Cellularization over group-rings with applications to the Eilenberg-Moore

spectral-sequence

Thesis submitted for the degree of

”Doctor of Philosophy” by

Shoham Shamir

Cellularization over group-rings with applications to the Eilenberg-Moore

spectral-sequence

Thesis submitted for the degree of

”Doctor of Philosophy” by

Shoham Shamir

This work was carried out under the supervision of Prof. Emmanuel Farjoun

Abstract

This work studies cellularization (cellular approximation) in the stable homotopy category. Cellularization is a type of co-localization and thus dual to localization of spectra as defined in [1]. Dwyer and Greenlees showed in [6] that local cohomology for commutative rings can be formulated in terms of cellularization. Continuing along these lines we show various properties of cellulariztion that arise as analogs of known properties of local cohomology, even over non-commutative rings. Much of this work is done over group-rings. The group-rings we consider are quite general and include, for example, the chains of the loop-space of a topological space.

Results of Dwyer, Greenlees and Iyengar from [7] enable us to apply the cellular-approximation techniques we developed to the question of convergence of the Eilenberg-Moore cohomology spectral sequence for a fibration. By applying the results obtained for cellularization over group-rings we generalize one known convergence result and give a different proof for another.

Contents

1 Introduction 2 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Description of Results . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Layout of this Work . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Setting and Conventions . . . . . . . . . . . . . . . . . . . . . 9 1.5 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Nullification and Cellularization 12

3 R/I-Cellularization 18 3.1 Cellularization when R is a classical ring . . . . . . . . . . . . 29

4 Change of Rings 33

5 Cellularization in Group-Rings 38

6 Two Examples 45

7 Cellularization over Abelian and Nilpotent Groups 51 7.1 Finite Cyclic Groups . . . . . . . . . . . . . . . . . . . . . . . 51 7.2 Finite Abelian Groups . . . . . . . . . . . . . . . . . . . . . . 53 7.3 Finite p-groups . . . . . . . . . . . . . . . . . . . . . . . . . . 56 7.4 Finite Nilpotent Groups . . . . . . . . . . . . . . . . . . . . . 59

8 Relation with the Eilenberg-Moore Spectral Sequence 61

A Proof of Lemma 8.4 70

Chapter 1

Introduction

1.1 Background

The concept of cellular approximation originated in the work of J.H.C. Whitehead (see e.g. [22]). Whitehead showed how every topological space can be approximated (up to a weak equivalence) by a CW-complex. This can be thought of as a ”cellular approximation by spheres”.

The more general framework for topological spaces is the following. Given a pointed topological space A, the A-cellularization of another pointed space X is the closest approximation of X built from A using pointed homotopy colimits. This notion of A-cellularization was first treated systematically by Dror-Farjoun [11] and Chachólski [3]. Also fundamental to this subject is the work of Hirschhorn [15]. Dwyer and Greenlees translated this concept to an algebraic setting in [6]. Further work on cellularization was done by Dwyer,Greenlees and Iyengar in [7], here in the wider setting of 𝕊-algebras.

Let R be a ring and let A and C be ℤ-graded chain-complexes over R. We say C is A-cellular over R, if Ext_{R}^{∗}(A, N) = 0 implies Ext_{R}^{∗}(C, N) = 0 for every R-chain complex N. The following, equivalent definition of Acellularity is perhaps less mysterious. Let D_{R} denote the derived category of R-modules. The localizing subcategory generated by A is the smallest triangulated subcategory of D_{R} containing A which is closed under isomorphisms, translations (i.e. suspensions of chain-complexes), coproducts and direct-summands (retracts). Then C is A-cellular if and only if C belongs to the localizing subcategory generated by A ([7, 4.1]).

If C belongs to the localizing subcategory generated by A, we say that A builds C. If C can be constructed from A in a finite number of steps using triangles (i.e. mapping-cones), suspensions, retracts and finite direct sums, we say that A finitely builds C. It is a simple matter to show that if A finitely builds C then A builds C.

The A-cellularization of a complex X is a map of R-complexes: Cell_{A}^{R}X → X such that Cell_{A}^{R}X is the closest approximation of X by an A-cellular complex. The term ”closest approximation” has two equivalent meanings, one is a universal property: the map Cell_{A}^{R}X → X is terminal in D_{R} among all the maps from A-cellular objects to X. The second meaning is that the induced map: Ext_{R}^{∗}(A, Cell_{A}^{R}X) → Ext_{R}^{∗}(A, X) is an isomorphism.

The following result of Dwyer and Greenlees ([6]) shows cellularization can be used to define local cohomology. Suppose R is a commutative ring and I is a finitely generated ideal of R then Dwyer and Greenlees showed the homology groups of Cell_{R/I}^{R}X are the local cohomology groups at I of X, denoted H_{I}^{∗}(X).

The concept of A-cellularization extends effortlessly to the stable homotopy category, which is the setting used here. We take R to be an 𝕊-algebra (ring spectrum) in the sense of [10], and we work in the category of left Rmodules. One can also take R to be an ordinary ring (or a DGA), so long as the term ”R-module” is understood to be an unbounded chain-complex over R, i.e. a differential graded R-module. In the present work, often A will also be an 𝕊-algebra with an 𝕊-algebra map R → A making A into an R-module. The homotopy fiber of this map will be denoted I and we use the notation R/I for A. Again, one can take R to be an ordinary ring, I a two-sided ideal of R and R/I as the usual quotient ring.

Before proceeding to describe the results, we briefly go over some necessary conventions. We use ⊗_{R} to denote the smash product of spectra over R. If R is a ring this is equivalent to the usual tensor product. Similarly Hom_{R}(−, −) stands for the function spectrum of R-modules. The functors Hom_{R} and ⊗_{R} are taken in the derived sense, i.e. we always take appropriate resolutions before applying them.

1.2 Description of Results

Cell_{R/I}^{R} Construction

If R is a commutative ring and I ⊆ R is a finitely generated ideal, then the I-local cohomology of an R-module M is given by the formula (see e.g. [17]):

H_{I}^{∗}(M) = colim_{n}Ext_{R}^{∗}(R/I^{n}, M)

We give, under some restrictions, an analogous construction for R/Icellularization, where R → R/I is a map of 𝕊-algebras and I is the homotopy fiber of this map. Dwyer, Greenlees and Iyengar defined in [7] the notion of proxy-smallness (definition 2.7 below). In the present context R/I is called proxy-small if there is an R-module K (called a Koszul complex for R/I) such that R finitely builds K, R/I finitely builds K and K builds R/I (see example 2.9). For a proxy-small R/I we use the generalized Adams resolution (defined in [19]) to construct R/I-cellularization. This Adams resolution for R/I over R is a tower of 𝕊-algebras:

R/I ← R/I^{⊗2} ← R/I^{⊗3} ← ⋯

Where R/I^{⊗n} is the homotopy cofiber of the map \underbrace{I ⊗ I ⋯ ⊗ I}_{n times} → R.

(Theorem 3.4). If R/I is proxy-small then for every R-module M, the R/I-cellularization of M is canonically equivalent to:

hocolim_{n}Hom_{R}(R/I^{⊗n}, M)

This theorem allows us to build the R/I-cellularization of an R-module X from R/I-modules. Although Cell_{R/I}X is built from R/I-modules, it is usually not an R/I-module. The simplest example for this is when R is a commutative ring, I an ideal of R and M is an I-nilpotent R-module, namely I^{n}M = 0 for some n. Then M is built from R/I-modules, but M is not an R/I-module, it is rather an R/I^{n}-module.

Independence of Base

Independence of base for local cohomology of commutative Noetherian rings is the following property: if R → S is a map of such rings and I is an ideal of R then for every S module M: the I-local cohomology of M as an R module is isomorphic to the IS-local cohomology of M as an S-module. Here are the analogous statements for cellularization:

(Lemma 4.2). Let R → S be a map of 𝕊-algebras.

1. ( Strong independence of base) Let A be an S-module, if S ⊗_{R} A is Acellular as an S-module, then for any S-module X: Cell_{A}^{R}X ≃ Cell_{A}^{S}X as R-modules.

2. ( Weak independence of base) Let A be an R-module, if S ⊗_{R} A is Acellular as an R-module, then for any S-module X: Cell_{A}^{R}X ≃ Cell_{S⊗_{R}A}^{S}X as R-modules.

One can easily recover the original independence of base property from the weak independence of base above. Our main application of these statements is to group rings, but before we can demonstrate this we require some notation. If G is a discrete or simplicial group and k is a commutative 𝕊algebra we denote by k[G] the group-ring k ⊗_{𝕊} Σ^{∞}G_{+}. Observe that k is a k[G]-module. We use the notation Cell^{G} for Cell_{k}^{k[G]} whenever k is understood from the context. The following example shows the application of strong independence of base to group-rings, as well as some relation with the Eilenberg-Moore spectral sequence.

Example 1.1. Let B be a finite, connected, nilpotent space with an abelian fundamental group and let k be a commutative ring. We denote by ΩB the Kan loop-group of B, which is a simplicial group equivalent to the loopspace of B. Consider the map of 𝕊-algebras: k[ΩB] → k[π_{1}(B)] induced by the projection ΩB → π_{1}(B). Then this map has the following strong independence of base property:

Cell^{ΩB}(M) ≃ Cell^{π_{1}(B)}(M)

for every k[π_{1}(B)]-module M. To prove this last statement we must show that k[π_{1}(B)] ⊗_{k[ΩB]} k is a k-cellular k[π_{1}(B)]-module, since this would give the desired strong independence of base property. As we will see in chapter 5, the chain-complex of the universal cover of B, with coefficients in k, is equivalent to k[π_{1}(B)] ⊗_{k[ΩB]} k as k[π_{1}(B)]-modules. We denote this complex by C_{∗}(B̃; k). Since B is a nilpotent space, by [14, II.2.18] the homology groups H_{∗}(B̃; k) are nilpotent k[π_{1}(B)]-modules and hence k-cellular. Since B is a finite space, C_{∗}(B̃; k) is bounded and therefore built by it’s homology groups over k[π_{1}(B)] (see the proof of proposition 3.12). We conclude that C_{∗}(B̃; k) is a k-cellular k[π_{1}(B)]-module.

The group π_{1}(B) is a finitely generated abelian group, so by [6], kcellularization of a k[π_{1}(B)]-module M computes the local cohomology of M at the augmentation ideal I of k[π_{1}(B)], hence:

π_{n}(Cell^{ΩB}(M)) ≅ H_{I}^{−n}(M)

Now consider a covering fibration F → E → B. Since F is a π_{1}(B)- space, the cochain complex of F with coefficients in k, denoted C^{∗}(F; k), is a k[π_{1}(B)]-module (this cochain complex is simply ∏_{π_{0}(F)} k). As we will see in chapter 8, the Eilenberg-Moore cohomology spectral sequence for this fibration (with coefficients in k), converges to Cell^{ΩB}C^{∗}(F; k) (see [10, IV.4.1] for convergence conditions of this spectral sequence). We conclude that the Eilenberg-Moore spectral sequence for this fibration, converges to the I-local cohomology groups of the k[π_{1}(B)]-module C^{∗}(F; k).

Here is another application of independence of base. We say a cellularization functor Cell_{A}^{R} is trivial if it is equivalent to the identity functor. The following result is an easy consequence of independence of base:

(Lemma 5.6). Let N → G → Q be a short exact sequence of topological (or simplicial) groups. If Cell^{Q} is trivial then for every G-module X there is an equivalence of N-modules:

Cell^{G}(X) ≃ Cell^{N}(X)

If, in addition, Cell^{N} is trivial, then Cell^{G} is also trivial.

The Eilenberg-Moore Spectral Sequence

As mentioned above, the machinery of cellularization can be used to understand the Eilenberg-Moore spectral sequence. The Eilenberg-Moore cohomology spectral sequence for a fibration F → E → B, with coefficients in a commutative 𝕊-algebra k, has the form:

E_{p,q}^{2} = Tor_{p,q}^{H^{∗}(B;k)}(H^{∗}(E; k), k) ⇒ π_{p+q}(C^{∗}(E; k) ⊗_{C^{∗}(B;k)} k)

where the notation C^{∗}(X; k) stands for the function spectrum F_{𝕊}(Σ^{∞}X, k), i.e. the ”cochains” of X with coefficients in k. Thus, the Eilenberg-Moore spectral sequence computes the homotopy groups of C^{∗}(E; k) ⊗_{C^{∗}(B;k)} k. We will not discuss the convergence properties of this spectral sequence, as this subject appears in literature, see e.g. [10, IV.4.1].

The dual, homological, spectral sequence has been studied extensively. We mention three results which motivate some of the research in this work. In these results the ground ring k is a classical commutative ring. The first is the original result, due to Eilenberg and Moore [9]. Their result states that if the base space B is simply connected, then the homological spectral sequence strongly converges to the homology of the fiber space F. The second result is a generalization of Dwyer [4]. Dwyer showed that the Eilenberg-Moore homology spectral sequence strongly converges to the homology of F if and only if the fundamental group of the base space π_{1}(B) acts nilpotently on the homology groups of the fiber H_{∗}(F; k).

The third result we mention is also due to Dwyer [5]. It is known as the exotic convergence. This result can be roughly stated in the following way. Suppose B is a nilpotent space satisfying certain finiteness conditions. Then the Eilenberg-Moore homology spectral sequence converges to the homology of F up to a nilpotent filtration. Where a nilpotent filtration of H_{n}(F; k) is a filtration: H_{n}(F; k) = M_{0} ⊇ M_{1} ⊇ ⋯ such that M_{i}/M_{i+1} has a trivial π_{1}(B) action. Note that ∩_{i}M_{i} need not be zero. We also mention the works of Bousfield [2] and Shipley [21] which extended these results to more general homotopy pullback diagrams.

All three results mentioned above depend on the action of π_{1}(B) on the homology groups of F. This curious role of π_{1}(B) in such convergence results has motivated much of the research in this work. This role is explained in section 8.

The relation between the 𝕊-module C^{∗}(E; k) ⊗_{C^{∗}(B;k)} k, the fiber F and cellularization is given by the proposition below, which is a direct consequence of [7]. For a simplicial group G, the notation kG stands for the 𝕊-algebra C_{∗}(G; k) = k ∧_{𝕊} Σ^{∞}G_{+}.

(Proposition). Let F → E → B be a fibration, suppose B is connected and all spaces are pointed. Let k be a commutative 𝕊-algebra, if k is proxy-small as a k[ΩB]-module then:

C^{∗}(E; k) ⊗_{C^{∗}(B;k)} k ≃ Cell^{ΩB}C^{∗}(F; k)

This happens, for example, if B is a finite space or if B is the classifying space of a finite nilpotent group.

Using cellularization arguments and the results we have for cellularization over group-rings, we get the following pair of theorems.

(Theorem 8.1). Fix a prime p. Let N be a finite nilpotent group and let P ⊆ N be the p-Sylow subgroup of N, so N ≅ P × H with the order of |H| being prime to p. Let F → E → BN be a fibration over the classifying space of N. If k is any commutative 𝕊-algebra such that π_{0}(k) is an 𝔽_{p}-algebra, then:

C^{∗}(E; k) ⊗_{C^{∗}(B;k)} k ≃ C^{∗}(F_{h(H)})

where F_{h(H)} is the homotopy orbit space of F with respect to the H-action on F. In particular, if N = P then

C^{∗}(E; k) ⊗_{C^{∗}(B;k)} k ≃ C^{∗}(F; k)

(Theorem 8.2). Fix a prime p. Let B be a finite connected nilpotent space with a finite fundamental group N = π_{1}(B). Let P ⊆ N be the p-Sylow subgroup of N, so N ≅ P × H. Given a fibration of pointed spaces F → E → B, then:

Tor_{−n}^{C^{∗}(B;𝔽_{p})}(C^{∗}(E; 𝔽_{p}), 𝔽_{p}) = H^{n}(F; 𝔽_{p})^{H}

where H^{n}(F; 𝔽_{p})^{H} are the fixed points of the H-action on H^{n}(F; 𝔽_{p}).

The first theorem deals with general multiplicative cohomology theories with characteristic p and thus generalizes a result of Kriz from [18], showing convergence of the Eilenberg-Moore cohomology spectral sequence for a fibration over B(ℤ/p), with coefficients in Morava K-theories.

The second result concerns only mod-p cohomology. This result is hardly surprising as it is dual to the aforementioned result of Dywer ([5]) concerning convergence of the Eilenberg-Moore homology spectral sequence. However, we give a different proof, which uses only cellularization arguments.

1.3 Layout of this Work

We begin by introducing the notions of cellularization and nullification in chapter 2. We recall their basic properties and give elementary examples. We also recall the definition of proxy-smallness and show A-cellularization commutes with homotopy colimits of telescopes when A is proxy-small.

Next, in chapter 3, we specialize to R/I-cellularization for R-modules. Here we give the construction for Cell_{R/I}(M) when R/I is proxy-small mentioned above. One interesting implication of this construction is that although Cell_{R/I}(M) is not an R/I-module in general, it can be built from R/I-modules. The end of this chapter deals with the case where R is a classical ring and I is an ideal of R. In this case we show some connections of R/I-cellularization with the I-power torsion functor, denoted Γ_{I}.

In chapter 4 we prove the independence of base lemmas. We give examples where there is independence of base and where it does not hold. Starting from chapter 5 we focus on group-rings, concentrating our attention on k-cellularization over a group-ring k[G]. In this chapter we give criteria for existence of the independence of base property over group-rings. Chapter 6 gives two examples where such a condition for independence of base is satisfied.

Chapter 7 deals with cellularization over abelian and nilpotent groups, giving proofs of several simple, but necessary, results. In chapter 8 we finally come to the Eilenberg-Moore spectral sequence, applying the machinery we constructed earlier to prove the desired convergence results. The appendix gives the deferred proof of a lemma from chapter 8.

1.4 Setting and Conventions

We work in the category of left R-modules where R is an 𝕊-algebra in the sense of [10] and 𝕊 stands for the sphere spectrum. Alternately, R can be a symmetric ring-spectrum in the sense of [16]. As noted above, one can also take R to be a DGA (ring) and take the category of R-modules to be the category of differential graded left modules (unbounded chain complexes of left R-modules).

We follow [7] in both terminology and notation. Thus, as in [7], we use the notation − ⊗_{R} − for the smash product of R-modules. Similarly we use Hom_{R}(−, −) to denote the function spectrum of R-modules. All functors in this paper are taken in the derived sense, in particular − ⊗_{R} − and Hom_{R}(−, −). This means that we always assume to have replaced our modules by appropriate resolutions before applying the functor in question. A few times in this work we need to use the strict, non-derived, version of the Hom functor, in these cases we will use the notation ^{S}Hom_{R}(−, −).

The homotopy groups of an R-module M are the stable homotopy groups of its underlying spectrum. If one chooses to work in the category of chaincomplexes over a ring R, then homotopy groups should be understood as homology groups of chain-complexes. Recall that for a ring R the 𝕊-algebra HR is the appropriate Eilenberg-Mac Lane 𝕊-algebra. Note that a result of Schwede and Shipley ([20, 5.1.6]), shows the model category of HR-modules is Quillen equivalent to the projective model category of unbounded chain complexes over R, denoted Ch_{R}.

We denote the A-cellularization of an R-module M by Cell_{A}^{R}M, or by Cell_{A}M whenever R is clear from the context. The derived (or homotopy) category of R-modules is denoted by D_{R}. Since this is a triangulated category, we sometimes use the term triangle to indicate a homotopy cofibration sequence. If f : M → N is a map of R-modules, then Cone(f) denotes the mapping cone of f. An R-module M is called connective if π_{i}(M) = 0 for all i < 0. We say that M is bounded above (bounded below) if there exists some index n such that π_{i}(M) = 0 for all i > n (i < n).

Always k will denote a commutative 𝕊-algebra, which is our ”ground ring”. Given a based space X the chains of X with coefficients in k is the spectrum:

C_{∗}(X; k) = k ⊗_{𝕊} Σ^{∞}X

When k is understood from the context we will simply use the notation C_{∗}(X). If X has no base point we add a disjoint base point to X before taking the chains, i.e. C_{∗}(X; k) = k ⊗_{𝕊} Σ^{∞}X_{+}. Similarly, the cochains of based space X with coefficients in k is the function spectrum:

C^{∗}(X; k) = Hom_{𝕊}(Σ^{∞}X, k)

Spaces are simplicial sets. The notation Ω stands for the Kan loop-group functor. For a based space X, the space ΩX is a simplicial group naturally weakly equivalent to the loop-space of X.

In many cases the ring R we work over will be a group-ring. If G is a discrete or simplicial group and k is a commutative 𝕊-algebra, the group-ring kG is C_{∗}(G; k), which is an 𝕊-algebra. If k is a commutative ring and G a discrete group, this boils down to the usual group-ring kG. Whenever R is a group-ring kG the notation Cell^{G}(−) will stand for Cell_{k}^{kG}(−). Since we consider kG as an 𝕊-algebra, the category of kG-modules has the ”weak” model category structure on it. This means a map of kG-modules is a weak equivalence if it induces an isomorphism on the homotopy groups. Therefore a map f : X → Y of G-spaces that yields an isomorphism on homotopy groups will induce a weak equivalence C_{∗}(f; k) : C_{∗}(X; k) → C_{∗}(Y; k) of kG-modules.

For a simplicial group G, we denote by WG a free contractible G-space. The construction of WG is functorial. The classifying space of G is denoted in the usual manner by BG.

1.5 Acknowledgements

I am grateful to my supervisor Prof. Emmanuel Dror-Farjoun for his guidance, support and for many useful suggestions. Most notably, it was he who led me to consider the Eilenberg-Moore spectral sequence in the context of cellular approximations. Part of this work was done while visiting the Centre De Recerca Mátematica in Barcelona under a Marie Curie scholarship. I would like to thank Prof. Carles Casacuberta who enabled this visit and guided me during my stay.

I would also like to thank the referees for carefully reading this work and for their helpful comments. The third referee also deserves a special thanks for his remarks, which were essential in making the foundations of this work sound. Moreover, one of his remarks helped correct a mistake in the proof of proposition 3.18.

Nullification and Cellularization

In this section we introduce the concepts of cellularization and nullification and give some of their elementary properties. The definition of proxysmallness from [7] is also quoted in this section, as well as some properties of cellularization of proxy-small modules.

We open with the definitions of cellularization and nullification. Recall the category we work in is the category of left R-modules, where R is an 𝕊-algebra.

Definition 2.1. Fix an R-module A.

• An R-module N is called A-null if Hom_{R}(A, N) ≃ 0.

• An R-module M is called A-cellular if Hom_{R}(M, N) ≃ 0 for every A-null module N.

• A map f : X → Y is called an A-equivalence if the induced map:

Hom_{R}(A, f) : Hom_{R}(A, X) → Hom_{R}(A, Y)

is an equivalence.

• A-nullification is an augmented functor η : id → P_{A} such that P_{A}(X) is A-null for every R-module X and η_{X} is initial among all maps in D_{R} from X to A-null objects.

• A-cellularization is a coaugmented functor μ : Cell_{A}^{R} → id such that Cell_{A}^{R}X is A-cellular and the map μ_{X} is an A-equivalence. We will use the notation Cell_{A} whenever the ring R is clear from the context.

Remark 2.2. Let S = {A_{i}}_{i∈I} be a set of R-modules. We say an R-module N is S-null if Hom_{R}(A_{i}, N) ≃ 0 for every i ∈ I. We say an R-module M is S-cellular if Hom_{R}(M, N) ≃ 0 for every S-null R-module N. It is a simple exercise to show that an R-module M is S-cellular if and only if M is ⊕_{i∈I}A_{i}-cellular.

The basic properties of cellularization and nullification are summed up in the following lemma. These properties are well known and all but the last are easy to prove. The proof of the last property is explaind in remark 2.6 below.

Lemma 2.3. Let A be an R-module.

1. Let X → Y → Z be a triangle. If two of the three are A-null (Acellular) then so is the third.

2. If f : X → Y is an A-equivalence then the cone of f is A-null and Cell_{A}f is an equivalence.

3. If f : C → C′ is an A-equivalence between A-cellular modules, then f is an equivalence.

4. If C is A-cellular then C is also A ⊕ B-cellular for any R-module B.

5. A module C is A-cellular if and only if every A-equivalence is also a C-equivalence.

6. The homotopy colimit of A-cellular modules is also A-cellular.

7. Transitivity of cellularity: If A′′ is A′-cellular and A′ is A-cellular then A′′ is also A-cellular.

8. A map f : C → M is the A-cellularization of M if and only if C is A-cellular and f is terminal among all maps in D_{R} from A-cellular objects to X.

9. For every R-module X there is a triangle:

Cell_{A}X → X → P_{A}X

In particular, the homotopy fiber of the A-nullification map: X → P_{A}X is the A-cellularization of X.

10. The nullification functor P_{A} and the cellularization functor Cell_{A} exist and are unique up to equivalence. These functors are exact, i.e. preserve triangles.

Example 2.4. Assume k is a commutative ring and R is a k-algebra. For simplicity, we will work in the category Ch_{R} of unbounded chain complexes over R in this example. As noted before, the model category of HR-modules (in the brave new algebra sense) is Quillen-equivalent to the model category Ch_{R} of unbounded chain complexes over R (see [20, 5.1.6]). So this example in fact applies to both settings.

Let I be a two sided ideal of R and consider the R-module R/I. An R/I-null R-complex is a chain complex X such that Ext_{R}^{∗}(R/I, X) = 0. As we will see below (in lemma 3.1), every R/I-complex is R/I-cellular as an Rcomplex. The following argument shows that, in addition, every I-nilpotent R-module is R/I-cellular. Suppose M is an I-nilpotent R-module, then there is a filtration of M:

0 = M_{0} ⊆ M_{1} ⊆ M_{2} ⊆ ⋯ ⊆ M_{n} = M

where M_{i+1}/M_{i} are R/I-modules. We show M_{i} is R/I-cellular by induction on i. The module M_{1} is R/I-cellular since it is an R/I-module. In the short exact sequence:

M_{i} → M_{i+1} → M_{i+1}/M_{i}

both M_{i} and M_{i+1}/M_{i} are R/I-cellular, the first by induction and the second because it is an R/I-module. The short exact sequence M_{i} → M_{i+1} → M_{i+1}/M_{i} is also a triangle, therefore, by property 1 of lemma 2.3, M_{i+1} is also R/I-cellular.

This can be generalized to certain chain-complexes. Let X be a bounded above chain-complex, i.e. there is an integer t such that X_{n} is zero for n > t. An argument of Dwyer and Greenlees [6, proof of proposition 5.2] shows X is ⨁_{n} H_{n}(X)-cellular (we quote their argument in the proof of proposition 3.12). If, in addition to being bounded above, every homology group of X is I-nilpotent then, by property 7 of lemma 2.3 (transitivity of cellularity), X is R/I-cellular.

The full subcategory of A-cellular modules has a well known alternate description. Before we can give this description we must recall some definitions.

Definition 2.5. A full subcategory of R − Mod is called thick if it is closed under equivalences, triangles and retracts (direct summands). Closure under triangles means that given a triangle X → Y → Z where two of the modules belong to the subcategory, so does the third. In particular a thick subcategory is closed under suspensions and finite coproducts. A thick subcategory is called localizing if in addition it is closed under arbitrary coproducts.

Let A be an R-module, the smallest thick subcategory containing A is called the thick subcategory generated by A. The localizing subcategory generated by A is similarly defined. Following [7], we say an R-module B is finitely built by A if B belongs to the thick subcategory generated by A. An R-module which is finitely built by R is also called a small R-module. Finally, we say B is built by A if B belongs to the localizing subcategory generated by A.

It is easy to see that any R-module built from A is A-cellular. In fact, an R-module is A-cellular if and only if it is built from A, see [7, 4.1]. Thus, the full subcategory of A-cellular R-modules is the localizing subcategory generated by A. Similarly, if S = {A_{i}}_{i∈I} is a set of R-modules, then an R-module M is S-cellular (see remark 2.2) if and only if M belongs to the localizing subcategory generated by the R-module ⊕_{A∈S}A. For this reason, we will say an R-module M is built from a set of R-modules S if M is (⊕_{A∈S}A)-cellular. Note that the localizing subcategory generated by ⊕_{A∈S}A is the smallest localizing subcategory which contains S.

Remark 2.6. As noted in [7, 4.1], for every R-module M there is an Aequivalence M′ → M where M′ is built by A. This can be seen from the arguments of Dror-Farjoun ([11]) or, alternately, using the results of Hirschhorn ([15, 5.1.5]). From this property one easily sees that every A-cellular module is built from A. For suppose that M is A-cellular. Then the map M′ → M above is an A-equivalence between A-cellular modules and hence an equivalence. This implies M ≃ M′ is built from A.

Given any R-module M, the construction of the A-equivalence M′ → M mentioned above is indeed a functor. Together with the fact that M′ is Acellular, this proves the last property of lemma 2.3. The exactness of the cellularization functor follows from the observation that if X → Y → Z is a triangle then the map Cone(Cell_{A}X → Cell_{A}Y) → Z is an A-cellularization of Z and hence equivalent to Cell_{A}Z. Note that, as usual, one must take appropriate resolutions of X, Y and Z before applying the A-cellularization functor. Similarly, one must first replace an R-module M by an appropriate equivalent resolution (fibrant replacement) M̃ to get an augmentation map Cell_{A}M̃ → M̃.

In general we have little understanding of the A-cellularization functor. But, there is a case where we have slightly more control over Cell_{A}. In [7], Dwyer, Greenlees and Iyengar showed that when A has a certain property, called proxy-smallness, then A-cellulariztion is given by the following formula:

Cell_{A}X ≃ Hom_{R}(A, X) ⊗_{End_{R}(A)} A

We will also make use of this property, its definition is given below.

Definition 2.7. Call an R-module A proxy-small if there exists an R-module K such that K is finitely built from R, K is also finitely built from A and K builds A. The module K is called a Koszul complex associated to A.

Remark 2.8. Suppose A is proxy-small with an associated Koszul complex K. Because K and A build each other, a module M is built from A if and only if M is built from K. This is an easy consequence of transitivity of cellularity (property 7 in lemma 2.3). Moreover, property 5 of lemma 2.3 shows that a map is a K-equivalence if and only if it is an A-equivalence. Therefore K and A have the same cellularization and nullification functors. This means we can use K instead of A in our computations. The advantage of K over A is that K is a small R-module. We shall use this advantage in the next section.

Example 2.9 ([7, example 5.9] ). Here is an example of proxy-smallness of Dwyer,Greenlees and Iyengar from [7, example 5.9]. Let G be a finite group. Recall SU(n) is the special unitary group of n × n Hermitian matrices of determinant 1. It is a Lie group. There is an embedding of G into SU(n) for large enough n. This embedding makes SU(n) into a G-space (with no base point). Let k be a commutative ring. Dwyer, Greenlees and Iyengar showed that C_{∗}(SU(n); k) is a Koszul complex for k over the group-ring kG.

Here is another nice property of A-cellularization when A is proxy-small, proved by Dwyer and Greenlees, in [6, proposition 4.3].

Proposition 2.10. If A is a proxy-small R-module then Cell_{A}R is an Rbimodule and for every R-module X:

Cell_{A}X ≃ (Cell_{A}R) ⊗_{R} X

Remark 2.11. In [6] this was proved in the case where R is a ring and A is small R-complex. As noted by the authors, the proof works just as well when R is an 𝕊-algebra and A is a small R-module. If A is proxy-small, let K be a Koszul complex for A, then there is a natural equivalence of coaugmented functors: Cell_{K} ≃ Cell_{A}. Since K is small the proposition is true for K and therefore also true for A.

We end this section with yet another property of A-cellularization, for A proxy-small, showing that A-cellularization commutes with homotopy colimits of telescopes.

Lemma 2.12. Assume A is proxy-small and let X_{0} → X_{1} → ⋯ be a telescope, then

hocolim_{n}(Cell_{A}X_{n}) ≃ Cell_{A}(hocolim_{n}X_{n})

Proof. The functor Cell_{A}(−) ≃ Cell_{A}(R) ⊗_{R} − is a left adjoint and therefore commutes with homotopy colimits.

R/I-Cellularization

In this section we specialize to R/I-cellularization of R-modules. The main result of this section is theorem 3.4, where we give a construction for R/Icellularization when R/I is proxy-small. We then consider two specific cases of R/I-cellularization. In the first case R/I is Hπ_{0}(R) and we give, in proposition 3.12, a result dual to a result of Bousfield ([1, lemma 3.3]). In the second case R/I is a retract of R and we show that in this case R/Icellularization is simply the coaugmented functor Hom_{R}(R/I, −) (proposition 3.15). Towards the end of this section we explore the case of R/Icellularization when R is a classical ring and I is a two-sided ideal of R. This last part represents an attempt to generalize some results of Dwyer and Greenlees from [6] to non-commutative rings.

The setting we consider is the following: let k be a commutative 𝕊-algebra and let R be a k-algebra. Fix a fibration sequence of R-modules:

I ⟶^{i} R ⟶^{a} T

where T is a k-algebra, I is a k-algebra without a unit and both maps are k-algebra maps. This is the definition of an extension of T by I given by Lazarev in [19]. We use the notation R/I for T. Note that I is naturally an R-bimodule through the map i and R/I = T is an R-bimodule via a. We further remark that I can always be replaced by an equivalent cofibrant Rbimodule. Similarly, the map a : R → R/I can be replaced by an equivalent k-algebra map a′ : R → T′ which makes T′ into a cofibrant R-bimodule. Details of such constructions can be found in [10, VII].

Our first goal is to give an alternate construction for R/I-cellularization of R-modules. We start with the following simple lemma.

Lemma 3.1. Every R/I-module is R/I-cellular as an R-module.

Proof. Let M be an R/I-module and let N be an R-module that is R/I-null, we must show that Hom_{R}(M, N) ≃ 0. Using standard ⊗ ↔ Hom adjunctions (see [10, III.6]) yields:

Hom_{R}(M, N) ≃ Hom_{R}(R/I ⊗_{R/I} M, N)
≃ Hom_{R/I}(M, Hom_{R}(R/I, N))
≃ Hom_{R/I}(M, 0) ≃ 0

The construction of R/I-cellularization we are about to give uses the generalized Adams resolution, defined in [19]. We recall this definition below.

Definition 3.2. Denote by I^{⊗n} the tensor product: \underbrace{I ⊗_{R} I ⋯ ⊗_{R} I}_{n times} and set

I^{⊗0} = R. Recall i is the map I ⟶^{i} R. Define R/I^{⊗n} to be the cone of the map:

i ⊗ i ⊗ ⋯ ⊗ i : I^{⊗n} → R

Thus, there is a triangle:

I^{⊗n} → R → R/I^{⊗n}

The map: \underbrace{1 ⊗_{R} ⋯ ⊗_{R} 1}_{n times} ⊗i : I^{⊗n+1} → I^{⊗n} induces (via the mapping cone

functor) a map:

R/I^{⊗n+1} → R/I^{⊗n}

which completes the commutative diagram:

[figure: I^{⊗n+1}; i^{⊗n+1}; 1^{⊗n}⊗i; R; R/I^{⊗n+1}; I^{⊗n}; i^{⊗n}; R; R/I^{⊗n}]

Note that the map I^{⊗n} → R is a map of R_{e} ≅ R ⊗_{k} R^{op} modules (i.e. Rbimodules), thus its cone: R/I^{⊗n} is also an R_{e}-module, in other words it is an R-bimodule. By similar arguments, the maps R/I^{⊗n+1} → R/I^{⊗n} are maps of R-bimodules.

The resulting tower of R-bimodules:

⋯ → R/I^{⊗n} → ⋯ → R/I^{⊗2} → R/I

is called a generalized Adams resolution by Lazarev ([19, theorem 7.1]).

Remark 3.3. Moreover, in [19] Lazarev shows that R/I^{⊗n} is a k-algebra whose R-bimodule structure is given by the k-algebra map: R → R/I^{⊗n}. Lazarev also shows the maps: R/I^{⊗n+1} → R/I^{⊗n} are k-algebra maps. For our needs the R-bimodule structure on R/I^{⊗n} will suffice.

Theorem 3.4. If R/I is proxy-small then:

Cell_{R/I}(−) ≃ hocolim_{n}Hom_{R}(R/I^{⊗n}, −)

and

P_{R/I}(−) ≃ hocolim_{n}Hom_{R}(I^{⊗n}, −)

The maps: R → R/I^{⊗n} induce maps: Hom_{R}(R/I^{⊗n}, −) → Hom_{R}(R, −) which give the natural transformation Cell_{R/I} → id. Similarly, the natural transformation id → P_{R/I} is induced by the maps: I^{⊗n} → R.

Remark 3.5. We need to explain why Hom_{R}(R/I^{⊗n}, X) and Hom_{R}(I^{⊗n}, X) are naturally R-modules for every (left) R-module X. Recall, that both R/I^{⊗n} and I^{⊗n} are R-bimodules. The functors Hom_{R}(R/I^{⊗n}, −) and Hom_{R}(I^{⊗n}, −) are taken in the category of left R-modules. Therefore, the right R-action on R/I^{⊗n} gives a natural left R-module structure on Hom_{R}(R/I^{⊗n}, X). The same argument applies for Hom_{R}(I^{⊗n}, X).

Remark 3.6. The equivalence: Cell_{R/I}X ≃ hocolim_{n}Hom_{R}(R/I^{⊗n}, −) is reminiscent of the usual construction for I-local cohomology: if R is a commutative ring and I is a finitely generated ideal of R, then the I-local cohomology groups of an R-module M are:

H_{I}^{∗}M = colim_{n}Ext_{R}^{∗}(R/I^{n}, M)
≅ H_{−∗}(hocolim_{n}Hom_{R}(R/I^{n}, M))

In fact, if I is a principal ideal generated by an element x and x is not a zero divisor, then the two constructions are the same. Here is the reason why: the ideal I in this case is isomorphic to R as an R-module, so the triangle I^{⊗n} → R → R/I^{⊗n} is isomorphic to the short exact sequence:

R ⟶^{x^{n}} R → R/(x^{n})

and therefore R/I^{⊗n} ≅ R/I^{n}.

Before proving theorem 3.4 we need a different formulation for the intended construction of R/I-nullification. We give this construction in the following definition.

Definition 3.7. Given an R-module X we define a telescope:

P_{0}X → P_{1}X → ⋯ → P_{n}X → ⋯

inductively. Set P_{0}X = X and

P_{n+1}X = Cone(Hom_{R}(R/I, P_{n}(X)) ⟶^{a^{∗}} P_{n}(X))

(recall a is the map a : R → R/I).

Because the cone of the map:

Hom_{R}(R/I, P_{n}(X)) → P_{n}(X)

is equivalent to Hom_{R}(I, P_{n}(X)), we see (by induction on n) that P_{n}(X) is equivalent to Hom_{R}(I^{⊗n}, X). Moreover, these equivalences yield a map of telescopes:

\xymatrix{P_{0}X \ar[r] \ar[d] & P_{1}X \ar[r] \ar[d] & ⋯ \ar[r] & P_{n}X \ar[r] \ar[d] & ⋯ \\ X \ar[r] & Hom_{R}(I, X) \ar[r] & ⋯ \ar[r] & Hom_{R}(I^{⊗n}, X) \ar[r] & ⋯}

in which every vertical map is an equivalence. Denote by P (X) the homotopy colimit: hocolim_{n}P_{n}X. The maps: X → P_{n}X induce a map X → P (X), which is clearly equivalent to the map X → hocolim_{n}Hom_{R}(I^{⊗n}, X). We will use the sequence of lemmas below to show P (X) is the R/I-nullification of X.

Lemma 3.8. Let a : R → R/I be a map of k-algebras (as above), then there exists a map m : R/I ⊗_{R} R/I → R/I in the derived category D_{R} such that the following maps are equal in D_{R}:

m(a ⊗_{R} 1) = m(1 ⊗_{R} a) = id_{R/I}

This lemma essentially says that R/I is almost an R-ring, i.e. there is a multiplication map (in the derived category D_{R}) m : R/I ⊗_{R} R/I → R/I for which a is a left and right unit. Of course, to get such a map on the model category level, one must replace R/I with some equivalent 𝕊-algebra.

Proof. The map a induces a functor a^{∗} : R/I − Mod → R − Mod which is right adjoint to the functor a_{∗} = R/I ⊗_{R} −. If A is an R-module and B is an R/I-module then the adjoint of a map f : R/I ⊗_{R} A → B is the composition (a^{∗}f) ∘ (a ⊗_{R} 1) : A → a^{∗}B.

Let μ : R/I ⊗_{R} R/I → R/I be the map of R/I-modules that is adjoint to the identity map id_{R/I} via the adjunction above. Set m = a^{∗}μ, then the composition:

R/I ⟶^{a⊗1} R/I ⊗_{R} R/I ⟶^{m} R/I

is the map adjoint to μ, hence it is the identity. This proves m(a⊗_{R}1) ≃ id_{R/I}, i.e. a is a left unit for m. To show m(1 ⊗_{R} a) ≃ id_{R/I} we need only repeat the proof in the category of right R-modules.

Lemma 3.9. For any R-module X:

hocolim_{n}Hom_{R}(R/I, P_{n}X) ≃ 0

Proof. From the previous lemma we see that R/I is a retract (in D_{R}) of R/I ⊗_{R} R/I, via the map a ⊗ 1. Thus for any R-module Y, the R-module Hom_{R}(R/I, Y) is a retract of Hom_{R}(R/I ⊗_{R} R/I, Y), via the map (a ⊗ 1)^{∗}. The adjunction Hom_{R}(R/I ⊗_{R} R/I, Y) ≃ Hom_{R}(R/I, Hom_{R}(R/I, Y)) gives the following commutative diagram:

\xymatrix{Hom_{R}(R/I ⊗_{R} R/I, Y) \ar[rr]^{≃} \ar[dr]_{(a⊗1)^{∗}} &  & Hom_{R}(R/I, Hom_{R}(R/I, Y)) \ar@{-->}[dl]^{Hom_{R}(R/I,a^{∗})} \\  & Hom_{R}(R/I, Y)}

So applying the functor Hom_{R}(R/I, −) to the map

a^{∗} : Hom_{R}(R/I, Y) → Y

yields a map:

Hom_{R}(R/I, Hom_{R}(R/I, Y)) → Hom_{R}(R/I, Y)

that has a left inverse in D_{R}.

There is a natural bijection between homotopy classes of maps Σ^{m}R/I → Y and elements of π_{m}(Hom_{R}(R/I, Y)). We have shown that π_{m}(Hom_{R}(R/I, Y)) is a retract of the group π_{m}(Hom_{R}(R/I, Hom_{R}(R/I, Y))), therefore every map Σ^{m}R/I → Y splits, in D_{R}, through Hom_{R}(R/I, Y).

Consider the triangle Hom_{R}(R/I, P_{n}X) → P_{n}X → P_{n+1}X. Every map Σ^{m}R/I → P_{n}X splits through Hom_{R}(R/I, P_{n}X) and is therefore killed in P_{n+1}X, hence:

hocolim_{n}Hom_{R}(R/I, P_{n}X) ≃ 0

Lemma 3.10. Let X_{0} → X_{1} → ⋯ be a telescope. Let K be a small Rmodule finitely built from R/I. If hocolim_{n}Hom_{R}(R/I, X_{n}) ≃ 0 then:

Hom_{R}(K, hocolim_{n}X_{n}) ≃ 0

Proof. Since K is small, we have that: Hom_{R}(K, hocolim_{n}X_{n}) ≃ hocolim_{n}Hom_{R}(K, X_{n}). So it is sufficient to show that hocolim_{n}Hom_{R}(K, X_{n}) ≃ 0. Consider the full subcategory C of R−Mod generated by modules M such that hocolim_{n}Hom_{R}(M, X_{n}) ≃ 0. This subcategory contains R/I and is closed under suspensions, finite coproducts, retracts and equivalences. It is also closed under triangles, because a triangle:

A → B → C

gives rise to a triangle of telescopes:

Hom_{R}(C, X_{n}) → Hom_{R}(B, X_{n}) → Hom_{R}(A, X_{n})

and hence to a triangle of homotopy colimits:

hocolim_{n}Hom_{R}(C, X_{n}) → hocolim_{n}Hom_{R}(B, X_{n}) → hocolim_{n}Hom_{R}(A, X_{n})

Thus C contains every module finitely built from R/I, in particular K ∈ C.

Proof of theorem 3.4. We start by showing P (X) is the R/I-nullification of X, for any R-module X. Lemma 3.10 shows P (X) is K-null and thus R/Inull (since K builds R/I). It remains only to prove that the natural transformation η : id → P has the desired universal property. Assume N is an R/I-null module. Thus Hom_{R}(R/I, N) ≃ 0 and the map N = P_{0}N → P_{1}N is an equivalence. Continuing by induction we see that P_{n}N ≃ N and the maps η_{n} : P_{n}N → P_{n+1}N are equivalences. Therefore the map η : N → P (N) is also an equivalence.

Now given a map f : X → N, applying the functor P yields a map P (f) : P (X) → P (N) ≃ N. Therefore, in D_{R}, we get that f = η_{N}^{−1}∘P (f)∘η_{X} and f splits uniquely (again in D_{R}) through η_{X} : X → P (X), as in the following diagram:

\xymatrix{X \ar[r]^{η_{X}} \ar[d]_{f} & P (X) \ar[d]^{P (f)} \ar@{-->}[dl] \\ N \ar[r]^{η_{N}}_{≃} & P (N)}

which proves the desired universal property.

Next we prove the formula for R/I-cellularization. From lemma 2.3 we see that Cell_{R/I}X is the homotopy fiber of the map X → P (X). Consider the following commutative diagram of functors, where the rows are triangles:

[figure: Hom_{R}(R/I^{⊗n}, −); Hom_{R}(R, −) = id; Hom_{R}(I^{⊗n}, −); Hom_{R}(R/I^{⊗n+1}, −); id; Hom_{R}(I^{⊗n+1}, −)]

Taking homotopy colimit yields the following triangle of functors:

hocolim_{n}Hom_{R}(R/I^{⊗n}, −) → id → P (−)

When R/I is proxy-small, we use the construction of R/I-cellularization to show every R/I-cellular module has a (possibly infinite) filtration whose successive quotients are R/I-modules:

Theorem 3.11. Assume R/I is proxy-small as an R-module. Then an Rmodule X is R/I-cellular if and only if there exists a telescope: 0 = X_{0} → X_{1} → ⋯ such that X ≃ hocolim_{n}X_{n} and Cone(X_{n} → X_{n+1}) is equivalent (as an R-module) to an R/I-module, for every n.

Proof. The ”if” part is simple. Suppose we have such a telescope 0 = X_{0} → X_{1} → · · ·. We show, by induction on n, that X_{n} is R/I-cellular. In the triangle:

X_{n} → X_{n+1} → Cone(X_{n} → X_{n+1})

the module X_{n} is R/I-cellular (by the induction assumption) and Cone(X_{n} → X_{n+1}) is an R/I-module and therefore R/I-cellular. Hence X_{n+1} is also R/Icellular. The homotopy colimit of R/I-cellular modules is also R/I-cellular.

Now assume X is R/I-cellular, therefore X is equivalent to its R/Icellularization:

X ≃ hocolim_{n}Hom_{R}(R/I^{⊗n}, X)

We need only show that the successive cones of the telescope:

0 → Hom_{R}(R/I, X) → Hom_{R}(R/I^{⊗2}, X) → ⋯

are R/I-modules. First we show that Hom_{R}(R/I, X) is an R/I-module. As we saw, the (left) R-module structure of Hom_{R}(R/I, X) comes from the right R-module structure on R/I. But R/I is a right R/I-module and the right R-module structure comes from this R/I-module structure. Hence Hom_{R}(R/I, X) is an R/I-module.

To understand the other successive cones in the telescope above, we need the following definitions, given by Lazarev in [19]. Define I^{⊗n}/I^{⊗n+1} by the following triangle:

I^{⊗n+1} ≃ I^{⊗n} ⊗_{R} I ⟶^{1⊗···⊗1⊗i} I^{⊗n} → I^{⊗n}/I^{⊗n+1}

Clearly I^{⊗n}/I^{⊗n+1} is an R-bimodule, since it is a cone in the category of R-bimodules. Recall the map R/I^{⊗n+1} → R/I^{⊗n} is induced by the map 1 ⊗ ⋯ ⊗ 1 ⊗ i : I^{⊗n+1} → I^{⊗n} (using the mapping cone functor). Simple diagram chasing reveals that the homotopy fiber of the map R/I^{⊗n+1} → R/I^{⊗n} is I^{⊗n}/I^{⊗n+1}. Therefore, the cone of the map

Hom_{R}(R/I^{⊗n}, X) → Hom_{R}(R/I^{⊗n+1}, X)

is equivalent to Hom_{R}(I^{⊗n}/I^{⊗n+1}, X).

The module I^{⊗n}/I^{⊗n+1} has another description given in [19]: applying the functor I^{⊗n} ⊗_{R} − to the triangle: I → R → R/I gives a natural equivalence:

I^{⊗n}/I^{⊗n+1} ≃ I^{⊗n} ⊗_{R} R/I

Thus I^{⊗n}/I^{⊗n+1} is naturally equivalent to an R − R/I-bimodule. Since I^{⊗n}/I^{⊗n+1} is equivalent to a right R/I-module, we get that Hom_{R}(I^{⊗n}/I^{⊗n+1}, X) is a left R/I-module.

Having proved the main theorem of this section, we turn to consider two specific cases of R/I-cellularization. The results we will get will come into play in sections 7 and 8.

In the first case we take R to be a connective 𝕊-algebra. An 𝕊-algebra R is called connective if π_{n}(R) = 0 for all n < 0. Recall that for an abelian group A, the appropriate Eilenberg-MacLane 𝕊-module is denoted HA. By [10, IV.3.1], if R is connective then there exists map of 𝕊-algebras R → Hπ_{0}(R) which yields an isomorphism on the π_{0}-level. In particular, Hπ_{0}(R) is an R-module. We show that when R is connective, every bounded above Rmodule is Hπ_{0}(R)-cellular. This result is dual to a result of A.K. Bousfield from [1, lemma 3.3] over 𝕊.

Proposition 3.12. Let R be a connective 𝕊-algebra, then every R-module whose homotopy groups are bounded above is Hπ_{0}(R)-cellular.

Proof. Let us denote the 𝕊-algebra Hπ_{0}(R) by R_{0}. We begin by making two simple observations. First, the 𝕊-algebra map R → R_{0} implies that for every π_{0}(R)-module P, the R_{0}-module HP is also an R-module. Second, if M is an R-module then clearly π_{i}(M) is a π_{0}(R)-module and therefore Hπ_{i}(M) is an R_{0}-module.

Next we quote an argument of Dwyer and Greenlees [6, proof of proposition 5.2] showing that if M is a bounded-above R-module then M is ⊕_{n}Hπ_{n}(M)- cellular. Connectivity of R implies the existence of Postnikov sections in the category of R-modules (see [7, lemma 3.2]). Following Dwyer and Greenlees ([6]), we make the following definitions. Let M⟨−∞, j⟩ be the j-th Postnikov section of an R-module M. Thus, there is a natural map of R-modules p_{j} : M → M⟨−∞, j⟩ such that π_{k}(p_{j}) is an isomorphism for k ≤ j and π_{m}(M⟨−∞, j⟩) is zero for m > j. Define M⟨i, ∞⟩ to be the homotopy fiber of the map M → M⟨−∞, i − 1⟩, yielding a triangle:

M⟨i, ∞⟩ ⟶^{c_{i}} M ⟶^{p_{i−1}} M⟨−∞, i − 1⟩

The map π_{k}(c_{i}) is an isomorphism for k ≥ i and π_{m}(M⟨i, ∞⟩) is zero for m < i. Define M⟨i, j⟩ (where i ≤ j) to be

(M⟨i, ∞⟩)⟨−∞, j⟩

(this is equivalent to (M⟨−∞, i⟩)⟨j, ∞⟩ ). The homotopy cofibration sequence below is especially useful:

(∗) M⟨j + 1, j + 1⟩ → M⟨i, j + 1⟩ → M⟨i, j⟩

Note that M⟨i, i⟩ ≃ Hπ_{i}(M).

Let M be an R-module. If π_{n}(M) is non-zero only in a bounded range, say between i and j, then M ≃ M⟨i, j⟩. A simple induction argument, using the sequence (∗) above, shows M⟨i, j⟩ is built from the modules M⟨n, n⟩ ≃ Hπ_{n}(M) for i ≤ n ≤ j. These are all R_{0}-modules and in particular they are R_{0}-cellular, hence M is also R_{0}-cellular.

Next, suppose M is bounded above, so there is an index j such that π_{n}(M) = 0 for all n > j. In this case M⟨i, j⟩ ≃ M⟨i, ∞⟩ and we can build a telescope:

M⟨j, j⟩ → M⟨j − 1, j⟩ → ⋯ → M⟨i, j⟩ → ⋯

with obvious maps from this telescope to M. It is easy to see that the map

hocolim_{(i→−∞)}M⟨i, j⟩ → M

is an equivalence. Since every M⟨i, j⟩ is R_{0} cellular, so is M.

Example 3.13. Let k be a commutative ring (in the classical sense), let G be a simplicial group and let X be a G-space. Denote by G_{0} the discrete group π_{0}(G). The map of 𝕊-algebras kG → kG_{0} is induced by the map of simplicial groups G → π_{0}(G). The group-ring kG is connective, the kG-module C^{∗}(X; k) is coconnective, therefore, by proposition 3.12 above, C^{∗}(X; k) is kG_{0}-cellular. In fact, C^{∗}(X; k) is built from the cohomology groups H^{n}(X; k) which are kG_{0}-cellular as kG-modules.

Example 3.14. Let B be a finite connected nilpotent space with a fundamental group G = π_{1}(B). A result of Hilton, Mislin and Roitberg ([14, II.2.18]) shows that G operates nilpotently on the modules H_{∗}(B̃), where B̃ is the universal cover of B. Since B is a finite space we see C_{∗}(B̃; ℤ) is a bounded above ℤG-module (i.e. a bounded above ℤG-chain complex). From the proof of proposition 3.12 we see that C_{∗}(B̃; ℤ) is built by its homology groups. Every H_{n}(B̃) is a nilpotent ℤG-module and hence ℤ-cellular. Therefore C_{∗}(B̃; ℤ) is a ℤ-cellular ℤG-chain complex.

If k is any commutative ring (in the classical sense), then C_{∗}(B̃; k) ≃ k ⊗_{ℤ} C_{∗}(B̃; ℤ) and it is easy to see that C_{∗}(B̃; k) is a k-cellular kG-chain complex. Moreover, this result can be generalized for any 𝕊-algebra k that is a commutative, connective and bounded above, by the following argument. Using the Atiyah-Hirzebruch spectral sequence (see [10, IV.3.7]) we see that the groups π_{n}(C_{∗}(B̃; k)) are nilpotent π_{0}(kG)-modules. Since k is bounded above, C_{∗}(B̃; k) is also bounded above. This, coupled with the fact that k is connective, shows C_{∗}(B̃; k) is built from the k-modules Hπ_{n}(C_{∗}(B̃; k)) (see the proof of proposition 3.12). Therefore C_{∗}(B̃; k) is a k-cellular kG-module.

The second special case we consider is one in which R/I is a retract of R in the derived category D_{R}. In this case R/I-cellularization has a simple formula, given in proposition 3.15 below. Proposition 3.15. Let I → R ⟶^{a} R/I be as above. Suppose there exists in D_{R} a map b : R/I → R such that ab = id_{R/I} (in D_{R}). Then R/Icellularization is the coaugmented functor:

Hom_{R}(R/I, −) ⟶^{a^{∗}} id

Proof. We start by showing R/I ⊗_{R} R/I ≃ R/I. Consider the map:

b ⊗ 1 : R/I ⊗_{R} R/I → R/I

clearly (a ⊗ 1)(b ⊗ 1) = id_{R/I} in D_{R} i.e. b ⊗ 1 is a right inverse (in D_{R}) for a ⊗ 1. The map m from lemma 3.8 is a left inverse for a ⊗ 1. Therefore (b ⊗ 1) = m in D_{R} and a ⊗ 1 : R/I → R/I ⊗_{R} R/I is an equivalence.

Given an R-module X, consider the map a_{X}^{∗} : Hom_{R}(R/I, X) → X. Since Hom_{R}(R/I, X) is an R/I-module, it is R/I-cellular. All that is left is to show a_{X}^{∗} is an R/I-equivalence. Applying the functor Hom_{R}(R/I, −) to this map gives:

Hom_{R}(R/I, a_{X}^{∗}) : Hom_{R}(R/I, Hom_{R}(R/I, X)) → Hom_{R}(R/I, X)

Using the adjunction:

Hom_{R}(R/I, Hom_{R}(R/I, −)) ≃ Hom_{R}(R/I ⊗_{R} R/I, −)

we see the map Hom_{R}(R/I, a_{X}^{∗}) is equal in D_{R} to the map:

Hom_{R}(R/I ⊗_{R} R/I, X) ⟶^{(a⊗1)^{∗}} Hom_{R}(R/I, X)

which is an equivalence. Example 3.16. Let G be a finite group. We work over the group-ring R = ℚG. Consider the augmentation map a : ℚG → ℚ, where ℚ has a trivial G-action. The map a has as a left inverse the map b : ℚ → ℚG defined by:

1 ↦ \frac{1}{|G|} ∑_{g∈G} g

Hence, for any ℚG-module M:

Cell_{ℚ}M ≃ Hom_{ℚG}(ℚ, M)

Since ℚ is projective as a ℚG-module, we see that Hom_{ℚG}(ℚ, M) is equal to the fixed points of the G-action on M, denoted M^{G}.

3.1 Cellularization when R is a classical ring

Until the end of this section R is a (classical) ring and I is a two-sided ideal of R. To facilitate some of the definitions, we will work in the category Ch_{R} of unbounded chains-complexes over R. So until the end of this section, an R-module means an ordinary R-module.

We begin by recalling some algebraic definitions. The I-power torsion functor on R-modules, denoted Γ_{I}, is defined by:

Γ_{I}M = {m ∈ M|I^{n}m = 0 for some n}

where M is an R-module. Clearly Γ_{I}M is a submodule of M, hence Γ_{I} is a coaugmented functor. The I-local cohomology of M is the right-derived functors of Γ_{I}, applied to M, i.e. H_{I}^{n}(M) = R^{n}Γ_{I}(M). An R-module M is called an I-power torsion module if Γ_{I}M = M. Equivalently, M is an I-power torsion module if there is a sequence of submodules of M:

0 = M_{0} ⊆ M_{1} ⊆ M_{2} ⊆ ⋯ ⊆ M_{n} ⊆ ⋯

such that M = ∪_{n}M_{n} and M_{i+1}/M_{i} is an R/I-module for all i ≥ 0. In particular, every I-nilpotent module is an I-power torsion module.

For a commutative ring R and a finitely generated ideal I ⊆ R, Dwyer and Greenlees showed in [6] the following two results:

1. Cell_{R/I} calculates the local cohomology groups at I ([6, proposition

6.10]), i.e.

π_{−n}(Cell_{R/I}M) ≅ R^{n}Γ_{I}(M) = H_{I}^{n}(M)

where H_{I}^{∗}(M) are the local cohomology groups of M at the ideal I. If R is Noetherian then Cell_{R/I} calculates the total right derived functor of Γ_{I} (see e.g. [13]).

2. An R-chain complex X is R/I-cellular if and only if π_{n}(X) is an Ipower torsion module for every n ([6, proposition 6.11]).

The results of this section represent an attempt to generalize these results to the non-commutative case, an attempt which is only partly successful. We start with the following consequence of theorem 3.11. Recall R is any classical ring and I is a two-sided ideal of R.

Proposition 3.17. For every R/I-cellular R-chain complex X, the homotopy groups π_{n}(X) are I-power torsion modules.

Proof. Let C denote the full subcategory of R-chain complexes whose objects are chain complexes X such that π_{n}(X) is an I-power torsion module for every n. Clearly C is closed under weak equivalences and suspensions.

A submodule of an I-power torsion module is an I-power torsion module. Similarly a quotient of an I-power torsion module is also an I-power torsion module. This implies C is closed under retracts. It also implies C is closed under triangles, since every triangle gives rise to a long exact sequence of homotopy groups. It is also clear that C is closed under arbitrary coproducts (since the coproduct of I-power torsion modules is an I-power torsion module).

Thus C is a localizing subcategory of R-modules. Since R/I ∈ C, the subcategory C contains the localizing subcategory generated by R/I - which is the full subcategory of R/I-cellular complexes.

Proposition 3.18. Suppose R/I is proxy-small as an R-module, then for every R-module M, π_{0}(Cell_{R/I}M) ≅ Γ_{I}M.

Proof. First we show the map π_{0}(Cell_{R/I}M) → M is an injection. Let P_{0}M → P_{1}M → ⋯ be the telescope constructed in definition 3.7. We show, by induction on n, that π_{i}(P_{n}M) = 0 for i > 0. This is certainly true for n = 0. Given n ≥ 0, consider the triangle:

Hom_{R}(R/I, P_{n}M) → P_{n}M → P_{n+1}M

By induction, π_{i}(P_{n}M) = 0 for i > 0. This implies that

hom_{D_{R}}(Σ^{i}R/I, P_{n}M) = Ext_{R}^{−i}(R/I, P_{n}M) = 0

for i > 0 (this can be seen, for example, by taking a free resolution of R/I). We conclude that, for i > 0,

π_{i}(Hom_{R}(R/I, P_{n}M)) ≅ hom_{D_{R}}(Σ^{i}R/I, P_{n}M) = 0

The triangle:

P_{n}M⟨−1, −∞⟩ → P_{n}M → π_{0}(P_{n}M)

gives rise to a long exact sequence:

Ext_{R}^{0}(R/I, P_{n}M⟨−1, −∞⟩) → Ext_{R}^{0}(R/I, P_{n}M) → Ext_{R}^{0}(R/I, π_{0}(P_{n}M)) → ⋯ Arguments similar to those above show Ext_{R}^{0}(R/I, P_{n}M⟨−1, −∞⟩) = 0. Thus, the map Ext_{R}^{0}(R/I, P_{n}M) → π_{0}(P_{n}M) is an injection. From the long exact sequence of homotopy groups for the triangle Hom_{R}(R/I, P_{n}M) → P_{n}M → P_{n+1}M we see that π_{i}(P_{n+1}M) = 0 for i > 0.

Recall that P_{R/I}M is equivalent to the homotopy colimit of the telescope P_{0}M → P_{1}M → · · ·. Therefore π_{i}(P_{R/I}M) = 0 for i > 0. The triangle Cell_{R/I}M → M → P_{R/I}M gives a long exact sequence:

0 = π_{1}(P_{R/I}M) → π_{0}(Cell_{R/I}M) → M → ⋯

showing the map π_{0}(Cell_{R/I}M) → M is an injection.

Before continuing we make the following observation. If M′ is a submodule of M that is R/I-cellular then the injection M′ → M splits through Cell_{R/I}M (by the universal property of Cell_{R/I}). In particular, the map M′ → M splits through π_{0}(Cell_{R/I}M).

Recall that any I-nilpotent module is R/I-cellular (example 2.4). The module Γ_{I}M is the colimit of the telescope:

0 = M_{0} → M_{1} → M_{2} → ⋯

where M_{n} = {m ∈ M|I^{n}m = 0}. Since the maps M_{n} → M_{n+1} are inclusions, we get that Γ_{I}M ≃ hocolim_{n}M_{n}. Every module M_{n} is I-nilpotent, hence Γ_{I}M is the homotopy colimit of R/I-cellular modules, so it is also R/Icellular. We conclude that π_{0}(Cell_{R/I}M) contains Γ_{I}M.

By the previous proposition π_{0}(Cell_{R/I}M) is an I-power torsion module. It is easy to see that any I-power torsion submodule of M is a submodule of Γ_{I}M. Since π_{0}(Cell_{R/I}M) is a submodule of M, it must be a submodule of Γ_{I}M.

Remark 3.19. Two things remain unclear. First, under what conditions is the full subcategory of R/I-cellular complexes equal to the full subcategory of complexes whose homotopy groups are I-power torsion modules? We have only shown that the former is contained in the latter. Dwyer and Greenlees showed that the two categories are equal when R is commutative and I is a finitely generated ideal of R.

Second, under what conditions is Cell_{R/I} the derived functor of Γ_{I}? Again, when R is commutative and I is a finitely generated ideal of R we know from Dwyer and Greenlees results that this is true. If R is any ring, R/I is proxysmall as an R-module and M is an R-module (in the usual sense), then proposition 3.18 shows Cell_{R/I}M is coconnective (i.e. π_{i}(Cell_{R/I}M) = 0 for i > 0) and π_{0}(Cell_{R/I}M) = Γ_{I}(M), which is still insufficient to answer our questions.

Change of Rings

Suppose we are given a map of 𝕊-algebras R → S and an S-module A. Given another S-module X, there are two A-cellularization functors we can apply to X: Cell_{A}^{S}(X) which is A-cellularization over S and Cell_{A}^{R}(X) which is Acellularization over R. In this section we explore the relations between these functors. We give, in lemma 4.2, conditions under which the two cellularizations are equivalent. We start with the following basic lemma, which is simply a reformulation of lemma 3.1.

Proposition 4.1. Let R → S be a map of 𝕊-algebras and let X be an Acellular S-module, then X is also A-cellular as an R-module.

We have mentioned before, in section 3.1, the relation between cellularization and local cohomology in commutative rings. For local cohomology there is the independence of base property, which says the following: let R → S be a map of commutative Noetherian rings and let I be an ideal of R. Then for every S-module M, the I-local cohomology groups of M, as an R-module, are isomorphic to the IS-local cohomology of groups M as an S-module, where IS is the ideal of S generated by I (see e.g. [17, 1.10]). By a result of Dwyer and Greenlees ([6, proposition 6.10]) this implies π_{∗}(Cell_{R/I}^{R}M) ≅ π_{∗}(Cell_{S/IS}^{S}M) as R-modules. The next lemma gives analogs of this property in terms of cellularization.

Lemma 4.2. Let R → S be a map of 𝕊-algebras.

1. ( Strong independence of base) Let A be an S-module. If S ⊗_{R} A is Acellular as an S-module, then for any S-module X: Cell_{A}^{R}X ≃ Cell_{A}^{S}X as R-modules. Moreover, the natural map Cell_{A}^{S}X → X is the Acellularization of X as an R-module.

2. ( Weak independence of base) Let A be an R-module. If S ⊗_{R} A is A-cellular as an R-module, then for any S-module X: Cell_{A}^{R}X ≃ Cell_{S⊗_{R}A}^{S}X as R-modules. Moreover, the natural map Cell_{S⊗_{R}A}^{S}X → X is the A-cellularization of X as an R-module.

In addition, these equivalences are natural in D_{R}.

Although this lemma is quite simple, it is useful in analyzing cellularization over group-rings, as we will see in the next section.

Proof. First we prove the strong independence of base property. Suppose S ⊗_{R} A is A-cellular as an S-module. We will prove that Cell_{A}^{S}X → X is the A-cellularization of X as an R-module. By the previous proposition (4.1), Cell_{A}^{S}X is A-cellular as an R-module. We need only show that the map μ_{X} : Cell_{A}^{S}X → X is an A-equivalence of R-modules. Using the usual adjunctions between ⊗ and Hom (see [10, III.6.3]) we get:

Hom_{R}(A, μ_{X}) ≃ Hom_{R}(A, Hom_{S}(S, μ_{X})) ≃ Hom_{S}(S ⊗_{R} A, μ_{X})

The map μ_{X} is an A-equivalence of S-modules. Since S ⊗_{R} A is A-cellular as an S-module, property 5 of lemma 2.3 shows μ_{X} is also an S⊗_{R}A-equivalence, thus proving Hom_{R}(A, μ_{X}) is an equivalence.

Now we prove the weak independence of base property. Let A be an R-module and suppose S ⊗_{R} A is A-cellular as an R-module. As we saw, Cell_{S⊗_{R}A}^{S}X is also S ⊗_{R} A-cellular as an R-module. Transitivity of cellularity (property 7 of lemma 2.3) implies Cell_{S⊗_{R}A}^{S}X is also A-cellular as an R-module. We are left with showing the map ν_{X} : Cell_{S⊗_{R}A}^{S}X → X is an A-equivalence of R-modules. As above, the usual adjunctions give the equivalence:

Hom_{R}(A, ν_{X}) ≃ Hom_{S}(S ⊗_{R} A, ν_{X})

Since Hom_{S}(S⊗_{R}A, ν_{X}) is an equivalence, we see that ν_{X} is an A-equivalence of R-modules.

Naturality of the equivalences we proved is a simple consequence of the universal property of cellularization.

Let A be an R-bimodule and let M be an R-module, then the R-module A ⊗_{R} M is A-cellular. This is easily seen from the adjunction: Hom_{R}(A ⊗_{R} M, N) ≃ Hom_{R}(M, Hom_{R}(A, N)). However, if M is an R-bimodule and A is an R-module, then M ⊗_{R} A need not be A-cellular. An example of this is given below, in 4.4. One obvious case in which M ⊗_{R} A is A-cellular, is when R is a commutative 𝕊-algebra, since M ⊗_{R} A is equivalent to A ⊗_{R} M when R is commutative. This proves the following lemma.

Lemma 4.3. Let R → S be a map of 𝕊-algebras as above. If R is a commutative 𝕊-algebra and A is an R-module then S ⊗_{R} A is A-cellular as an R-module.

We see that, whenever R is commutative, then for any map R → S of 𝕊-algebras and for any S-module A there is the weak independence of base property. Next we give two examples, the first showing a case where there is no independence of base and the second giving an example of strong independence of base.

Example 4.4. Let Σ_{3} be the symmetric group on three elements. There is a short exact sequence of groups:

C_{3} → Σ_{3} → C_{2}

where C_{n} is the cyclic group of n elements. Let R be the group ring ℤΣ_{3}, let S be the group-ring ℤC_{2} and let A be the S-module ℤ with trivial C_{2}-action. As we will see in the next section, S ⊗_{R}A is the chains of the classifying space of C_{3}, namely C_{∗}(BC_{3}; ℤ). Recall H_{1}(BC_{3}; ℤ) ≅ ℤ/3 (the group H_{1}(BC_{3}; ℤ) is π_{1}(C_{∗}(BC_{3}; ℤ)) in our usual notation). From lemma 6.9 below, we see that the C_{2} action on H_{1}(BC_{3}; ℤ) is simply switching the two generators of ℤ/3.

Let I be the augmentation ideal of S, i.e. the kernel of the map ℤC_{2} → ℤ. Had S ⊗_{R} A been A-cellular as an S-module then, by proposition 3.17, H_{1}(BC_{3}; ℤ) would have been an I-power torsion module (as an S-module). In particular, the elements of H_{1}(BC_{3}; ℤ) would have been I-nilpotent. Let C_{2} = {1, x}, then I is generated by the element 1 − x. The action of 1 − x on H_{1}(BC_{3}; ℤ) ≅ ℤ/3 is multiplication by −1. Therefore, the action of 1 − x on the generators of H_{1}(BC_{3}; ℤ) cannot be nilpotent and S ⊗_{R} A is not Acellular as an S-module. Moreover, this argument also shows that S ⊗_{R} A is not A-cellular even as an R-module.

Example 4.5. Fix some n and consider the following fibration of pointed spaces:

S^{n} → ℝℙ^{n} → BC_{2}

Since all the spaces are connected, this fibration gives rise to a short exact sequence of simplicial groups:

ΩS^{n} → Ωℝℙ^{n} → ΩBC_{2}

Let R be the group-ring ℤ[Ωℝℙ^{n}], recall this is C_{∗}(Ωℝℙ^{n}; ℤ). Let S be the group ring ℤ[ΩBC_{2}]. The weak equivalence of simplicial groups ΩBC_{2} ≃ C_{2} induces an equivalence between the derived category of S-modules and the derived category of ℤC_{2}-modules (see [10, theorem III.4.2]). So we will allow ourselves to identify S-modules and ℤC_{2}-modules.

Let A be the S-module ℤ with trivial C_{2}-action. Given a short exact sequence of simplicial groups N → G → Q, there is a well known equivalence: C_{∗}(BN; k) ≃ kQ ⊗_{kG} k (a detailed explanation is given in section 5). Taking N = ΩS^{n}, G = Ωℝℙ^{n} and Q = ΩBC_{2} we get:

S ⊗_{R} ℤ ≃ C_{∗}(S^{n}; ℤ)

The action of C_{2} on S^{n} is by the antipodal map (see lemma 6.7 and the preceding discussion). If n is odd this map is homotopy equivalent to the identity. Thus C_{∗}(S^{n}; ℤ) is actually a ℤ-module and in particular it is ℤcellular as an S-module. In this case we have the strong independence of base property.

Note, however, that if n is even, the antipodal map induces a non-trivial action on the reduced homology groups of S^{n}. Thus π_{n}(C_{∗}(S^{n}; ℤ)) ≅ ℤ is a non-trivial C_{2}-module, the C_{2}-action being multiplication by −1. We will show this C_{2}-module, which we denote by ℤ̃, is not ℤ-cellular as an S-module.

Let I be the augmentation ideal of ℤC_{2}. The ring ℤC_{2} is commutative and the augmentation ideal I is finitely generated, so by [6, proposition 6.11] a ℤC_{2}-chain complex is ℤ-cellular if and only if all it’s homology groups are I-power torsion modules. It is easy to see that ℤ̃ is not an I-power torsion module and therefore C_{∗}(S^{n}; ℤ) described above is not ℤ-cellular as a ℤC_{2}- module.

Remark 4.6. We can recover the original independence of base property for commutative Noetherian rings from lemma 4.2. Let f : R → S be a map of commutative Noetherian rings and let I be an ideal of R. Given an S-module M, we need to show that the I-local cohomology groups of M as an R-module are isomorphic to the IS-local cohomology groups of M as an S-module. By a result of Dwyer and Greenlees from [6, proposition 6.11] the homology groups of Cell_{R/I}^{R}M are the I-local cohomology groups of M. So it will suffice to show that Cell_{R/I}^{R}M ≃ Cell_{S/IS}^{S}M.

The ideal I is finitely generated by elements r_{1}, ..., r_{n}, let r denote the sequence (r_{1}, ..., r_{n}). The classical Koszul complex for the sequence r is:

K_{r} = K_{r_{1}} ⊗_{R} ⋯ ⊗_{R} K_{r_{n}}

where K_{r_{i}} is the complex r_{i} : R → R with the two copies of R being in dimensions 0 and −1. Another result of Dwyer and Greenlees ([6, proposition 6.1]) shows K_{r} and R/I build each other over R. Hence R/I-cellularization is equivalent to K_{r}-cellularization and in particular Cell_{R/I}^{R}M ≃ Cell_{K_{r}}^{R}M.

The ideal IS is generated by the elements f(r_{1}), ..., f(r_{n}), let f(r) denote the sequence (f(r_{1}), ..., f(r_{n})). As above, the complex K_{f(r)} and S/IS build each other. Hence, it is enough to show that:

Cell_{K_{r}}^{R}M ≃ Cell_{K_{f(r)}}^{S}M

The complex K_{f(r)} is clearly isomorphic to the complex S ⊗_{R} K_{r}. By the weak independence of base property (which holds for commutative rings by lemma 4.3) we get:

Cell_{K_{r}}^{R}M ≃ Cell_{S⊗_{R}K_{r}}^{S}M

Cellularization in Group-Rings

Fix a commutative 𝕊-algebra k and let G be a simplicial (or discrete) group. In this section the ring R we work over is the group ring kG (which is C_{∗}(G; k)). To avoid excessive notation we will refer to kG-modules as Gmodules, as long as the base ring k is understood from the context. Similarly, we will use the notation Hom_{G} and ⊗_{G} instead of Hom_{kG} and ⊗_{kG}.

When working with simplicial groups it is perhaps more natural to work with symmetric spectra ([16]) than with 𝕊-algebras. However, the realization of a simplicial group is a simplicial monoid. So, if one opts to work with 𝕊algebras, the group-ring kG is in fact the 𝕊-algebra C_{∗}(|G|; k), where |G| is the realization of G ([10, IV.7.8]).

The motivation for working with simplicial over topological monoid comes from the following phenomenon. For a fibration sequence of pointed and connected spaces F → E → B, applying the Kan loop-group functor yields a short exact sequence of simplicial groups:

1 → ΩF → ΩE → ΩB → 1

This follows from [12, V.6.9] using the adjunction of the loop group functor and the classifying space functor described in [12, V.5]. Note that for every n we have a short exact sequence of the groups of n-simplices. This property be used in section 8 and is our main reason for using simplicial groups.

Let us begin by relating this setting to that of section 3. The augmentation map:

a : kG → k

induced by the map G → pt, is a map of k-algebras. Taking the homotopy fiber of a gives the following fibration sequence:

I → kG → k

where I is analogous to the augmentation ideal. The maps in this sequence are k-algebra maps and I is equivalent to a k-algebra without a unit. This shows we are in the setting of section 3, where R = kG and R/I is k. Our interest lies in k-cellularization over kG, which we denote by Cell^{G} whenever k is understood. We say Cell^{G} is trivial if there is a natural equivalence of coaugmented functors Cell^{G} ≃ id.

Throughout this section we fix a short exact sequence of simplicial groups:

N → G → Q

The aim of this section is to relate Cell^{N} and Cell^{Q} with Cell^{G}. The main result here is theorem 5.8, where we show that, under a certain condition, Cell^{G} can be constructed from Cell^{N} and Cell^{Q} together with some extra data.

Before starting we need a way to relate the theory of G-modules to the theory of G-spaces. For a left G-space X, the spectrum Σ^{∞}X_{+} is a Σ^{∞}G_{+}- module ([10, theorem IV.7.8]). We add a disjoint base-point since the basepoint needs to have a trivial G-action. Clearly, C_{∗}(X; k) is a C_{∗}(G; k)-module (recall that when taking C_{∗}(−; k) of a space we add disjoint base-point if the space is not pointed).

A map of left G-spaces will be called an equivalence if it is a weak equivalence of spaces, i.e. induces an isomorphism on all homotopy groups and all base points. These are the equivalences in the ”weak” (or naive) model category structure on G-spaces. The point is that if f : X → Y is such an equivalence of G spaces and k is any commutative 𝕊-algebra, then the induced map:

C_{∗}(X; k) → C_{∗}(Y; k)

is a map of left G-modules that induces an isomorphism on all homotopy groups. Hence it is an equivalence of G-modules. So this weak notion of an equivalence of G-spaces suffices for our purposes. An appropriate model category structure is described, for example, in Goerss and Jardine’s book ([12, V.2]).

The main ingredient we need is the following result from [10, proposition IV.7.5 & theorem IV.7.8], connecting the derived product −×_{G}− of G-spaces (meaning the bar construction) with the tensor product of their chains over kG.

Lemma 5.1 ([10, IV.7.5 & IV.7.8] ). Let X be a right G-space and Y a left G space, then:

C_{∗}(X ×_{G} Y; k) ≃ C_{∗}(X; k) ⊗_{kG} C_{∗}(Y; k)

Proof. It is enough to prove this theorem for k = 𝕊. The general case follows easily by applying k ⊗_{𝕊}− to the required equivalence. Thus we assume k = 𝕊 and set R = C_{∗}(G; k).

For a right R-module M and a left R-module N, there is a bar construction described in [10, IV.7.2]. The realization of this bar construction is denoted by B(M, R, N). By [10, IV.7.5] the natural map: B(M, R, N) → M ^{S}⊗_{R} N is an equivalence if M is a cell R-module, where ^{S}⊗_{R} denotes the strict, not derived, tensor product. Recall that when M is a cell R-module then M ^{S}⊗_{R} N is equivalent to the derived tensor product.

Let B(X, G, Y) denote the realization of the usual bar construction for G-spaces. By [10, IV.7.8] there is an isomorphism of 𝕊-modules:

C_{∗}(X ×_{G} Y; k) ≅ B(C_{∗}(X; k), R, C_{∗}(Y; k))

To complete the proof we use the following standard argument. Let F be a cell C_{∗}(G; k)-module together with a weak equivalence F → C_{∗}(X; k) by a map. By [10, X.2.4] the map

B(F, R, C_{∗}(Y; k)) → B(C_{∗}(X; k), R, C_{∗}(Y; k))

is an equivalence. Now we can apply [10, IV.7.5] to get that B(F, R, C_{∗}(Y; k)) ≃ F ^{S}⊗_{R}C_{∗}(Y; k). Clearly, the strict tensor product F ^{S}⊗_{R}C_{∗}(Y; k) is equivalent to the derived tensor product C_{∗}(X; k) ⊗_{R} C_{∗}(Y; k).

Remark 5.2. One can also apply the model category structure on G-spaces described in [12, V.2] to get this result. By [12, lemma V.2.7 and corollary V.2.10], if X is a cofibrant G-space then X = colim_{n}X_{n} where X_{−1} = ∅ and for each X_{n} there is a pushout diagram:

\xymatrix{∐_{α} ∂Δ^{n} × G \ar[r] \ar[d] & X_{n−1} \ar[d] \\ ∐_{α} Δ^{n} × G \ar[r] & X_{n}}

where α runs over the non-degenerate n-simplices of X/G and the left vertical map is induced by the cofibration ∂Δ^{n} → Δ^{n}. From this it is a simple exercise to show that given a cofibrant right G-space X and a left G-space Y, the simplicial set B(X, G, Y) which is the realization of the bar construction, is naturally equivalent to the strict quotient X ×_{G} Y. Thus we see that in order to compute the derived product − ×_{G} − one must take a cofibrant replacement of either the left or the right G-space.

On the other hand, for a cofibrant right G-space X, the right kG-module C_{∗}(X; k) is a CW kG-module in much the same way. Thus C_{∗}(X; k) = colim_{n}C_{∗}(X_{n}; k) where X_{−1} = 0 and for each X_{n} there is a pushout diagram:

\xymatrix{∐_{α} ∂Δ^{n} ⊗ kG \ar[r] \ar[d] & C_{∗}(X_{n−1}; k) \ar[d] \\ ∐_{α} Δ^{n} ⊗ kG \ar[r] & C_{∗}(X_{n}; k)}

(Note that for a space A and an 𝕊-module M there is an 𝕊-module A ⊗ M, see [10, VII]). Since the map ∂Δ^{n} ⊗ kG → Δ^{n} ⊗ kG is a cofibration of kG-modules, this pushout diagram is also a homotopy pushout diagram. Again, standard arguments show that C_{∗}(X; k) ⊗_{G} C_{∗}(Y; k) is equivalent to C_{∗}(X ×_{G} Y; k), where X is a cofibrant G-space. Note that in this case C_{∗}(X; k) is also a CW KG-module by the argument above.

We begin by showing Cell^{N} is equivalent to Cell_{kQ}^{G}. Note the ring homomorphism kN → kG makes kG into an G − N bimodule.

Lemma 5.3. The module kG ⊗_{kN} k is k-cellular as an N-module.

Proof. It suffices to show that kG ⊗_{kN} k is equivalent to kQ as a left kGmodule. Because the structure of a KG-module on KQ comes from the maps of 𝕊-algebras kG → kQ. Hence the structure of KQ as an N-module comes from the composition kN → kG → kQ of 𝕊-algebras. But the map N → Q factors as N → pt → Q. Therefore the map kN → kQ factors through kN → k → kQ. Thus, as a left N-module, kQ is built by k.

The result we cited from [10] shows:

kG ⊗_{kN} k ≃ C_{∗}(G ×_{N} pt; k)

The G-space G ×_{N} pt (or rather the realization of the bar construction B(G, N, pt)) is indeed equivalent to Q as a left G-space. This can be seen by the following argument. Consider the bar construction B_{∗}(G, N, pt) as a bisimplicial set. Thus B_{p,t}(G, N, pt) are the t simplicies of the p’th level of the bar construction. The simplicial set B_{∗,t}(G, N, pt) is a discrete bar construction for the groups N_{t} and G_{t} and the N_{t} module pt_{t} = pt. This simplicial set is readily seen to be equivalent to the constant simplicial set Q_{t} as G_{t}-spaces.

Note that one can also use remark 5.2 above to show that G ×_{N} pt ≃ Q as left N-spaces. Since G is clearly a cofibrant N-space (see [12, corollary V.2.10]), we see that the derived product G ×_{N} pt is simply G/N (i.e. the non-derived variant) which is Q.

Corollary 5.4. For every G-module X, there is a natural weak equivalence of N-modules:

Cell^{N}(X) ≃ Cell_{kQ}^{G}(X)

In fact, the natural map Cell_{kQ}^{G}(X) → X is the k-cellularization of X as an N-module.

Proof. The previous lemma shows we have weak independence of base property (lemma 4.2), i.e.:

Cell^{N}(X) ≃ Cell_{kG⊗_{N} k}^{G}(X)

As above: kG ⊗_{N} k ≃ C_{∗}(G ×_{N} pt; k). There is an obvious equivalence of G-spaces: G ×_{N} pt ≃ Q and therefore kG ⊗_{N} k ≃ kQ as G-modules.

Example 5.5. Suppose N is a connected simplicial group and k is a commutative ring (in the classical sense), then kN is a connective 𝕊-algebra and π_{0}(kN) ≅ k. By proposition 3.12, every N-module whose homotopy groups are bounded above is k-cellular. Therefore, every bounded-above G-module X is kQ-cellular.

Lemma 5.6. If Cell^{Q} is trivial then for every G-module X there is a natural equivalence of N-modules:

Cell^{G}(X) ≃ Cell^{N}(X)

In fact, the natural map Cell^{G}(X) → X is the k-cellularization of X as an N-module. Moreover, if, in addition, Cell^{N} is trivial, then Cell^{G} is also trivial.

Proof. Since Cell^{Q} ≃ id we conclude that kQ is k-cellular as a Q-module. Since k is clearly Q-cellular as a Q-module, we see k and kQ build each other (over Q). Proposition 4.1 implies k and kQ build each other also over G. Therefore there is a natural equivalence of coaugmented functors: Cell^{G} ≃ Cell_{kQ}^{G}. By the previous corollary, for every G-module X:

Cell^{N}(X) ≃ Cell_{kQ}^{G}(X)

which proves the first part of the lemma.

For the second part, note the natural map Cell^{G}(X) → X of G-modules is the k-cellularization of X as an N-module. Since Cell^{N} is trivial, this map is an equivalence of N-modules, hence it is also an equivalence of Gmodules. By standard arguments this implies there is a natural equivalence of coaugmented functors Cell^{G} ≃ id.

Next we give a condition under which Cell^{Q}(X) ≃ Cell^{G}(X) for every Qmodule X. First we must recall some facts about classifying spaces of groups. The classifying space of the group N, denoted BN, is the space pt ×_{N} WN where WN is any contractible space with a free N action. In particular WG is a contractible space with a free N action (since it has a free G-action). Note that for any G-space X there is an equivalence: pt ×_{N} X ≃ Q ×_{G} X. These observations show that:

BN ≃ pt ×_{N} WG ≃ Q ×_{G} WG

Hence BN is naturally a Q-space. Moreover, the result from [10] cited above (lemma 5.1), implies the following well known equivalence:

C_{∗}(BN; k) ≃ kQ ⊗_{kG} k

Lemma 5.7. Assume C_{∗}(BN; k) is a k-cellular Q-module, then for any Qmodule X:

Cell^{Q}(X) ≃ Cell^{G}(X)

Proof. Since C_{∗}(BN; k) ≃ kQ ⊗_{G} k, the result follows from strong independence of base (lemma 4.2)

We now come to the main result of this section:

Theorem 5.8. Assume k is proxy-small as an N-module and that C_{∗}(BN; k) is a k-cellular Q-module. Then for every G-module X there exists a telescope:

X_{0} → X_{1} → ⋯

of kG-modules such that:

1. The homotopy colimit of this telescope is equivalent to Cell^{N}(X) as N-modules.

2. The successive mapping cones M_{i} = Cone(X_{i} → X_{i+1}) and M_{0} = X_{0} are Q-modules.

3. Cell^{G}(M_{i}) ≃ Cell^{Q}(M_{i})

4. The homotopy colimit of the induced telescope

Cell^{G}(X_{0}) → Cell^{G}(X_{1}) → ⋯

is equivalent to Cell^{G}X.

Remark 5.9. If k is proxy-small as a G-module and Q is a finite group, then k is also proxy-small as an N-module, because in such a case G is small as an N-module (it is the free module generated by the set Q).

Before proving the theorem, we need the following lemma:

Lemma 5.10. There is a natural equivalence of coaugmented functors:

Cell^{G}(Cell_{kQ}^{G}(−)) ≃ Cell^{G}(−)

Proof. It is sufficient to prove that the map Cell_{kQ}^{G}(X) → X is a k-equivalence. The module k is a Q-module and as such is built from Q (over Q). Therefore k is also built from Q over G, i.e. k is a Q-cellular G-module. Since the map Cell_{kQ}^{G}(X) → X is a kQ-equivalence, it is also a k-equivalence.

Proof of theorem 5.8. The telescope X_{0} → X_{1} → ⋯ comes from applying theorem 3.11 to get a telescope whose homotopy colimit is Cell_{kQ}^{G}(X). The second property comes from that theorem and corollary 5.4. The rest of the theorem is an immediate consequence of lemmas 5.7 and 5.10.

Two Examples

The purpose of this section is to give two examples in which C_{∗}(BN; k) is a kcellular Q-module, where N → G → Q is a short exact sequence of simplicial groups and k a commutative 𝕊-algebra. Recall that whenever C_{∗}(BN; k) is a k-cellular Q-module we have the following strong independence of base property (lemma 5.7):

Cell^{G}(X) ≃ Cell^{Q}(X) for every Q-module X

Along the way we give a more explicit description to the Q-module structure of C_{∗}(BN; k) in two cases. In the first case BN → BG → BQ is a principal fibration and in the second case Q is a discrete group. All the conventions and notations of the previous section apply here as well, in particular by a G-module we mean a kG-module.

We start with the case where:

BN → BG → BQ

is a principal fibration. In this case BN is equivalent to a simplicial group P. Moreover, it is known that the action of Q on BN comes from a canonical map of simplicial groups: Q → P. Below we recall the definition of a principal fibration and go over the details of this last statement.

Definition 6.1. We say a fibration sequence F → E → B is a principal fibration if there is a simplicial group P and a map f : B → BP such that:

E → B → BP

is a homotopy fibration sequence. In particular, F is equivalent to P and we have the following commutative diagram:

\xymatrix{F \ar[r] \ar[d]^{≃} & E \ar[r] \ar[d] & B \ar[d]^{f} \\ P \ar[r] & ∗ \ar[r] & BP}

where ∗ is a contractible space and the right hand square is a homotopy pull-back square.

Remark 6.2. Note that any connected pointed space X can be considered as a classifying space of a simplicial group, because X is equivalent to the classifying space of the loop-group of X, i.e. X ≃ B(ΩX).

As mentioned above, when BN → BG → BQ is a principal P -fibration, then the Q-action on BN is induced by a map of simplicial groups Q → P. Next we give a precise statement of this well known result. Its proof is deferred till after example 6.4.

Proposition 6.3. Let P be a simplicial group and let:

P → E → B

be a principal P -fibration, determined by the map f : B → BP. Consider the short exact sequence of simplicial groups:

ΩP → ΩE → ΩB

Then B(ΩP) ≃ P as ΩB-spaces, where the ΩB-action on P comes from the homomorphism of groups Ωf : ΩB → P. In particular, the k[ΩB]-module structure on kP comes from the map of 𝕊-algebras: k[ΩB] → kP.

Before giving the proof of this proposition, we present the first of our two examples.

Example 6.4. Let G be a discrete, finite group. Take an embedding of G into the group SU(n). There is always such an embedding for a large enough n. This embedding makes SU(n) into a G-space. Consider the principal fibration sequence:

SU(n) → SU(n)/G → BG

This gives rise to a short exact sequence:

ΩSU(n) → Ω(SU(n)/G) → ΩBG

Recall that ΩBG and G are equivalent as simplicial groups. From now on we will not differentiate between them.

Proposition 6.3 above shows the G-action on B(ΩSU(n)) ≃ SU(n) comes from the embedding G → SU(n). Since SU(n) is connected, the G-action on SU(n) is homotopically trivial, meaning that for every g ∈ G the map g : SU(n) → SU(n) is homotopy equivalent to the identity map. Let k be any commutative ring (in the classical sense). We see that the reduced homology groups of SU(n)_{+} with coefficients in k, H̃(SU(n)_{+}; k), have a trivial G-action and in particular they are nilpotent kG-modules. Note that C_{∗}(SU(n); k) is a bounded-above kG-module. This implies C_{∗}(SU(n); k) is built from the kG-modules π_{n}(C_{∗}(SU(n); k)) = H̃_{n}(SU(n)_{+}; k) (see the proof of proposition 3.12). We conclude that C_{∗}(SU(n); k) is k-cellular as a kG-module.

Let R denote the group-ring k[Ω(SU(n)/G)]. The map Ω(SU(n)/G) → G induces a map of 𝕊-algebras R → kG. Since C_{∗}(SU(n); k) is k-cellular as a kG-module, the results of the preceding section (lemma 5.7) show that for every kG-module X:

Cell_{k}^{R}(X) ≃ Cell^{G}(X)

In particular Cell_{k}^{R}(kG) ≃ Cell^{G}(kG). We will return to this example in section 8, example 8.6.

To prove proposition 6.3 we need two lemmas. The first lemma recalls a well known implication of BN → BG → BQ being a principal fibration.

Lemma 6.5. Let BN → BG → BQ be a principal fibration. Then there is the following commutative diagram of simplicial groups:

\xymatrix{N \ar[r] & G \ar[r] \ar[d]^{≃} & Q \ar[d]^{≃} \\  & G′ \ar[r] & Q′ \ar[r] & P}

and ΩP ≃ N as simplicial groups.

Proof. By the definition of a principal fibration, there is a simplicial group P such that BG → BQ → BP is a homotopy fibration sequence. Applying the Kan loop-group functor to this sequence gives the desired diagram.

The next lemma is also known, perhaps in a different formulation.

Lemma 6.6. Assume we have the following diagram of simplicial groups with exact rows:

\xymatrix{ΩP \ar[r] & G \ar[r] \ar[d]^{≃} & Q \ar[d]^{≃} \\  & G′ \ar[r] & Q′ \ar[r] & P′}

Suppose also P′ ≃ P (this implies B(ΩP) ≃ P′). Then B(ΩP) ≃ P′ also as Q-spaces, where the Q-action on P′ comes from the homomorphism of

groups: Q ⟶^{≃} Q′ → P′.

Proof. There is an obvious equivalence: B(ΩP) ≃ P ≃ P′. We must show that B(ΩP) ≃ P′ as Q-spaces. The Q-action on B(ΩP) comes from the equivalence:

Q_{hG} ≃ Q ×_{G} W G ≃ W G/ΩP ≃ B(ΩP)

where Q_{hG} is the G-homotopy orbit space of Q. The commuting square:

\xymatrix{G \ar[r] \ar[d]^{≃} & Q \ar[d]^{≃} \\ G′ \ar[r] & Q′}

implies that Q_{hG} ≃ Q′_{hG′} as Q-spaces.

Note that Q′ is a free G′-space, hence the G′-homotopy orbit space of Q′ is equivalent to the usual G′ orbit space of Q, yielding the following equivalence of Q′-spaces

Q′_{hG′} ≃ Q′_{G′} = Q′/G′ ≅ P′

This shows B(ΩP) ≃ P′ as Q-modules. Moreover, the Q-module structure we got on N′ is indeed induced by the composition Q → Q′ → N′, which completes the proof.

Proof of proposition 6.3. Consider the short exact sequence of simplicial groups:

ΩP → ΩE → ΩB

Since P → E → B is a principal P -fibration, we are in the same setting as lemma 6.6 (we see this from lemma 6.5). The result follows.

We now turn to the second case, where Q is a discrete group. As before, N → G → Q is a short exact sequence of simplicial groups, but we no longer assume BN → BG → BQ is a principal fibration. Since Q is discrete, we have that π_{0}(N) → π_{0}(G) → Q is a short exact sequence of discrete groups and π_{i}(N) → π_{i}(G) is an isomorphism for i > 0.

Consider the following homotopy fibration sequence:

Q → BN → BG

This fibration is equivalent to the covering fibration:

Q → WG/N → WG/G

The group π_{1}(WG/G) ≅ π_{0}(G) acts on WG/N by deck transformations. The stabilizer of this action is π_{0}(N), hence we get an action of Q on WG/N. On the other hand, we have an obvious action of Q on Q ×_{G} WG and it is this action we have used before. Since WG/N is isomorphic to Q ×_{G} WG, we get two Q-actions on the same space. Both these actions are the same, as stated in the following lemma.

Lemma 6.7. Suppose N → G → Q is a short exact sequence of simplicial groups and Q is a discrete group. Then the obvious action of Q on Q×_{G} WG and the action of Q on WG/N coming from the covering fibration Q → WG/N → WG/G are the same.

The proof of this lemma is an easy exercise and is therefore omitted. We now come to the second example of this section.

Example 6.8. Let B be a finite, connected, based, nilpotent space with a finite fundamental group and let B̃ denote the universal covering space of B. Let k be any commutative 𝕊-algebra. Consider the following short exact sequence of simplicial groups:

ΩB̃ → ΩB → π_{1}(B)

From lemma 6.7 above, we see that the action of π_{1}(B) on the classifying space of ΩB̃ is simply the usual action of π_{1}(B) on B̃.

From example 3.14 we see that C_{∗}(B̃; k) is a k-cellular k[π_{1}(B)]-module. Hence, by lemma 5.7, for every k[π_{1}(B)]-module X:

Cell^{ΩB}(X) ≃ Cell^{π_{1}(B)}(X)

We end this section with a simple result, which gives a somewhat more concrete understanding of the action of Q on BN. In this case N, G and Q are all discrete groups. The proof of the next lemma is easy and therefore omitted.

Lemma 6.9. Let N → G → Q be a short exact sequence of discrete groups. Recall that H_{1}(BN; ℤ) is the abelian group A = N/[N, N]. Let p be the map G ⟶^{p} Q and let a denote the projection N ⟶^{a} A. Then the induced action of Q on A is given by:

p(g) · a(n) = a(g^{−1}ng)

for g ∈ G and n ∈ N.

Cellularization over Abelian and Nilpotent Groups

We describe in some simple cases the k-cellularization functor over discrete abelian and nilpotent groups, where k is a commutative 𝕊-algebra. Most of the results we give are obvious generalizations of known results. If A is a finite abelian group and k is a commutative ring then k-cellularization of kA-modules has been described by Dwyer and Greenlees in [6]. Their results show k-cellularization over kA is equivalent to Greenlees and May’s homotopical J-power torsion functor, defined in [13]. We generalize their result to any commutative 𝕊-algebra k.

For p-groups we also show that k-cellularization over a finite p-group is trivial if π_{0}(k) is an 𝔽_{p}-algebra. Finally we give conditions under which k-cellularization can be easily computed over finite nilpotent groups.

7.1 Finite Cyclic Groups

We start with the case of a cyclic group C of order n with a generator g. Let k be a commutative ring. Dwyer and Greenlees ([6]) show the chain complex kC ⟶^{1−g} kC, concentrated in dimensions 0 and 1, is a Koszul complex for k over kC. Indeed in this case there is a triangle: Σk → K → k, where K is the Koszul complex described above. Using this Koszul complex Dwyer and Greenlees ([6]) gave the following explicit description for k-cellularization of kC-modules, Cell_{k}(X) is the homotopy fiber of the map X → X[(1 − g)^{−1}], where X[(1 − g)^{−1}] is the homotopy colimit of the telescope:

X ⟶^{1−g} X ⟶^{1−g} ⋯

Now let k be a commutative 𝕊-algebra. Our goal is to similarly show that the mapping cone of the map kC ⟶^{1−g} kC is a Koszul complex for k over kC. Later, we will use this to get a similar description of k-cellularization over kC.

Proposition 7.1. Let C be as above. Then the kC-module K = Cone(kC ⟶^{1−g} kC) is a Koszul complex for k. Moreover, we have the following triangle:

Σk → K → k

Proof. We start by proving this for the case where k is the sphere spectrum 𝕊. By [10, IV.3.1] there is a map of 𝕊-algebras 𝕊C → Hπ_{0}(𝕊C). Note that Hπ_{0}(𝕊C) ≅ HℤC. Clearly HℤC ⊗_{𝕊C} K is the cone of the map ℤC ⟶^{1−g} ℤC and hence:

π_{i}(HℤC ⊗_{𝕊C} K) = \{ \begin{matrix}ℤ & i = 0, 1 \\ 0 & otherwise \end{matrix}

The augmentation map 𝕊C → 𝕊 induces a map f : K → 𝕊. There is also a map 𝕊 → 𝕊C of 𝕊C-modules, which is defined by 1 ↦ ∑_{c∈C} c. One can easily check this is indeed a map of 𝕊C-modules and it induces a map g : Σ𝕊 → K. Let F be the homotopy fiber of the map f. Then clearly:

π_{i}(HℤC ⊗_{𝕊C} F) = \{ \begin{matrix}ℤ & i = 1 \\ 0 & otherwise \end{matrix}

Since the composition 𝕊 → 𝕊C ⟶^{1−g} 𝕊C is zero, one sees that the map g induces a map g′ : Σ𝕊 → F. It is easy to check that the induced map

HℤC ⊗_{𝕊C} Σ𝕊 → HℤC ⊗_{𝕊C} F

is an equivalence. Consider the mapping cone Cone(g′). This 𝕊C-module is bounded below, simply because of its construction. In addition, HℤC ⊗_{𝕊C} Cone(g′) ≃ 0. Hence, by [10, IV.3.6], Cone(g′) ≃ 0 and the map g′ is an equivalence, yielding the desired triangle:

Σ𝕊 → K → 𝕊

This triangle shows K is finitely built from 𝕊.

For the general case, simply note that: k ≃ k ⊗_{𝕊}𝕊 and kC ≃ k ⊗_{𝕊}𝕊C. So applying the functor k ⊗_{𝕊} − to the triangle Σ𝕊 → K → 𝕊 gives the desired result.

7.2 Finite Abelian Groups

Let A be a finite abelian group and let k be a commutative 𝕊-algebra. We begin by describing a Koszul complex for k over kA. Later we use this Koszul complex to show k-cellularization over a finite abelian group is equivalent to Greenlees and May’s homotopical J-power torsion functor. Both the Koszul complex and the description of k-cellularization are proved in [6] for a commutative ring k. The generalization given here is self-evident, even though the proofs are long (which indicates the shortcomings of the author rather than the difficulty of the proofs).

Lemma 7.2. Suppose an abelian group A is a product of two groups A_{1}×A_{2}. Let E_{i} be a Koszul complex for k over kA_{i}. Then E_{1}⊗_{k}E_{2} is a Koszul complex for k over kA.

Proof. Since k finitely builds E_{1} over kA_{1}, it also finitely builds E_{1} over kA. The module E_{2} is finitely built from k over kA_{2} hence E_{1} ⊗_{k} E_{2} is finitely built from E_{1} ⊗_{k} k ≃ E_{1} over kA. It is easy to see this implies that E_{1} ⊗_{k} E_{2} is finitely built from k over kA.

To show E_{1} ⊗_{k} E_{2} is finitely built from kA we note that:

E_{1} ⊗_{k} E_{2} ≃ (E_{1} ⊗_{kA_{1}} kA) ⊗_{kA} (kA ⊗_{kA_{2}} E_{2})

By the same arguments as above E_{1}⊗_{kA_{1}} kA is finitely built from kA (because E_{1} is finitely built by kA). Similarly (E_{1} ⊗_{kA_{1}} kA)⊗_{kA} (kA⊗_{kA_{2}} E_{2}) is finitely built from E_{1} ⊗_{kA_{1}} kA over kA. As above we see that (E_{1} ⊗_{kA_{1}} kA) ⊗_{kA} (kA ⊗_{kA_{2}} E_{2}) is finitely built from kA.

Finally, we must show k is built from E_{1} ⊗_{k} E_{2}. Let N be a E_{1} ⊗_{k} E_{2}-null module, then:

Hom_{kA}(E_{1} ⊗_{k} E_{2}, N) ≃ Hom_{kA_{1}}(E_{1}, Hom_{kA_{2}}(E_{2}, N))

(this equivalence can be deduced for example from [10, lemma III.6.5] using the fact that kA ≅ kA_{1} ⊗_{k} kA_{2}). We see that Hom_{kA_{2}}(E_{2}, N) is E_{1}-null as a kA_{1}-module. Since E_{1} builds k over kA_{1} we get that:

Hom_{kA_{1}}(k, Hom_{kA_{2}}(E_{2}, N)) ≃ 0

By an equivalence like the one above we get:

Hom_{kA}(E_{2}, N) ≃ Hom_{kA}(k ⊗_{k} E_{2}, N) ≃ Hom_{kA_{1}}(k, Hom_{kA_{2}}(E_{2}, N)) ≃ 0 Thus N is an E_{2}-null kA-module. Since E_{2} builds k over kA_{2}, it also builds k over kA, hence N is also k-null.

Let A be a finite abelian group. We write A as a direct product of cyclic groups

A = ∏_{i=1}^{n} C_{i}

Let g_{i} be the generator of C_{i}. We also denote by g_{i} the appropriate element in A = ∏_{i=1}^{n} C_{i}. Let L_{i} be the mapping cone of the map kA ⟶^{g_{i}−1} kA.

Lemma 7.3. The kA-module: L = L_{1} ⊗_{kA} L_{2} ⊗_{kA} ⋯ ⊗_{kA} L_{n} is a Koszul complex for k over kA.

Proof. Let E_{i} be the Koszul complex for k over kC_{i} described in proposition 7.1. Then L_{i} is equivalent to the induced module kA ⊗_{kC_{i}} E_{i}. Using induction and the preceding lemma (7.2) we get that the module:

E = E_{1} ⊗_{k} ⋯ ⊗_{k} E_{n}

is a Koszul complex for k over kA. All that is left is to show that E ≃ L, but this easily follows from the following observation: Let M_{1} be a C_{1}-module and let M_{2} be a C_{2}-module then:

(kA ⊗_{kC_{1}} M_{1}) ⊗_{kA} (kA ⊗_{kC_{2}} M_{2}) ≃ kA ⊗_{k[C_{1}×C_{2}]} (M_{1} ⊗_{k} M_{2})

There is a version of local cohomology for modules over commutative 𝕊algebras defined by Greenlees and May in [13], which is called homotopical J-power torsion functor. We recall the definition below.

Definition 7.4. Let R be a commutative 𝕊-algebra and let J be a finitely generated ideal of R_{∗}. Suppose G = {x_{1}, ..., x_{n}} is a set of generators for J. Let R[1/x_{i}] be the homotopy colimit of the telescope R ⟶^{x_{i}} R ⟶^{x_{i}} · · ·. Define K(x_{i}) by the fiber sequence: K(x_{i}) → R → R[1/x_{i}] and let K(J) be the product:

K(J) = K(x_{1}) ⊗_{R} ⋯ ⊗_{R} K(x_{n})

For any R-module X, define Γ_{J}(X) = K(J) ⊗_{R} X. There is an obvious map K(J) → R which make Γ_{J} into a coaugmented functor.

Let A be a finite abelian group and let J ⊆ (kA)_{∗} be the augmentation ideal of kA_{∗} (note that (k[A])_{∗} ≅ k_{∗}[A]). If k is a (classical) commutative ring then a result of Dwyer and Greenlees ([6, proposition 6.10]), shows Γ_{J} is equivalent to k-cellularization. Here is a generalization of their result to any commutative 𝕊-algebra k.

Corollary 7.5. For a finite abelian group A and a commutative 𝕊-algebra k there is a natural equivalence of coaugmented functors:

Cell^{A} ≃ Γ_{J}

where J is the augmentation ideal of kA_{∗}.

Proof. Let the modules L and L_{1}, ..., L_{n} be as in lemma 7.3. Define x_{i} to be the element (g_{i} − 1) ∈ J and let G be the set {x_{1}, ..., x_{n}}. Then G is a set of generators for J.

We recall some notation from [13]. Let K_{s}(x_{i}) denote the homotopy fiber of the map

kA ⟶^{(x_{i})^{s}} kA

Thus K_{1}(x_{i}) is equivalent to Σ^{−1}L_{i} and it is also the dual of L_{i}, i.e. :

K_{1}(x_{i}) ≃ L_{i}^{#} = Hom_{A}(L_{i}, kA)

Define K_{1}(G) to be the tensor product K_{1}(x_{1}) ⊗_{A} ⋯ ⊗_{A} K_{1}(x_{n}). It is easy to see that L^{#} ≃ K_{1}(G) and that Σ^{−n}L ≃ K_{1}(G). Therefore, by lemma 7.3, K_{1}(G) is a Koszul complex for k over kA.

Next, we show that K_{s}(G) is built from L. Consider the following commutative diagram whose rows are triangles:

[figure: K_{s}(x_{i}); K_{s+1}(x_{i}); K_{1}(x_{i}); kA; (x_{i})^{s}; kA; (x_{i})^{s+1}; 0; kA; (x_{i}); kA; ΣK_{1}(x_{i})]

By induction we see that K_{s}(x_{i}) is built from K_{1}(x_{i}). Therefore, for any Amodule M, the module K_{s}(x_{i})⊗_{A}M is built from K_{1}(x_{i})⊗_{A}M. In particular the module

K_{s}(x_{1}) ⊗_{A} (K_{1}(x_{2}) ⊗_{A} ⋯ ⊗_{A} K_{1}(x_{n}))

is built from K_{1}(G), so it is also built from L. Similarly, the module:

K_{s}(x_{1}) ⊗_{A} K_{s}(x_{2}) ⊗_{A} (K_{1}(x_{3}) ⊗_{A} ⋯ ⊗_{A} K_{1}(x_{n}))

is built from

K_{s}(x_{1}) ⊗_{A} (K_{1}(x_{2}) ⊗_{A} ⋯ ⊗_{A} K_{1}(x_{n}))

and therefore is also built by L. Continuing this way we see that for every s the module:

K_{s}(G) ≃ K_{s}(x_{1}) ⊗_{A} ⋯ ⊗_{A} K_{s}(x_{n})

is built from L. By [13, lemma 3.6], the module K(J) is the homotopy colimit of the modules K_{s}(G), therefore K(J) is L-cellular.

Let M be an A-module, since Γ_{J}(M) = K(J) ⊗_{A} M we see that Γ_{J}(M) is L-cellular, because for every L-null module N:

Hom_{A}(Γ_{J}(M), N) ≃ Hom_{A}(M, Hom_{A}(K(J), N)) ≃ 0

We are left with showing the map Γ_{J}(M) → M is an L-equivalence. Note that L is small and therefore has duality properties:

Hom_{A}(L, K(J) ⊗_{A} M) ≃ L^{#} ⊗_{A} K(J) ⊗_{A} M

By [13, lemma 3.6 & lemma 3.8] the map L^{#} ⊗_{kA} K(J) → L^{#} is an equivalence, yielding:

L^{#} ⊗_{A} K(J) ⊗_{A} M ≃ L^{#} ⊗_{A} M ≃ Hom_{A}(L, M)

as required.

7.3 Finite p-groups

Fix a prime p and let k be a commutative 𝕊-algebra such that π_{0}(k) is an 𝔽_{p}- algebra. Note that, in this case, the homotopy groups π_{∗}(k) are 𝔽_{p}-modules. In particular the multiplication by p map: k ⟶^{·p} k is equivalent to zero (the multiplication by p map is adding the identity map of k to itself p times). We show that if P is a finite p-group then k-cellularization over kP is trivial, i.e. it is the identity functor. We further show that in this setting k is a proxy-small kP -module. We start with the following simple result.

Lemma 7.6. If P = C_{p} and π_{0}(k) is an 𝔽_{p}-algebra then k-cellularization over kP is the identity.

Proof. Let J be the augmentation ideal of (kP)_{∗} as above. We saw that kcellularization is tensoring with K(J) (defined in 7.4). Let g be a generator of C_{p}. The triangle:

K(J) → kP → kP [1/(g − 1)]

implies that k-nullification is tensoring with kP [1/(g − 1)]. Recall kP [1/(g − 1)] is the homotopy colimit of the telescope:

kP ⟶^{g−1} kP ⟶^{g−1} ⋯

Composing the map ·(g −1) with itself p times is equivalent to multiplication by the element (g − 1)^{p} in (kP)_{∗}. Because p-multiplication in k_{∗} is zero, we get that:

(g − 1)^{p} = g^{p} − 1^{p} = 1 − 1 = 0

Thus, for every module in the telescope above, the map to a far enough successor is equivalent to zero. This implies the homotopy colimit of this telescope is equivalent to zero and therefore K(J) ≃ kP.

Proposition 7.7. Let P be a finite p-group and k a commutative 𝕊-algebra such that π_{0}(k) is an 𝔽_{p}-algebra. Then k-cellularization over kP is the identity.

Proof. We prove this by induction on the order of P. Since the center of every p-group is non-trivial, there is a short exact sequence of groups:

C_{p} → P → P′

From the previous lemma k-cellularization over kC_{p} is the identity and from the induction assumption k-cellularization over kP′ is also the identity. Lemma 5.6 shows that Cell^{P} is also trivial.

Proposition 7.8. Let P be a finite p-group and k a commutative 𝕊-algebra such that π_{0}(k) is an 𝔽_{p}-algebra. Then k finitely builds kP in the category of kP -modules and therefore k is a proxy-small kP -module.

Proof. We start with the case where P is the cyclic group with p elements C_{p}. Let g be a generator of C_{p} and let K_{1} be the homotopy fiber of the map kC_{p} ⟶^{1−g} kC_{p}. From proposition 7.1 we see K_{1} is finitely built by k. Let K_{n}

be the homotopy fiber of the map kC_{p} ⟶^{(1−g)^{n}} kC_{p} and consider the following commutative diagram whose rows and columns are triangles:

[figure: K_{n}; K_{n+1}; K_{1}; kC_{p}; (1−g)^{n}; kC_{p}; (1−g)^{n+1}; 0; kC_{p}; 1−g; kC_{p}; ΣK_{1}]

From the top row we see, by induction on n, that K_{n} is finitely built from K_{1}. Since the map (1 − g)^{p} is equivalent to the zero map, the module K_{p} is equivalent to kC_{p} ⊕ Σ^{−1}kC_{p}. In particular, K_{p} finitely builds kC_{p} and hence k finitely builds kC_{p}.

Now suppose P is any finite p-group. We will prove the claim by induction on the size of P. Since P is a finite p-group, it is nilpotent and therefore has a short exact sequence:

N ↪ P ↠ C_{p}

By the induction assumption, k finitely builds kN in the category of kNmodules. By applying the functor kP ⊗_{kN} − to the recipe of constructing kN from k, we see that kP ⊗_{kN} k ≃ kC_{p} finitely builds kP ⊗_{kN} kN ≃ kP in the category of kP -modules. Another, longer, way of saying this is the following: The image of the thick subcategory generated by k under the functor kP ⊗_{kN} − is contained in the thick subcategory generated by kP ⊗_{kN} k ≃ kC_{p}. This image must contain kP ⊗_{kN} kN ≃ kP and therefore kC_{p} finitely builds kP in the category of kP -modules.

We saw that k finitely builds kC_{p} in the category of kC_{p}-modules, so clearly k finitely builds kC_{p} in the category of kP -modules. We conclude k finitely builds kP in the category of kP -modules.

Finally, kP is a Koszul-complex for k, since kP is a small kP -module, k finitely builds kP, and clearly kP builds k.

7.4 Finite Nilpotent Groups

Let N be a finite nilpotent group. As before, p is a prime number and k is a commutative 𝕊-algebra such that π_{0}(k) is an 𝔽_{p}-algebra. Let P be a p-Sylow subgroup of N and let H = N/P. Since N is nilpotent, it is isomorphic to the group P × H.

Proposition 7.9. With N,P,and k as above, there are natural equivalences of coaugmented functors on kN-modules:

Cell^{N}(−) ≃ Hom_{H}(k, −) ≃ Hom_{N}(kP, −)

Proof. Because N is nilpotent, it is the product of P and H, yielding a short exact sequence of groups: H → N → P. As we saw in proposition 7.7 Cell^{P} ≃ id, so by lemma 5.6:

Cell^{N} ≃ Cell^{H}

Since P is a p-Sylow subgroup, the order of H is prime to p. This implies the map: k ⟶^{·|H|} k (multiplication by the order of H) is an equivalence. Let b : k → kH be the map defined by sending 1 to Σ_{h∈H}h. It is easy to check that this is indeed a map of kH-modules. The composition k → kH → k is the multiplication by |H| map, hence an equivalence. Thus the augmentation map kH → k has a right inverse, showing k is a retract of kH. Now we use proposition 3.15, showing:

Cell^{H} ≃ Hom_{H}(k, −) ≃ Hom_{N}(kP, −)

The last equivalence comes from an obvious adjunction.

Remark 7.10. Suppose that k = 𝔽_{p}. Then k is a projective H-module (because it is a retract of kH). This implies that the derived functor: Hom_{H}(k, −) is equivalent to the non-derived one ^{S}Hom_{H}(k, −). So for a k-module X (a chain-complex), Cell^{N} is the fixed points of the H-action on X, denoted X^{H}.

In fact, the proof of proposition 7.9 implies a slightly more general result:

Proposition 7.11. Let H → G → P be a short exact sequence of finite groups, with P being a p-group. Suppose that the order of H is prime to p and that π_{0}(k) is an 𝔽_{p}-algebra, then:

Cell^{G}(−) ≃ Hom_{H}(k, −)

We can also show that k is proxy-small as a kN-module.

Proposition 7.12. Let N be a finite nilpotent group and k a commutative 𝕊-algebra such that π_{0}(k) is an 𝔽_{p}-algebra. Then k is a proxy-small kNmodule.

Proof. As before let P be the p-Sylow subgroup of N and let H = N/P so N ≅ P × H. We will show that kP is a Koszul-complex for k over kN. From the proof of proposition 7.9 above we see that k is a retract of kH. Applying the functor kN ⊗_{kH} − to this shows kN ⊗_{kH} k ≃ kP is a retract of kN ⊗_{kH} kH ≃ kN. Hence kP is a small kN-module.

From proposition 7.8 we see k finitely builds kP over kP and therefore k also finitely builds kP over kN. Clearly kP builds k over kN, since k is a kP -module. Therefore kP is a Koszul-complex for k over kN.

Relation with the Eilenberg-Moore Spectral Sequence

In this chapter we consider a fibration sequence F → E → B, where E and B are always assumed to be connected spaces. Taking the pullback of the diagram:

\xymatrix{□ \ar@{-->}[r] \ar@{-->}[d] & E \ar[d] \\ WΩB \ar[r] & B}

gives a ΩB-space equivalent to F. So, without loss of generality, we will assume F is a ΩB-space. Let k be a commutative 𝕊-algebra. Throughout this chapter the chains of spaces are always taken with coefficients in k, therefore we omit k from the notation C_{∗}(−; k). Let R be the group-ring k[ΩB], hence C_{∗}(F) and C^{∗}(F) are R-modules.

The results of [7] imply that, under some conditions on the space B, C^{∗}(E) ⊗_{C^{∗}(B)} k is equivalent to Cell_{k}C^{∗}(F). These conditions on B are specified in lemma 8.5. By analyzing the k-cellularization of C^{∗}(F) over R we obtain the following results, which are the main results of this chapter:

Theorem 8.1. Fix a prime p. Let N be a finite nilpotent group and let P ⊆ N be the p-Sylow subgroup of N, so N ≅ P × H with the order of |H| being prime to p. Let F → E → BN be a fibration over the classifying space of N. If k is any commutative 𝕊-algebra such that π_{0}(k) is an 𝔽_{p}-algebra, then:

C^{∗}(E; k) ⊗_{C^{∗}(B;k)} k ≃ C^{∗}(F_{h(H)})

where F_{hH} is the homotopy orbit space of F with respect to the H-action on F. In particular, if N = P then

C^{∗}(E; k) ⊗_{C^{∗}(B;k)} k ≃ C^{∗}(F; k)

Theorem 8.2. Fix a prime p. Let B be a finite connected nilpotent space with a finite fundamental group N = π_{1}(B). Let P ⊆ N be the p-Sylow subgroup of N, so N ≅ P × H. Given a fibration of pointed spaces F → E → B, then:

Tor_{−n}^{C^{∗}(B;𝔽_{p})}(C^{∗}(E; 𝔽_{p}), 𝔽_{p}) = H^{n}(F; 𝔽_{p})^{H}

where H^{n}(F; 𝔽_{p})^{H} are the fixed points of the H-action on H^{n}(F; 𝔽_{p}).

We now explain how these results relate to the Eilenberg-Moore spectral sequence. The maps E → B and pt → B induce maps of k-algebras C^{∗}(B) → C^{∗}(E) and C^{∗}(B) → k. These maps make C^{∗}(E) and k into C^{∗}(B)-bimodules. We consider C^{∗}(E) as a right C^{∗}(B)-module and k as a left C^{∗}(B)-module. The Eilenberg-Moore spectral sequence is of the form:

E_{p,q}^{2} = Tor_{p,q}^{π_{∗}(C^{∗}(B))}(π_{∗}(C^{∗}(E)), π_{∗}(k)) ⇒ π_{p+q}(C^{∗}(E) ⊗_{C^{∗}(B)} k)

The version of the Eilenberg-Moore spectral sequence we consider is the one given in [10, IV.4.1]. Its convergence properties are also described there and we will not discuss them in this work. We see that theorems 8.1 and 8.2 above describe the E^{∞} term of this spectral sequence.

As noted in the introduction, the first result (theorem 8.1) generalizes a result of Kriz from [18], showing convergence of the Eilenberg-Moore cohomology spectral sequence for a fibration over B(ℤ/p), with coefficients in Morava K-theory. The second result (theorem 8.2) is the dual of Dywer’s exotic convergence of the Eilenberg-Moore spectral sequence ([5]), concerning convergence of the Eilenberg-Moore homology spectral sequence.

We begin by recalling some well known properties we require.

Lemma 8.3. Let F → E → B be as above and let R be the 𝕊-algebra k[ΩB].

1. Hom_{R}(k, k) ≃ C^{∗}(B).

2. k ⊗_{R} C_{∗}(F) ≃ C_{∗}(E).

Proof. For the first equivalence, note that pt ×_{ΩB} pt is equivalent to the classifying space of ΩB which is B. Therefore k ⊗_{R} k ≃ C_{∗}(B; k) and

C^{∗}(B) ≃ Hom_{k}(C_{∗}(B; k), k) ≃ Hom_{k}(k ⊗_{R} k, k)
≃ Hom_{R}(k, Hom_{k}(k, k)) ≃ Hom_{R}(k, k)

For the second equivalence, recall the homotopy orbit space of the ΩBspace F is F_{hΩB} = WΩB ×_{ΩB} F, where WΩB is a contractible space with a free ΩB-action. There is a weak equivalence E ≃ F_{hΩB}, in other words E ≃ pt ×_{ΩB} F when ×_{ΩB} is taken in the derived sense (this is known as the Borel correspondence). This implies k ⊗_{R} C_{∗}(F) ≃ C_{∗}(E).

Note that Hom_{R}(k, C^{∗}(F)) is a right Hom_{R}(k, k)-module by the composition pairing:

Hom_{R}(k, C^{∗}(F)) ⊗_{k} Hom_{R}(k, k) → Hom_{R}(k, C^{∗}(F))

(see [10, III.6.12]). Hence Hom_{R}(k, C^{∗}(F)) is equivalent to a right C^{∗}(B)- module. As we saw above, C^{∗}(E) is also a right C^{∗}(B)-module.

Lemma 8.4. The modules Hom_{R}(k, C^{∗}(F)) and C^{∗}(E) are equivalent as right C^{∗}(B)-modules.

The proof of this lemma is an exercise in formal manipulations of the definitions of the two module structures, therefore the proof is deferred to appendix A.

Several results from [7] conclude that, under some conditions, C^{∗}(E)⊗_{C^{∗}(B)} k is equivalent to the k-cellularization of C^{∗}(F) over R (recall R is the groupring k[ΩB]). These conditions are given below:

Lemma 8.5. If any one of the three conditions below holds, then k is proxysmall as an R-module and C^{∗}(E) ⊗_{C^{∗}(B)} k ≃ Cell_{k}C^{∗}(F).

1. k is a commutative 𝕊-algebra and B is a finite space.

2. k is a commutative 𝕊-algebra such that π_{0}(k) is an 𝔽_{p} algebra and B is an Eilenberg-Mac Lane space of type K(N, 1) where N is a finite nilpotent group.

3. k is a commutative ring (in the classical sense) and B ≃ K(G, 1), where G is a finite group.

Proof. By the previous lemma (8.4):

C^{∗}(E) ⊗_{C^{∗}(B)} k ≃ Hom_{R}(k, C^{∗}(F)) ⊗_{End_{R}(k)} k

If k is proxy-small as an R-module, then, by [7, theorem 4.10]:

Hom_{R}(k, C^{∗}(F)) ⊗_{End_{R}(k)} k ≃ Cell_{k}C^{∗}(F)

Every one of conditions above ensures k is a proxy-small R-module. The first condition by [7, proposition 5.3], the second condition by proposition 7.12 and the third condition by [7, example 5.9].

Example 8.6. We continue with example 6.4. Recall G is a finite group and G → SU(n) is an embedding of groups. Consider the fibration sequence:

G → SU(n) → SU(n)/G

The base space is a finite space, so we see that:

C^{∗}(SU(n)) ⊗_{C^{∗}(SU(n)/G)} k ≃ Cell_{k}^{R}(kG)

Where R is the group ring k[Ω(SU(n)/G)]. In example 6.4 we saw there is a strong independence of base with respect to the map R → kG and the module k. In particular:

Cell_{k}^{R}(kG) ≃ Cell^{G}(kG)

This implies the Eilenberg-Moore cohomology spectral sequence for this fibration abuts to Cell^{G}kG (whenever the Eilenberg-Moore spectral sequence discussed above converges to π_{∗}(C^{∗}(E) ⊗_{C^{∗}(B)} k) ).

Proof of theorem 8.1. By [10, theorem III.4.2] the equivalence ΩB ≃ N induces an equivalence of the derived categories:

D_{k[ΩB]} ≃ D_{kN}

which means we can work in the category of N-modules instead of R-modules. To do that we replace F by an equivalent N-space, also denoted F. Since C^{∗}(E) ⊗_{C^{∗}(B)} k ≃ Cell^{N}C^{∗}(F), the result follows from proposition 7.9. Note that:

Hom_{k[H]}(k, C^{∗}(F)) ≃ Hom_{k}(C_{∗}(F) ⊗_{k[H]} k, k) ≃ C^{∗}(F_{h(H)})

The equivalence C_{∗}(F) ⊗_{k[H]} k ≃ F_{h(H)} follows from a result of [10] (quoted in lemma 5.1).

Our goal now is to prove theorem 8.2, so for the rest of this chapter we assume k is the Eilenberg-Mac Lane spectrum of some commutative ring, i.e. k = Hk where k is a commutative ring. We will also assume that B is a finite nilpotent space. Recall R is the group-ring k[ΩB]. To further analyze Cell_{k}^{R} we will decompose the group ΩB using a short exact sequence and then apply the results of chapter 5. The decomposition of ΩB is achieved using the following fibration:

B̃ → B → K(π_{1}(B), 1)

where B̃ is the universal cover of B. This is a fibration of connected spaces, hence taking their Kan loop-group spaces yields a short exact sequence of simplicial groups:

ΩB̃ → ΩB → ΩK(π_{1}(B), 1)

Let Q denote the simplicial group ΩK(π_{1}(B), 1) and set G = π_{1}(B). The map μ′ : Q → π_{0}(Q) = G is a weak equivalence, hence so is the induced map μ : C_{∗}(Q; k) → kG. Again, by [10, theorem III.4.2], we see μ induces an equivalence of the derived categories:

D_{kG} ⟶^{μ^{∗}} D_{kQ}

implying we can work over kG instead of kQ. Moreover, the map φ : R → kG, induced by the map ΩB → π_{0}(ΩB) = G, is equal to the composition

R ⟶^{ψ} kQ ⟶^{μ} kG

This last observation shows we can consider kG-modules as R-modules via the map of 𝕊-algebras φ. Before continuing we note one more point that will make our proofs easier. The results of Scwede and Shipley ([20, 5.1.6]) show that the projective model category structure on the category Ch_{kG} of unbounded chain complexes over kG is Quillen equivalent to the model category of kG-modules.

The next result notes an interesting (and rather obvious) property of C^{∗}(F). We will not use this result explicitly, but it does help explain the ideas behind theorem 8.2.

Lemma 8.7. Let F → E → B be a fibration where E and B are connected. Suppose k = Hk where k is a commutative ring (in the classical sense). Recall R = k[ΩB]. Then C^{∗}(F) is k-cellular as a k[ΩB̃]-module. Moreover, this implies C^{∗}(F) is kG-cellular as an R-module.

Proof. The 𝕊-algebra k[ΩB̃] is connective. Because B̃ is simply connected we get that π_{0}k[ΩB̃] = k. Clearly π_{n}(C^{∗}(F)) is zero for all n > 0. By proposition 3.12 we see C^{∗}(F) is k-cellular as a k[ΩB̃]-module. Corollary 5.4 implies C^{∗}(F) is kG-cellular as an R-module.

We see that C^{∗}(F) is built from kG-modules. In fact C^{∗}(F; k) is built from the Eilenberg-Mac Lane spectra of the cohomology groups H^{∗}(F; k), which are kG-modules (see the proof of proposition 3.12). In other words, C^{∗}(F; k) is built from the kG modules H^{∗}(F; k). From now on we will not distinguish between kG-modules and kG-modules.

To go further and show that Cell_{k}^{R}C_{∗}(F) is built from Cell^{G}H^{∗}(F; k), we need to have a strong independence of base property for the map R → kG and the kG-module k. This will reduce the analysis of Cell_{k}^{R}C_{∗}(F) to analysis of k-cellularization over kG (which is a discrete group-ring). The next lemma describes a sufficient condition for the existence of such an independence of base property.

Lemma 8.8. Let F → E → B be a fibration where E and B are connected. Let k be as above, R = k[ΩB] and G = π_{1}(B). Suppose C_{∗}(B̃) is a k-cellular G-module, then for every G-module M:

Cell_{k}^{R}(M) ≃ Cell^{G}(M)

i.e. strong independence of base. This condition is satisfied, for example, when B is a finite nilpotent space.

Proof. The space B̃ is the classifying space of the group k[Ω(B̃)]. Since C_{∗}(B̃) is a k-cellular G-module, by lemma 5.7 there is the following strong independence of base:

Cell_{k}^{R}(M) ≃ Cell^{G}(M)

for every G-module M. Example 3.14 shows that C_{∗}(B̃) is a k-cellular Gmodule whenever B is a finite nilpotent space.

Proof of theorem 8.2. Set k = H𝔽_{p}. As we saw before C^{∗}(E) ⊗_{C^{∗}(B)} k is the k-cellularization of C^{∗}(F) over R = k[ΩB]. From example 3.14 and by the previous lemma, we see that for every N-module X, Cell_{k}^{R}X ≃ Cell^{N}X. Recall that N ≅ P × H, so proposition 7.9 shows that:

Cell^{N}X ≃ Hom_{k[H]}(k, X)

The module k is projective as a k[H]-module because the order of H is prime to p (see the proof of proposition 7.9 and the succeeding remark). Hence Hom_{k[H]}(k, X) is the strict fixed points of the H action on X, i.e. X^{H}.

We use the notation from the proof of proposition 3.12. Recall that C^{∗}(F)⟨−n, −n⟩ = H^{n}(F; k), this is an N-module, hence:

Cell_{k}^{R}C^{∗}(F)⟨−n, −n⟩ ≃ H^{n}(F; k)^{H}

We prove, by induction on i, that for every 0 ≤ n ≤ i:

π_{−n}(Cell_{k}^{R}C^{∗}(F)⟨−i, 0⟩) ≅ H^{n}(F; k)^{H}

For the induction step consider the triangle:

C^{∗}(F)⟨−i, 0⟩ → C^{∗}(F)⟨−i − 1, 0⟩ → C^{∗}(F)⟨−i − 1, −i − 1⟩

applying the k-cellularization functor (over R) to this triangle yields the following triangle:

Cell_{k}^{R}(C^{∗}(F)⟨−i, 0⟩) → Cell_{k}^{R}(C^{∗}(F)⟨−i−1, 0⟩) → Cell_{k}^{R}(C^{∗}(F)⟨−i−1, −i−1⟩)

The long exact sequence of homotopy groups, for this last triangle, shows that

π_{−n}(Cell_{k}^{R}C^{∗}(F; k)⟨−i − 1, 0⟩) ≅ H^{n}(F; k)^{H}

for i + 1 ≥ n ≥ 0.

As noted in the proof of proposition 3.12: C^{∗}(F) ≃ hocolim_{i}C^{∗}(F)⟨−i, 0⟩. By lemma 2.12 k-cellulariztion commutes with homotopy colimits of telescopes (because k is proxy-small), hence:

Cell_{k}^{R}C^{∗}(F; k) ≃ hocolim_{i}Cell_{k}^{R}(C^{∗}(F; k)⟨−i, 0⟩)

yielding:

π_{−n}(Cell_{k}^{R}C^{∗}(F; k)) ≅ π_{−n}(Cell_{k}^{R}C^{∗}(F; k)⟨−n, 0⟩) ≅ H^{n}(F; k)^{H}

The proof of theorem 8.2 can be used to give a spectral sequence converging (conditionally) to Cell_{k}^{R}C^{∗}(F).

Proposition 8.9. Let F → E → B be a fibration where E and B are connected and let k be a commutative ring (in the classical sense). Suppose B is a finite nilpotent space and set N = π_{1}(B). There exists a spectral sequence:

E_{p,q}^{1} = π_{2p+q}(Cell^{N}H^{−p}(F; k)) ⇒ π_{p+q}(Cell_{k}^{R}C^{∗}(F))

Convergence is as in the usual case for the spectral sequence of the homotopy colimit of chain-complexes.

Proof. Since k is a ring, R is connective with π_{0}(R) = kN. We saw that

C^{∗}(F) ≃ hocolim_{i}C^{∗}(F)⟨−i, 0⟩

which implies:

Cell_{k}^{R}C^{∗}(F) ≃ hocolim_{i}Cell_{k}^{R}(C^{∗}(F)⟨−i, 0⟩)

because k is a proxy-small R-module (lemma 2.12). This gives a spectral sequence whose E^{2}-term is:

E_{p,q}^{1} = π_{p}(Cone(Cell_{k}C^{∗}(F)⟨−p + 1, 0⟩ → Cell_{k}C^{∗}(F)⟨−p, 0⟩))

for p = 0, E_{0,q}^{1} = π_{q}(Cell_{R}^{k}C^{∗}(F)⟨0, 0⟩). The k-cellularization functor is exact, hence:

Cone(Cell_{k}C^{∗}(F)⟨−p + 1, 0⟩ → Cell_{k}C^{∗}(F)⟨−p, 0⟩) ≃
≃ Cell_{k}Cone(C^{∗}(F)⟨−p + 1, 0⟩ → C^{∗}(F)⟨−p, 0⟩)
≃ Cell_{k}(Σ^{−p}H^{p}(F; k))

Lemma 8.8 shows that Cell_{k}^{R}H^{p}(F; k) ≃ Cell^{N}H^{p}(F; k) for every p (since H^{p}(F; k) is an N-module). This yields the desired E^{1}-page.

Remark 8.10. If, in addition to the conditions above in proposition 8.9, the action of N on H^{∗}(F; k) is nilpotent, then H^{n}(F; k) is a k-cellular N-module (see 2.4). Since C^{∗}(F) is built from H^{n}(F; k), it must also be k-cellular and therefore:

C^{∗}(E) ⊗_{C^{∗}(B)} k ≃ C^{∗}(F)

These conditions are slightly weaker than the conditions given by Dwyer ([5]) for the covergence of the Eilenberg-Moore homology spectral sequence for a fibration.

For the rest of this chapter we give different proofs for known results using the machinery we have set up.

Proposition 8.11. Let F → E → B be a fibration sequence with E and B connected and let k be a commutative ring. Suppose B is a finite, simply connected space. Then:

π_{−n}(C^{∗}(E) ⊗_{C^{∗}(B)} k) ≅ H^{n}(F; k)

Proof. Since B is a finite space, we see that

C^{∗}(E) ⊗_{C^{∗}(B)} k ≃ Cell_{k}^{R}C^{∗}(F)

By the preceding lemma, C^{∗}(F) is k[π_{1}(B)]-cellular. Since B is simply connected, it means C^{∗}(F) is k-cellular.

The next result below is an immediate consequence of Shipley’s paper [21]. We give here a different proof, using k-cellularity arguments. This result is also a corollary of theorem 8.2.

Proposition 8.12. Let B be a finite space. If k = 𝔽_{p} for some prime p and G = π_{1}(B) is a finite p-group then C^{∗}(F) is k-cellular as an R-module. In particular:

C^{∗}(E) ⊗_{C^{∗}(B)} k ≃ C^{∗}(F)

Proof. By proposition 7.7, Cell^{G} ≃ id. Using lemma 5.6 we see that Cell_{k}^{R} ≃ Cell_{k}^{k[ΩB̃]}. By lemma 8.7, C^{∗}(F) is k-cellular as a k[ΩB̃]-module, hence it is also k-cellular as an R-module.

Appendix A

Proof of Lemma 8.4

Note that a proof of lemma 8.4, for the case k = H𝔽_{p}, can be found in [8] - in the course of the proof of lemma 2.10 there. These arguments of Dwyer and Wilkerson from [8] readily generalize to an arbitrary 𝕊-algebra k. We give here a more detailed, and slightly different, argument.

Recall that F → E → B is a fibration sequence, F is a ΩB space, k is a commutative 𝕊-algebra and R denotes the group-ring k[ΩB]. In lemma 8.3 we saw that Hom_{R}(k, k) ≃ C^{∗}(B) and k ⊗_{R} C_{∗}(F) ≃ C_{∗}(E). The map of k-algebras C^{∗}(B) → C^{∗}(E) makes C^{∗}(E) into a right C^{∗}(B)-module. While the composition pairing:

Hom_{R}(k, C^{∗}(F)) ⊗_{k} Hom_{R}(k, k) → Hom_{R}(k, C^{∗}(F))

makes Hom_{R}(k, C^{∗}(F)) into a right Hom_{R}(k, k)-module (see [10, III.6.12]). Our goal is to prove that Hom_{R}(k, C^{∗}(F)) is equivalent to C^{∗}(E) as right C^{∗}(B)-modules.

We start by showing Hom_{R}(k, C^{∗}(F)) and C^{∗}(E) are equivalent as kmodules. For that we need to recall some basic constructions on modules over group-rings. Let G denote the simplicial group ΩB. Given two kGmodules X and Y, we consider X ⊗_{k} Y as a kG-module with a diagonal kG-action. Formally, X ⊗_{k} Y is a kG ⊗_{k} kG-module. The diagonal map G → G × G gives a map of k-algebras kG → kG ⊗_{k} kG, which gives the kG-module structure on X ⊗_{k} Y.

Similarly Hom_{k}(X, Y) is a kG-module. Here we must first translate the left kG-module structure on X to a right module structure via the inverse map G ⟶^{?^{−1}} G and then take a ”diagonal” action. Let X, Y and Z be kGmodules, then we have the following adjunction (recall R = kG):

Hom_{R}(X, Hom_{k}(Y, Z)) ≃ Hom_{R}(X ⊗_{k} Y, Z)

In particular:

Hom_{R}(k, C^{∗}(F)) ≃ Hom_{R}(k, Hom_{k}(C_{∗}(F), k))
≃ Hom_{R}(k ⊗_{k} C_{∗}(F), k)
≃ Hom_{R}(C_{∗}(F), k)

Now we apply the usual adjunctions to give the desired equivalence:

Hom_{R}(C_{∗}(F), k) ≃ Hom_{R}(C_{∗}(F), Hom_{k}(k, k))
≃ Hom_{k}(k ⊗_{R} C_{∗}(F), k)
≃ Hom_{k}(C_{∗}(R), k) ≃ C^{∗}(E)

Next, we consider the origin of the k-algebra structure on C^{∗}(E). For a space A, the k-algebra structure on C^{∗}(A) is induce by the diagonal map A → A × A. Consider the diagonal map Δ : F → F × F. Note that F × F is a G-space by the diagonal action. As we saw before pt ×_{G} F ≃ E. It is easy to see that pt ×_{G} (F × F) ≃ E × E. Moreover, applying the homotopy orbit space functor to the map Δ gives a map

pt ×_{G} Δ : pt ×_{G} F → pt ×_{G} (F × F)

which is equivalent to the diagonal map E → E × E. Applying the cochains functor to Δ and pt ×_{G} Δ we see that the multiplication on C^{∗}(F) is induce by C^{∗}(Δ) and the multiplication map on C^{∗}(E) is induced by C^{∗}(pt ×_{G} Δ).

We now recall some basic constructions on function k-modules. This material can be found in [10], proposition III.6.12 and the preceding discussion. We will be using the adjunction:

hom_{R}(L ⊗_{k} M, N) ≅ hom_{k}(L, Hom_{R}(M, N))

where L is a k-module and M and N are R-modules. Note that the isomorphism above is natural in every variable. The evaluation map:

ϵ : Hom_{R}(M, N) ⊗_{k} M → N

is a map of R-modules which is adjoint to the identity map: 1 : Hom_{R}(M, N) → Hom_{R}(M, N) of k-modules. Note that the left R-module structure on Hom_{R}(M, N)⊗_{k} M comes from the R-module structure on M (using the fact that k is a commutative 𝕊-algebra).

Given R-modules M, N and P, consider the composition:

Hom_{R}(N, P) ⊗_{k} Hom_{R}(M, N) ⊗_{K} M ⟶^{1⊗ϵ} Hom_{R}(N, P) ⊗_{k} N ⟶^{ϵ} P

The adjoint of this map ϵ(1 ⊗_{k} ϵ) is the composition pairing:

π : Hom_{R}(N, P) ⊗_{k} Hom_{R}(M, N) → Hom_{R}(M, P)

If we take M = k for the evaluation map above, we get the map:

ϵ : Hom_{R}(k, N) → N

(note we have used the isomorphism Hom_{R}(k, N) ⊗_{k} k ≅ Hom_{R}(k, N)). This map is adjoint to the identity map Hom_{R}(k, N) = Hom_{R}(k, N). Thus we get a natural transformation: ϵ : Hom_{R}(k, −) → Id.

Remark A.1. The augmentation map a : R → k induces a map:

a^{∗} : Hom_{R}(k, N) → N

This natural transformation is equivalent to ϵ above.

Now we can explicitly build the multiplication map on C^{∗}(E) from the multiplication map on C^{∗}(F). Let:

m : C^{∗}(F) ⊗_{k} C^{∗}(F) → C^{∗}(F)

be the multiplication map on C ∗ (F), recall this map is C^{∗}(Δ). Let:

u : k → C^{∗}(F)

be the unit map of C^{∗}(F). The unit map u is induced by the map F → pt. Recall that m(u ⊗ 1_{F}) = m(1_{F} ⊗ u) = 1_{F}. Consider the map:

Hom_{R}(k, C^{∗}(F)) ⊗_{k} Hom_{R}(k, C^{∗}(F)) ⟶^{ϵ⊗ϵ} C^{∗}(F) ⊗_{k} C^{∗}(F)

The adjoint of this map is a map:

Hom_{R}(k, C^{∗}(F)) ⊗_{k} Hom_{R}(k, C^{∗}(F)) → Hom_{R}(k, C^{∗}(F) ⊗_{k} C^{∗}(F))

which we will denote by ad(ϵ⊗ϵ). The multiplication map on Hom_{R}(k, C^{∗}(F)) is given by the composition:

Hom_{R}(k, C^{∗}(F))⊗_{k}Hom_{R}(k, C^{∗}(F)) ⟶^{ad(ϵ⊗ϵ)} Hom_{R}(k, C^{∗}(F)⊗_{k}C^{∗}(F)) ⟶^{m_{∗}} Hom_{R}(k, C^{∗}(F))

We denote this map by μ. The map m_{∗} is clearly C^{∗}(pt ×_{G} Δ). Note the map ad(ϵ ⊗ ϵ) is in fact the map:

C^{∗}(E) ⊗_{k} C^{∗}(E) → C^{∗}(E × E)

used in constructing the multiplication on cohomology of spaces. We conclude that the map μ described above gives the multiplication on C^{∗}(E).

Next, we describe the right C^{∗}(B)-module structure on C^{∗}(E) coming from the map C^{∗}(B) → C^{∗}(E) of k-algebras. We will call this the first module structure. Applying the functor Hom_{R}(k, −) to the unit map u of C^{∗}(F) gives a map:

C^{∗}(B) ≃ Hom_{R}(k, k) ⟶^{u_{∗}} Hom_{R}(k, C^{∗}(F)) ≃ C^{∗}(E)

This map is equivalent to the given map C^{∗}(B) → C^{∗}(E) (because applying the G-homotopy orbit space functor to the map F → pt gives the map E → B). Hence, the first C^{∗}(B)-module structure on C^{∗}(E) is the composition:

Hom_{R}(k, C^{∗}(F))⊗_{k}Hom_{R}(k, k) ⟶^{1⊗u_{∗}} Hom_{R}(k, C^{∗}(F))⊗_{k}Hom_{R}(k, C^{∗}(F)) ⟶^{μ} Hom_{R}(k, C^{∗}(F))

The second right Hom_{R}(k, k)-module structure on Hom_{R}(k, C^{∗}(F)) comes from the composition pairing:

Hom_{R}(k, C^{∗}(F)) ⊗_{k} Hom_{R}(k, k) ⟶^{π} Hom_{R}(k, C^{∗}(F))

We will call this the second module structure.

The first module structure μ(1 ⊗ u_{∗}) is adjoint to the composition:

Hom_{R}(k, C^{∗}(F))⊗_{k}Hom_{R}(k, k) ⟶^{1⊗u_{∗}} Hom_{R}(k, C^{∗}(F))⊗_{k}Hom_{R}(k, C^{∗}(F)) ⟶^{ϵ⊗ϵ} F ⊗_{k}F ⟶^{m} F

The second module structure π is adjoint to the composition:

Hom_{R}(k, C^{∗}(F)) ⊗_{k} Hom_{R}(k, k) ⟶^{1⊗ϵ} Hom_{R}(k, C^{∗}(F)) ⊗_{k} k ⟶^{ϵ⊗1} F ⊗ k ≃ F

So, all that remains is to show that, in the derived category D_{k}:

m(ϵ ⊗ ϵu_{∗}) = (ϵ ⊗ 1)(1 ⊗ ϵ) = ϵ ⊗ ϵ

Since ϵ is a natural transformation we see that ϵu_{∗} = uϵ. Hence:

m(ϵ ⊗ ϵu_{∗}) = m(ϵ ⊗ uϵ)

= m(1 ⊗ u)(ϵ ⊗ ϵ) = 1_{C^{∗}(F)}(ϵ ⊗ ϵ) = ϵ ⊗ ϵ

and we are done.

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