LOOP SPACE HOMOLOGY OF A SMALL CATEGORY

C. BROTO, R. LEVI, AND B. OLIVER

Abstract. In a 2009 paper, Dave Benson gave a description in purely algebraic terms of the mod p homology of Ω(BG_{p}^{∧}), when G is a finite group, BG_{p}^{∧} is the p-completion of its classifying space, and Ω(BG_{p}^{∧}) is the loop space of BG_{p}^{∧}. The main purpose of this work is to shed new light on Benson’s result by extending it to a more general setting. As a special case, we show that if C is a small category, |C| is the geometric realization of its nerve, R is a commutative ring, and |C|_{R}^{+} is a “plus construction” for |C| in the sense of Quillen (taken with respect to R-homology), then H_{∗}(Ω(|C|_{R}^{+}); R) can be described as the homology of a chain complex of projective RC-modules satisfying a certain list of algebraic conditions that determine it uniquely up to chain homotopy. Benson’s theorem is now the case where C is the category of a finite group G, R = 𝔽_{p} for some prime p, and |C|_{R}^{+} = BG_{p}^{∧}.

Introduction

Let G be a finite group, and let BG_{p}^{∧} denote its classifying space after p-completion in the sense of Bousfield and Kan [BK]. In general, the higher homotopy groups π_{i}(BG_{p}^{∧}) for i ≥ 2 can be nonvanishing, and hence the loop space Ω(BG_{p}^{∧}) is interesting in its own right. These spaces are the subject of several papers by the second author (e.g., [L, Theorem 1.1.4]). In particular, the homology of Ω(BG_{p}^{∧}) with p-local coefficients is known to have some very interesting properties, as described in [CL, § 2]. This helped to motivate the question of whether the homology of Ω(BG_{p}^{∧}) admits a purely algebraic definition (e.g., in [CL, § 2.6]).

In [Be2], Benson answered this question by showing that H_{∗}(Ω(BG_{p}^{∧}); k), for a field k of characteristic p, is isomorphic to the homology of what he called a “left k-squeezed resolution for G”: a chain complex of projective kG-modules satisfying certain axioms. He also showed that any two such complexes are chain homotopy equivalent, and hence have the same homology. The k-homology of Ω(BG_{p}^{∧}) is thus determined by the axioms of a squeezed resolution.

Our original aim in this paper was to check whether Benson’s concept of a squeezed resolution can be formulated in a more categorical context. This was motivated in part by the problem of identifying p-compact groups in the sense of Dwyer and Wilkerson: loop spaces with finite mod p homology and p-complete classifying spaces (see Section 5 for more discussion). When doing this, we discovered that in fact, squeezed resolutions can be defined in a much more general setting, where we call them Ω-resolutions to emphasize the connection

1991 Mathematics Subject Classification. Primary 55R35. Secondary 55R40, 20D20. Key words and phrases. Classifying space, Loop space, Small category, p-completion, Finite groups,

Fusion.

C. Broto acknowledges financial support from the Spanish Ministry of Economy through the “María de Maeztu” Programme for Units of Excellence in R&D (MDM-2014-0445) and FEDER-MINECO Grant MTM2016-80439-P and from the Generalitat de Catalunya through AGAUR Grant 2017SGR1725.

B. Oliver is partially supported by UMR 7539 of the CNRS. R. Levi and B. Oliver were partly supported by FEDER-MINECO Grant MTM2016-80439-P during

several visits to the Universitat Autònoma de Barcelona.

The authors also thank the University of Aberdeen for its support during visits by two of us.

to loop spaces. In this setting, Benson’s result can be generalized to a statement about plus constructions (in the sense of Quillen) on nerves of small categories.

When C is a small category, we let |C| denote the geometric realization of the nerve of C. If R is a commutative ring, then an RC-module is a (covariant) functor C ⟶ R-mod, and a morphism of RC-modules is a natural transformation of functors. When π is a group, we let B(π) be the category with one object ∘_{π} and End(∘_{π}) = π. If X is a connected CW complex and H ⊴ π_{1}(X), then a plus construction for X with respect to R and H means a space |C|_{R}^{+} together with a map κ: |C| ⟶ |C|_{R}^{+} such that π_{1}(κ) is surjective with kernel H and H_{∗}(κ; R) is an isomorphism; Quillen’s construction shows that this exists whenever H_{1}(H; R) = 0 (see Lemma A.3). Finally, a group G is R-perfect if H_{1}(G; R) = 0.

Theorem A. Fix a commutative ring R, a small connected category C, a group π, and a functor θ: C ⟶ B(π) such that π_{1}(|θ|): π_{1}(|C|) ⟶ π is surjective. Set H = Ker(π_{1}(|θ|)), and assume that H is R-perfect. Then there is an Ω-resolution

⋯ ⟶^{∂_{3}} C_{2} ⟶^{∂_{2}} C_{1} ⟶^{∂_{1}} C_{0} ⟶^{ε} θ^{∗}(Rπ) ⟶ 0

(a chain complex of RC-modules satisfying conditions listed in Definition 1.5 or Lemma 3.10), and H_{∗}(C_{∗}, ∂_{∗}) ≅ H_{∗}(Ω(|C|_{R}^{+}); R) for each such (C_{∗}, ∂_{∗}) and each plus construction |C|_{R}^{+} for |C| with respect to R and H.

Theorem A will be stated in a more precise form as Theorem 4.4. Upon restricting to the case where R = 𝔽_{p} for a prime p, C = B(G) for some finite group G, and π = G/O^{p}(G) ≅ π_{1}(BG_{p}^{∧}) (the largest p-group quotient of G), we recover Benson’s theorem, since BG_{p}^{∧} is a plus construction on BG = |B(G)| with respect to the ring 𝔽_{p} and the subgroup O^{p}(G).

As another special case of Theorem A, let (S, F, L) be a p-local compact group in the sense of [BLO, Definition 4.2]. Thus S is a discrete p-toral group (an extension of (ℤ/p^{∞})^{r} by a finite p-group), F is a saturated fusion system over S, and L is a centric linking system associated to F. Set π = π_{1}(|L|_{p}^{∧}): a finite p-group by [BLO, Proposition 4.4]. By Theorem 4.7 or 4.4 applied with L in the role of C, H_{∗}(Ω(|L|_{p}^{∧}); 𝔽_{p}) can be described in terms of Ω-resolutions. As noted above, our original motivation for this work was the search for new conditions sufficient to guarantee that Ω(|L|_{p}^{∧}) has finite homology, and hence that |L|_{p}^{∧} is a p-compact group in the sense of Dwyer and Wilkerson [DW]. We did not succeed in doing this, but our attempt to do so is what led to this more general setting. Also, we do construct some examples in Propositions 5.5, 5.6, and 5.11 of explicit Ω-resolutions of finite length (in fact, of minimal length) for certain p-compact groups.

It turns out that Ω-resolutions can be defined in much greater generality than that needed

in Theorem A. Let (A \overrightarrow{⟵}_{θ^{∗}}^{θ_{∗}} B) be an Ω-system: a pair of abelian categories and additive

functors such that θ_{∗} is left adjoint to θ^{∗}, θ_{∗}θ^{∗} ≅ Id_{B}, and θ^{∗} is exact (Definition 1.1). In this situation, for a projective object X in B, an Ω-resolution of X is a chain complex of projective objects in A augmented by a morphism to θ^{∗}(X) which satisfies certain axioms listed in Definition 1.5. In particular, these axioms are minimal conditions needed to ensure the uniqueness of Ω-resolutions up to chain homotopy equivalence (Proposition 1.6). However, while each such X has at most one Ω-resolution up to homotopy, we have examples that show that it need not have any in this very general situation. The examples in Theorem A are the special case where A = RC-mod, B = Rπ-mod, and θ_{∗} is left Kan extension with respect to the functor θ. Another large family of examples, where we show that Ω-resolutions exist but haven’t yet found a geometric interpretation of their homology, is described in the following proposition (and in more detail in Proposition 3.5).

Proposition B. Let θ: C ⟶ D be a functor between small categories that is bijective on objects and surjective on morphism sets, and which has the following property:

for each c, c′ ∈ Ob(C) and each φ, φ′ ∈ Mor_{C}(c, c′) such that θ_{c,c′}(φ) =

θ_{c,c′}(φ′), there is α ∈ Aut_{C}(c′) such that θ_{c}(α) = Id_{θ(c′)} and φ = αφ′.

Then for each commutative ring R, (RC-mod \overrightarrow{⟵}_{θ^{∗}}^{θ_{∗}} RD-mod) is an Ω-system, where

θ_{∗} is defined by left Kan extension. Furthermore, projective objects in RD-mod all have Ωresolutions if and only if Ker[θ_{c}: Aut_{C}(c) ⟶ Aut_{D}(θ(c))] is R-perfect for each c ∈ Ob(C).

We begin in Section 1 by defining Ω-resolutions in our most general setting and proving their uniqueness. In Section 2, we find some necessary conditions, and some sufficient conditions, for their existence. We then restrict in Section 3 to the special case of RC-modules, and construct examples where Ω-resolutions do or do not exist (Propositions 3.5 and 3.12). Our results connecting the homology of certain Ω-resolutions to the homology of loop spaces are shown in Section 4, where Theorem A is stated and proved in a slightly more precise form as Theorem 4.4. Afterwards, we look in Section 5 at some detailed examples of Ω-resolutions arising from p-local compact groups in which the maximal torus is normal.

Notation: For each small category C, |C| denotes its geometric realization. When C is a small category and R is a commutative ring, we let RC-mod denote the category of “RC-modules”: covariant functors from C to R-mod. When C is an abelian category, we write 𝒫(C) to denote the class of projective objects in C. For a group G, we write G^{ab} = G/[G, G] for the abelianization, and let B(G) denote the category with one object ∘_{G} and End_{B(G)}(∘_{G}) ≅ G.

1. Ω-systems and Ω-resolutions

We begin by defining Ω-resolutions and proving their uniqueness in a very general setting. We do not prove any results about the existence of Ω-resolutions in this section, but leave that for Sections 2 and 4.

Definition 1.1. An Ω-system (A, B; θ_{∗}, θ^{∗}) consists of a pair of abelian categories A and B, together with additive functors

A \overrightarrow{⟵}_{θ^{∗}}^{θ_{∗}} B

such that

(OP1) θ_{∗} is left adjoint to θ^{∗};

(OP2) θ_{∗} is a retraction in the sense that the counit of the adjunction 𝔟: θ_{∗} ∘ θ^{∗} ⟶ Id_{B} is an isomorphism; and

(OP3) θ^{∗} sends epimorphisms in B to epimorphisms in A.

It will be important, in the situation of Definition 1.1, to know that θ^{∗}(B) is a full subcategory of A. In fact, this holds without assuming condition (OP3).

Lemma 1.2. Let A and B be a pair of categories, together with functors

A \overrightarrow{⟵}_{θ^{∗}}^{θ_{∗}} B

such that θ_{∗} is left adjoint to θ^{∗}, and such that the counit 𝔟: θ_{∗} ∘ θ^{∗} ⟶ Id_{B} of the adjunction is an isomorphism of functors. Then the image θ^{∗}(B) is a full subcategory of A.

Proof. Fix B, B′ ∈ Ob(B) and α ∈ Mor_{A}(θ^{∗}(B), θ^{∗}(B′)). Let 𝔞: Id_{A} ⟶ θ^{∗} ∘θ_{∗} be the unit of the adjunction. Upon applying 𝔞 to α, one obtains the commutative square in the following diagram in A:

\xymatrix{ & θ^{∗}(B) \ar[r]^{α} \ar[d]^{𝔞_{θ^{∗}(B)}} \ar[dl]_{_{Id_{θ^{∗}(B)}}} & θ^{∗}(B′) \ar[d]_{𝔞_{θ^{∗}(B′)}} \ar[dr]^{_{Id_{θ^{∗}(B′)}}} \\ θ^{∗}(B) & θ^{∗}θ_{∗}θ^{∗}(B) \ar[l]_{θ^{∗}(𝔟_{B})}^{≅} \ar[r]^{θ^{∗}θ_{∗}(α)} & θ^{∗}θ_{∗}θ^{∗}(B′) \ar[r]^{θ^{∗}(𝔟_{B′})}_{≅} & θ^{∗}(B′) .}

The two triangles commute by, e.g., [Mac1, § V.5], and 𝔟_{B} and 𝔟_{B′} are isomorphisms in B by assumption. Hence α = θ^{∗}(𝔟_{B′} ∘ θ_{∗}(α) ∘ 𝔟_{B}^{−1}) is a morphism in θ^{∗}(B), and so θ^{∗}(B) is a full subcategory of A. □

The following is one family of Ω-systems to which we will frequently refer. More examples will be given in Section 3.

Example 1.3. Fix a commutative ring R, a pair of groups G and π, and a surjective homomorphism θ: G ⟶ π. Let RG-mod and Rπ-mod be the categories of (left) RG- and Rπ-modules, respectively, and let

RG-mod \overrightarrow{⟵}_{θ^{∗}}^{θ_{∗}} Rπ-mod

be the functors defined as follows. For each RG-module M, set θ_{∗}(M) = Rπ ⊗_{RG} M where Rπ is regarded as a right RG-module via θ. For each Rπ-module N, set θ^{∗}(N) = N regarded as an RG-module via θ. Then (RG-mod, Rπ-mod; θ_{∗}, θ^{∗}) is an Ω-system.

Proof. If M and N are RG- and Rπ-modules, respectively, then there is an obvious natural bijection Hom_{RG}(M, θ^{∗}(N)) ≅ Hom_{Rπ}(θ_{∗}(M), N). Thus (OP1) holds: θ_{∗} is left adjoint to θ^{∗}. Conditions (OP2) and (OP3) are clear. □

The following properties of Ω-systems follow easily from the basic properties of abelian categories and adjoint functors (See, e.g., [Mac1, § IV-V].).

Lemma 1.4. The following hold for each Ω-system (A, B; θ_{∗}, θ^{∗}).

(a) The functor θ^{∗} is exact, and θ_{∗} is right exact.

(b) The functor θ_{∗} sends projectives to projectives.

(c) A sequence in B is exact if and only if its image under θ^{∗} is exact in A. A morphism in B is an isomorphism, an epimorphism, or a monomorphism if and only if the same is true in A of its image under θ^{∗}.

Proof. (a,b) Since θ_{∗} is left adjoint to θ^{∗}, θ_{∗} is right exact and θ^{∗} is left exact. By (OP3), θ^{∗} also preserves epimorphisms, and hence is exact. Since θ_{∗} has a right adjoint that is exact, it sends projectives to projectives.

(c) The exactness of θ^{∗} implies that it sends the kernel, cokernel, and image of each morphism ψ in B to the kernel, cokernel, and image in A of θ^{∗}(ψ). Also, if φ ∈ Iso_{A}(θ^{∗}(N), θ^{∗}(N′)) for N, N′ in B, then since θ^{∗}(B) is a full subcategory of A and θ_{∗}θ^{∗} is naturally isomorphic to the identity, φ = θ^{∗}(ψ) for some ψ ∈ Iso_{B}(N, N′). Hence a sequence in B whose image under θ^{∗} is exact in A is also exact in B, and if θ^{∗}(φ) is a monomorphism or epimorphism in A, then φ is a monomorphism or epimorphism, respectively, in B. This proves the “if” statements, and the converse in all cases holds by the exactness of θ^{∗}. □

We are now ready to define Ω-resolutions.

Definition 1.5. Let (A, B; θ_{∗}, θ^{∗}) be an Ω-system. For a projective object X in B, an Ωresolution of X with respect to (A, B; θ_{∗}, θ^{∗}) is a chain complex

R = (⋯ ⟶^{∂_{3}} P_{2} ⟶^{∂_{2}} P_{1} ⟶^{∂_{1}} P_{0} ⟶^{ε} θ^{∗}(X) ⟶ 0)  (1)

in A such that

(Ω-1) P_{n} is projective in A for all n ≥ 0;

(Ω-2) θ_{∗}(R) is exact; and

(Ω-3) H_{n}(P_{∗}, ∂_{∗}) is isomorphic to an object in θ^{∗}(B) for each n ≥ 0, and ε induces an isomorphism H_{0}(P_{∗}, ∂_{∗}) ≅ θ^{∗}(X).

If an Ω-resolution R exists as above, then we set H_{∗}^{Ω}(A, B; X) = θ_{∗}(H_{∗}(P_{∗}, ∂_{∗})): the image under θ_{∗} of the homology of the complex (P_{∗}, ∂_{∗}).

There are, in fact, many Ω-systems for which Ω-resolutions do not exist. In the situation of Example 1.3, when R is a field and θ: G ⟶ π is a surjection of groups, we will see in Example 2.9 that a nonzero projective object in B has an Ω-resolution if and only if H_{1}(Ker(θ); R) = 0. However, whenever X does have at least one Ω-resolution, the next proposition implies that H_{∗}^{Ω}(A, B; X) is unique up to natural isomorphism.

Proposition 1.6. Let (A, B; θ_{∗}, θ^{∗}) be an Ω-system. Let X and Y be projective objects in B, and let f ∈ Mor_{B}(X, Y) be a morphism. Let

⋯ ⟶^{∂_{2}} P_{1} ⟶^{∂_{1}} P_{0} ⟶^{ε} θ^{∗}(X) ⟶ 0 and ⋯ ⟶^{∂′_{2}} P′_{1} ⟶^{∂′_{1}} P′_{0} ⟶^{ε′} θ^{∗}(Y) ⟶ 0

be chain complexes in A, where the first satisfies conditions (Ω-1) and (Ω-2) in Definition 1.5 and the second satisfies condition (Ω-3). Then there are morphisms f_{n} ∈ Mor_{A}(P_{n}, P′_{n}) which make the following diagram commute:

\xymatrix{⋯ \ar[r]^{∂_{3}} & P_{2} \ar[r]^{∂_{2}} \ar[d]_{f_{2}} & P_{1} \ar[r]^{∂_{1}} \ar[d]_{f_{1}} & P_{0} \ar[r]^{ε} \ar[d]_{f_{0}} & θ^{∗}(X) \ar[r] \ar[d]_{θ^{∗}(f)} & 0 \\ ⋯ \ar[r]^{∂′_{3}} & P′_{2} \ar[r]^{∂′_{2}} & P′_{1} \ar[r]^{∂′_{1}} & P′_{0} \ar[r]^{ε′} & θ^{∗}(Y) \ar[r] & 0}

Furthermore, {f_{n}}_{n∈ℕ} is unique up to chain homotopy.

Proof. For each i ≥ 0, θ_{∗}(P_{i}) is projective in B by Lemma 1.4(b) and since P_{i} is projective. Also, θ_{∗}θ^{∗}(X) ≅ X by (OP2), and X is projective in B by assumption. So by (Ω-2), θ_{∗}(P_{∗}) ⟶ θ_{∗}θ^{∗}(X) → 0 is an exact sequence of objects projective in B, and hence splits in each degree.

Existence of f_{∗}: Since P_{0} is projective, and the augmentation ε′: P′_{0} ⟶ θ^{∗}(Y) is onto by (Ω-3), θ^{∗}(f) ∘ ε lifts to a homomorphism f_{0} : P_{0} ⟶ P′_{0}.

Assume, for some n ≥ 0, that f_{i} has been constructed for all 0 ≤ i ≤ n, where f_{n−1} ∘ ∂_{n} = ∂′_{n} ∘ f_{n}. Then f_{n} ∘ ∂_{n+1} sends P_{n+1} into Ker(∂′_{n}), and hence induces a homomorphism χ: P_{n+1} ⟶ H_{n}(P′_{∗}, ∂′_{∗}). By assumption, H_{n}(P′_{∗}, ∂′_{∗}) ≅ θ^{∗}(B) for some B in B, and hence there are natural bijections

Mor_{A}(P_{i}, H_{n}(P′_{∗}, ∂′_{∗})) ≅ Mor_{A}(P_{i}, θ^{∗}(B)) ≅ Mor_{B}(θ_{∗}(P_{i}), B)

for i = n, n + 1, n + 2. Since the complex (θ_{∗}(P_{∗}), θ_{∗}(∂_{∗})) ⟶ θ_{∗}(θ^{∗}X) → 0 is exact and split and χ ∘ ∂_{n+2} = 0 by construction, θ_{∗}(χ) factors through Im(θ_{∗}(∂_{n+1})) and extends to θ_{∗}(P_{n}). By adjointness, there is φ: P_{n} ⟶ H_{n}(P′_{∗}, ∂′_{∗}) such that χ = φ ∘ ∂_{n+1}.

Since P_{n} is projective, φ lifts to a morphism φ̃: P_{n} ⟶ Ker(∂′_{n}). An easy diagram chase now shows that Im((f_{n}−φ̃)∘∂_{n+1}) ≤ Im(∂′_{n+1}). Hence, upon replacing f_{n} by f_{n}−φ̃, f_{n−1}∘∂_{n} = ∂′_{n} ∘ f_{n} still holds (where f_{−1} = f if n = 0) and Im(f_{n} ∘ ∂_{n+1}) ≤ Im(∂′_{n+1}). Upon using the projectivity of P_{n+1} again, one can lift f_{n} ∘ ∂_{n+1} to a homomorphism f_{n+1} : P_{n+1} ⟶ P′_{n+1} such that f_{n} ∘ ∂_{n+1} = ∂′_{n+1} ∘ f_{n+1}. We now continue inductively.

Uniqueness of f_{∗}: Let f′_{∗} and f′′_{∗} be two homomorphisms covering f, and set t_{∗} = f′_{∗} − f′′_{∗}. Thus t_{∗}: (P_{∗}, ∂_{∗}) ⟶ (P′_{∗}, ∂′_{∗}) is a homomorphism covering X ⟶^{0} Y, and we must construct a chain homotopy D: P_{∗} ⟶ P′_{∗} of degree +1 such that D ∘ ∂ + ∂′ ∘ D = t_{∗}.

Set D_{−1} = 0: θ^{∗}(X) ⟶ P′_{0}. Since ε′ ∘ t_{0} = 0, and the sequence

P′_{1} ⟶^{∂′_{1}} P′_{0} ⟶^{ε′} θ^{∗}(Y) ⟶ 0

is exact by condition (Ω-3), t_{0} lifts to a homomorphism D_{0} : P_{0} ⟶ P′_{1}. The rest of the proof is carried out using arguments similar to those used to show existence. □

When (A, B; θ_{∗}, θ^{∗}) is an Ω-system and X ∈ 𝒫(B) has an Ω-resolution, there is a spectral sequence that links the Ω-homology of X to higher derived functors of θ_{∗}.

Proposition 1.7. Let (A, B; θ_{∗}, θ^{∗}) be an Ω-system, and assume A has enough projectives. Let X be a projective object in B that has an Ω-resolution. Then there is a first quadrant spectral sequence E_{∗,∗}^{r} in B such that

E_{i,j}^{2} ≅ (L_{i}θ_{∗})(θ^{∗}(H_{j}^{Ω}(A, B; X))) and E_{i,j}^{∞} ≅ \begin{cases}X & if (i, j) = (0, 0) \\ 0 & if (i, j) ≠ (0, 0). \end{cases}

Proof. Let (P_{∗}, ∂_{∗}) be an Ω-resolution of X. Let {Q_{ij}}_{i,j≥0} be a proper projective resolution of (P_{∗}, ∂_{∗}); i.e., a double complex of projective objects in A, where for each j, the sequences

(i) 0 ⟵ P_{j} ⟵ Q_{0,j} ⟵ Q_{1,j} ⟵ ⋯,

(ii) 0 ⟵ H_{j}(P_{∗}) ⟵ H_{j}(Q_{0,∗}) ⟵ H_{j}(Q_{1,∗}) ⟵ ⋯, and

(iii) 0 ⟵ Z_{j}(P_{∗}) ⟵ Z_{j}(Q_{0,∗}) ⟵ Z_{j}(Q_{1,∗}) ⟵ ⋯

are all projective resolutions (see [Be1, Definition 3.6.1]). Proper projective resolutions exist by [Mac2, Proposition XII.11.6] (see also, [Be1, Lemma 3.6.2]).

Consider the two spectral sequences associated to the double complex θ_{∗}(Q_{∗,∗}). Since each row Q_{∗,j} is a projective resolution of the projective object P_{j}, θ_{∗}(Q_{∗,j}) is a resolution of θ_{∗}(P_{j}), and thus Ē_{0,j}^{1} ≅ θ_{∗}(P_{j}), while Ē_{i,j}^{1} = 0 if i ≥ 1. Since P_{∗} is an Ω resolution of X, we now obtain Ē_{0,0}^{2} ≅ X, while Ē_{i,j}^{2} = 0 if (i, j) ≠ (0, 0).

Now consider the other spectral sequence E_{i,j}^{r}, where we first take homology of the columns. By (ii), H_{j}(Q_{i,∗}) and Z_{j}(Q_{i,∗}) are projective for each i, j ≥ 0, so B_{j}(Q_{i,∗}) is also projective, and all sequences involved in the homology of Q_{i,∗} split. In other words,

E_{i,j}^{1} = H_{j}(θ_{∗}(Q_{i,∗})) ≅ θ_{∗}(H_{j}(Q_{i,∗}))

for all i, j ≥ 0. By (ii) again, the j-th row in the E^{1}-page is obtained by applying θ_{∗} to a projective resolution of H_{j}(P_{∗}), and so

E_{i,j}^{2} ≅ (L_{i}θ_{∗})(H_{j}(P_{∗}, ∂_{∗})) ≅ (L_{i}θ_{∗})(θ^{∗}(H_{j}^{Ω}(A, B; X))).

Since Ē_{i,j}^{∞} = 0 for all (i, j) ≠ (0, 0), the two spectral sequences have isomorphic E^{∞}-pages, and this proves the proposition. □

We finish the section with the following observation.

Remark 1.8. If (A, B; θ_{∗}, θ^{∗}) and (B, C; η_{∗}, η^{∗}) are two Ω-systems, then their composite (A, C; η_{∗}θ_{∗}, θ^{∗}η^{∗}) is easily seen to be an Ω-system. In other words, there is a category whose objects are the small abelian categories and whose morphisms are isomorphism classes of Ωsystems. One obvious question is whether there is a natural way to construct Ω-resolutions for the composite Ω-system from Ω-resolutions for the two factors, and if so, what connection there is (if any) between the homology groups of these three complexes.

2. The existence of Ω-resolutions

We saw in the last section that Ω-resolutions, when they exist, are unique up to chain homotopy. The question of when they do exist is more complicated, and in this section, we give some necessary conditions and some sufficient conditions for that to happen. When doing this, the following slightly more general form of Definition 1.5 will be needed: one that allows for truncated resolutions.

Definition 2.1. Let (A, B; θ_{∗}, θ^{∗}) be an Ω-system. For X ∈ 𝒫(B) and 1 ≤ n ≤ ∞, an Ω_{n}-resolution of X is a chain complex

R_{n} = \begin{cases}(P_{n} ⟶^{∂_{n}} ⋯ ⟶^{∂_{3}} P_{2} ⟶^{∂_{2}} P_{1} ⟶^{∂_{1}} P_{0} ⟶^{ε} θ^{∗}(X) ⟶ 0) & if n < ∞ \\ (⋯ ⟶^{∂_{3}} P_{2} ⟶^{∂_{2}} P_{1} ⟶^{∂_{1}} P_{0} ⟶^{ε} θ^{∗}(X) ⟶ 0) & if n = ∞ \end{cases}

(1)

in A such that

(Ω_{n}-1) P_{i} ∈ 𝒫(A) for all 0 ≤ i ≤ n (for all i ≥ 0 if n = ∞);

(Ω_{n}-2) θ_{∗}(R_{n}) is exact;

(Ω_{n}-3) H_{i}(P_{∗}, ∂_{∗}) is isomorphic to an object in θ^{∗}(B) for each 0 ≤ i < n, and ε induces an isomorphism H_{0}(P_{∗}, ∂_{∗}) ≅ θ^{∗}(X); and

(Ω_{n}-4) if n < ∞, the inclusion Im(∂_{n}) ≤ P_{n−1} induces a monomorphism θ_{∗}(Im(∂_{n})) ⟶ θ_{∗}(P_{n−1}).

In particular, an Ω_{∞}-resolution is the same as an Ω-resolution (Definition 1.5).

Lemma 2.2. Condition (Ω_{n}-4) can be replaced by the following equivalent condition:

(Ω_{n}-4′) If n < ∞, the sequence θ_{∗}(Ker(∂_{n})) ⟶^{θ_{∗}(incl)} θ_{∗}(P_{n}) ⟶^{θ_{∗}(∂_{n})} θ_{∗}(P_{n−1}) is exact.

Proof. The sequence θ_{∗}(Ker(∂_{n})) ⟶ θ_{∗}(P_{n}) ⟶ θ_{∗}(Im(∂_{n})) → 0 is exact since θ_{∗} is right exact by Lemma 1.4(a). Hence the sequence in (Ω_{n}-4′) is exact if and only if the inclusion of Im(∂_{n}) in P_{n−1} induces a monomorphism θ_{∗}(Im(∂_{n})) ⟶ θ_{∗}(P_{n−1}). □

Lemma 2.3. Fix an Ω-system (A, B; θ_{∗}, θ^{∗}), and a projective object X ∈ 𝒫(B). Let P_{m} ⟶ ⋯ ⟶ P_{0} ⟶ X → 0 be an Ω_{m}-resolution of X for some 1 < m ≤ ∞. Then for each 1 ≤ n < m, the truncation P_{n} ⟶ ⋯ ⟶ P_{0} ⟶ X → 0 is an Ω_{n}-resolution of X.

Proof. Conditions (Ω_{n}-1)–(Ω_{n}-3) in Definition 2.1 follow immediately from (Ω_{m}-1)–(Ω_{m}-3), so we need only prove that (Ω_{n}-4) holds. Consider the following commutative diagram:

\xymatrix{θ_{∗}(P_{n+1}) \ar[r]^{θ_{∗}(∂_{n+1})} \ar[drr]_{0} & θ_{∗}(P_{n}) \ar[rr]^{θ_{∗}(∂_{n})} \ar@{->>}[dr]_{θ_{∗}(∂_{n}^{∗})} &  & θ_{∗}(P_{n−1}) \\  &  & θ_{∗}(Im(∂_{n})) \ar[ur]_{θ_{∗}(incl)}}  (2)

where ∂_{n}^{∗}: P_{n} ⟶ Im(∂_{n}) is the restriction of ∂_{n}. The row in (2) is exact by (Ω_{m}-2) and since m > n, and θ_{∗}(∂_{n}^{∗}) is an epimorphism since θ_{∗} is right exact. Hence Ker(θ_{∗}(∂_{n}^{∗})) = Ker(θ_{∗}(∂_{n})) and θ_{∗}(incl) is a monomorphism, and (Ω_{n}-4) holds. □

Definition 2.4. When (A, B; θ_{∗}, θ^{∗}) is an Ω-system, we say that θ^{∗}(B) is closed under subobjects in A if for each monomorphism A_{1} ⟶ A_{2} in A, A_{1} is isomorphic to an object of θ^{∗}(B) if A_{2} is. Similarly, we say that θ^{∗}(B) is closed under extensions in A if for each short exact sequence 0 → M′ ⟶ M ⟶ M′′ → 0 in A, M is isomorphic to an object in θ^{∗}(B) if M′ and M′′ are isomorphic to objects in θ^{∗}(B).

We will show that for each Ω-system (A, B; θ_{∗}, θ^{∗}) in which θ^{∗}(B) is closed under subobjects and extensions, all projectives in B have Ω-resolutions (Proposition 2.8).

Lemma 2.5. Let (A, B; θ_{∗}, θ^{∗}) be an Ω-system, where A has enough projectives.

(a) The following are equivalent:

(a.i) θ^{∗}(B) is closed under subobjects in A.

(a.ii) For each M in A, the unit morphism 𝔞_{M}: M ⟶ θ^{∗}θ_{∗}(M) is an epimorphism.

(b) If either condition (a.i) or (a.ii) holds, then the following two conditions are equivalent:

(b.i) θ^{∗}(B) is closed under extensions in A.

(b.ii) For each N in B, (L_{1}θ_{∗})(θ^{∗}(N)) = 0.

(c) If 0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0 is an extension in A, where M′ and M′′ are in θ^{∗}(B) but M is not isomorphic to an object in θ^{∗}(B), then (L_{1}θ_{∗})(M′′) ≠ 0.

Proof. (a.i ⟹ a.ii) Fix an object M in A, and consider the unit morphism 𝔞_{M}: M ⟶ θ^{∗}θ_{∗}(M). Since θ^{∗}(B) is closed under subobjects, Im(𝔞_{M}) ≅ θ^{∗}(B) for some B in B. Since θ^{∗}(B) is a full subcategory of A by Lemma 1.2, each morphism in Mor_{A}(θ^{∗}(B), θ^{∗}θ_{∗}(M)) lies in θ^{∗}(B). Thus 𝔞_{M} factors as a composite

𝔞_{M}: M ⟶^{g} θ^{∗}(B) ⟶^{θ^{∗}(ψ)} θ^{∗}θ_{∗}(M)

for some ψ ∈ Mor_{B}(B, θ_{∗}(M)), where g is surjective and θ^{∗}(ψ) is injective.

Let γ ∈ Mor_{B}(θ_{∗}(M), B) be the morphism adjoint to g. Then ψ ∘ γ = Id_{θ_{∗}(M)} since 𝔞_{M} is adjoint to the identity, and thus ψ is surjective. So θ^{∗}(ψ) is also surjective by (OP3), and hence 𝔞_{M} is surjective.

(a.ii ⟹ a.i) Now assume that M ⟶^{𝔞_{M}} θ^{∗}θ_{∗}(M) is an epimorphism for each M in A. Let M_{1} ⟶^{f} M_{2} be a monomorphism in A, where M_{2} is in θ^{∗}(B) and hence 𝔞_{M_{2}} is an isomorphism. From the commutative square

\xymatrix{M_{1} \ar[r]^{f} \ar[d]_{𝔞_{M_{1}}} & M_{2} \ar[d]_{𝔞_{M_{2}}}^{≅} \\ θ^{∗}θ_{∗}(M_{1}) \ar[r]^{θ^{∗}θ_{∗}(f)} & θ^{∗}θ_{∗}(M_{2}) ,}

we see that θ^{∗}θ_{∗}(f) ∘ 𝔞_{M_{1}} = 𝔞_{M_{2}} ∘ f is a monomorphism, and hence that 𝔞_{M_{1}} is a monomorphism. Since 𝔞_{M_{1}} is also an epimorphism, we have M_{1} ≅ θ^{∗}θ_{∗}(M_{1}) ∈ Ob(θ^{∗}(B)).

(c) Let 0 → M′ ⟶^{α} M ⟶^{β} M′′ → 0 be a short exact sequence in A, where M′, M′′ ∈ Ob(θ^{∗}(B)) and M is not isomorphic to any object in θ^{∗}(B). Consider the following commutative diagram with exact rows:

\xymatrix{0 \ar[r] & M′ \ar[r]^{α} \ar[d]_{𝔞_{M′}}^{≅} & M \ar[r]^{β} \ar[d]_{𝔞_{M}} & M′′ \ar[r] \ar[d]_{𝔞_{M′′}}^{≅} & 0 \\  & θ^{∗}θ_{∗}(M′) \ar[r]^{θ^{∗}θ_{∗}(α)} & θ^{∗}θ_{∗}(M) \ar[r]^{θ^{∗}θ_{∗}(β)} & θ^{∗}θ_{∗}(M′′) \ar[r] & 0 .}

Here, 𝔞_{M′} and 𝔞_{M′′} are isomorphisms since M′ and M′′ are in θ^{∗}(B), while 𝔞_{M} is not an isomorphism since M is not isomorphic to any object in θ^{∗}(B). Thus θ^{∗}θ_{∗}(α) is not injective in A, so θ_{∗}(α) is not injective in B (Lemma 1.4(c)), and (L_{1}θ_{∗})(M′′) ≠ 0.

(b) The implication (b.ii ⟹ b.i) follows immediately from (c), and it remains to prove the converse. So assume that (a.ii) holds, and that θ^{∗}(B) is closed under extensions in A. Fix M in θ^{∗}(B), and let

0 ⟶ K ⟶^{α} P ⟶^{β} M ⟶ 0

be a short exact sequence in A where P is projective. Set K_{0} = Ker(𝔞_{K}), and consider the following commutative diagram:

\xymatrix{ &  & 0 \ar[r] & K/K_{0} \ar[r]^{α̂} \ar[d]_{𝔞̂_{K}}^{≅} & P/α(K_{0}) \ar[r]^{β̂} \ar@{->>}[d]_{𝔞̂_{P}} & M \ar[r] \ar[d]_{𝔞_{M}}^{≅} & 0 \\ 0 \ar[r] & θ^{∗}((L_{1}θ_{∗})(M)) \ar[rr] &  & θ^{∗}θ_{∗}(K) \ar[r]^{θ^{∗}θ_{∗}(α)} & θ^{∗}θ_{∗}(P) \ar[r]^{θ^{∗}θ_{∗}(β)} & θ^{∗}θ_{∗}(M) \ar[r] & 0 .} (3)

Here, 𝔞_{M} is an isomorphism since M is in θ^{∗}(B), 𝔞̂_{K} and 𝔞̂_{P} are epimorphisms since 𝔞_{K} and 𝔞_{P} are surjective by (a.ii), and 𝔞̂_{K} is injective by construction. The top row is exact by construction, and the bottom row since (L_{1}θ_{∗})(P) = 0 (P is projective) and θ^{∗} is exact.

Now, K/K_{0} ≅ θ^{∗}θ_{∗}(K) and M are both isomorphic to objects of θ^{∗}(B), and the same holds for P/α(K_{0}) since θ^{∗}(B) is closed under extensions. Thus there is an object N in B, a surjective morphism f: P ⟶ θ^{∗}(N) with kernel α(K_{0}), and a morphism ν: θ^{∗}(N) ⟶ θ^{∗}θ_{∗}(P) such that ν ∘ f = 𝔞_{P}. Since θ^{∗}(B) is a full subcategory of A (Lemma 1.2), ν = θ^{∗}(χ) for some χ ∈ Mor_{B}(N, θ_{∗}(P)).

Let φ ∈ Mor_{B}(θ_{∗}(P), N) be adjoint to f. Then

[figure: P; 𝔞_{P}; f; θ^{∗}θ_{∗}(P); θ^{∗}(φ); θ^{∗}(N); ν=θ^{∗}(χ); is adjoint to; θ_{∗}(P); _{Id_{θ∗(P)}}; φ; θ_{∗}(P); φ; N ,; χ]

so θ^{∗}(φ) ∘ 𝔞_{P} = f and ν ∘ f = 𝔞_{P}. Since f and 𝔞_{P} are both surjective, ν and θ^{∗}(φ) are isomorphisms (and inverses to each other). So in diagram (3), 𝔞̂_{P} is an isomorphism, θ^{∗}θ_{∗}(α) is injective, and thus (L_{1}θ_{∗})(M) = 0. □

The next proposition provides one tool for showing that Ω-resolutions do not exist in certain cases.

Proposition 2.6. For each Ω-system (A, B; θ_{∗}, θ^{∗}) for which A has enough projectives, and each X ∈ 𝒫(B), there is an Ω_{1}-resolution of X if and only if (L_{1}θ_{∗})(θ^{∗}(X)) = 0.

Proof. Assume that (L_{1}θ_{∗})(θ^{∗}(X)) = 0. Let R_{1} = (P_{1} ⟶^{∂_{1}} P_{0} ⟶^{ε} θ^{∗}(X) → 0) be an exact sequence in A, where P_{0}, P_{1} ∈ 𝒫(A). Then the sequence

0 ⟶ Im(∂_{1}) ⟶ P_{0} ⟶ θ^{∗}(X) ⟶ 0

is short exact, and since (L_{1}θ_{∗})(θ^{∗}(X)) = 0, the induced morphism θ_{∗}(Im(∂_{1})) ⟶ θ_{∗}(P_{0}) is a monomorphism. So R_{1} is an Ω_{1}-resolution of X, where (Ω_{1}-2) holds since θ_{∗} is right exact.

Conversely, if R_{1} = (P_{1} ⟶^{∂_{1}} P_{0} ⟶^{ε} θ^{∗}(X) → 0) is an Ω_{1}-resolution of X, then R_{1} is exact by (Ω_{n}-3). Since P_{0} ∈ 𝒫(A), the sequence

0 ⟶ (L_{1}θ_{∗})(θ^{∗}(X)) ⟶ θ_{∗}(Im(∂_{1})) ⟶^{θ_{∗}(incl)} θ_{∗}(P_{0}) ⟶^{θ_{∗}(ε)} θ_{∗}(θ^{∗}(X)) ⟶ 0

is exact. Since θ_{∗}(incl) is a monomorphism by condition (Ω_{1}-4), (L_{1}θ_{∗})(θ^{∗}(X)) = 0. □

The following lemma gives conditions for extending an Ω_{n}-resolution to an Ω_{n+1}-resolution.

Lemma 2.7. Fix an Ω-system (A, B; θ_{∗}, θ^{∗}), where A has enough projectives, and let X be a projective in B. Let R_{n} = (P_{n} ⟶^{∂_{n}} P_{n−1} ⟶ ⋯ ⟶ P_{0} ⟶^{ε} θ^{∗}(X) → 0) be an Ω_{n}-resolution of X, for some 1 ≤ n < ∞.

(a) The natural morphism f_{0} : θ_{∗}(Ker(∂_{n})) ⟶ Ker(θ_{∗}(∂_{n})) is a split epimorphism.

(b) If P_{n+1} ∈ 𝒫(A), and ∂_{n+1} ∈ Mor_{A}(P_{n+1}, P_{n}) are such that ∂_{n} ∘ ∂_{n+1} = 0, then the complex R_{n+1} = (P_{n+1} ⟶^{∂_{n+1}} P_{n} ⟶ ⋯ ⟶^{ε} θ^{∗}(X) → 0) is an Ω_{n+1}-resolution of X if and only if (b.i) Ker(∂_{n})/Im(∂_{n+1}) is in B; and

(b.ii) the composite θ_{∗}(Im(∂_{n+1})) ⟶^{θ_{∗}(incl)} θ_{∗}(Ker(∂_{n})) ⟶^{f_{0}} Ker(θ_{∗}(∂_{n})) is an isomorphism.

(c) The resolution R_{n} extends to an Ω_{n+1}-resolution if and only if for some splitting s of f_{0}, the composite

𝔞_{Ker(∂_{n})}^{[s]} : Ker(∂_{n}) ⟶^{𝔞_{Ker(∂n)}} θ^{∗}θ_{∗}(Ker(∂_{n})) ⟶^{θ^{∗}(χ^{[s]})} θ^{∗}(θ_{∗}(Ker(∂_{n}))/Im(s))

(where χ^{[s]} is the natural projection) and the induced map

(L_{1}θ_{∗})(𝔞_{Ker(∂_{n})}^{[s]}): (L_{1}θ_{∗})(Ker(∂_{n})) ⟶ (L_{1}θ_{∗})(θ^{∗}(θ_{∗}(Ker(∂_{n}))/Im(s)))

are both epimorphisms. If R_{n} does extend to R_{n+1} as in (b), then for some splitting s of f_{0}, Im(∂_{n+1}) = Ker(𝔞_{Ker(∂_{n})}^{[s]}), and

θ_{∗}(H_{n}(P_{∗}, ∂_{∗})) ≅ θ_{∗}(Ker(∂_{n}))/Im(s) ≅ Ker(f_{0}) ≅ (L_{1}θ_{∗})(Im(∂_{n})) .  (4)

Proof. (a) Since θ_{∗} is right exact, we have the following commutative diagram in B

[figure: θ_{∗}(Ker(∂_{n})); f_{0}; θ_{∗}(P_{n}); θ_{∗}(Im(∂_{n})); f_{1}; ≅; 0; 0; Ker(θ_{∗}(∂_{n})); θ_{∗}(P_{n}); Im(θ_{∗}(∂_{n})); 0]  (5)

with exact rows. By condition (Ω_{n}-4), f_{1} is a monomorphism (hence an isomorphism), and so f_{0} is an epimorphism. Also, Ker(θ_{∗}(∂_{n})) ∈ 𝒫(B) since the sequence θ_{∗}(P_{∗}) ⟶ θ^{∗}(X) → 0 is an exact sequence of projective objects, and hence f_{0} splits.

(b) Assume that R_{n+1} = (P_{n+1} ⟶^{∂_{n+1}} P_{n} ⟶ ⋯) is an Ω_{n+1}-resolution of X, and set J = Im(∂_{n+1}) ≤ Ker(∂_{n}). Consider the following commutative diagram

[figure: θ_{∗}(P_{n+1}); θ_{∗}(∂_{n+1}^{∗}); f_{3}; θ_{∗}(J); f_{4}; θ_{∗}(incl); θ_{∗}(Ker(∂_{n})); f_{0}; Im(θ_{∗}(∂_{n+1})); Ker(θ_{∗}(∂_{n})) ,]

where ∂_{n+1}^{∗}: P_{n+1} ⟶ J is the restriction of ∂_{n+1} and is surjective, and f_{3} is the restriction of θ_{∗}(∂_{n+1}). By condition (Ω_{n+1}-4) on R_{n+1}, the morphism θ_{∗}(J) ⟶ θ_{∗}(P_{n}) is a monomorphism. Hence f_{4} is a monomorphism, and is an isomorphism since f_{3} is an epimorphism. This proves (b.ii), and (b.i) follows from condition (Ω_{n+1}-3).

Conversely, assume that (b.i) and (b.ii) hold. In particular, R_{n+1} satisfies (Ω_{n+1}-3), and it satisfies (Ω_{n+1}-1) (P_{n+1} is projective) by assumption. Condition (Ω_{n+1}-4) (that θ_{∗}(Im(∂_{n+1})) injects into θ_{∗}(P_{n})) follows from (b.ii).

It remains to prove (Ω_{n+1}-2); i.e., the exactness of θ_{∗}(R_{n+1}). Since θ_{∗}(R_{n}) is exact, we need only show that Im(θ_{∗}(∂_{n+1})) = Ker(θ_{∗}(∂_{n})). Consider the following diagram:

\xymatrix{P_{n+1} \ar@{->>}[r]^{∂_{n+1}^{∗}} \ar[d]^{𝔞_{P_{n+1}}} & Im(∂_{n+1}) \ar[r]^{i} \ar[d]^{𝔞_{Im(∂_{n+1})}} & P_{n} \ar[d]^{𝔞_{Pn}} \\ θ^{∗}θ_{∗}(P_{n+1}) \ar@{->>}[r]^{θ^{∗}θ_{∗}(∂_{n+1}^{∗})} \ar[rr]_{θ^{∗}θ_{∗}(∂_{n+1})} & θ^{∗}θ_{∗}(Im(∂_{n+1})) \ar[r]^{θ^{∗}θ_{∗}(i)} & θ^{∗}θ_{∗}(P_{n})}

where ∂_{n+1}^{∗} is surjective by definition and θ^{∗}θ_{∗}(∂_{n+1}^{∗}) is surjective since θ^{∗}θ_{∗} is right exact. Hence Im(θ^{∗}θ_{∗}(∂_{n+1})) = Im(θ^{∗}θ_{∗}(i)), and so Im(θ_{∗}(∂_{n+1})) = Im(θ_{∗}(i)) by Lemma 1.4(c). Finally, Im(θ_{∗}(i)) = Ker(θ_{∗}(∂_{n})) by the following diagram

\xymatrix{θ_{∗}(Im(∂_{n+1})) \ar[r]^{θ_{∗}(i′)} \ar@{-->}[drr]_{θ_{∗}(i)} & θ_{∗}(Ker(∂_{n})) \ar[r]^{f_{0}} \ar[dr]^{θ_{∗}(i′′)} & Ker(θ_{∗}(∂_{n})) \ar[d]^{incl} \\  &  & θ_{∗}(P_{n})}

and since f_{0} ∘ θ_{∗}(i′) is an isomorphism by (b.ii).

(c) Assume first that R_{n} does extend to an Ω_{n+1}-resolution

R_{n+1} = (P_{n+1} ⟶^{∂_{n+1}} P_{n} ⟶ ⋯) .

Set J = Im(∂_{n+1}). By (b.i) and (b.ii), Ker(∂_{n})/J is isomorphic to an object in θ^{∗}(B), and the composite

\xymatrix{θ_{∗}(J) \ar[r]_{θ_{∗}(i_{2})} \ar[rr]^{≅} & θ_{∗}(Ker(∂_{n})) \ar[r]^{f_{0}} & Ker(θ_{∗}(∂_{n})) \ar@{-->}[l]^{s}}

is an isomorphism. Set s = θ_{∗}(i_{2}) ∘ (f_{0} ∘ θ_{∗}(i_{2}))^{−1}: a splitting for f_{0}.

Consider the following commutative diagram:

\xymatrix{0 \ar[r] & J \ar[r]^{i_{2}} \ar[d]_{𝔞_{J}} & Ker(∂_{n}) \ar[r]^{^{pr^{2}}} \ar[d]_{𝔞_{Ker(∂n)}} \ar[dr]^{ω} & Ker(∂_{n})/J \ar[r] \ar[d]_{f_{2}}^{≅} & 0 \\ 0 \ar[r] & θ^{∗}θ_{∗}(J) \ar[r]^{θ^{∗}θ_{∗}(i_{2})} & θ^{∗}θ_{∗}(Ker(∂_{n})) \ar[r]^{θ^{∗}θ_{∗}(pr_{2})} & θ^{∗}θ_{∗}(Ker(∂_{n})/J) \ar[r] & 0 .} (6)

Here, f_{2} = 𝔞_{Ker(∂_{n})/J} is an isomorphism since Ker(∂_{n})/J is isomorphic to an object in θ^{∗}(B) by (b.i). The bottom row of (6) is exact since θ^{∗}θ_{∗} is right exact and θ_{∗}(i_{2}) is a monomorphism by (b.ii) (and θ^{∗} is left exact). Also, Im(θ_{∗}(i_{2})) = Im(s), and hence

(Ker(∂_{n}) ⟶^{𝔞_{Ker(∂n)}^{[s]}} θ^{∗}(θ_{∗}(Ker(∂_{n}))/Im(s))) ≅ (Ker(∂_{n}) ⟶^{ω} θ^{∗}θ_{∗}(Ker(∂_{n})/J))

≅ (Ker(∂_{n}) ⟶^{pr_{2}} Ker(∂_{n})/J) .

Thus 𝔞_{Ker(∂_{n})}^{[s]} is an epimorphism, and (L_{1}θ_{∗})(𝔞_{Ker(∂_{n})}^{[s]}) ≅ (L_{1}θ_{∗})(pr_{2}) is also an epimorphism

since θ_{∗}(i_{2}) is injective. This also proves that Im(∂_{n+1}) = J = Ker(𝔞_{Ker(∂_{n})}^{[s]}), and proves (4) except for the isomorphism Ker(f_{0}) ≅ (L_{1}θ_{∗})(Im(∂_{n})) which follows from (5).

Conversely, assume, for some splitting s of f_{0}, that 𝔞_{Ker(∂_{n})}^{[s]} and (L_{1}θ_{∗})(𝔞_{Ker(∂_{n})}^{[s]}) are both epimorphisms. Set J = 𝔞_{Ker(∂_{n})}^{−1}(θ^{∗}(Im(s))) ≤ Ker(∂_{n}), and consider the following commutative diagram:

\xymatrix{0 \ar[r] & J \ar[r]^{i_{2}} \ar[d] & Ker(∂_{n}) \ar[r]^{^{pr^{2}}} \ar[d]_{𝔞_{Ker(∂n)}} & Ker(∂_{n})/J \ar[r] \ar[d]_{f_{3}}^{≅} & 0 \\ 0 \ar[r] & θ^{∗}(Im(s)) \ar[r] & θ^{∗}θ_{∗}(Ker(∂_{n})) \ar[r]^{χ^{[s]}} & θ^{∗}(θ_{∗}(Ker(∂_{n}))/Im(s)) \ar[r] & 0 .} (7)

The left square in (7) is a pullback square by definition of J, so f_{3} is a monomorphism, and f_{3} is an epimorphism since χ^{[s]} ∘ 𝔞_{Ker(∂_{n})} = 𝔞_{Ker(∂_{n})}^{[s]} is an epimorphism.

In particular, Ker(∂_{n})/J is isomorphic to an object in θ^{∗}(B), and (b.i) holds. Also, f_{2} is an isomorphism in (6), and the bottom row in (6) is exact since (L_{1}θ_{∗})(𝔞_{Ker(∂_{n})}^{[s]}) is an epimorphism. Upon comparing (6) and (7), we see that Im(s) = θ_{∗}(i_{2})(θ_{∗}(J)), and (b.ii) now follows since Im(s) is the image of a splitting of f_{0}. So R_{n} extends to an Ω_{n+1}-resolution by (b). □

The next proposition is our most general result on the existence of Ω-resolutions.

Proposition 2.8. Fix an Ω-system (A, B; θ_{∗}, θ^{∗}), where A has enough projectives. Assume that θ^{∗}(B) is closed under subobjects and extensions in A. Then each X ∈ 𝒫(B) admits an Ω-resolution. Furthermore, for n ≥ 0, each Ω_{n}-resolution of X extends to an Ω-resolution of X.

Proof. This follows immediately from Proposition 2.6, Lemma 2.7(c), and Lemma 2.5. □

Under the assumptions of Example 1.3, we can now describe exactly under what conditions there are Ω-resolutions. Recall, for a commutative ring R, that a group G is R-perfect if H_{1}(G; R) = 0.

Example 2.9. Fix a commutative ring R and a surjective homomorphism θ: G ⟶ π of groups. Let (RG-mod, Rπ-mod; θ_{∗}, θ^{∗}) be the Ω-system of Example 1.3. Then θ^{∗}(Rπ-mod) is closed under subobjects.

(a) If Ker(θ) is R-perfect, then θ^{∗}(Rπ-mod) is closed under extensions in RG-mod. So by Proposition 2.8, Ω-resolutions exist of all projective objects in Rπ-mod.

(b) If Ker(θ) is not R-perfect, then θ^{∗}(Rπ-mod) is not closed under extensions, and for each nonzero object X in Rπ-mod that is free as an R-module, (L_{1}θ_{∗})(θ^{∗}(X)) ≠ 0. So by Proposition 2.6, no nonzero projective object in Rπ-mod that is free as an R-module has an Ω-resolution.

Proof. Set K = Ker(θ), and note that an RG-module M is isomorphic to an object in θ^{∗}(Rπ-mod) if and only if K acts trivially on M. Thus θ^{∗}(Rπ-mod) is closed under subobjects. If H_{1}(K; R) = 0, then θ^{∗}(Rπ-mod) is closed under extensions by Lemma A.1, and the existence of Ω-resolutions follows from Proposition 2.8.

If H_{1}(K; R) ≠ 0, then for each nonzero object X in Rπ-mod that is free as an R-module, (L_{1}θ_{∗})(θ^{∗}(X)) ≅ H_{1}(K; θ^{∗}(X)) ≠ 0 since H_{1}(K; R) ≠ 0 and K acts trivially on θ^{∗}(X). Thus θ^{∗}(Rπ-mod) is not closed under extensions in RG-mod by Lemma 2.5(b). If in addition, X is projective in Rπ-mod, Proposition 2.6 implies that it has no Ω-resolution. □

Note that the “squeezed resolutions” defined and studied by Benson [Be2] are Ω-resolutions in the context of Example 2.9(a), when G is a finite group and K = O^{p}(G).

Remark 2.10. Proposition 2.8 gives some general conditions for the existence of Ω-resolu-tions: conditions which are satisfied by the Ω-systems of Example 1.3 (as just seen), and also by the much larger family of examples to be described in Proposition 3.5(b). However, they do not hold for the family of examples constructed in Proposition 3.12(a), even though Ω-resolutions are shown to exist in those cases (at least for certain projective objects) in Proposition 4.3. This suggests that there should be a more general existence result that covers all of these cases.

In fact, there are two questions of this type that one can ask. First, of course, we want to find conditions as general as possible on an Ω-system (A, B; θ_{∗}, θ^{∗}) that imply the existence of Ω-resolutions of all projectives in B. But we will see in Example 3.6 that there are Ωsystems for which some nonzero projectives have Ω-resolutions and others do not, and so we would also like to find more general conditions on a pair ((A, B; θ_{∗}, θ^{∗}), X), for X ∈ 𝒫(B), that imply the existence of an Ω-resolution of X.

3. Ω-systems of functor categories

We next look at a large family of examples of Ω-systems and Ω-resolutions involving functor categories; especially categories of RC-modules for a small category C and a commutative ring R. At the end of the section, in Propositions 3.5 and 3.12, we give two large families of examples of Ω-systems where we can say fairly precisely in which cases Ω-resolutions exist.

We refer to [Mac1, § II.6 and § X.3] for the definitions and properties of overcategories and left Kan extensions. As usual, when A and C are categories and C is small, A^{C} denotes the functor category whose objects are the functors C ⟶ A, and whose morphisms are the natural transformations of functors.

In the following proposition, we define a functor θ: C ⟶ D between small categories to be quasisurjective if it is surjective on objects, and each morphism in D is a composite of morphisms θ(φ) and θ(ψ)^{−1} for φ, ψ ∈ Mor(C) such that θ(ψ) is an isomorphism in D. In other words, D is generated by the image of θ together with inverses of isomorphisms in the image of θ.

Proposition 3.1. Let A be an abelian category with colimits, and let θ: C ⟶ D be a quasisurjective functor between small categories. For each d in D, let I(θ↓d) be the full subcategory of θ↓d with objects (c, φ) for φ ∈ Iso_{D}(θ(c), d), and assume that all objects (c, Id_{d}) for c ∈ θ^{−1}(d) lie in the same connected component of I(θ↓d). Let θ^{∗}: A^{D} ⟶ A^{C} be composition with θ, and let θ_{∗}: A^{C} ⟶ A^{D} be left Kan extension along θ. Then (A^{C}, A^{D}; θ_{∗}, θ^{∗}) is an Ω-system.

Proof. Conditions (OP1) and (OP3) are clear. So the only difficulty is to show that condition (OP2) holds: that the counit 𝔟: θ_{∗}θ^{∗} ⟶ Id_{A^{D}} associated to the adjunction is an isomorphism.

Fix a functor α: D ⟶ A and an object d in D, and let \underline{α}(\underline{d}): θ↓d ⟶ A be the constant functor sending all objects to α(d). Let α_{d}: θ↓d ⟶ A be the functor that sends an object (c, φ), where φ ∈ Mor_{D}(θ(c), d), to (θ^{∗}α)(c) = α(θ(c)). Let α_{∗}: α_{d} ⟶ \underline{α}(\underline{d}) be the natural transformation of functors that sends (c, φ) to α(φ) ∈ Mor_{A}(α_{d}(c, φ), α(d)). Then

𝔟(α)(d) = colim_{θ↓d}(α_{∗}): (θ_{∗}θ^{∗}(α))(d) = colim_{θ↓d}(α_{d}) ⟶ α(d) ,

and we must show that this is an isomorphism (for all α and d).

To see this, choose d̂∈ Ob(C) such that θ(d̂) = d. For each (c, φ) in θ↓d, let ι_{(c,φ)} be the natural morphism from α_{d}(c, φ) = α(θ(c)) to the colimit. Set

β = ι_{(d̂,Id_{d})} : α(d) = α_{d}(d̂, Id_{d}) ⟶ colim(α_{d}) = (θ_{∗}θ^{∗}(α))(d).

Then 𝔟(α)(d) ∘ β = Id_{α(d)}, and it remains to show that β ∘ 𝔟(α)(d) is also the identity. This means showing, for each object (c, φ) in θ↓d, that β ∘ 𝔟(α)(d) ∘ ι_{(c,φ)} = ι_{(c,φ)} . Since

𝔟(α)(d) ∘ ι_{(c,φ)} = α(φ): α_{d}(c, φ) = α(θ(c)) ⟶ α(d) = α_{d}(d̂, Id_{d}),

we are reduced to showing, for each (c, φ), that

ι_{(d̂,Id_{d})} ∘ α(φ) = ι_{(c,φ)} .  (1)

We now claim the following:

(i) Equation (1) holds for (d̂, Id_{d}).

(ii) If there is χ ∈ Mor_{θ↓d}((c, φ), (c′, φ′)), then

(ii.1) if (1) holds for (c′, φ′), then it also holds for (c, φ); and

(ii.2) if θ(χ) is an isomorphism, then (1) holds for (c, φ) if and only if it holds for (c′, φ′).

(iii) If θ(c) = θ(c′) and φ ∈ Mor_{D}(θ(c), d), then (1) holds for (c, φ) if and only if it holds for (c′, φ).

Point (i) is clear. If χ ∈ Mor_{θ↓d}((c, φ), (c′, φ′)), then ι_{(c′,φ′)} ∘ α(θ(χ)) = ι_{(c,φ)} by definition of colimits, and (ii) follows immediately from this. Point (iii) follows from (ii.2) and the assumption that (c, Id_{θ(c)}) and (c′, Id_{θ(c)}) are in the same connected component of I(θ↓θ(c)).

Now let (c, φ) be arbitrary. Since θ is quasisurjective, for some m ≥ 1, there are objects θ(c) = d_{0}, d_{1}, …, d_{m} = d in D and morphisms φ_{i} ∈ Mor_{D}(d_{i−1}, d_{i}) for 1 ≤ i ≤ m such that φ = φ_{m}∘⋯∘φ_{2}∘φ_{1}, and for each i, either φ_{i} ∈ θ(Mor(C)) or φ_{i} ∈ Iso(D) and φ_{i}^{−1} ∈ θ(Mor(C)). Since (1) holds for (d̂, Id_{d}) by (i), it also holds for (c, Id_{d}) for all c ∈ θ^{−1}(d) = θ^{−1}(d_{m}) by (iii). If φ_{m} ∈ θ(Mor(C)), then (1) holds for (c, φ_{m}) for some c ∈ θ^{−1}(d_{m−1}) by (ii.1), while if φ_{m} ∈ Iso(D) and φ_{m}^{−1} ∈ θ(Mor(C)), then (1) holds for (c, φ_{m}) for some c ∈ θ^{−1}(d_{m−1}) by (ii.2). In either case, (1) holds for (c, φ_{m}) for all c ∈ θ^{−1}(d_{m−1}) by (iii). Upon continuing this argument, we see by downward induction that for each 1 ≤ i ≤ m, (1) holds for (c, φ_{m}∘⋯∘φ_{i}) for all c ∈ θ^{−1}(d_{i−1}). In particular, (1) holds for (c, φ) (the case i = 1). □

We now specialize to the case where A = R-mod: the category of modules over a commutative ring R.

Definition 3.2. Let C be a small category, and let R be a commutative ring.

(a) An RC-module is a covariant functor M: C ⟶ R-mod, and a morphism of RCmodules is a natural transformation of functors. Let RC-mod denote the category of RC-modules.

(b) Let θ: C → D be a functor between small categories. When M is an RC-module, let θ_{∗}(M) denote the left Kan extension of M along θ. When N is an RD-module, let θ^{∗}(N) = N ∘ θ be the RC-module induced by composition with θ.

(c) An RC-module M is locally constant on C if it sends all morphisms in C to isomorphisms of R-modules.

(d) An RC-module M is essentially constant if M is isomorphic to a constant RC-module; i.e., isomorphic to a functor C ⟶ R-mod that sends each object to the same R-module V and each morphism to Id_{V}.

The next lemma characterizes essentially constant modules in terms of an action of π_{1}(|C|).

Lemma 3.3. Assume C is a small category, and let R be a commutative ring.

(a) If M is a locally constant RC-module, then for each object c_{0} in C, there is a unique

homomorphism

M_{#} : π_{1}(|C|, c_{0}) ⟶ Aut_{R}(M(c_{0}))

with the following property: for each sequence

σ = (c_{0} ⟶^{f_{1}} c_{1} ⟵^{f_{2}} c_{2} ⟶^{f_{3}} ⋯ ⟵^{f_{2m}} c_{2m} = c_{0})

of morphisms in C (m ≥ 1), beginning and ending at c_{0}, regarded as a loop in |C|,

M_{#}([σ]) = M(f_{2m})^{−1} ∘ M(f_{2m−1}) ∘ ⋯ ∘ M(f_{2})^{−1} ∘ M(f_{1}) ∈ Aut_{R}(M(c_{0})) .

(b) If C is connected, then a locally constant RC-module M is essentially constant if and only if M_{#} (as defined in (a)) is the trivial homomorphism for some object c_{0} in C.

Proof. (a) Let Is(R-mod) be the category of R-modules with only isomorphisms as morphisms, and regard M as a functor M: C ⟶ Is(R-mod). This induces a map between the geometric realizations, and hence a homomorphism of fundamental groups

M_{#} : π_{1}(|C|, c_{0}) ⟶ π_{1}(|Is(R-mod)|, M(c_{0})) ≅ Aut_{R}(M(c_{0})) .

For each sequence σ as described above, M_{#} sends the class [σ] ∈ π_{1}(|C|, c_{0}) to

M(c_{0}) ⟶^{M(f_{1})} M(c_{1}) ⟵^{M(f_{2})} M(c_{2}) ⟶^{M(f_{3})} ⋯ ⟵^{M(f_{2m)}} M(c_{2m}) = M(c_{0}) ,

regarded as a loop in |Is(R-mod)|, and this is homotopic to the composite

M(f_{2m})^{−1} ∘ M(f_{2m−1}) ∘ ⋯ ∘ M(f_{1})^{−1} ∘ M(f_{0}) ∈ Aut_{R}(M(c_{0}))

when also regarded as a loop in |Is(R-mod)|.

(b) If M is isomorphic to a constant functor, then it clearly sends all morphisms to isomorphisms, and sends a loop σ as above to a sequence whose composite is the identity. Thus for each c_{0} in C, the homomorphism M_{#} defined in (a) is trivial. It remains to prove the converse.

Assume that M is locally constant, and that for some object c_{0} in C, the homomorphism M_{#} defined in (a) is trivial. Set M_{C} = colim_{C}(M). We claim that the natural morphism ι_{c}: M(c) ⟶ M_{C} is an isomorphism for each object c. Once this has been shown, the ι_{c} define an isomorphism of functors from M to the constant functor with value M_{C}.

For each pair of objects c, d and each φ ∈ Mor_{C}(c, d), we have ι_{c} = ι_{d} ∘M(φ), where M(φ) is an isomorphism since M is locally constant. Thus Im(ι_{c}) = Im(ι_{d}) whenever Mor_{C}(c, d) ≠ ∅, and so Im(ι_{c}) = Im(ι_{d}) for each pair of objects c, d since C is connected. So ι_{c} is surjective for each c in C.

For each object d in C, since C is connected, there is a sequence

c_{0} ⟶^{f_{1}} c_{1} ⟵^{f_{2}} c_{2} ⟶^{f_{3}} ⋯ ⟵^{f_{2m}} c_{2m} = d

(m ≥ 1) of morphisms in C connecting c_{0} to d. Set

η_{d} = M(f_{1})^{−1} ∘ M(f_{2}) ∘ ⋯ ∘ M(f_{2m−1})^{−1} ∘ M(f_{2m}): M(d) ⟶^{≅} M(c_{0}) .

Then η_{d} is independent of the choice of the f_{i} since M_{#} = 1. This independence of the choice of sequence of morphisms also implies that for each pair of objects d and d′ and each morphism φ ∈ Mor_{C}(d, d′), we have η_{d′} ∘ M(φ) = η_{d}. We thus get a natural morphism η: M_{C} ⟶ M(c_{0}) such that η ∘ ι_{d} = η_{d} for each d, and ι_{d} is injective for each d in C since η_{d} is. We already showed that ι_{d} is surjective for each d, so ι is a natural isomorphism of functors from M to the constant functor \underline{M_{C}}. □

The following description of certain projective RC-modules will be needed later.

Lemma 3.4. Let R be a commutative ring, and let C be a small category. For each object c in C, let F_{c}^{RC} be the RC-module that sends an object d to R(Mor_{C}(c, d)) (the free R-module with basis Mor_{C}(c, d)); and sends a morphism φ ∈ Mor_{C}(d, d′) to composition with φ. Then F_{c}^{RC} is projective, and for each RC-module M, evaluation at Id_{c} ∈ F_{c}^{RC}(c) defines a bijection Mor_{RC}(F_{c}^{RC}, M) ≅ M(c).

Proof. The bijection Mor_{RC}(F_{c}^{RC}, M) ≅ M(c) is clear. In particular, Mor_{RC}(F_{c}^{RC}, −) is an exact functor, and so F_{c}^{RC} is projective. □

We now restrict further to two different cases: one where θ: C ⟶ D is bijective on objects, and the second where D is the category of a group. In each of these cases, we are able to get much more precise results about the existence of Ω-resolutions. In both cases, for a commutative ring R, we call a group G R-perfect if H_{1}(G; R) = 0; i.e., if (G/[G, G])⊗_{ℤ}R = 0.

3.1. Functors bijective on objects.

We begin with the case where θ is bijective on objects. When R is a commutative ring and one additional technical assumption holds, we can say quite precisely in which cases there always exist Ω-resolutions.

Proposition 3.5. Fix a commutative ring R. Let θ: C ⟶ D be a functor between small categories that is bijective on objects and surjective on morphism sets. Then

(a) (RC-mod, RD-mod; θ_{∗}, θ^{∗}) is an Ω-system, and the subcategory θ^{∗}(RD-mod) is closed under subobjects in RC-mod.

For each object c in C, set

K_{c} = Ker[θ_{c}: Aut_{C}(c) ⟶ Aut_{D}(θ(c))],

and assume that θ has the following property:

for each pair of objects c, c′ in C, and each pair of morphisms φ, φ′ ∈ Mor_{C}(c, c′)

such that θ_{c,c′}(φ) = θ_{c,c′}(φ′), there is some α ∈ K_{c′} such that φ = αφ′.

(2)

Then the following hold.

(b) If K_{c} is R-perfect for each c ∈ Ob(C), then θ^{∗}(RD-mod) is closed under extensions, and hence all projectives in RD-mod have Ω-resolutions.

(c) If K_{c} is not R-perfect for some c ∈ Ob(C), then θ^{∗}(RD-mod) is not closed under extensions, and there is a projective object X in RD-mod that does not have an Ωresolution.

Proof. (a) Since θ is surjective on objects and morphisms, it is quasisurjective. Since it is bijective on objects, the condition on I(θ↓d) in Proposition 3.1 holds for all objects d in D, and so (RC-mod, RD-mod; θ_{∗}, θ^{∗}) is an Ω-system by that proposition.

Since θ is bijective on objects and surjective on morphisms, an RC-module M is isomorphic to an object in θ^{∗}(RD-mod) if and only if it has the following property: if φ, ψ ∈ Mor_{C}(c, c′) are such that θ(φ) = θ(ψ) (some c, c′ ∈ Ob(C)), then M_{c,c′}(φ) = M_{c,c′}(ψ). In particular, θ^{∗}(RD-mod) is closed under subobjects.

(b,c) Now assume that (2) holds. For each RC-module M, let M_{K} be the RD-module defined by setting, for each d ∈ Ob(D) and c ∈ θ^{−1}(d),

(M_{K})(d) ≅ M(c)_{K_{c}} =^{def} M(c)/⟨α(x) − x | x ∈ M(c), α ∈ K_{c}⟩.

For each morphism φ ∈ Mor_{C}(c, c′) and each α ∈ K_{c}, θ_{c,c′}(φ ∘ α) = θ_{c,c′}(φ), so by (2), there is β ∈ K_{c′} such that φ ∘α = β ∘φ. Hence for each x ∈ M(c), φ_{∗}(x) and φ_{∗}(α(x)) are in the same orbit of K_{c′}. It follows that φ_{∗} ∈ Mor_{R}(M(c), M(c′)) induces a homomorphism between the quotient modules M_{K}(c) and M_{K}(c′). So by (2) and since θ is surjective on morphisms, there is a unique functor M_{K} on D such that the natural surjections M(c) ⟶ M_{K}(θ(c)) define a morphism of RC-modules M ⟶ θ^{∗}(M_{K}), and hence a morphism of RD-modules θ_{∗}(M) ⟶ M_{K}.

By (2) and the surjectivity of θ again, we have a natural bijection Mor_{RC}(M, θ^{∗}N) ≅ Mor_{RD}(M_{K}, N) for each RC-module M and each RD-module N, and thus M_{K} ≅ θ_{∗}(M). We have now shown that

for each RC-module M and each c ∈ Ob(C), the natural morphism

𝔞_{M}: M(c) ⟶ (θ^{∗}θ_{∗}M)(c) = (θ_{∗}M)(θ(c))

induces an isomorphism M(c)_{K_{c}} ≅ (θ_{∗}M)(θ(c)).

(3)

(b) Assume K_{c} is R-perfect for each c ∈ Ob(C). Let 0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0 be an extension of RC-modules such that M′ and M′′ are in θ^{∗}(RD-mod). For each c ∈ Ob(C), K_{c} acts trivially on M′(c) and on M′′(c), and hence also acts trivially on M(c) by Lemma A.1. So M ≅ θ^{∗}(θ_{∗}(M)) by (3).

This proves that θ^{∗}(RD-mod) is closed under extensions in RC-mod, and hence by Proposition 2.8 that Ω-resolutions exist of all projectives in RD-mod.

(c) Assume, for some object c_{0} in C, that K_{c_{0}} is not R-perfect, and set d_{0} = θ(c_{0}). Let F_{c_{0}}^{RC} and F_{d_{0}}^{RD} be the projective RC- and RD-modules defined in Lemma 3.4; thus

F_{c_{0}}^{RC}(c) = R(Mor_{C}(c_{0}, c)) and F_{d_{0}}^{RD}(d) = R(Mor_{D}(d_{0}, d))

for each c ∈ Ob(C) and d ∈ Ob(D). Since θ is surjective on morphisms, there is a natural surjection of RC-modules χ: F_{c_{0}}^{RC} ⟶ θ^{∗}(F_{d_{0}}^{RD}) that sends φ ∈ F_{c_{0}}^{RC}(c) to θ(φ) ∈ θ^{∗}(F_{d_{0}}^{RD})(c) = F_{d_{0}}^{RD}(θ(c)).

Set Q_{0} = Ker(χ), and consider the exact sequence

0 ⟶ (L_{1}θ_{∗})(θ^{∗}(F_{d_{0}}^{RD})) ⟶ θ_{∗}(Q_{0}) ⟶ θ_{∗}(F_{c_{0}}^{RC}) ⟶ F_{d_{0}}^{RD} ⟶ 0.

Here, θ_{∗}(F_{c_{0}}^{RC}) ≅ F_{d_{0}}^{RD} by (3) and since Mor_{C}(c_{0}, c)/K_{c} ≅ Mor_{D}(d_{0}, θ(c)) for each c in C by (2). Thus (L_{1}θ_{∗})(θ^{∗}F_{d_{0}}^{RD}) ≅ θ_{∗}(Q_{0}). We will show that θ_{∗}(Q_{0})(d_{0}) ≠ 0; then (L_{1}θ_{∗})(θ^{∗}F_{d_{0}}^{RD}) ≠ 0, so F_{d_{0}}^{RD} has no Ω-resolution by Proposition 2.8, and θ^{∗}(RD-mod) is not closed under extensions by Lemma 2.5(b).

Set End_{C}^{(1)}(c_{0}) = Aut_{C}(c_{0}), and let End_{C}^{(2)}(c_{0}) be its complement (as a set) in End_{C}(c_{0}). Set End_{D}^{(i)}(d_{0}) = θ_{c_{0}}(End_{C}^{(i)}(c_{0})) for i = 1, 2. Thus

End_{C}(c_{0}) = End_{C}^{(1)}(c_{0}) ⨿ End_{C}^{(2)}(c_{0}) and End_{D}(d_{0}) = End_{D}^{(1)}(d_{0}) ⨿ End_{D}^{(2)}(d_{0}) :

the first by definition, and the second by (2) and since θ_{c_{0}} is surjective. Set U_{C}^{(i)} = R(End_{C}^{(i)}(c_{0})) and U_{D}^{(i)} = R(End_{D}^{(i)}(d_{0})) (i = 1, 2), so that F_{c_{0}}^{RC}(c_{0}) = U_{C}^{(1)} ⊕ U_{C}^{(2)} and F_{d_{0}}^{RD}(d_{0}) = U_{D}^{(1)} ⊕ U_{D}^{(2)}. Thus Q_{0}(c_{0}) = Q_{0}^{(1)} ⊕ Q_{0}^{(2)} where Q_{0}^{(i)} is the kernel of the surjection U_{C}^{(i)} ⟶ U_{D}^{(i)}.

By (3), we must show that Q_{0}(c_{0})_{Kc_{0}} ≅ θ_{∗}(Q_{0})(d_{0}) ≠ 0, and to do this, it suffices to show that (Q_{0}^{(1)})_{Kc_{0}} ≠ 0. Set A = R[Aut_{C}(c_{0})] ≅ U_{C}^{(1)}, and identify it with the group ring. We can also identify Q_{0}^{(1)} = I: the 2-sided ideal in A generated as an R-module by the elements g − h for g, h ∈ Aut_{C}(c_{0}) such that gh^{−1} ∈ K_{c_{0}}. Then X_{K_{c_{0}}} = X/IX for each A-module X. In particular, (Q_{0}^{(1)})_{Kc_{0}} ≅ I/I^{2}.

Consider the short exact sequence 0 → I → A → A/I → 0 of R[K_{c_{0}}]-modules. Since A is projective, this induces an isomorphism I/I^{2} ≅ H_{1}(K_{c_{0}}; A/I). Since K_{c_{0}} is not R-perfect and acts trivially on the free R-module A/I ≅ R[Aut_{D}(d_{0})], we now conclude that I/I^{2} ≠ 0. □

Example 3.6. In the situation of Proposition 3.5(c), there can also be nonzero RC-modules that do have Ω-resolutions. For example, fix a prime p, set R = 𝔽_{p}, and assume that Ob(C) = Ob(D) = {x, y}, where End_{D}(x) = End_{C}(y) = End_{D}(y) = {Id} and End_{C}(x) ≅ C_{p}, and each category has a unique morphism from x to y and none from y to x. Then the unique functor θ: C ⟶ D satisfies the hypotheses of Proposition 3.5, and K_{x} ≅ C_{p} is not R-perfect while K_{y} = 1 is. Set X = F_{y}^{RD}; then θ^{∗}(X) ≅ F_{y}^{RC} is projective as an RC-module, so X has as Ω-resolution the sequence 0 ⟶ F_{y}^{RC} ⟶^{ε} θ^{∗}(F_{y}^{RD}) ⟶ 0.

3.2. Categories over a group.

The other large family of examples we consider are those where D = B(π) for a group π.

Definition 3.7. A category over a group π consists of a pair (C, θ), where C is a nonempty small connected category, and θ: C ⟶ B(π) is a functor such that the homomorphism π_{1}(|C|) ⟶ π induced by θ is surjective.

For example, if G and π are groups, and θ: B(G) ⟶ B(π) is the functor induced by a surjective homomorphism G ⟶ π, then (B(G), θ) is a category over π.

As another example, one that helped motivate this work, let (S, F, L) be a p-local compact group as defined in [BLO]. Set π = π_{1}(|L|_{p}^{∧}). Then π is a finite p-group, and there is a natural functor θ: L ⟶ B(π) whose restriction to B(S) is surjective. It follows from properties of linking systems that (L, θ) is a category over π. We refer to the introduction to Section 5 for more details.

Lemma 3.8. Let R be a commutative ring, and let (C, θ) be a category over π. Then

(a) the overcategory θ↓∘_{π} is connected, (RC-mod, Rπ-mod; θ_{∗}, θ^{∗}) is an Ω-system, and the projection |θ↓∘_{π}| ⟶ |C| is a covering space with covering group π.

For an RC-module M,

(b) θ_{∗}(M) ≅ colim_{θ↓∘_{π}}(M); and

(c) M ≅ θ^{∗}(N) for some Rπ-module N if and only if M is locally constant on C and essentially constant on θ↓∘_{π}.

Proof. (a) Since π acts freely on |θ↓∘_{π}| with orbit space |C|, the projection to |C| is a covering space with covering group π. In particular, |θ↓∘_{π}| is connected since π_{1}(|C|) surjects onto π. Also, θ is quasisurjective since π_{1}(|C|) surjects onto π, and so (RC-mod, Rπ-mod; θ_{∗}, θ^{∗}) is an Ω-system by Proposition 3.1.

(b) By definition of left Kan extension, θ_{∗}(M) = colim_{θ↓∘_{π}}(M).

(c) Assume M is locally constant on C and essentially constant on θ↓∘_{π}. We claim that the natural morphism 𝔞_{M}: M ⟶ θ^{∗}θ_{∗}(M) is an isomorphism. This means showing, for each c in C, that the natural morphism from M(c) to θ_{∗}(M) is an isomorphism. By (b), this is equivalent to showing that the natural morphism ξ_{c}: M(c) ⟶ M_{θ↓∘_{π}} is an isomorphism for each c. But this holds since by assumption, the restriction of M to θ↓∘_{π} is isomorphic to a constant functor.

Conversely, if M ≅ θ^{∗}(N), then M is locally constant on C, and isomorphic to a constant functor on θ↓∘_{π}. □

Lemma 3.9. Let R be a commutative ring, let (C, θ) be a category over a group π, and set H = Ker[π_{1}(|θ|): π_{1}(|C|) ⟶ π].

(a) If H is R-perfect, then (L_{1}θ_{∗})(θ^{∗}X) = 0 for each Rπ-module X; and

(b) if H is not R-perfect, then (L_{1}θ_{∗})(θ^{∗}X) ≠ 0 for each Rπ-module X ≠ 0 that (as an R-module) contains R as a direct summand.

Proof. For each Rπ-module X, (L_{1}θ_{∗})(θ^{∗}X) ≅ (L_{1}(colim_{θ↓∘_{π}}))(θ^{∗}X) as R-modules by Lemma

3.8(b), and (L_{1}(colim_{θ↓∘_{π}}))(θ^{∗}X) ≅ H_{1}(|θ↓∘_{π}|; X) since the two sides are homology groups of

the same chain complex by [GZ, Appendix II, Proposition 3.3]. Here, the homology is with untwisted coefficients since H ≅ π_{1}(|θ↓∘_{π}|) (Lemma 3.8(a)) acts trivially on the Rπ-module X. Thus (L_{1}θ_{∗})(θ^{∗}X) = 0 if and only if H_{1}(|θ↓∘_{π}|; X) = 0. Points (a) and (b) now follow since H ≅ π_{1}(|θ↓∘_{π}|) is R-perfect if and only if H_{1}(|θ↓∘_{π}|; R) ≅ H_{1}(H; R) = 0. □

Since a category over a group π gives rise to an Ω-system, we can now work with Ωresolutions in this situation.

Lemma 3.10. Let (C, θ) be a category over a group π, and let R be a commutative ring. A complex of RC-modules

… ⟶ P_{2} ⟶^{∂_{2}} P_{1} ⟶^{∂_{1}} P_{0} ⟶^{ε} θ^{∗}(Rπ) ⟶ 0

is an Ω-resolution of θ^{∗}(Rπ) with respect to the Ω-system (RC-mod, Rπ-mod; θ_{∗}, θ^{∗}) of Proposition 3.1 if and only if

(1) P_{n} is a projective RC-module for each n ≥ 0;

(2) the complex θ_{∗}(P_{∗}) is acyclic, and ε induces an isomorphism H_{0}(P_{∗})(c) ≅ Rπ for each c ∈ Ob(C); and

(3) for each n ≥ 0, H_{n}(P_{∗}) is locally constant on C and essentially constant on θ↓∘_{π}.

Proof. By Lemma 3.8(c), (3) is equivalent to the first statement in (Ω-3) (that H_{n}(P_{∗}, ∂_{∗}) is isomorphic to an object in θ^{∗}(Rπ-mod)). The equivalence of (1) with (Ω-1), and of (2) with (Ω-2) and the second part of (Ω-3) (that H_{0}(P_{∗}, ∂_{∗}) ≅ X), is clear. □

By Proposition 1.6, if R is a commutative ring and (C, θ) is a category over a group π, and there is at least one Ω-resolution of Rπ, then all Ω-resolutions are chain homotopy equivalent to each other. This allows us to define “Ω-homology” in this situation.

Definition 3.11. Let (C, θ) be a category over a group π. For a commutative ring R, if there is an Ω-resolution (P_{∗}, ∂_{∗}) of θ^{∗}(Rπ) with respect to (C, θ), then we define

H_{∗}^{Ω}(C, θ; R) = θ_{∗}(H_{∗}(P_{∗}, ∂_{∗})).

The following proposition is a first step towards determining for which categories over π the free module Rπ has an Ω-resolution. In the next section, we will show that Ωresolutions of Rπ do exist in all cases not excluded here. Recall that C is an EI-category if all endomorphisms of objects in C are automorphisms.

Proposition 3.12. Fix a commutative ring R. Let (C, θ) be a category over a group π, and set H = Ker[π_{1}(|θ|): π_{1}(|C|) ⟶ π]. Thus (RC-mod, Rπ-mod; θ_{∗}, θ^{∗}) is an Ω-system by Lemma 3.8(a).

(a) If H is R-perfect, then (L_{1}θ_{∗})(θ^{∗}(X)) = 0 for each X in Rπ-mod, and θ^{∗}(Rπ-mod) is closed under extensions in RC-mod. If C = B(G) for a group G, then θ^{∗}(Rπ-mod) is closed under subobjects in RC-mod, and each projective Rπ-module has an Ω-resolution. If C is an EI-category with more than one isomorphism class, then θ^{∗}(Rπ-mod) is not closed under subobjects in RC-mod.

(b) If H is not R-perfect, then θ^{∗}(Rπ-mod) is not closed under extensions in RC-mod, and the projective Rπ-module Rπ does not have an Ω-resolution. More generally, if X is a nonzero projective Rπ-module that is free as an R-module, then X has no Ω-resolution.

Proof. (a) Assume that H is R-perfect. Then (L_{1}θ_{∗})(θ^{∗}(X)) = 0 for each Rπ-module X by Lemma 3.9(a). Hence by Lemma 2.5(c), θ^{∗}(Rπ-mod) is closed under extensions in RC-mod. If C ≅ B(G) for some G, then θ is surjective on morphisms, H = Ker[G ⟶ π], and θ^{∗}(Rπ-mod) is closed under subobjects in RC-mod by Proposition 3.5(a).

Assume C is an EI-category with more than one isomorphism class, and let x, y ∈ Ob(C) be a pair of nonisomorphic objects. At least one of the sets Mor_{C}(x, y) and Mor_{C}(y, x) must be empty; we can assume that Mor_{C}(x, y) = ∅. Let \underline{R} be the constant RC-module with value R, and let M ≤ \underline{R} be the submodule where M(c) = 0 if Mor_{C}(x, c) = ∅ and M(c) = R otherwise. Then M(c) = R and M(c′) = 0 imply that Mor_{C}(c, c′) = ∅; thus M is well defined as a submodule of \underline{R}. Also, M(x) = R so M ≠ 0, and M(y) = 0 so M is properly contained in \underline{R}. Since C is connected, M is not locally constant, and hence not isomorphic to an object in θ^{∗}(Rπ-mod). So θ^{∗}(Rπ-mod) is not closed under subobjects in RC-mod.

(b) Fix an object c_{0} in C, and set G = π_{1}(|C|, c_{0}) for short. Let η: G ⟶ π be the homomorphism induced by |θ|: |C| ⟶ |B(π)| = Bπ. Thus η is surjective and H = Ker(η).

Assume H is not R-perfect. By Lemma 3.9(b), for each nonzero Rπ-module X that is free as an R-module, (L_{1}θ_{∗})(θ^{∗}X) ≠ 0. So X has no Ω-resolution by Proposition 2.6, and it remains to show that θ^{∗}(Rπ-mod) is not closed under extensions in RC-mod.

Set N_{0} = H^{ab} ⊗_{ℤ} R ≅ H_{1}(H; R), regarded as an R-module, and let χ: H ⟶ N_{0} be the homomorphism χ(h) = [h] ⊗ 1. Since H is not R-perfect, N_{0} ≠ 0, and χ is not the trivial homomorphism. Let M_{0} be the RH-module with underlying R-module N_{0} × N_{0}, where h ∈ H acts via the matrix ( \begin{matrix}1 & χ(h) \\ 0 & 1 \end{matrix} ). Thus there is a submodule M′_{0} = \{(x, 0) | x ∈ R\} ≤ M_{0} such that H acts trivially on M′_{0} and on M_{0}/M′_{0}.

Now set M = RG⊗_{RH} M_{0}. Thus M is an RG-module, and contains a submodule M′ such that M′ and M/M′ are both isomorphic to η^{∗}(Rπ).

We now use this to construct a counterexample to θ^{∗}(Rπ-mod) being closed under extensions. For each c ∈ Ob(C), choose a path ϕ̃_{c} in |θ↓∘_{π}| from (c_{0}, Id) to (c, Id) (|θ↓∘_{π}| is connected by Lemma 3.8(a)), and let ϕ_{c} be its image in |C|. In particular, let ϕ̃_{c_{0}} and ϕ_{c_{0}} be the constant paths at (c_{0}, Id) and c_{0}, respectively. Define a functor θ̃: C ⟶ B(G) by sending each object in C to the unique object ∘_{G}, and by sending each morphism ω ∈ Mor_{C}(c, c′) to the class of the loop ϕ_{c}·ω·ϕ_{c′}^{−1} (where we compose paths from left to right). We claim that

(i) π_{1}(|θ̃|): π_{1}(|C|, c_{0}) ⟶ π_{1}(B(G), ∘_{G}) = G is the identity on G; and

(ii) θ = B(η) ∘ θ̃.

Point (i) is immediate from the definition of θ̃ (and since ϕ_{c_{0}} is the constant path). Point (ii) holds since the paths ϕ_{c} all lift to |θ↓∘_{π}| and hence are sent to trivial loops in B(π), and since η: G ⟶ π is induced by θ.

Now, θ̃^{∗}(M) is an RC-module with submodule θ̃^{∗}(M′), such that by (ii),

θ̃^{∗}(M′) ≅ θ̃^{∗}(η^{∗}(Rπ)) ≅ θ^{∗}(Rπ) and θ̃^{∗}(M)/θ̃^{∗}(M′) ≅ θ̃^{∗}(M/M′) ≅ θ^{∗}(Rπ).

Thus θ̃^{∗}(M′) and θ̃^{∗}(M)/θ̃^{∗}(M′) are both isomorphic to objects in θ^{∗}(Rπ-mod). As for θ̃^{∗}(M), by (i), the homomorphism

(θ̃^{∗}(M))_{#} : G = π_{1}(|C|, c_{0}) ⟶ Aut_{R}(θ̃^{∗}(M)(c_{0})) = Aut_{R}(M)

of Lemma 3.3(a) is just the given action of G on the RG-module M. So its restriction to H = π_{1}(|θ↓∘_{π}|) is nontrivial, and by Lemma 3.3(b), θ̃^{∗}(M) is not essentially constant on θ↓∘_{π}. By Lemma 3.8(c), it is not isomorphic to an object in θ^{∗}(Rπ-mod), and thus θ^{∗}(Rπ-mod) is not closed under extensions in RC-mod. □

Note that Proposition 2.8 does not apply under the hypotheses of Proposition 3.12(a), although at least Ω_{1}-resolutions exist by Proposition 2.6. If C is the category of a group, then Ω-resolutions always exist by Proposition 3.5(b). We will show in Theorem 4.4 that in fact, Ω-resolutions of Rπ always exist in under the hypotheses of Proposition 3.12(a).

Example 3.13. In the situation of Proposition 3.12(a), if C is not an EI-category, then θ^{∗}(Rπ-mod) can fail to be closed under subobjects even when C has only one object, and can be closed under subobjects even when C has more than one isomorphism class of object:

(a) Set R = ℤ, π = ℤ, and C = B(ℕ), and let θ: B(ℕ) ⟶ B(π) be the inclusion. Then (C, θ) is a category over π. Let N be the Rπ-module with underlying group ℚ, where π = ℤ acts via n(x) = 2^{n}x. Let M be the Rℕ-module with underlying group ℤ, where n ∈ ℕ acts in the same way. Thus M is a submodule of θ^{∗}(N), but is not isomorphic to an object in θ^{∗}(Rπ-mod).

(b) Let C be a category with two objects x and y, where End_{C}(x) = {0_{x}, 1_{x}}, End_{C}(y) = {0_{y}, 1_{y}}, and there are unique morphisms 0_{xy} ∈ Mor_{C}(x, y) and 0_{yx} ∈ Mor_{C}(y, x). Composition is defined by multiplication of the labels 0 or 1. Set π = ℤ, and let θ: C ⟶ B(ℤ) be the functor that sends all endomorphisms to 0 and the other two morphisms to 1 and −1, respectively. Via generators and relations, one checks that θ induces an isomorphism π_{1}(|C|) ≅ ℤ. We are thus in the situation of Proposition 3.12(a) with H = 1. An RC-module M is isomorphic to an object in θ^{∗}(Rπ-mod) if and only if all endomorphisms induce the identity, in which case the other two morphisms induce inverse isomorphisms between M(x) and M(y). So θ^{∗}(Rπ-mod) is closed under subobjects in this case.

4. Homology of loop spaces of categories over groups

We next show, in the situation of Proposition 3.12(a), that Ω-resolutions of Rπ with respect to (C, θ) do exist, and that their homology is the R-homology of a certain loop space (Theorem 4.4). For example, when k is a field of characteristic p for some prime p and π is a finite p-group, the homology of the Ω-resolution is isomorphic to H_{∗}(Ω(|C|_{p}^{∧}); k) (Theorem 4.7).

Throughout this section, we work mostly with simplicial sets and their realizations, referring to [GJ, Chapter I] and [Cu] for the definitions and basic properties that we use. In particular, Kan fibrations of simplicial sets (called “fibre maps” by Curtis) play an important role here, and we refer to [GJ, § I.3] and [Cu, Definition 2.5] for their definitions. We let |K| denote the geometric realization of a simplicial set K, let C_{∗}(K) denote its simplicial chain complex, and write H_{∗}(K) = H_{∗}(C_{∗}(K)) (≅ H_{∗}(|K|)). Thus |C| = |N (C)| when C is a small category and N (C) is its nerve. Note that if f: E ⟶ K is a Kan fibration and μ: L ⟶ K is a simplicial map, then the pullback of f along μ is also a Kan fibration.

For a small category C, a C-diagram of simplicial sets is a functor from C to simplicial sets, and a morphism of C-diagrams is a natural transformation of such functors. Let \underline{K} denote the constant C-diagram that sends each object to the simplicial set K, and let \underline{f}: \underline{K} ⟶ \underline{L} denote the morphism induced by a map f: K ⟶ L of simplicial sets.

Let EC denote the C-diagram of simplicial sets where EC(c) = N (Id_{C}↓c), and a morphism φ in C induces a map between spaces EC(−) by composition with φ. Then |EC(c)| is contractible for each c in C, so |EC| is the “C-CW-approximation” of the trivial (point) Cspace in the sense of [DL, Definitions 3.6 and 3.8]. The forgetful functors Id_{C}↓c → C induce a natural transformation η: EC → \underline{N (C)}.

For each Kan fibration f: K ⟶ N (C), let μ: E_{f} ⟶ EC denote the pullback of K along η. Thus E_{f} is the C-diagram of simplicial sets that sends an object c in C to the pullback E_{f}(c) of the system

K ⟶^{f} N (C) ⟵^{η_{c}} EC(c).

Lemma 4.1. Fix a commutative ring R and a small category C, and let f: K → N (C) be a Kan fibration. Then for each n ≥ 0, the RC-module C_{n}(E_{f}; R) is projective, and the morphism ω: E_{f} ⟶ \underline{K} induces an isomorphism colim_{C}(C_{∗}(E_{f}; R)) ≅ C_{∗}(K; R).

Proof. For each n ≥ 0 and each object c ∈ C, C_{n}(EC; R)(c) has as basis the set of all chains (c_{0} → c_{1} → ⋯ → c_{n} → c). So in the notation of Lemma 3.4, the RC-module C_{n}(EC; R) is the direct sum of one copy of F_{c_{n}}^{RC} for each n-simplex (c_{0} → c_{1} → ⋯ → c_{n}) in N (C). In particular, it is projective, and since colim_{C}(F_{c_{n}}^{RC}) ≅ R, the natural transformation η: EC ⟶ \underline{N (C)} induces an isomorphism colim_{C}(C_{n}(EC; R)) ≅ C_{n}(N (C); R).

This proves the lemma when f is the identity fibration, and the general case is similar. An n-simplex in the pullback E_{f}(c) is a pair (σ, c_{0} → ⋯ → c_{n} → c) where σ ∈ K_{n} is such that f(σ) = (c_{0} → ⋯ → c_{n}). Hence the RC-module C_{n}(E_{f}; R) is the direct sum of copies of F_{c_{n}}^{RC}, one for each pair (σ, c_{0} → ⋯ → c_{n}) as above, hence is projective, and

colim_{C}(C_{n}(E_{f}; R)) ≅ C_{n}(K; R). □

We next define plus constructions, which play an important role in this section.

Definition 4.2. Fix a commutative ring R, a connected CW complex X, and a normal subgroup H ⊴ π_{1}(X). An R-plus construction for (X, H) consists of a CW complex X_{R}^{+} together with a map κ: X ⟶ X_{R}^{+}, such that π_{1}(κ) is surjective with kernel H, and H_{∗}(κ; N) is an isomorphism for each R[π_{1}(X)/H]-module N.

A few results about R-plus constructions are collected in the appendix. For example, we show there that (X, H) has an R-plus construction if and only if H is R-perfect (i.e., H_{1}(H; R) = 0), and that the R-completion of a space in the sense of Bousfield and Kan is an R-plus construction under certain hypotheses.

For n ≥ 0, let Δ^{n} denote the n-simplex as a simplicial set, and let v_{0}, …, v_{n} be its vertices. For 0 ≤ k ≤ n, let Λ_{k}^{n} ⊆ Δ^{n} be the simplicial subset whose realization is the union of all proper (closed) faces in Δ^{n} containing v_{k}. Thus a Kan fibration is a simplicial map f: K ⟶ L with the following lifting property: for each 0 ≤ k ≤ n, each σ: Δ^{n} ⟶ L, and each τ: Λ_{k}^{n} ⟶ K such that f ∘ τ = σ|_{Λ_{k}^{n}}, there is a simplicial map σ̃: Δ^{n} ⟶ K such that σ̃|_{Λ_{k}^{n}} = τ and f ∘ σ̃ = σ. A Kan complex is a simplicial set K for which the (unique) map to Δ^{0} is a Kan fibration; equivalently, a simplicial set for which each simplicial map Λ_{k}^{n} ⟶ K extends to Δ^{n} (see [GJ, § I.3], [Cu, Definition 1.12], or [GZ, § IV.3]). For example, for each space X, the singular simplicial set S.(X) is a Kan complex [GJ, Lemma I.3.3].

For any connected simplicial set K with basepoint x_{0} ∈ K_{0}, let P(K) = P(K, x_{0}) be the simplicial set of paths in K based at x_{0}. Thus an n-simplex in P(K) is a map of simplicial sets Δ^{1} ×Δ^{n} ⟶ K that sends {v_{0}}×Δ^{n} to x_{0} (more precisely, to the image of x_{0} under the degeneracy map K_{0} ⟶ K_{n}). Let e = e_{K}: P(K) ⟶ K denote the path-loop fibration over K: the simplicial map that sends an n-simplex Δ^{1} × Δ^{n} ⟶ K to the image of {v_{1}} × Δ^{n}. If K is a Kan complex, then e_{K}: P(K) ⟶ K is a Kan fibration and |P(K)| is weakly contractible (see [GJ, Lemma I.7.5]). Thus the fibre of e_{K} over x_{0} is the loop simplicial set Ω(K, x_{0}) based at x_{0} [GJ, p. 31]. Using the fact that the realization of a Kan fibration is a Serre fibration (see [GJ, Theorem I.10.10]), one can show that |Ω(K, x_{0})| is weakly equivalent to Ω(|K|, x_{0}).

If f: K ⟶ L is a Kan fibration, and χ: L̂ ⟶ L is an arbitrary simplicial map, then the pullback f̂: K̂ ⟶ L̂ is defined levelwise: K̂_{n} is the pullback (as a set) of f_{n}: K_{n} ⟶ L_{n} along χ_{n}: L̂_{n} ⟶ L_{n}. It is immediate from the definitions that f̂ is also a Kan fibration. By [GZ, Theorem III.3.1], pullbacks commute with geometric realization; i.e., |K̂| is the pullback of |K| ⟶ |L| along |L̂|. Note, however, that this requires that the pullbacks of realizations be taken in the category of compactly generated Hausdorff spaces (called “Kelley spaces” in [GZ]).

Proposition 4.3. Fix a commutative ring R and a group π, let (C, θ) be a category over π,

and set H = Ker[π_{1}(|C|) ⟶^{π_{1}(|θ|)} π]. Assume that H is R-perfect, let κ: |C| ⟶ |C|_{R}^{+} be an R-plus construction for (|C|, H), and let κ̂: N (C) ⟶ S.(|C|_{R}^{+}) be the simplicial map adjoint to κ. Fix an object c_{0} in C, regarded as a vertex in N (C), set x_{0} = κ̂(c_{0}), and let e = e_{S.(|C|_{R}^{+})} be the path-loop fibration over S.(|C|_{R}^{+}) based at x_{0}. Let ν: AC ⟶ N (C) be the pullback of e along κ̂, and let μ: E_{ν} ⟶ EC denote the fibration of C-diagrams of simplicial sets obtained as the pullback of ν along η. We thus have, for each object c in C, the following diagram of simplicial sets with pullback squares

\xymatrix{E_{ν}(c) \ar[r] \ar[d]^{μ_{c}} & AC \ar[d]^{ν} \ar[r] & P(S.(|C|_{R}^{+}), x_{0}) \ar[d]^{e} \\ EC(c) \ar[r]^{η_{c}} & N (C) \ar[r]^{κ̂} & S.(|C|_{R}^{+}) .}  (1)

Then the following hold.

(a) For each n ≥ 0, C_{n}(E_{ν}; R) is a projective RC-module.

(b) The complex θ_{∗}(C_{∗}(E_{ν}; R)) ≅ colim_{θ↓∘_{π}}(C_{∗}(E_{ν}; R)) is acyclic, and ε induces an isomor-

phism H_{0}(θ_{∗}(C_{∗}(E_{ν}; R))) ≅ Rπ.

(c) For each n ≥ 0, H_{n}(E_{ν}; R) is locally constant on C and essentially constant on θ↓∘_{π}.

(d) For each object c in C, |E_{ν}(c)| is weakly equivalent to Ω(|C|_{R}^{+}).

In particular, by (a)–(c), C_{∗}(E_{ν}; R) is an Ω-resolution of Rπ with respect to (C, θ).

Proof. We write C_{∗}(−) = C_{∗}(−; R) and H_{∗}(−) = H_{∗}(−; R) for short, and refer to diagram (1), where by construction, μ_{c}, ν, and e are all Kan fibrations with fibre Ω(S.(|C|_{R}^{+})) ≅ S.(Ω(|C|_{R}^{+})). Then AC is R-acyclic since P(S.(|C|_{R}^{+}), x_{0}) is contractible and H_{∗}(κ̂; N) is an isomorphism for each Rπ-module N. Point (a) follows from Lemma 4.1, applied with AC and ν in the roles of K and f, and point (d) holds since each EC(c) is the nerve of a category with final object and hence contractible.

Let σ: θ↓∘_{π} ⟶ C be the forgetful functor, and define ÃC, ν̃, and Eν̃ as pullbacks in the following diagram (for each object (c, g) in θ↓∘_{π}):

[figure: Eν̃(c, g); ≅; E_{ν}(c); μ_{c}; ÃC; ν̃; AC; ν; E(θ↓∘_{π})(c, g); Eσ_{(c,g)}; ≅; EC(c); η_{c}; N (θ↓∘_{π}); N (σ); N (C)]

Note that Eσ_{(c,g)} is an isomorphism of simplicial sets since for each morphism φ ∈ Mor_{C}(c′, c) and each g ∈ π, there is a unique g′ ∈ π such that φ ∈ Mor_{θ↓∘_{π}}((c′, g′), (c, g)). Hence Eν̃(c, g) ≅ E_{ν}(c). So by Lemma 3.8(b), and Lemma 4.1 applied with θ↓∘_{π} in the role of C,

H_{∗}(θ_{∗}(C_{∗}(E_{ν}))) ≅ H_{∗}(colim_{θ↓∘_{π}} σ^{∗}(C_{∗}(E_{ν}))) ≅ H_{∗}(colim_{θ↓∘_{π}} C_{∗}(Eν̃)) ≅ H_{∗}(C_{∗}(ÃC)) ≅ H_{∗}(ÃC) .

But |θ↓∘_{π}| is the covering space of |C| with fundamental group H and covering group π (Lemma 3.8(a)), the image of π_{1}(|AC|) in π_{1}(|C|) is contained in H = Ker(π_{1}(|κ|)) since it vanishes in π_{1}(|C|_{R}^{+}), and hence |ÃC| ≅ π × |AC|. Since |AC| is R-acyclic, this proves (b): θ_{∗}(C_{∗}(E_{ν})) is acyclic and H_{0}(θ_{∗}(C_{∗}(E_{ν}))) ≅ Rπ.

For each object c in C, let F (c) = ν^{−1}(c) be the fibre of ν over the vertex c in N (C). Via homotopy lifting, this is extended to a homotopy functor F from C to simplicial sets, and this in turn defines a locally constant graded RC-module M_{∗} = H_{∗}(F). For each c in C, the action of π_{1}(|C|, c) on M_{∗}(c) = H_{∗}(F (c)) described in Lemma 3.3(a) is the usual action of the fundamental group of the base on the homology of a fibre, and since ν is a pullback of e, this action factors through π_{1}(|C|_{R}^{+}) ≅ π. So M_{∗} is essentially constant on θ↓∘_{π} by Lemma 3.3(b). Also, since each EC(c) contracts to the vertex (c, Id_{c}) in a natural way, where η_{c}(c, Id_{c}) = c, we have homotopy equivalences E_{ν}(c) ≃ F (c) natural in C up to homotopy. So H_{∗}(E_{ν}) ≅ M_{∗} as RC-modules, and this proves (c).

Thus by Lemma 3.8(c), H_{∗}(E_{ν}) ≅ θ^{∗}(N_{∗}) for some graded Rπ-module N_{∗}. Since θ_{∗} is right exact,

N_{0} ≅ θ_{∗}θ^{∗}(N_{0}) ≅ θ_{∗}(H_{0}(E_{ν})) ≅ H_{0}(θ_{∗}(C_{∗}(E_{ν}))) ≅ Rπ,

and so H_{0}(E_{ν}) ≅ θ^{∗}(Rπ). This defines a surjective homomorphism ε: C_{0}(E_{ν}) ⟶ θ^{∗}(Rπ), and finishes the proof that (C_{∗}(E_{ν}), ∂_{∗}) ⟶ θ^{∗}(Rπ) ⟶ 0 is an Ω-resolution of Rπ. □

Upon combining Proposition 4.3 with Lemma A.3, we get the following theorem.

Theorem 4.4. Fix a commutative ring R and a group π. Let (C, θ) be a category over π,

set H = Ker[π_{1}(|C|) ⟶^{π_{1}(|θ|)} π], and assume that H is R-perfect. Then

(a) (|C|, H) admits an R-plus construction;

(b) the free Rπ-module Rπ has Ω-resolutions with respect to (C, θ); and

(c) for each R-plus construction |C|_{R}^{+} for (|C|, H), H_{∗}^{Ω}(C, θ; R) ≅ H_{∗}(Ω(|C|_{R}^{+}); R).

In the special case where π ≅ π_{1}(|C|), this takes the form:

Corollary 4.5. Let C be a small, connected category, and set π = π_{1}(|C|). Then there is a functor θ: C ⟶ B(π) such that π_{1}(|θ|) is an isomorphism. For such θ, the 4-tuple (RC-mod, Rπ-mod; θ_{∗}, θ^{∗}) is an Ω-system, the free module ℤπ has an Ω-resolution with respect to (RC-mod, Rπ-mod; θ_{∗}, θ^{∗}), and

H_{∗}^{Ω}(C, θ; ℤ) ≅ H_{∗}(Ω(|C|); ℤ).

It is not hard to find examples of small categories that are not groupoids, but whose geometric realizations are aspherical. This leads to the following curious situation.

Remark 4.6. Let θ: C ⟶ B(π) be as in Corollary 4.5, and assume in addition that θ induces a homotopy equivalence |C| ≃ Bπ but is not an equivalence of categories. Then H_{i}^{Ω}(C, θ; ℤ) = 0 for i > 0 by the corollary, and hence each Ω-resolution of θ^{∗}(ℤπ) is a projective resolution in the usual sense in the category ℤC. Equivalently, by condition (Ω-2) in Definition 1.5, (L_{i}θ_{∗})(ℤπ) = 0 for each i > 0.

The R-plus construction of (N (C), H) as defined in Definition 4.2 is not in general unique, not even up to homotopy. However, in certain cases, we can choose it to be a completion or a fibrewise completion of |C| in the sense of Bousfield and Kan. Recall [BK, III.5.1] that for R ⊆ ℚ, a group π is R-nilpotent if it has a central series for which each quotient is an R-module.

Theorem 4.7. Let (C, θ) be a category over a group π, and set H = Ker[π_{1}(|C|) ⟶^{π_{1}(|θ|)} π].

(a) Assume that R is a subring of ℚ or R = 𝔽_{p} for some prime p, and that H is R-perfect. Let |C|^{∧} be the fibrewise R-completion of |C| over Bπ. Then

H_{∗}^{Ω}(C, θ; R) ≅ H_{∗}(Ω(|C|^{∧}); R).

(b) If R ⊆ ℚ is such that H is R-perfect, and π is R-nilpotent with nilpotent action on H_{i}(|θ↓∘_{π}|; R) for each i, then

H_{∗}^{Ω}(C, θ; R) ≅ H_{∗}(Ω(|C|_{R}^{∧}); R),

where |C|_{R}^{∧} is the R-completion of |C|.

(c) If for some prime p, k is a field of characteristic p, π is a finite p-group, and H is

p-perfect, then

H_{∗}^{Ω}(C, θ; k) ≅ H_{∗}(Ω(|C|_{p}^{∧}); k)

where |C|_{p}^{∧} is the p-completion of |C|.

Proof. By Lemma A.4, the natural map from |C| to |C|^{∧}, |C|_{R}^{∧}, or |C|_{p}^{∧} is an R- or k-plus construction for (|C|, H) under the hypotheses of (a), (b), or (c), respectively. So this theorem follows as a special case of Theorem 4.4(c). □

The following corollary includes the case proven by Benson in [Be2]. Note that when G is a finite group, its quotient by the maximal normal p-perfect subgroup is always a p-group.

Corollary 4.8. Fix a prime p. Let G be a (possibly infinite) discrete group, and let O^{p}(G) be the maximal normal p-perfect subgroup of G. Set π = G/O^{p}(G), let χ: G ⟶ π be the natural surjection, and assume that π is a finite p-group. Then for each field k of characteristic p, H_{∗}(Ω(BG_{p}^{∧}); k) ≅ H_{∗}^{Ω}(B(G), B(χ); k): the homology of an Ω-resolution of kπ with respect to (B(G), B(χ)) as a category over π.

Proof. This is just Theorem 4.7(c) when C = B(G). □

The results in this section lead in a natural way to the following question.

Question 4.9. Are there more general conditions on an Ω-system (A, B; θ_{∗}, θ^{∗}) and X ∈ 𝒫(B) under which H_{∗}^{Ω}(A, B; X), or a functorial image, describes the homology of a space (e.g., of a loop space)? In particular, can the homology of the Ω-resolutions of Proposition 3.5(b) be realized as the homology of some space determined by the Ω-systems?

5. Examples: Ω-resolutions for some p-local compact groups

One problem that motivated this work was that of finding a way to characterize the p-compact groups among the more general p-local compact groups. As already noted in the introduction, we did not succeed in doing so. The aim of this section is to give some very simple examples that demonstrate how complicated this problem can be, for example, by analyzing some p-local compact groups that are not p-compact. We also give some results, and one explicit computation, that follow from knowing that Ω-resolutions determine homology of loop spaces without having to explicitly construct the resolutions themselves.

Throughout this section, we fix a prime p and a field k of characteristic p. We first recall some definitions. A p-compact group consists of a loop space X and its classifying space BX, such that X ≃ Ω(BX), H_{∗}(X; 𝔽_{p}) is finite (in particular, H_{n}(X; 𝔽_{p}) = 0 for n large enough), and BX is p-complete. This concept was first introduced by Dwyer and Wilkerson [DW], and developed by them and others in several papers. If G is a compact Lie group whose group of components π_{0}(G) is a p-group, then Ω(BG_{p}^{∧}) is a p-compact group, but this need not be the case if π_{0}(G) is not a p-group. Every p-compact group contains a maximal torus with properties very similar to those of maximal tori in compact Lie groups.

A p-local compact group consists of a discrete p-toral group S (i.e., an extension of a discrete p-torus (ℤ/p^{∞})^{r} for some r ≥ 0 by a finite p-group), together with a fusion system F over S and a linking system L associated to F. We refer to [BLO, Definitions 2.2 and 4.1] for the precise definitions of fusion and linking systems in this context; here, we just note that F and L are categories, Ob(F) is the set of subgroups of S, each morphism in F is a homomorphism between subgroups, and there is a functor L ⟶ F that is an inclusion on objects and surjective on each morphism set. The classifying space of such a triple (S, F, L) is the p-completed space |L|_{p}^{∧}. By [BLO, §§ 9–10], each compact Lie group G or p-compact group X has a maximal discrete p-toral subgroup S (unique up to conjugacy), together with a fusion system F and a linking system L such that |L|_{p}^{∧} is homotopy equivalent to BG_{p}^{∧} or BX, respectively.

By [BLO, Proposition 4.4], for each p-local compact group (S, F, L), the fundamental group of the classifying space |L|_{p}^{∧} is a finite p-group. So as a special case of Theorem 4.7(c), we get:

Theorem 5.1. Let (S, F, L) be a p-local compact group, and set π = π_{1}(|L|_{p}^{∧}). Then there is θ: L ⟶ B(π) such that (L, θ) is a category over π and

H_{∗}(Ω(|L|_{p}^{∧}); 𝔽_{p}) ≅ H_{∗}^{Ω}(L, θ; 𝔽_{p}).

If Γ is an extension of a discrete p-torus by a finite p-group, then BΓ _{p}^{∧} is the classifying space of a p-compact group. In contrast, if Γ is an extension of a discrete p-torus by an arbitrary finite group, then BΓ _{p}^{∧} need not be the classifying space of a p-compact group (nor the p-completion of BG for a compact Lie group G), but it is always the classifying space of a p-local compact group. For example, if p is an odd prime and r ≥ 2, and Γ = (ℤ/p^{∞})^{r} ⋊C_{2} where C_{2} inverts (ℤ/p^{∞})^{r}, then Ω(BΓ _{p}^{∧}) is not a p-compact group since its mod p homology is nonvanishing in arbitrarily large degrees (see Example 5.10 for the case r = 2).

What we want to do now is to give some explicit examples of such Ω-resolutions. We focus on p-local compact groups associated to extensions of discrete p-tori by finite groups, especially by those of order prime to p.

Proposition 5.2. Let T ⊴ Γ be a pair of groups such that T ≅ (ℤ/p^{∞})^{r} for some r ≥ 1 and Γ/T is finite. Then there is a p-local compact group (S, F, L) associated to Γ, where T ≤ S ≤ Γ and S/T ∈ Syl_{p}(Γ/T), and where |L|_{p}^{∧} ≃ BΓ _{p}^{∧}. If, in addition, Ω(BΓ _{p}^{∧}) is a p-compact group, then O^{p}(Γ/T) has order prime to p.

Proof. Embed T in GL_{r}(ℂ) as the subgroup of diagonal matrices of p-power order. Then via induction, Γ embeds as a subgroup of GL_{r·|Γ/T |}(ℂ), and hence is a linear torsion group in the sense of [BLO, § 8]. So by [BLO, Theorem 8.10], it has an associated p-local compact group (S, F, L), where T ≤ S ≤ Γ, S/T ∈ Syl_{p}(Γ/T), and BΓ _{p}^{∧} ≃ |L|_{p}^{∧}. Also, F = F_{S}(Γ): the fusion system over S whose morphisms are those homomorphisms between subgroups of S induced by conjugation in Γ.

Now assume that Ω(BΓ _{p}^{∧}) is a p-compact group; i.e., that H_{∗}(Ω(BΓ _{p}^{∧}); 𝔽_{p}) is finite. Let O^{p}(Γ) ⊴ Γ be such that O^{p}(Γ) ≥ T and O^{p}(Γ)/T = O^{p}(Γ/T), and set π = Γ/O^{p}(Γ) (a finite p-group). By the mod-R fibre lemma [BK, II.5.2.iv] and since Bπ is p-complete [BK, VI.3.4 and VI.5.4], BO^{p}(Γ)_{p}^{∧} is the homotopy fibre of the natural map BΓ _{p}^{∧} ⟶ B(Γ/O^{p}(Γ)), and hence is equivalent to the covering space of BΓ _{p}^{∧} with covering group π. So Ω(BO^{p}(Γ)_{p}^{∧}) also has finite mod p homology. We can thus assume that Γ = O^{p}(Γ) is p-perfect, and hence that Ω(BΓ _{p}^{∧}) is connected.

For a finite p-group Q, set Rep(Q, L) = Hom(Q, S)/∼, where ρ_{1} ∼ ρ_{2} if ρ_{1} = αρ_{2} for some α ∈ Iso_{F}(ρ_{2}(Q), ρ_{1}(Q)). In other words, it is the set Γ -conjugacy classes in Hom(Q, S). Let [BQ, BΓ _{p}^{∧}] be the set of homotopy classes of maps BQ ⟶ BΓ _{p}^{∧}. By [BLO, Theorem 6.3(a)], there is a bijection Rep(Q, L) ⟶ [BQ, BΓ _{p}^{∧}] that sends the conjugacy class of a homomorphism ρ to the homotopy class of Bρ.

By [DW, Proposition 5.6] and since Ω(BΓ _{p}^{∧}) is connected, for each n ≥ 1 and each f: BC_{p^{n}} ⟶ BΓ _{p}^{∧}, f extends (up to homotopy) to a map from BC_{p^{n+1}} to BΓ _{p}^{∧}. Hence each ρ ∈ Hom(C_{p^{n}}, S) extends, up to Γ -conjugacy, to some ρ ∈ Hom(C_{p^{n+1}}, S). Since T ⊴ Γ and S/T is finite, we conclude that S = T, and thus that Γ/T has order prime to p. □

In fact, whenever T ⊴ Γ are such that T is a discrete p-torus and Γ/T is finite, Ω(BΓ _{p}^{∧}) is a p-compact group if and only if O^{p}(Γ/T) has order prime to p and Aut_{O^{p}(Γ/T)}(T) is generated by pseudoreflections on T. The necessity of this last condition was shown by Dwyer and Wilkerson [DW, Theorem 9.7.ii]. Conversely, Clark and Ewing [CE, Corollary, p. 426] showed that if O^{p}(Γ/T) has order prime to p and its action is generated by pseudoreflections, then H^{∗}(BΓ _{p}^{∧}; 𝔽_{p}) is a polynomial algebra over 𝔽_{p}, and hence the (co)homology of its loop space is finite.

Remark 5.3. Assume T ⊴ Γ are as in Proposition 5.2, where in addition, p ∤ |Γ/T | and the conjugation action of Γ/T on T is faithful (i.e., C_{Γ} (T) = T). Then S = T, Ob(L) = {T }, and Aut_{L}(T) = Γ, so that L ≅ B(Γ), π_{1}(|L|) ≅ Γ, and π_{1}(|L|_{p}^{∧}) = 1 since Γ is p-perfect. Thus by Proposition 3.12 and Theorem 4.7(b), the Ω-system associated to (S, F, L) is (kΓ -mod, k-mod; θ_{∗}, θ^{∗}), where θ_{∗}(M) = colim_{Γ}(M) for a kΓ -module M and θ^{∗} sends a k-module N to the corresponding kΓ -module with trivial action; and H_{∗}(Ω(|L|_{p}^{∧}); k) is the homology of an Ω-resolution of k.

Note also, in the situation of Remark 5.3, that since |Γ/T | has order prime to p, the group T is uniquely |Γ/T |-divisible. Hence H^{i}(Γ/T; T) = 0 for all i > 0, and Γ must be a semidirect product: Γ ≅ T ⋊ H where H ≅ Γ/T.

Along the lines of Remark 4.6, we also note the following:

Remark 5.4. Let Γ be a linear torsion group: a subgroup of GL_{n}(K), for some field K of characteristic different from p, all of whose elements have finite order. By [BLO, Theorem 8.10], there is a p-local compact group (S, F, L), where S ≤ Γ is a maximal discrete ptoral subgroup and |L|_{p}^{∧} ≃ BΓ _{p}^{∧}. Set π = π_{1}(BΓ _{p}^{∧}) ≅ π_{1}(|L|_{p}^{∧}) (a finite p-group by [BLO, Proposition 4.4]), and let θ: B(Γ) ⟶ B(π) and η: L ⟶ B(π) be functors that induce these isomorphisms. By Theorem 4.7(c), H_{∗}^{Ω}(B(Γ), θ; k) ≅ H_{∗}^{Ω}(L, η; k); i.e., Ω-resolutions with respect to these two different Ω-systems have the same homology.

Throughout the rest of the section, whenever Γ is a group whose maximal normal p-perfect subgroup O^{p}(Γ) is such that π =^{def} Γ/O^{p}(Γ) is a finite p-group, we write “Ω-resolution of kπ with respect to Γ ” to mean an Ω-resolution of kπ with respect to the category (B(Γ), θ) over π or the Ω-system (kΓ -mod, kπ-mod; θ_{∗}, θ^{∗}), where θ: B(Γ) ⟶ B(π) is the natural projection.

5.1. Ω-resolutions with respect to discrete p-tori.

Let T ⊴ Γ be a pair of groups, where T ≅ (ℤ/p^{∞})^{r} is a discrete p-torus of rank r ≥ 1 and Γ/T is finite of order prime to p. Thus Γ = T ⋊ H for some finite subgroup H ≤ Γ of order prime to p. We regard the group ring kT as a left kΓ -module, where for t ∈ T, h ∈ H, and x ∈ kT, t(x) = tx and h(x) = hxh^{−1}. We will construct complexes of projective kΓ -modules which, as complexes of kT -modules, are Ω-resolutions of k with respect to T. The kΓ -module structure on these complexes will be used in the next two subsections.

Set V = Ω_{1}(T) ≅ (C_{p})^{r}. For each n ≥ 1, set T_{n} = Ω_{n}(T) ≅ (C_{p^{n}})^{r} (thus V = T_{1}), and regard kT_{n} as a subring of kT. Let I(kT) ≤ kT and I(kT_{n}) ≤ kT_{n} be the augmentation ideals.

For each n ≥ 1, and each kT_{n}-module M and proper submodule M_{0} < M, M/M_{0} has a nontrivial quotient module with trivial T_{n}-action. Hence I(kT_{n})·(M/M_{0}) < M/M_{0}, and so M_{0} + I(kT_{n})·M < M.

For each n ≥ 0, let φ_{n}: V ⟶ I(kT_{n})/I(kT_{n})^{2} be the map φ_{n}(t) = [t − 1]. This is a homomorphism of groups, and extends to a kH-linear isomorphism k ⊗_{𝔽_{p}} V ≅ I(kT_{n})/I(kT_{n})^{2}. Lift φ_{n} to an 𝔽_{p}H-linear homomorphism φ̃_{n}: V ⟶ I(kT_{n}) (the ring 𝔽_{p}H is semisimple since p ∤ |H|), and extend that to a kΓ -linear homomorphism

φ_{n}: kT ·V =^{def} kT ⊗_{𝔽_{p}} V ⟶ kT

by setting φ_{n}(ξ ⊗ v) = ξ·φ̃_{n}(v) for ξ ∈ kT and v ∈ V. Since φ_{n} induces an isomorphism k⊗V ≅ I(kT_{n})/I(kT_{n})^{2} by assumption, φ_{n}(kT_{n}·V)+I(kT_{n})^{2} = I(kT_{n}), and so φ_{n}(kT_{n}·V) = I(kT_{n}) by the last paragraph (applied with φ_{n}(kT_{n}·V) and I(kT_{n}) in the roles of M_{0} and M). Hence Im(φ_{n}) = kT ·I(kT_{n}).

In particular, for each n ≥ 1, Im(φ_{n}) ≤ I(kT)·Im(φ_{n+1}). Since kT ·V is projective as a kΓ - module, there is a kΓ -linear homomorphism ψ_{n}: kT ·V ⟶ kT ·V such that φ_{n+1} ∘ ψ_{n} = φ_{n}. Note that

ψ_{n}(kT ·V) ≤ I(kT)·V.  (1)

Let Λ_{R}^{m}(M) denote the m-th exterior power over a commutative ring R of an R-module M. Define, for each 0 ≤ m ≤ r and each n ≥ 1,

D_{m} = Λ_{kT}^{m} (kT ·V ) ≅ kT ⊗_{𝔽_{p}} Λ_{𝔽_{p}}^{m}(V) ,

regarded as a kΓ -module. In particular, D_{0} = kT. For each n ≥ 1 and each 1 ≤ m ≤ r, define a boundary map ∂_{m}^{(n)} : D_{m} ⟶ D_{m−1} by setting

∂_{m}^{(n)}(v_{1} ∧ v_{2} ∧ ⋯ ∧ v_{m}) = ∑_{i=1}^{m}(−1)^{i−1}φ_{n}(v_{i})·v_{1} ∧ ⋯ v̂_{i} ⋯ ∧ v_{m}

for v_{1}, …, v_{m} ∈ kT ·V. Then

D^{(n)} =^{def} (D_{∗}, ∂_{∗}^{(n)}) = (0 ⟶ D_{r} ⟶^{∂_{r}^{(n)}} D_{r−1} ⟶ ⋯ ⟶ D_{1} ⟶^{∂_{1}^{(n)}} kT ⟶ 0)

is a chain complex of projective kΓ -modules, and {Λ^{m}(ψ_{n})}_{m=0}^{r} defines a morphism of chain complexes Ψ^{(n)} : D^{(n)} ⟶ D^{(n+1)} where Λ^{0}(ψ_{n}) = Id_{kT}.

We claim that

for each n ≥ 1, T_{n} acts trivially on the homology of D^{(n)}.  (2)

To see this, fix n ≥ 1, 0 ≤ m ≤ r, and x ∈ D_{m} = Λ_{kT}^{m} (kT ·V) such that ∂_{m}^{(n)}(x) = 0. For each v ∈ V,

∂_{m+1}^{(n)}(x ∧ v) = ∂_{m}^{(n)}(x) ∧ v + (−1)^{m}φ_{n}(v)·x = (−1)^{m}φ_{n}(v)·x.

Thus φ_{n}(v)·[x] = 0 in H_{m}(D^{(n)}) for each v ∈ V, so Im(φ_{n}) annihilates H_{∗}(D^{(n)}). Since Im(φ_{n}) ≥ I(kT_{n}), we now conclude that T_{n} acts trivially.

As usual, whenever ψ_{∗}: (C_{∗}, ∂_{∗}) ⟶ (C′_{∗}, ∂′_{∗}) is a morphism of chain complexes, the mapping cone of ψ is the chain complex

C_{ψ} = (⋯ \underrightarrow{( \begin{matrix}∂′_{4} & −ψ_{3} \\ 0 & ∂_{3} \end{matrix} )} C′_{3} ⊕ C_{2} \underrightarrow{( \begin{matrix}∂′_{3} & ψ_{2} \\ 0 & ∂_{2} \end{matrix} )} C′_{2} ⊕ C_{1} \underrightarrow{( \begin{matrix}∂′_{2} & −ψ_{1} \\ 0 & ∂_{1} \end{matrix} )} C′_{1} ⊕ C_{0} ⟶^{(∂′_{1},ψ_{0})} C′_{0}).

The signs are chosen so that C′_{∗} ≤ C_{ψ} as chain complexes, and C_{ψ}/C′_{∗} ≅ ΣC_{∗}. See [W, § 1.5] for more details.

Let D be the mapping cone of the chain map

Ψ = (Id − ⊕Ψ^{(n)}): ⨁_{n=1}^{∞} D^{(n)} ⟶ ⨁_{n=1}^{∞} D^{(n)}

(see [W, § 1.5]). More explicitly,

Ψ(x_{1}, x_{2}, x_{3}, …) = (x_{1}, x_{2} − Ψ^{(1)}(x_{1}), x_{3} − Ψ^{(2)}(x_{2}), …).

Since Ψ is injective, H_{∗}(D) ≅ H_{∗}(Coker(Ψ)), where Coker(Ψ) ≅ colim(D^{(n)}, Ψ^{(n)}). So

H_{∗}(D) ≅ colim(H_{∗}(D^{(1)}) ⟶^{H_{∗}(Ψ^{(1)})} H_{∗}(D^{(2)}) ⟶^{H_{∗}(Ψ^{(2)})} H_{∗}(D^{(3)}) ⟶ ⋯) ,

since colimits are exact. Also, T acts trivially on H_{∗}(D) by (2).

Now, H_{∗}(k ⊗_{kT} D) is isomorphic to the homology of the cokernel of the chain map

Ψ = (Id − ⊕Ψ^{(n)}): ⨁_{n=1}^{∞} k ⊗_{kT} D^{(n)} ⟶ ⨁_{n=1}^{∞} k ⊗_{kT} D^{(n)} .

Recall that D^{(n)} = (Λ_{kT}^{m} (kT ·V), ∂_{m}^{(n)})_{m=0}^{r} and Ψ^{(n)} = {Λ^{m}(ψ_{n})}_{m=0}^{r}. Here, Λ^{0}(ψ_{n}) = Id, while by (1), (Λ^{m}(ψ_{n}))(Λ_{kT}^{m} (kT ·V)) ≤ I(kT)·Λ_{kT}^{m} (kT ·V) for m > 0. So each Ψ^{(n)} is zero in positive degrees and the identity in degree 0, and hence the quotient complex Coker(Ψ) is zero in positive degrees and isomorphic to k in degree 0. Thus k ⊗_{kT} D is acyclic with H_{0}(k ⊗_{kT} D) ≅ k. Also,

H_{0}(D) ≅ k ⊗_{kT} H_{0}(D) ≅ H_{0}(k ⊗_{kT} D) ≅ k:  (3)

the first isomorphism since T acts trivially on H_{0}(D) and the second since (k ⊗_{kT} −) is right exact.

We have now proven:

Proposition 5.5. For k, Γ, and D as above, D is a chain complex of length r + 1 of projective kΓ -modules, and D ⟶^{ε} k ⟶ 0 is an Ω-resolution of k with respect to T.

5.2. Ω-resolutions with respect to the Sullivan spheres.

We now restrict to the special case of Remark 5.3 where p is odd and r = 1. Thus T ⊴ Γ where T ≅ ℤ/p^{∞}, p ∤ |Γ/T |, and C_{Γ} (T) = T. Then Aut(T) ≅ (ℤ_{p})^{×} ≅ C_{p−1} × ℤ_{p}, and since Γ/T is finite and acts faithfully on T, it must be cyclic of order dividing p − 1. Again, we will construct explicit Ω-resolutions of k with respect to Γ.

Spaces BΓ _{p}^{∧} for Γ of this form are the simplest and oldest examples constructed of pcompact groups (other than compact Lie groups). They were originally constructed from the space K(ℤ_{p}, 2) (≃ B(ℤ/p^{∞})_{p}^{∧}), by taking the Borel construction B(p, m) of the faithful action of a cyclic group C_{m} (for m|(p − 1)) on K(ℤ_{p}, 2). The p-completion of B(p, m) is a classifying space for the p-completed sphere (S^{2m−1})_{p}^{∧}, and hence B(p, m)_{p}^{∧} is the classifying space of a p-compact group that is often referred to as a “Sullivan sphere”. What we will show is that not only do these spaces have finite dimensional homology, but also that the associated Ω-systems have Ω-resolutions of finite length.

Write Γ = T ⋊ H ≅ ℤ/p^{∞} ⋊ H, where H is cyclic of order m|(p − 1). Let χ: H ⟶ 𝔽_{p}^{×} be the injective homomorphism such that hth^{−1} = t^{χ(h)} for all h ∈ H and all t ∈ V = Ω_{1}(T). Set kT_{(1)} = kT ·V = kT ⊗_{k} V as a kΓ -module, which we identify with kT but with H-action h(x) = χ(h)·hxh^{−1} for h ∈ H and x ∈ kT. More generally, for arbitrary j ≥ 0, we write

k_{(j)} = V^{⊗j} and kT_{(j)} = kT ⊗_{k} k_{(j)}

as kΓ -modules. Thus k_{(j)} ≅ k and kT_{(j)} ≅ kT as kT -modules, but h ∈ H acts on the first via multiplication by χ(h)^{j} and on the second via that and conjugation. We also write kT = kT_{(0)} and k = k_{(0)} for short.

Let φ_{n} ∈ Hom_{kΓ} (kT_{(1)}, kT) and ψ_{n} ∈ Hom_{kΓ} (kT_{(1)}, kT_{(1)}) be as above, and set μ_{n} = φ_{n}(1) and ν_{n} = ψ_{n}(1). Then for all n ≥ 1,

μ_{n+1}ν_{n} = μ_{n} and kT_{n}·μ_{n} = I(kT_{n}),  (4)

the first since φ_{n+1} ∘ ψ_{n} = φ_{n} and the second since φ_{n}(kT_{n}·V) = I(kT_{n}). Also,

h(μ_{n}) = χ(h)·μ_{n}  (5)

for all n ≥ 1 and h ∈ H since φ_{n} is kΓ -linear.

The complex D of Proposition 5.5 has the form

D = (0 ⟶ ⨁_{n=1}^{∞} kT_{(1)}·a_{n} ⟶^{∂_{2}} ⨁_{n=1}^{∞} kT_{(1)}·a_{n} ⊕ ⨁_{n=1}^{∞} kT ·b_{n} ⟶^{∂_{1}} ⨁_{n=1}^{∞} kT ·b_{n} ⟶ 0),

where

∂_{2}(a_{n}) = −(a_{n} − ν_{n}a_{n+1}) + μ_{n}b_{n} ,  ∂_{1}(a_{n}) = μ_{n}b_{n} ,  and ∂_{1}(b_{n}) = b_{n} − b_{n+1}.

Since (⨁_{n=1}^{∞} kT ·b_{n} ⟶^{(b_{n}↦b_{n}−b_{n+1})} ⨁_{n=1}^{∞} kT ·b_{n}) is injective with cokernel kT, the complex D is equivalent to

D = (0 ⟶ ⨁_{n=1}^{∞} kT_{(1)}·a_{n} ⟶^{∂_{2}} ⨁_{n=1}^{∞} kT_{(1)}·a_{n} ⟶^{∂_{1}} kT_{(0)}·a_{0} ⟶ 0),

where this time

∂_{2}(a_{n}) = a_{n} − ν_{n}a_{n+1} and ∂_{1}(a_{n}) = μ_{n}a_{0}.

By (4) and (5), ∂_{1} and ∂_{2} are kΓ -linear and ∂_{1} ∘ ∂_{2} = 0.

We now want to identify H_{1}(D) more precisely, and use this complex to construct an Ω-resolution with respect to Γ. To do this, define elements σ_{n} (all n ≥ 1) and ν_{0} in kT_{n} by setting

σ_{n} = ∑_{t∈T_{n}} t (all n ≥ 1) and ν_{0} = σ_{1}.  (6)

To better understand the relation between the μ_{n}, ν_{n}, and σ_{n}, fix n ≥ 2 and a generator t_{n} ∈ T_{n}, and set X = t_{n}−1 ∈ I(kT_{n}). Then X^{p^{n}} = t_{n}^{p^{n}} −1 = 0 and {1, X, X^{2}, …, X^{p^{n}−1}} is a basis for kT_{n}, so kT_{n} ≅ k[X]/(X^{p^{n}}) as rings, and each ideal in kT_{n} is a power of I(kT_{n}) = (X). Thus (μ_{n}) = (X), (μ_{n−1}) = (t_{n}^{p} − 1) = (X)^{p}, and hence (ν_{n−1}) = (X)^{p−1} by (4). Also, (σ_{n}) = (X)^{p^{n}−1} and (σ_{n−1}) = (X)^{p(p^{n−1}−1)} (see (6)), so (ν_{n−1}σ_{n−1}) = (X)^{p^{n}−1} = k·σ_{n}. Thus for each n ≥ 2, ν_{n−1}σ_{n−1} = a_{n}·σ_{n} for some 0 ≠ a_{n} ∈ k. To simplify notation, we can replace the μ_{n} (all n ≥ 2) and ν_{n} (all n ≥ 1) by appropriate scalar multiples, and arrange that

for each n ≥ 1, ν_{n−1}σ_{n−1} = σ_{n} .  (7)

Each element in Coker(∂_{2}) is the class of ξ·a_{n} for some n and some ξ ∈ kT_{m}, and we can always arrange (modulo Im(∂_{2})) that m = n. If in addition, ∂_{1}(ξ·a_{n}) = 0, then μ_{n}ξ = 0, so I(kT_{n})·ξ = 0 by (4), and hence ξ = a·σ_{n} for some a ∈ k. Thus H_{1}(D) is generated by the classes [σ_{n}a_{n}] for n ≥ 1, where for each n, [σ_{n}a_{n}] = [σ_{n}ν_{n}a_{n+1}] = [σ_{n+1}a_{n+1}] by the definition of ∂_{2} and (7). Also, h(σ_{n}) = χ(h)σ_{n} in kT_{(1)}, so H_{1}(D) ≅ k_{(1)} as kΓ -modules. To summarize,

H_{0}(D) = ⟨[a_{0}]⟩ ≅ k and H_{1}(D) = ⟨[σ_{1}a_{1}]⟩ ≅ k_{(1)} .  (8)

Thus D is an Ω-resolution of k with respect to Γ if H = 1, but is not an Ω-resolution if H ≠ 1 since Γ acts nontrivially on H_{1}(D) (i.e., condition (Ω-3) fails). In this case, we construct an Ω-resolution by “pasting together” several copies of the above sequence.

Define a complex C_{∞} of projective kΓ -modules of infinite length

C_{∞} = (⋯ ⟶ ⨁_{n=0}^{∞} kT_{(3)}·a_{n} ⟶^{∂_{6}} ⨁_{n=1}^{∞} kT_{(3)}·a_{n} ⟶^{∂_{5}} ⨁_{n=0}^{∞} kT_{(2)}·a_{n} ⟶^{∂_{4}}

⨁_{n=1}^{∞} kT_{(2)}·a_{n} ⟶^{∂_{3}} ⨁_{n=0}^{∞} kT_{(1)}·a_{n} ⟶^{∂_{2}} ⨁_{n=1}^{∞} kT_{(1)}·a_{n} ⟶^{∂_{1}} kT_{(0)}·a_{0} ⟶ 0),

where ∂_{1}(a_{n}) = μ_{n}a_{0} (as in D), and for i ≥ 2,

∂_{i}(a_{n}) = \begin{cases}a_{n} − ν_{n}a_{n+1} & if i is even \\ μ_{n}(a_{0} + σ_{1}a_{1} + σ_{2}a_{2} + ⋯ + σ_{n−1}a_{n−1}) & if i is odd. \end{cases}

Here, it is understood that a_{0} = 0 in the terms of odd degree. By (4), (5), (6), and (7), all boundary maps are kΓ -linear and ∂_{i−1} ∘ ∂_{i} = 0 for all i ≥ 2.

For each j ≥ 1, let C_{j} ⊆ C_{∞} be the subcomplex consisting of all terms in C_{∞} of degree at most 2j − 1 together with the summands ⨁_{n=1}^{∞} kT_{(j)}·a_{n} in degree 2j (thus omitting only the summand kT_{(j)}·a_{0}). Thus C_{1} ≅ D. More generally, if we set C_{0} = 0, then for each j ≥ 0, C_{j+1}/C_{j} is isomorphic to the 2j-fold suspension of D tensored by k_{(j)}, and hence by (8) has homology isomorphic to k_{(j+1)} in degree 2j + 1 and k_{(j)} in degree 2j. If j ≥ 1, then the homology of C_{j+1}/C_{j} in degree 2j is represented by the class of a_{0} in that degree, ∂_{2j}(a_{0}) = −ν_{0}a_{1} = −σ_{1}a_{1}, and by (8) again, this represents the homology class in C_{j}/C_{j−1} of degree 2j − 1. Together, these observations imply that C_{∞} is acyclic, and that for each j ≥ 1,

H_{0}(C_{j}) ≅ k,  H_{2j−1}(C_{j}) ≅ k_{(j)},  and H_{i}(C_{j}) = 0 for i ≠ 0, 2j − 1.

Set R = C_{m} (recall m = |H|). We claim that R ⟶ k ⟶ 0 is an Ω-resolution with respect to Γ. Condition (Ω-1) clearly holds (each of the terms is projective), and Γ acts trivially on H_{∗}(R) since χ^{m} = 1. It remains to show (Ω-2): that k ⊗_{kΓ} R is acyclic. Since k ⊗_{kΓ} kT_{(i)} ≅ k whenever m|i and is zero otherwise,

k ⊗_{kΓ} R ≅ (0 ⟶ ⨁_{n=1}^{∞} k·a_{n} ⟶^{∂_{2m}=Id} ⨁_{n=1}^{∞} k·a_{n} ⟶ 0 ⟶ … ⟶ 0 ⟶ k ⟶ 0)

and is acyclic.

We have now shown:

Proposition 5.6. Let Γ = T ⋊ H, where T ≅ ℤ/p^{∞} and H acts on Ω_{1}(T) via an injective character χ: H ⟶ 𝔽_{p}^{×}. Set m = |H|. Then the complex R ⟶ k ⟶ 0 defined above is an Ω-resolution of k with respect to the Ω-system (kΓ -mod, k-mod; θ_{∗}, θ^{∗}), and H_{∗}(Ω(BΓ _{p}^{∧}); k) ≅ H_{∗}(S^{2m−1}; k).

Note that C_{j} is not an Ω-resolution when j > m since k ⊗_{kΓ} C_{j} has nonzero homology in degree 2m (the class of a_{0}), and is not an Ω-resolution when 1 ≤ j < m since Γ acts nontrivially on H_{2j−1}(C_{j}). Note also that since Ω(BΓ _{p}^{∧}) has nonzero homology in degree 2m − 1, R has the shortest possible length of any Ω-resolution of k with respect to (kΓ -mod, k-mod; θ_{∗}, θ^{∗}). The equivalence Ω(BΓ _{p}^{∧}) ≃ (S^{2m−1})_{p}^{∧} follows from [Su, pp. 103–105] (from the proof of the proposition), and since B(ℤ/p^{∞})_{p}^{∧} ≃ K(ℤ_{p}, 2) (see [BK, VI.2.1–2.2]).

We will see later (Proposition 5.11 and Remark 5.12) that there are similar constructions of Ω-resolutions when T ⊴ Γ are such that T ≅ (ℤ/p^{∞})^{r} for r > 1 and p ∤ |Γ/T |.

Remark 5.7. When H ≠ 1, the parameters μ_{n} and ν_{n} can be defined more explicitly as follows. Fix generators t_{n} ∈ T_{n} for each n ≥ 1, chosen so that (t_{n})^{p} = t_{n−1} when n ≥ 2, and set μ_{n} = ∑_{h∈H} χ(h)^{−1}t_{n}^{χ(h)} ∈ kT_{n} for all n ≥ 1. It is straightforward to check that h(μ_{n}) = χ(h)μ_{n} for h ∈ H, that kT_{n}·μ_{n} = I(kT_{n}), and that (μ_{1})^{p} = 0 while (μ_{n})^{p} = μ_{n−1} for n ≥ 2. Also, (μ_{n})^{p−1}σ_{n−1} = σ_{n}, so we can set ν_{n−1} = (μ_{n})^{p−1} for each n, and use these parameters to define the Ω-resolution R.

5.3. Groups with discrete p-tori of index prime to p.

We now make some more computations of Ω-homology in the situation of Proposition 5.2: this time by using the existence of Ω-resolutions without constructing them explicitly. The key to doing this is the following spectral sequence.

Lemma 5.8. Let T ⊴ Γ be a pair of groups such that T is p-perfect. Let (C_{∗}, ∂_{∗}) be a positively graded chain complex of kΓ -modules that are projective as kT -modules, and assume that T acts trivially on H_{∗}(C_{∗}, ∂_{∗}). Then there is a first quadrant spectral sequence of k[Γ/T]-modules of the form

E_{ij}^{2} = H_{i}(k ⊗_{kT} C_{∗}) ⊗_{k} H_{j}(Ω(BT _{p}^{∧}); k) ⟹ H_{i+j}(C_{∗})

where the action of Γ/T on H_{∗}(Ω(BT _{p}^{∧}); k) is that induced by conjugation on T.

Proof. Since T is p-perfect, BT is p-good and BT _{p}^{∧} is simply connected [BK, Proposition VII.3.2], and so Ω(BT _{p}^{∧}) is connected. Consider the following diagram of spaces

\xymatrix{L^{p}T \ar[r] \ar[d] & A^{p}T \ar[r] \ar[d]^{ν} & P(BT _{p}^{∧}) \ar[d] \\ ET \ar[r] & BT \ar[r] & BT _{p}^{∧}}

where BT = |B(T)| and ET is its universal covering space, P(BT _{p}^{∧}) is the space of paths in BT _{p}^{∧} originating at the image of the (unique) vertex in BT, both squares are pullbacks, and thus the vertical maps are all fibrations with fibre Ω(BT _{p}^{∧}). In particular, A^{p}T is the homotopy fibre of the completion map and is mod p acyclic, L^{p}T ≃ Ω(BT _{p}^{∧}) since ET ≃ ∗, and T acts freely on L^{p}T with orbit space A^{p}T.

Now, Γ acts on the right on all of these spaces via the conjugation action. More precisely, we identify the vertices in ET with T, and let Γ act on ET by setting x ∗ g = g^{−1}xg for x ∈ T and g ∈ Γ, in contrast to the free right action of T defined by x·t = xt. This induces actions of Γ on BT, BT _{p}^{∧}, and P(BT _{p}^{∧}), and hence on L^{p}T; and the actions of T and Γ on ET and on L^{p}T satisfy the relation ((x ∗ g)·t) ∗ g^{−1} = x·(gtg^{−1}) for g ∈ Γ and t ∈ T.

In particular, C_{∗}(L^{p}T; k) is a complex of k[T ⋊ Γ]-modules that are free as kT -modules, and C_{∗}(A^{p}T; k) ≅ C_{∗}(L^{p}T; k) ⊗_{kT} k is an acyclic complex of kΓ -modules. The action of Γ on H_{∗}(L^{p}T; k) restricts to the conjugation action of T = π_{1}(BT) on the fibre of ν over the basepoint, and this action is trivial since ν is pulled back from the simply connected space BT _{p}^{∧}. Thus the action of Γ on H_{∗}(L^{p}T; k) ≅ H_{∗}(Ω(BT _{p}^{∧}); k) factors through Γ/T.

Consider the complex C_{∗}(L^{p}T; k) ⊗_{kT} C_{∗}. This is a double complex of kΓ -modules, where g(x ⊗ y) = x ∗ g^{−1} ⊗ gy for g ∈ Γ. This action of Γ is well defined on the tensor product over kT, since for g ∈ Γ and t ∈ T,

g(xt ⊗ t^{−1}y) = (xt) ∗ g^{−1} ⊗ gt^{−1}y = (x ∗ g^{−1})·(gtg^{−1}) ⊗ gt^{−1}y
= x ∗ g^{−1} ⊗ (gtg^{−1})gt^{−1}y = x ∗ g^{−1} ⊗ gy = g(x ⊗ y)

by the relation shown above. As usual, we consider the two spectral sequences of kΓ -modules induced by this double complex.

If we first take homology in the left-hand factor, we obtain

E_{i,j}^{1} ≅ H_{j}(Ω(BT _{p}^{∧}); k) ⊗_{kT} C_{i} and E_{i,j}^{2} ≅ H_{j}(Ω(BT _{p}^{∧}); k) ⊗_{k} H_{i}(k ⊗_{kT} C_{∗}),

where the first isomorphism holds since each C_{i} is projective as a kT -module, and the second since T acts trivially on Ω(BT _{p}^{∧}). On the other hand, if we first take homology of the righthand factor, we obtain

Ē_{i,j}^{1} ≅ C_{j}(L^{p}T; k) ⊗_{kT} H_{i}(C_{∗}) ≅ (C_{j}(L^{p}T; k) ⊗_{kT} k) ⊗ H_{i}(C_{∗}),

the first isomorphism since each C_{j}(L^{p}T; k) is free as a kT -module, and the second since the action of T on the homology of C_{∗} is trivial. Then

Ē_{i,j}^{2} ≅ \begin{cases}H_{i}(C_{∗}) & if j = 0 \\ 0 & if j > 0 \end{cases}

since C_{∗}(L^{p}T; k) ⊗_{kT} k is isomorphic to C_{∗}(A^{p}T; k) and hence is acyclic. □

In the following lemma, when we say that a graded vector space over k is “finite dimensional”, we mean that it is finite dimensional in each degree and is nonzero in only finitely many degrees.

Lemma 5.9. Fix a pair of groups T ⊴ Γ, where T ≅ (ℤ/p^{∞})^{r} for some r ≥ 1 and Γ/T is finite. Let O^{p}(Γ) ⊴ Γ be the smallest normal subgroup such that Γ/O^{p}(Γ) is a finite p-group (thus O^{p}(Γ) ≥ T), set π = Γ/O^{p}(Γ), and let θ: B(Γ) ⟶ B(π) be the natural functor. Let (C_{∗}, ∂_{∗}) be an Ω-resolution of kπ with respect to the Ω-system (kΓ -mod, kπ-mod; θ_{∗}, θ^{∗}). Then H_{∗}(C_{∗}, ∂_{∗}) is finite dimensional if and only if H_{∗}(C_{∗} ⊗_{kT} k) is finite dimensional.

Proof. Since H_{∗}(Ω(BT _{p}^{∧}); k) is finite dimensional, Lemma 5.8 implies that if H_{∗}(C_{∗} ⊗_{kT} k) is finite dimensional, then so is H_{∗}(C_{∗}; k). Thus it remains to prove the converse. Since H_{∗}(−; k) ≅ H_{∗}(−; 𝔽_{p}) ⊗_{𝔽_{p}} k, it suffices to show this when k = 𝔽_{p}.

Consider the following diagram:

\xymatrix{L^{p}Γ \ar[r] \ar[d] & (L^{p}Γ)/T \ar[r] \ar[d] & A^{p}Γ \ar[r] \ar[d] & P(BΓ _{p}^{∧}) \ar[d] \\ EΓ \ar[r] & (EΓ)/T \ar[r] & BΓ \ar[r] & BΓ _{p}^{∧}}

where the squares are pullback squares. By Proposition 4.3, C_{∗}(L^{p}Γ; 𝔽_{p}) is an Ω-resolution of 𝔽_{p}π with respect to the given Ω-system. Hence by Proposition 1.6, we may assume C_{∗} = C_{∗}(L^{p}Γ; 𝔽_{p}), and so C_{∗} ⊗_{𝔽_{p}T} 𝔽_{p} ≅ C_{∗}(L^{p}Γ/T; 𝔽_{p}). From the above pullback diagram, we see that L^{p}Γ/T is the homotopy fibre of the map BT ≃ EΓ/T ⟶ BΓ _{p}^{∧}. Since π_{1}(BΓ _{p}^{∧}) is a finite p-group, it acts nilpotently on H_{∗}(L^{p}Γ/T; 𝔽_{p}), and hence by the mod-R fibre lemma of Bousfield and Kan [BK, Lemma II.5.1], (L^{p}Γ/T)_{p}^{∧} is the homotopy fibre of the map BT _{p}^{∧} ⟶ BΓ _{p}^{∧} induced by the inclusion. Since T is p-perfect and |Γ/T | is finite, BT _{p}^{∧} and BΓ _{p}^{∧} are p-complete, and hence (L^{p}Γ/T)_{p}^{∧} is also p-complete by the mod-R fibre lemma again.

Now assume that H_{∗}(C_{∗}; 𝔽_{p}) is finite dimensional, and hence that Ω(BΓ _{p}^{∧}) is a p-compact group. By Proposition 5.2, there is a p-local compact group (S, F, L) with T ≤ S ≤ Γ and |L|_{p}^{∧} ≃ BΓ _{p}^{∧}. So for each 1 ≠ t ∈ T, [BLO, Theorem 6.3] implies that the map B⟨t⟩ ⟶ BΓ _{p}^{∧} induced by the inclusion is not null homotopic. In the terminology of [DW, § 7], this means that the map BT _{p}^{∧} ⟶ BΓ _{p}^{∧}, regarded as a map of p-compact groups, has trivial kernel. So by [DW, Theorem 7.3], BT _{p}^{∧} ⟶ BΓ _{p}^{∧} is a monomorphism of p-compact groups, and by definition [DW, 3.2], the 𝔽_{p}-homology of its homotopy fibre (L^{p}Γ/T)_{p}^{∧} is finite dimensional. □

We now give some consequences of Lemmas 5.8 and 5.9. The first example can also be carried out using the Serre spectral sequence for the path-loop fibration of BΓ _{p}^{∧}, but the argument given here is slightly easier, and it illustrates nicely how the action of Γ/T on the spectral sequence of Lemma 5.8 can be exploited.

Example 5.10. Set Γ = T ⋊ C_{2}, where p is odd, T ≅ (ℤ/p^{∞})^{2}, and Γ/T ≅ C_{2} inverts T. Then

H_{i}(Ω(BΓ _{p}^{∧}); 𝔽_{p}) ≅ \begin{cases}𝔽_{p} & if i = 0 \\ 𝔽_{p}^{3} & if i = 3 \\ 𝔽_{p}^{4} & if i > 3 and i ∈ 3ℤ \\ 0 & otherwise. \end{cases}

Proof. To simplify notation, we do this over an arbitary field k of characteristic p. Fix an Ω-resolution (C_{∗}, ∂_{∗}) of k with respect to Γ, and let E be the spectral sequence of Lemma 5.8. Recall that this is a spectral sequence of k[Γ/T]-modules: the action of Γ/T plays a central role in the argument here.

Since H_{i}(C_{∗} ⊗_{kΓ} k) = 0 for i > 0 by (Ω-2), Γ/T ≅ C_{2} acts via −Id on E_{i,0}^{2} ≅ H_{i}(C_{∗} ⊗_{kT} k) for each i > 0. Also, since Γ acts trivially on H_{i}(C_{∗}) for all i by (Ω-3), Γ/T acts trivially on E_{i,j}^{∞} for all i, j.

We claim that E_{i,j}^{2} takes the following form:

[figure: |; k_{+}; 0; k_{−}^{2}; 0; 0; k_{−}^{2}; 0; 0; k_{−}^{2}; …; k_{−}^{2}; 0; k_{+}^{4}; 0; 0; k_{+}^{4}; 0; 0; k_{+}^{4}; …; k_{+}; 0; k_{−}^{2}; ≅; 0; 0; k_{−}^{2}; ≅; 0; 0; k_{−}^{2}; ≅; …]

where the subscripts (±) describe the action of Γ/T ≅ C_{2}, and where the pattern continues with E_{3k+2,0}^{2} ≅ k_{−}^{2} for i ≥ 0 and E_{i,0}^{2} = 0 when 0 < i ≢ 2 (mod 3). To see this, note first that E_{1,0}^{2} ≅ E_{1,0}^{∞} = 0 since Γ/T inverts E_{1,0}^{2} and fixes E_{1,0}^{∞}, and that the differential sends E_{2,0}^{2} isomorphically to E_{0,1}^{2} since the action on E^{∞} is trivial. Hence E_{i,j}^{2} is as described for i ≤ 2. Also, E_{3,0}^{2} = 0 and E_{4,0}^{2} = 0 since there are no terms inverted by Γ/T which their differentials could hit, and E_{5,0}^{2} ≅ k_{−}^{2} must be sent isomorphically to E_{2,2}^{2}.

Upon continuing in this way, we see inductively that E_{i,0}^{2} ≅ k_{−}^{2} for i = 3j + 2 (all j ≥ 0) and is zero in other positive degrees, and the differentials are as shown with one possible exception. We claim that the differential from E_{2,1}^{2} to E_{0,2}^{2} is surjective: this holds since H_{i}(BΓ _{p}^{∧}; k) ≅ (H_{i}(BT _{p}^{∧}; k))^{Γ/T} = 0 for i ≤ 3 implies that H_{i}(Ω(BΓ _{p}^{∧}); k) = 0 for i ≤ 2. Thus E_{i,j}^{∞} is zero in positive total degrees except E_{21}^{∞} ≅ k^{3} and E_{3i+2,1}^{∞} ≅ k^{4} (all i ≥ 1). □

The next proposition shows that in these examples, at least, whenever the homology of the loop space is bounded, we get an Ω-resolution of finite length.

Proposition 5.11. Let T ⊴ Γ be such that T ≅ (ℤ/p^{∞})^{r} for some r ≥ 1 and Γ/T is finite. If in addition, Ω(BΓ _{p}^{∧}) is a p-compact group, then there is an Ω-resolution of k with respect to Γ of finite length. More precisely, if N ≥ 1 is maximal such that H_{N}(Ω(BΓ _{p}^{∧}); k) ≠ 0, then there is an Ω-resolution of k with respect to Γ of length N + 1.

Proof. Since Ω(BΓ _{p}^{∧}) is a p-compact group, O^{p}(Γ/T) has order prime to p by Proposition 5.2. Let O^{p}(Γ) ⊴ Γ be such that O^{p}(Γ)/T = O^{p}(Γ/T), and set π = Γ/O^{p}(Γ). Since Ω(B(O^{p}(Γ))_{p}^{∧}) is homotopy equivalent to a connected component of Ω(BΓ _{p}^{∧}) (as shown in the proof of Proposition 5.2), Ω(BO^{p}(Γ)_{p}^{∧}) is also a p-compact group. If C ⟶ k ⟶ 0 is an Ω-resolution of k with respect to O^{p}(Γ), then Ind_{O^{p}(Γ)}^{Γ}(C) ⟶ kπ ⟶ 0 is an Ω-resolution of kπ with respect to Γ. So it suffices to prove the proposition when Γ = O^{p}(Γ); i.e., when Γ/T has order prime to p and π = 1.

Recall that r is the rank of T. By Proposition 5.5, there is an Ω-resolution D ⟶ k ⟶ 0 of k with respect to T of length r+1 which is also a chain complex of projective kΓ -modules.

Step 1: By Proposition 2.8, there is an Ω-resolution C ⟶ k ⟶ 0 of k with respect to Γ (possibly of infinite length). We will use this as a template for constructing an Ω-resolution of finite length N + 1.

By Lemma 5.9, the homology of k ⊗_{kT} C is finitely generated. Let

0 = m_{1} < m_{2} < ⋯ < m_{ℓ} = m

be the degrees in which H_{∗}(k ⊗_{kT} C) is nonzero. Thus ℓ is the number of distinct degrees in which this homology is nonzero. By the spectral sequence {E_{∗,∗}^{r}} of Lemma 5.8, H_{m+r}(C) ≅ E_{m,r}^{2} ≠ 0, and so N = m + r by Corollary 4.8.

Step 2: By Proposition 1.6, and since C satisfies condition (Ω-3) (Definition 1.5) as a complex of kT -modules, there is a kT -linear chain map

ψ_{∗}^{(0)} : D = H_{0}(k ⊗_{kT} C) ⊗_{k} Σ^{0}D ⟶ C

that induces a kΓ -linear isomorphism H_{0}(D) ⟶^{≅} H_{0}(C). By averaging, i.e., by replacing ψ_{∗}^{(0)} by the map x ↦ \frac{1}{|Γ/T} ∑_{gT ∈Γ/T} g(ψ_{∗}^{(0)}(g^{−1}x)) for x ∈ D, we can arrange that ψ_{∗}^{(0)} is

kΓ -linear without changing H_{0}(ψ_{∗}^{(0)}). Let C^{(1)} be the mapping cone of ψ_{∗}^{(0)} [W, § 1.5]; again a chain complex of projective kΓ -modules. Since the p-perfect group T acts trivially on the homology of D and of C, it also acts trivially on H_{i}(C^{(1)}) for each i (Lemma A.1). Also, the homology of k ⊗_{kT} C^{(1)} is isomorphic to that of k ⊗_{kT} C (as k-vector spaces), except in degree 0 = m_{1}.

Step 3: Let t be the minimum of all i such that H_{i}(C^{(1)}) ≠ 0. If t = ∞ (i.e., if C^{(1)} is exact), then the sequence splits, so k ⊗_{kT} C^{(1)} is also exact, and ℓ = 1.

Assume that C^{(1)} is not exact; i.e., that t < ∞. Then the exact sequence C_{t}^{(1)} ⟶ C_{t−1}^{(1)} ⟶ ⋯ ⟶ C_{0}^{(1)} ⟶ 0 of projective kΓ -modules splits. By this splitting, and since (k ⊗_{kT} −) is right exact and T acts trivially on H_{t}(C^{(1)}), we have H_{t}(k ⊗_{kT} C^{(1)}) ≅ H_{t}(C^{(1)}) ≠ 0, while H_{i}(k ⊗_{kT} C^{(1)}) = 0 for all i < t. Thus t = m_{2} and ℓ ≥ 2. By Proposition 1.6 again (and averaging), there is a kΓ -linear chain map

ψ_{∗}^{(1)} : H_{m_{2}}(k ⊗_{kT} C) ⊗_{k} Σ^{m_{2}}D ⟶ C^{(1)}

that induces an isomorphism in H_{m_{2}}(−). In other words, we shift D by degree m_{2}, tensor each term by the k-module

H_{m_{2}}(k ⊗_{kT} C) ≅ H_{m_{2}}(k ⊗_{kT} C^{(1)}) ≅ H_{m_{2}}(C^{(1)}),

and then map the resulting complex into C^{(1)}.

Let C^{(2)} be the mapping cone of ψ_{∗}^{(1)}. By the arguments used in Step 2, T acts trivially on H_{∗}(C^{(2)}), and H_{i}(k ⊗_{kT} C^{(2)}) ≅ H_{i}(k ⊗_{kT} C) for all i > m_{2} while H_{i}(k ⊗_{kT} C^{(2)}) = 0 for i ≤ m_{2}.

Step 4: We now repeat this procedure to obtain an increasing sequence

C ≤ C^{(1)} ≤ C^{(2)} ≤ ⋯ ≤ C^{(ℓ)}

of chain complexes of projective kΓ -modules, where for each 1 ≤ r ≤ ℓ, T acts trivially on H_{∗}(C^{(r)}), and H_{i}(k ⊗_{kT} C^{(r)}) ≅ H_{i}(k ⊗_{kT} C) for all i > m_{r} while H_{i}(k ⊗_{kT} C^{(r)}) = 0 for i ≤ m_{r}. Also, by the argument at the start of Step 3, H_{i}(C^{(r)}) = 0 for i < m_{r+1}, while H_{m_{r+1}}(C^{(r)}) ≅ H_{m_{r+1}}(k ⊗_{kT} C^{(r)}) ≅ H_{m_{r+1}}(k ⊗_{kT} C) ≠ 0.

In particular, C^{(ℓ)} is an exact sequence of projective kΓ -modules. Set

R = Σ^{−1}(C^{(ℓ)}/C).

Then C^{(ℓ)} is the mapping cone of a kΓ -linear chain map R ⟶ C, and Γ acts trivially on H_{∗}(R) ≅ H_{∗}(C). Also, k ⊗_{kΓ} R is acyclic since k ⊗_{kΓ} C is (and since the sequence k ⊗_{kΓ} C_{∗}^{(ℓ)} is exact), and so R is an Ω-resolution of k with respect to Γ of length m+r +1 = N +1. □

Note that the converse of Proposition 5.11 also holds: Ω(BΓ _{p}^{∧}) is a p-compact group if there is an Ω-resolution of finite length. As usual, this can be reduced to the case where Γ/T is p-perfect; i.e., where BΓ _{p}^{∧} is simply connected. If there is an Ω-resolution of k with respect to Γ of finite length, then H_{i}(Ω(BΓ _{p}^{∧}); 𝔽_{p}) = 0 for i large enough by Corollary 4.8. Also, H_{i}(BΓ _{p}^{∧}; 𝔽_{p}) ≅ H_{i}(BΓ; 𝔽_{p}) is finite for each i by the Serre spectral sequence for the fibration sequence BT ⟶ BΓ ⟶ B(Γ/T) (and since H_{∗}(BT; 𝔽_{p}) ≅ H_{∗}(B(S^{1})^{r}; 𝔽_{p}) and H_{∗}(B(Γ/T); 𝔽_{p}) are finite in each degree), and hence H_{i}(Ω(BΓ _{p}^{∧}); 𝔽_{p}) is finite for each i by [Se, Proposition 7] and since BΓ _{p}^{∧} is simply connected. So Ω(BΓ _{p}^{∧}) is a p-compact group.

Remark 5.12. By a closer inspection, one can say more about the Ω-resolution constructed in the proof of Proposition 5.11. By construction, R has a filtration whose successive quotients are the suspended complexes H_{m_{r}}(k ⊗_{kT} R) ⊗_{k} Σ^{m_{r}}D for 1 ≤ r ≤ ℓ, where D is the complex constructed in Proposition 5.5 and 0 = m_{1} < m_{2} < ⋯ < m_{ℓ} = m are the degrees in which H_{∗}(k ⊗_{kT} R) is nonzero.

Appendix A. R-perfect groups and R-plus constructions

Recall that for a group G, we write G^{ab} = G/[G, G] ≅ H_{1}(G; ℤ) for short. For a commutative ring R, we say that G is R-perfect if H_{1}(G; R) = 0; equivalently, if G^{ab} ⊗_{ℤ} R = 0. When p is a prime, G is p-perfect if it is 𝔽_{p}-perfect.

Lemma A.1. Fix a commutative ring R and an R-perfect group G. Let M_{0} ⊆ M be RGmodules such that G acts trivially on M_{0} and on M/M_{0}. Then G also acts trivially on M.

Proof. Let Aut_{R}^{0}(M) be the group of all R-linear automorphisms of M that induce the identity on M_{0} and on M/M_{0}. Then Aut_{R}^{0}(M) ≅ Hom_{R}(M/M_{0}, M_{0}) is abelian and has the structure of an R-module, and G acts on M via a homomorphism G ⟶ Aut_{R}^{0}(M). Each homomorphism from G to an R-module factors through G^{ab} ⊗_{ℤ} R = 0 and hence is trivial, so G acts trivially on M. □

We now turn our attention to plus constructions (see Definition 4.2). The following lemma will be needed when checking the condition in the definition about homology with twisted coefficients.

Lemma A.2. Fix a commutative ring R and a group π. Let f: X ⟶ Y be a map between connected spaces, and let η: π_{1}(Y) ⟶ π be a homomorphism such that η and η ∘ π_{1}(f) are both surjective. Let X̃ and Ỹ be the covering spaces of X and Y with fundamental groups Ker(η ∘ π_{1}(f)) and Ker(η), respectively, and assume that a covering map f̃: X̃ ⟶ Ỹ is an R-homology equivalence. Then H_{∗}(f; M) is an isomorphism for each Rπ-module M.

Proof. Let Ĉ_{∗} be the mapping cone of the chain map C_{∗}(f̃): C_{∗}(X̃; R) ⟶ C_{∗}(Ỹ; R) (see the remark just before Proposition 5.11). Since Ĉ_{∗} is an exact sequence of free Rπ-modules and is bounded below, Ĉ_{∗} ⊗_{Rπ} M is also exact, and hence H_{∗}(f; M) is an isomorphism. □

The conditions for the existence of an R-plus construction are essentially the same as those for the existence of the usual plus construction.

Lemma A.3. Let R be a commutative ring, let X be a connected CW complex, and let H ⊴ π_{1}(X) be a normal subgroup. Let X̃ be the covering space of X with fundamental group H. Then the following are equivalent:

(a) The pair (X, H) admits an R-plus construction.

(b) The subgroup H is R-perfect.

(c) H_{1}(X̃; R) = 0.

Proof. Set π = π_{1}(X)/H.

(a ⟹ c) Let κ: X ⟶ X_{R}^{+} be an R-plus construction for (X, H), and let X̃_{R}^{+} be the universal cover of X_{R}^{+}. Then by definition,

H_{1}(X̃; R) ≅ H_{1}(X; Rπ) ≅ H_{1}(X_{R}^{+}; Rπ) ≅ H_{1}(X̃_{R}^{+}; R) = 0 .

(b ⟺ c) This holds since H_{1}(X̃; R) ≅ π_{1}(X̃)^{ab} ⊗_{ℤ} R ≅ H_{1}(H; R). (c ⟹ a) This is essentially Quillen’s construction. Assume (c) holds, and attach 2-cells to X̃ in free π-orbits to obtain a free, simply connected π-space X̃_{0}^{+} ⊇ X̃. Then H_{2}(X̃_{0}^{+}; R) surjects onto the free Rπ-module H_{2}(X̃_{0}^{+}, X̃; R) since H_{1}(X̃; R) = 0 by assumption. By the Hurewicz theorem, H_{2}(X̃_{0}^{+}; R) ≅ π_{2}(X̃_{0}^{+}) ⊗_{ℤ} R, and hence free π-orbits of 3-cells can be attached to X̃_{0}^{+} to obtain a free π-space X̃_{R}^{+} with H_{∗}(X̃_{R}^{+}, X̃; R) = 0. Set X_{R}^{+} = X̃_{R}^{+}/π and let κ: X ⟶ X_{R}^{+} be the inclusion; then π_{1}(κ) is surjective with kernel H.

By Lemma A.2, and since the inclusion of X̃ in X̃_{R}^{+} is an R-homology equivalence, the inclusion of X in X_{R}^{+} induces an isomorphism in homology with coefficients in any Rπmodule. So X_{R}^{+} is an R-plus construction for (X, H). □

Under certain conditions, completion or fibrewise completion as defined by Bousfield and Kan gives another, more functorial way to construct plus constructions.

Lemma A.4. Let π be a group, and let θ: X ⟶ Bπ be a map of spaces where X is connected and π_{1}(θ) is onto. Set H = Ker(π_{1}(θ)).

(a) Assume that R is a subring of ℚ or R = 𝔽_{p} for some prime p, and also that H is R-perfect. Let θ̂: X^{∧} ⟶ Bπ be the fibrewise R-completion of X over Bπ. Then κ: X ⟶ X^{∧} is an R-plus construction for (X, H).

(b) Let X̃ be the covering space of X with covering group π and fundamental group H. Assume, for some R ⊆ ℚ, that H is R-perfect, and that π is R-nilpotent and has nilpotent action on H_{i}(X̃; R) for each i. Then the R-completion map κ: X ⟶ X_{R}^{∧} is an R-plus construction for (X, H).

(c) Assume, for some prime p, that π is a finite p-group and H is p-perfect. Then the p-completion map κ: X ⟶ X_{p}^{∧} is a k-plus construction for (X, H) for each field k of characteristic p.

Proof. We can assume that θ: X ⟶ Bπ is a fibration, with fibre F. Thus the inclusion F ⊆ X induces an isomorphism π_{1}(F) ≅ H. Let X̃ be the covering space of X with covering group π and fundamental group H; thus X̃ ≃ F.

(a) Since H is R-perfect, H_{1}(F; R) ≅ H_{1}(H; R) = 0 (Lemma A.3), and hence F _{R}^{∧} is simply connected by [BK, Lemma I.6.1] (applied with k = 1). Since F _{R}^{∧} is the fibre of the fibration θ̂ by [BK, Corollary I.8.3], θ̂ induces an isomorphism π_{1}(X^{∧}) ≅ π and X̃_{R}^{∧} is the universal cover of X^{∧}.

Since F _{R}^{∧} is simply connected, it is R-good by Proposition V.3.4 or VI.5.3 in [BK], and hence F is R-good by [BK, Proposition I.5.2]. So κ_{0} : F ⟶ F _{R}^{∧} is an R-homology equivalence. Hence X̃ ⟶ X̃_{R}^{∧} is also an R-homology equivalence, and by Lemma A.2, κ: X ⟶ X^{∧} induces an isomorphism in homology with coefficients in arbitrary Rπ-modules. So κ is an R-plus construction for (X, H).

(b) If R ⊆ ℚ and π is R-nilpotent, then Bπ is R-complete by [BK, Proposition V.2.2]. If, in addition, H is R-perfect and the action of π on H_{i}(X̃; R) (equivalently, on H_{i}(F; R)) is nilpotent for each i, then fibrewise completion over Bπ is the same as R-completion by the mod-R fibration lemma [BK, II.5.1], and the result follows from (a).

(c) If π is a finite p-group, then Bπ is p-complete by [BK, VI.3.4 and VI.5.4]. Hence fibrewise completion over Bπ is the same as p-completion by the mod-R fibration lemma [BK, II.5.1 and II.5.2.iv], and κ: X ⟶ X_{p}^{∧} is an 𝔽_{p}-plus construction by (a) (applied with R = 𝔽_{p}) when H is p-perfect. If k is an arbitrary field of characteristic p, then κ is also a k-plus construction since H_{∗}(−; k) ≅ H_{∗}(−; 𝔽_{p}) ⊗_{𝔽_{p}} k. □

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Departament de Matemàtiques, Universitat Autònoma de Barcelona, E–08193 Bellaterra, Spain

  E-mail address: broto@mat.uab.es

Department of Mathematical Sciences, University of Aberdeen, Fraser Noble 138, Aberdeen AB24 3UE, U.K.

  E-mail address: r.levi@abdn.ac.uk

Université Paris 13, Sorbonne Paris Cité, LAGA, UMR 7539 du CNRS, 99, Av. J.-B. Clément, 93430 Villetaneuse, France.

  E-mail address: bobol@math.univ-paris13.fr