⊗ M

where the horizontal differential is that of the two-sided bar construction on the symmetric groups Σ_{k}, and the vertical differential is that of the tree spaces T_{k} plus additional terms corresponding to the first and last faces in the monadic bar construction for C(B/A; M). There is a dual version for cohomology when B is projective.

This explicit complex gives one proof of Theorem 2.1, and also 3.2–3.5 below.

3.2 Theorem [14]. The edge p = 0 of the spectral sequence of 2.1 is precisely the complex used in defining the Harrison (co)homology [7] Harr^{∗}(B/A; M) of B (with a shift in degree). Therefore there are natural transformations

HΓ_{n}(B/A; M) → Harr_{n+1}(B/A; M) , HΓ^{n}(B/A; M) ← Harr^{n+1}(B/A; M)

when B is flat (resp. projective), which are isomorphisms when B contains a field of characteristic zero. □

It is known [10] that André/Quillen cohomology also coincides with Harrison theory under the same hypothesis of characteristic zero.

3.3 Proposition.

(1) HΓ_{0}(B/A; M) ≈ Ω_{B/A} ⊗_{B} M , HΓ^{0}(B/A; M) ≈ Der_{A}(B, M) ;

(2) HΓ^{1}(B/A; M) ≈ Exalcom_{A}(B, M) . □

(Here Exalcom_{A}(B, M), the module of infinitesimal A-algebra extensions of B by M, is as defined in [6], 0_{IV} §18.)

Thus Γ-(co)homology agrees with the theories of Harrison and André/Quillen in very low degrees. From H_{2} onwards Γ-(co)homology is different. The following example shows this, and reveals a non-trivial differential in the spectral sequence of 2.1.

3.4 Example. First take B = A = 𝔽_{2}. Then 1 ⊗ 1 ⊗ 1 ⊗ 1 is a non-bounding Harrison 4-cycle, by the calculation in ([2], §4). Thus Harr_{4}(𝔽_{2}/𝔽_{2}; 𝔽_{2}) ≉ 0, and by 3.2 our element 1 ⊗ 1 ⊗ 1 ⊗ 1 exists in E_{0,3}^{2}. Since transitivity theorems imply that HΓ_{3}(𝔽_{2}/𝔽_{2}; 𝔽_{2}) ≈ 0 (and similarly for André homology), this cycle must map by the only available differential d^{2} to a non-zero element of E_{1,1}^{2}. (The only such element is α ⊗ 1 ⊗ 1, where α generates H_{1}(Σ_{2}; 𝔽_{2}); for the module V_{2} is trivial). Thus HΓ_{3} ≉ Harr_{4}.

Now let us take B to be the polynomial algebra 𝔽_{2}[X], A = 𝔽_{2}, M = B/(X) ≈ 𝔽_{2}. A brief calculation with shuffles shows that E_{0,3}^{2} ≈ Harr_{4}(𝔽_{2}[X]/𝔽_{2}; 𝔽_{2}) contains no non-zero element of degree two in X. Therefore α ⊗ X ⊗ X ∈ E_{1,1}^{2} is an infinite cycle which is not in the image of d^{2} and therefore is not a boundary. So HΓ_{2}(𝔽_{2}[X]/𝔽_{2}; 𝔽_{2}) ≉ 0, and HΓ_{2} is not André’s H_{2}.

The next result is proved by methods like those which give the analogous result for André/Quillen homology. In the statement, C(B/A) denotes the Γ-cotangent complex in the derived category D_{B}^{+}.

3.5 Theorem. (1) Let B and C be A-algebras, with B flat over A. Then the complex C(B ⊗_{A} C/C) is quasi-isomorphic to C(B/A) ⊗_{A} C, so that

HΓ_{∗}(B ⊗_{A} C/C; M) ≈ HΓ_{∗}(B/A; M)

for any B ⊗_{A} C-module M.

(2) Let B and C be flat A-modules, and M a B ⊗_{A} C-module. Then there is a quasi-isomorphism

C(B ⊗_{A} C/A) ≃ C(B/A) ⊗_{A} C ⊕ C(C/A) ⊗_{A} B ,

and therefore HΓ_{∗}(B ⊗_{A} C/A; M) ≈ HΓ_{∗}(B/A; M) ⊕ HΓ_{∗}(C/A; M).

(3) If B is an étale A-algebra, then HΓ_{∗}(B/A; M) ≈ HΓ^{∗}(B/A; M) ≈ 0 for every B-module M. □

4. A Lie product in HΓ^{∗}(B/A; B)

For B a projective commutative A-algebra, the explicit bicomplex of the previous section may be used to construct a graded Lie product structure in Γ-cohomology.

This is compatible with the Lie product in Harrison cohomology ([5], §5.7), with which Γ-cohomology agrees in characteristic zero. The bicomplex for Γ-cohomology is

CΓ^{p,q}(B/A; B) = ⨁_{k≥1} Hom_{A}(B^{⊗k} ⊗ C_{p}(Σ_{k}) ⊗ C̃_{q−2}(T_{k}), B).

If f ∈ CΓ^{p,q} and g ∈ CΓ^{r,s} then we define f ∘ g in CΓ^{p+r,q+s} as follows. Consider a term (b_{1} ⊗ ⋯ ⊗ b_{k} ⊗ [σ_{1}| … |σ_{p+r}] ⊗ t), where t is an oriented (q + s − 2)-simplex in the tree space T_{k}, b_{i} ∈ B and σ_{i} ∈ Σ_{k}. Suppose that we can split the tree t at an internal edge, to get a part with m + 1 free edges (including that labelled 0) and q − 1 internal edges, and a part with k − m + 2 free edges and s − 1 internal edges. After some relabelling, this gives a tree t′, a (q − 2)-simplex in T_{m} and a tree t̄, an (s − 2)-simplex in T_{k−m+1}. Write t = t′ + t̄. Using the relabelling we split up our permutations in Σ_{k} into permutations, σ′_{1} …, σ′_{p+r} in Σ_{m} and σ̄_{1}, …, σ̄_{p+r} in Σ_{k−m+1}. We assign p permutations to f and r to g, exactly as in the Alexander-Whitney map for the two sided bar construction:

Δ[σ_{1}| … |σ_{n}] = ∑_{j=0}^{n}[σ_{1}| … |σ_{j}]σ_{j+1} … σ_{n} ⊗ σ_{1} … σ_{j}[σ_{j+1}| … |σ_{n}]

This is a strictly co-associative diagonal approximation, which is cocommutative up to homotopy. Write Δ[σ_{1}| … |σ_{n}] = ∑_{j=0}^{n} Δ_{1}^{j}[σ_{1}| … |σ_{n}] ⊗ Δ_{2}^{j}[σ_{1}| … |σ_{n}].

Then (f ∘ g)(b_{1} ⊗ ⋯ ⊗ b_{k} ⊗ [σ_{1}|σ_{2}| … |σ_{p+r}] ⊗ t) is given by

∑_{t=t′+t̄} f(b_{m_{1}} ⊗ ⋯ ⊗ g(b_{l_{1}} ⊗ ⋯ ⊗ b_{l_{u}} ⊗ Δ_{2}^{p}([σ̄_{1}|σ̄_{2}| … |σ̄_{p+r}]) ⊗ t̄)

⊗ ⋯ ⊗ b_{m_{v}} ⊗ Δ_{1}^{p}([σ′_{1}|σ′_{2}| … |σ′_{p+r}]) ⊗ t′)

where {l_{1}, …, l_{u}} is the ordered preimage of the set of labels of the part of the tree t which becomes t′, and {m_{1}, …, m_{v}} is its ordered complement.

Finally put [f, g] = f ∘g−(−1)^{(p+q)(r+s)}g∘f. The map [−, −] : CΓ^{p,q} ×CΓ^{r,s} → CΓ^{p+r,q+s} induces a graded Lie product in Γ-cohomology,

[−, −] : HΓ^{l}(B/A; B) × HΓ^{m}(B/A; B) → HΓ^{l+m}(B/A; B).

We have HΓ^{0}(B/A; B) = Der_{A}(B; B), and for 0-cocycles the Lie product above is the usual bracket of derivations.

Letting M be a B-module, for f ∈ CΓ^{l}(B/A; M), g ∈ CΓ^{m}(B/A; B), we can define f ∘ g ∈ CΓ^{l+m}(B/A; M) by the same formula as above. This makes HΓ^{∗}(B/A; M) a module over the graded Lie ring HΓ^{∗}(B/A; B). In particular HΓ^{n}(B/A; B) is a module over the ordinary Lie ring HΓ^{0}(B/A; B) = Der_{A}(B; B).

5. An analogue of the infinite delooping spectral sequence of H. Miller

Let X be an E_{∞} ring spectrum. The simplicial spectrum with nth term ℙ^{n+1}X has an extra degeneracy, given by the unit of the monad ℙ. All its face maps, and the standard degeneracy maps, are morphisms of E_{∞} ring spectra. Therefore its realization is an E_{∞} ring spectrum homotopy equivalent to X

(5.1)  ∫ ^{n} Δ^{n} ⋉ ℙ^{n+1}X ≃ X .

Let H𝔽_{p} denote the Eilenberg-Mac Lane spectrum for the field 𝔽_{p}, and R the Dyer-Lashof algebra for the prime p. Then H𝔽_{p∗}(ℙX) is the free allowable R-algebra generated by the module H𝔽_{p∗}X [3]. The equivalence (5.1) above thus implies that the simplicial algebra having sth algebra H𝔽_{p∗}(ℙ^{s+1}X) is a resolution of H𝔽_{p∗}X by free allowable R-algebras.

We now specialize to the discrete case in which X = HA, A being an augmented 𝔽_{p}-algebra. We form the absolute cotangent spectrum (i.e. we take the sphere spectrum as ground ring)

C(HA, H𝔽_{p}) = B(Ω_{H𝔽_{p}}, ℙ , HA) .

The bar filtration yields a homotopy spectral sequence

E_{s,t}^{1} ≈ π_{t}(Ω_{H𝔽_{p}}ℙ^{s}HA) ⟹ π_{s+t}C(HA; H𝔽_{p}) = HΓ_{s+t}(A; 𝔽_{p})

converging to the absolute or “topological” Γ-homology of A with coefficients in 𝔽_{p} (regarded as an A-module via the augmentation). We have

E_{s,∗}^{1} ≈ π_{∗}(H𝔽_{p} ∧ ℙ^{s}HA) ≈ H_{∗}(ℙ^{s}HA) ≈ 𝔽_{p} ⊗_{R} Q (H_{∗}(ℙ^{s+1}HA))

where H_{∗}(−) is homology with 𝔽_{p} coefficients, and Q(−) denotes indecomposables. This shows that (E^{1}, d^{1}) is obtained by applying the functor 𝔽_{p} ⊗_{R} Q(−), which is denoted Q(−) in [9], to the above free resolution of H𝔽_{p}(X), where X = HA. Therefore the spectral sequence becomes

(5.2)  E_{s,t}^{2} ≈ L_{s}Q(H_{∗}(HA))_{t} ⟹ HΓ_{s+t}(A; 𝔽_{p})

(where L_{s}Q is the sth left derived functor of Q), in close analogy with the main result of Miller [9]. This strongly reinforces the idea that the Γ-cotangent spectrum is to be regarded as the result of infinitely delooping, in some sense, the commutative algebra. Such ideas are in accord with the theory developed by Waldhausen and Schwede.

Calculations with the above spectral sequence are in progress at the time of writing.

References

[1] M. André, Méthode simpliciale en algèbre homologique et algèbre commutative, Lect. Notes in Math. #32, Springer-Verlag, New York–Heidelberg–Berlin, 1967.

[2] M. Barr, “Harrison homology, Hochschild homology and triples,” J. Algebra 8 (1963), 314–323.

[3] R.R. Bruner, J.P. May, J.E. McClure & M. Steinberger, H_{∞} ring spectra and their applications, Lect. Notes in Math. #1176, Springer-Verlag, New York– Heidelberg–Berlin, 1986.

[4] A.D. Elmendorf, I. Kriz, M.A. Mandell & J.P. May, “Rings, modules and algebras in stable homotopy theory,” 1995.

[5] M. Gerstenhaber & S.D. Schack, “A Hodge-type decomposition for commutative algebra cohomology,” J. Pure Appl. Algebra 48 (1987), 229–247.

[6] A. Grothendieck, “Éléments de géométrie algébrique,” Inst. Hautes Études Sci. Publ. Math. (1960–1967).

[7] D.K. Harrison, “Commutative algebras and cohomology,” Trans. Amer. Math. Soc. 104 (1962), 191–204.

[8] J.P. May, The geometry of iterated loop spaces, Lect. Notes in Math. #271, Springer-Verlag, New York–Heidelberg–Berlin, 1972.

[9] H.R. Miller, “A spectral sequence for the homology of an infinite delooping,” Pacific J. Math. 79 (1978), 139–155.

[10] D.G. Quillen, “On the (co-)homology of commutative rings,” in Proceedings of Symposia in Pure Mathematics, Alex Heller, ed. #17, American Mathematical Society, Providence, RI, 1970, 65–87.

[11] Alan Robinson, “Derived tensor products in stable homotopy theory,” Topology 22 (1983), 1–18.

[12] Alan Robinson & Sarah Whitehouse, “The tree representation of Σ_{n+1},” J. Pure Appl. Algebra (to appear).

[13] Stefan Schwede, “Spectra in model categories and applications to the algebraic cotangent complex,” Universität Bielefeld, Preprint.

[14] S.A. Whitehouse, Γ-(Co)homology of commutative algebras and some related representations of the symmetric group, Ph.D. thesis, University of Warwick, 1994 .

Mathematics Institute, University of Warwick, Coventry, England CV4 7AL E-mail address: car@uk.ac.warwick.maths sarah@uk.ac.warwick.maths