ENDOFINITENESS IN STABLE HOMOTOPY THEORY

HENNING KRAUSE AND ULRIKE REICHENBACH

Abstract. We study endofinite objects in a compactly generated triangulated category in terms of ideals in the category of compact objects. Our results apply in particular to the stable homotopy category. This leads, for example, to a new interpretation of stable splittings for classifying spaces of finite groups.

Introduction

A CW-complex is said to be endofinite if all stable homotopy groups π_{n}^{s}(X) = {S^{n}, X} are finite length modules over the ring {X, X} of stable self-maps of X. In this paper we study CW-complexes having this finiteness property. One motivation for this work is to understand stable splittings of certain spaces, for instance the classifying space BG of a finite group G. We shall give an algebraic interpretation of such splittings in terms of certain ideals in the category of finite spectra.

A natural framework for the study of endofinite objects are compactly generated triangulated categories. Thus we consider a triangulated category C, for example the stable homotopy category, which has arbitrary coproducts and is generated by a set of compact objects (an object X in C is compact if the representable functor Hom(X, −) preserves coproducts). An object X in C is said to be endofinite if the group Hom(C, X) is a finite length module over the endomorphism ring End(X) for all compact objects C in C. This finiteness condition has a number of interesting consequences. For example, in [15] it is shown that every endofinite object decomposes essentially uniquely into indecomposable objects with local endomorphism rings. Here, we shall study endofinite objects in terms of certain ideals in the full subcategory C_{0} of compact objects in C. Viewing C_{0} as a ring with several objects, we consider for every X in C the annihilator Ann X of X which is the ideal of maps ϕ in C_{0} such that Hom(ϕ, X) = 0. Two types of ideals ℑ in C_{0} are of particular importance. For the purpose of this introduction let us give an elementary definition; a conceptual description of these ideals can be found in Sections 1.2 and 1.3.

• ℑ is cohomological if for every sequence of maps in C_{0}

X′ ⟶^{ϕ′} X ⟶^{ϕ} Y ⟶^{ϕ′′} Y′′

such that ϕ′ is a fibre of ϕ′′ϕ, the map ϕ belongs to ℑ if ϕϕ′ and ϕ′′ϕ belong to ℑ. We shall see that the cohomological ideals in C_{0} are precisely the annihilator ideals of objects in C. In fact, for every cohomological ideal ℑ there exists a canonical object E_{ℑ} such that ℑ = Ann E_{ℑ}.

• ℑ is cofinite if for every X in C_{0} there exists n ∈ ℕ such that for every sequence of maps in C_{0}

X ⟶^{ϕ_{0}} X_{0} ⟶^{ϕ_{1}} X_{1} ⟶^{ϕ_{2}} … ⟶^{ϕ_{n}} X_{n}

Date: November 1998.

1991 Mathematics Subject Classification. 55P42, 55U35.

the composition of ϕ_{i−1} … ϕ_{0} with a fibre of ϕ_{i} … ϕ_{0} belongs to ℑ for some i ≤ n. Let us denote by Max C_{0} the set of maximal elements among the cofinite cohomological ideals ℑ ≠ C_{0} in C_{0}.

Theorem A. An object X in C is endofinite if and only if the annihilator of X is cofinite. Moreover the maps X ↦ Ann X and ℑ ↦ E_{ℑ} induce mutually inverse bijections between the set of isomorphism classes of indecomposable endofinite objects in C and Max C_{0}.

Any endofinite object X has an essentially unique decomposition X ≅ ∐_{i∈I} X_{i} into indecomposable endofinite objects, and the annihilator of X controls this decomposition. More precisely, an indecomposable object Y arises as a direct factor of an endofinite object X if and only if Ann X ⊆ Ann Y. This is a particular consequence of the following result:

Theorem B. Let X be an endofinite object in C with ℑ = Ann X. Then

ℑ = ⋂_{ℑ⊆𝔍∈Max C_{0}} 𝔍

and X has a unique decomposition

X ≅ ∐_{ℑ⊆𝔍∈Max C_{0}} X_{𝔍}

such that each X_{𝔍} is a non-empty coproduct of copies of E_{𝔍}. Moreover, Ann X ⊆ Ann Y holds for some object Y if and only if Y is isomorphic to a direct factor of a (co)product of copies of X.

Let us now explain some special results for the stable homotopy category of spectra. A fairly immediate consequence of Theorem B is the fact that we can work one prime at a time.

Corollary. Let X be an endofinite spectrum. Then the 𝔭-localization map X → X_{𝔭} is a split epimorphism for every prime ideal 𝔭 in ℤ. If X_{tor} denotes the fibre of the rational localization X → X_{(0)}, then X ≅ X_{(0)} ∨X_{tor} and X_{tor} ≅ ⋁_{𝔭}(X_{tor})_{𝔭} where 𝔭 runs through all non-zero prime ideals in ℤ.

Having developed a general theory of endofinite spectra, one may ask for conditions on a CW-complex X such that its suspension spectrum is endofinite. It turns out that this happens if and only if all stable homotopy groups of X are finite length modules over the ring of stable self-maps of X, hence if the CW-complex X is endofinite. Here is a useful criterion:

Theorem C. Let X be a connected CW-complex such that the singular homology groups H_{n}(X; ℤ) are finite for all n > 0. Then X is an endofinite CW-complex, and therefore its suspension spectrum is also endofinite.

Important examples of CW-complexes with finite homology are the classifying spaces of finite groups.

Corollary. The classifying space BG of a finite group G is endofinite.

It is a well-known consequence of Carlsson’s solution of the Segal conjecture [2] that BG splits stably into indecomposable spectra; see [9] for an alternative proof. We recover this result, and our ideal theoretic approach reduces the problem of finding such splittings to the category of finite spectra.

The concept of endofiniteness has its origin in representation theory of finite dimensional algebras. Following Crawley-Boevey, a module X over some associative ring Λ is endofinite if it has finite length when regarded as a module over its endomorphism ring End_{Λ}(X). Such modules have nice decomposition properties and they play a decisive role for the representation type of a finite dimensional algebra; see [3] for an excellent survey. Comparing the category of modules over some associative ring and the stable homotopy category of spectra, one finds that a number of results about endofinite objects carry over from one setting to the other. The reason for this is the fact that both categories can be embedded in an appropriate way into some abelian functor category. This embedding identifies the endofinite objects with certain injective objects having some nice extra properties. This enables us to use classical results about injective objects in abelian Grothendieck categories. However, the additional triangulated structure of the stable homotopy category led to the discovery of some genuine new phenomena. In fact, the ideal theoretic approach on which Theorems A and B are based, appears to be less natural for the category of modules over a ring.

This paper is organized as follows: In Section 1 we introduce cohomological and cofinite ideals for arbitrary triangulated categories and study their basic properties. For a compactly generated triangulated category C, the restricted Yoneda functor

C ⟶ Mod C_{0}, X ↦ H_{X} = Hom(−, X)|_{C_{0}}

is an important tool which translates the triangulated structure of C into the abelian structure of the category Mod C_{0} of C_{0}-modules. A detailed treatment of this functor can be found in [14], but we recall in Section 2 the basic results and prove that any cohomological ideal ℑ in C_{0} is the annihilator of some canonical object E_{ℑ} in C. The correspondence between endofinite objects and cofinite ideals is the main theme of Section 3 which contains the proofs of Theorem A and Theorem B. In the following two sections we apply the results from the previous ones to the stable homotopy category of spectra. In Section 4 we investigate 𝔭-localizations of endofinite spectra and introduce the endolength of a spectrum X which is a numerical invariant, i.e. a family (ℓ_{n}(X))_{n∈ℤ} of numbers ℓ_{n}(X) ∈ ℕ ∪ {∞} such that X is endofinite if and only if ℓ_{n}(X) is finite for all n ∈ ℤ. The final Section 5 is devoted to the criterion for a CW-complex to be endofinite which is formulated in Theorem C.

1. Cohomological ideals

In this section we introduce cohomological and cofinite ideals for arbitrary triangulated categories and study their basic properties. We use the Yoneda functor which embeds a triangulated category into an abelian category.

1.1. The Yoneda functor. Let C be any additive category. A C-module is by definition an additive functor C^{op} → Ab into the category Ab of abelian groups, and for C-modules M and N we denote by Hom(M, N) the class of natural transformations M → N. A sequence L → M → N of maps between C-modules is exact if the sequence L(X) → M(X) → N(X) is exact for all X in C. A C-module M is finitely generated if there exists an exact sequence Hom(−, X) → M → 0 for some X in C, and M is finitely presented if there exists an exact sequence Hom(−, X) → Hom(−, Y) → M → 0 with X and Y in C. Note that Hom(M, N) is a set for every finitely generated C-module M by Yoneda’s lemma. The finitely presented C-modules form an additive category with cokernels which we denote by mod C. It is well-known (and an easy exercise) that mod C is abelian if and only if every map Y → Z in C has a weak kernel X → Y, i.e. the sequence Hom(−, X) → Hom(−, Y) → Hom(−, Z) is exact. In particular, mod C is abelian if C is triangulated.

Suppose now that C is a triangulated category [18]. Any map ψ: Y → Z in C can be completed to a triangle

X ⟶^{ϕ} Y ⟶^{ψ} Z ⟶^{χ} X[1] (∗)

and we call the map ϕ a fibre of ψ and χ a cofibre of ψ. Recall that a functor F: C → A from C into an abelian category A is cohomological if for every triangle (∗) the sequence F (X) → F (Y) → F (Z) → F (X[1]) is exact. Examples of cohomological functors are the representable functors Hom(X, −): C → Ab and Hom(−, X): C^{op} → Ab for any X in C. Another important example is the fully faithful Yoneda functor

H: C ⟶ mod C, X ↦ H_{X} = Hom(−, X).

A map ϕ: X → Y is sent to the natural transformation H_{ϕ}: Hom(−, X) → Hom(−, Y). The Yoneda functor has the following well-known universal property:

Freyd’s Lemma. Every additive functor F: C → A into an abelian category A extends, up to isomorphism, uniquely to a right exact functor F′: mod C → A such that F = F′ ^{∘} H. The functor F′ is exact if and only if F is a cohomological functor.

Proof. See [6, Theorem 3.1].

In order to study the category mod C it is important to observe that any object F in mod C is of the form F = Im H_{ϕ} for some map ϕ: Y → Z in C. In fact, if H_{X} → H_{Y} → F → 0 is a presentation of F, then F = Im H_{ϕ} where ϕ: Y → Z denotes a cofibre of the map X → Y. Moreover, the representable functors H_{X} are projective objects in mod C by Yoneda’s lemma, and they are also injective since C is triangulated. Our next aim is a description of the exact sequences in mod C. To this end we make the following definition:

Definition 1.1. A sequence X′ →^{ϕ′} X →^{ϕ} Y →^{ϕ′′} Y′′ of maps in C is called cohomological exact if ϕ′ is a fibre of the composition ϕ′′ϕ.

The following two lemmas explain the relevance of the cohomological exact sequences.

Lemma 1.2. Let X′ →^{ϕ′} X →^{ϕ} Y →^{ϕ′′} Y′′ be a cohomological exact sequence in C. Then the induced sequence

0 ⟶ Im H_{ϕϕ′} ⟶ Im H_{ϕ} ⟶ Im H_{ϕ′′ϕ} ⟶ 0

is exact in mod C.

Proof. Clearly, the map Im H_{ϕ} → Im H_{ϕ′′ϕ} is an epimorphism, and Im H_{ϕϕ′} is the kernel of this map since ϕ′ is a fibre of ϕ′′ϕ.

Lemma 1.3. Let ε: 0 → F′ → F → F′′ → 0 be an exact sequence in mod C. Then there

exists a cohomological exact sequence X′ →^{ϕ′} X →^{ϕ} Y →^{ϕ′′} Y′′ in C such that the induced sequence

0 ⟶ Im H_{ϕϕ′} ⟶ Im H_{ϕ} ⟶ Im H_{ϕ′′ϕ} ⟶ 0

is isomorphic to ε.

Proof. Let F = Im H_{ϕ} for some map ϕ: X → Y. Taking a monomorphism F′′ → H_{Y′′}, the composition F → F′′ → H_{Y′′} extends to a map H_{Y} → H_{Y′′} since H_{Y′′} is injective. By Yoneda’s lemma, this map is induced by a map ϕ′′: Y → Y′′ and it follows that F′′ = Im H_{ϕ′′ϕ}. Taking for ϕ′: X′ → X a fibre of the composition ϕ′′ϕ, it follows from the preceding lemma that the induced sequence

0 ⟶ Im H_{ϕϕ′} ⟶ Im H_{ϕ} ⟶ Im H_{ϕ′′ϕ} ⟶ 0

is isomorphic to ε.

1.2. Cohomological ideals. Let C be a triangulated category. An ideal ℑ in C consists of subgroups ℑ(X, Y) in Hom(X, Y) for every pair of objects X, Y in C such that for all ϕ in ℑ(X, Y) and all maps α: X′ → X and β: Y → Y′ in C the composition βϕα belongs to ℑ(X′, Y′). Given an additive functor F: C → D, the ideal of maps ϕ in C satisfying F (ϕ) = 0 is called the annihilator of F and is denoted by Ann F. Note that any ideal ℑ in C is of the form ℑ = Ann F for some functor F: C → D since we can take for F the projection C → C/ℑ onto the additive quotient category C/ℑ.

Definition 1.4. An ideal in C is said to be cohomological if it is the annihilator of some cohomological functor C → A.

For example, if F: C → D is an exact functor between triangulated categories, then Ann F is a cohomological ideal since Ann F = Ann(H^{∘} F) where H: D → mod D denotes the Yoneda functor which is faithful.

Recall that a full subcategory S of an abelian category A is a Serre subcategory provided that for every exact sequence 0 → X′ → X → X′′ → 0 in A the object X belongs to S if and only if X′ and X′′ belong to S. The corresponding quotient category A/S is constructed as follows [7]: The objects of A/S are those of A and

Hom_{A/S}(X, Y) = \underrightarrow{lim} Hom_{A}(X′, Y/Y′)

with X′ ⊆ X, Y′ ⊆ Y and X/X′, Y′ ∈ S. Note that the maps in A/S form a set if one assumes that the subobjects of any fixed object in A form a set. Again the category A/S is abelian and there is canonically defined the quotient functor Q: A → A/S such that Q(X) = X; it is exact with Ker Q = S. Here the kernel Ker F of a functor F: A → B is, by definition, the full subcategory of all objects X in A such that F (X) ≅ 0. Now suppose that F: A → B is an exact functor. Then Ker F contains S if and only if F induces a (unique and exact) functor G: A/S → B such that F = G^{∘} Q. Moreover, G is faithful if and only if S = Ker F.

Proposition 1.5. The assignments

ℑ ↦ S_{ℑ} = {Im H_{ϕ} | ϕ ∈ ℑ} and S ↦ ℑ_{S} = {ϕ ∈ C | Im H_{ϕ} ∈ S}

induce mutually inverse bijections between the class of cohomological ideals in C and the class of Serre subcategories in mod C.

Proof. Let A = mod C. Suppose that ℑ = Ann F for some cohomological functor F: C → A′ and denote by F′: A → A′ the unique exact functor extending F which exists by Freyd’s lemma. Clearly, Ker F′ = S_{ℑ}, and this is a Serre subcategory of A since F′ is exact. Conversely, if S is a Serre subcategory of A, then ℑ_{S} = Ann F where F: C → A/S denotes the composition of the Yoneda functor C → A with the quotient functor A → A/S. It is easily checked that these constructions are mutually inverse.

As a consequence of this proposition, we obtain an internal characterization of cohomological ideals in terms of maps in C.

Corollary 1.6. An ideal ℑ in C is cohomological if and only if for every cohomological exact sequence in C

X′ ⟶^{ϕ′} X ⟶^{ϕ} Y ⟶^{ϕ′′} Y′′

the map ϕ belongs to ℑ provided that ϕϕ′ and ϕ′′ϕ are in ℑ.

Proof. Combine the description of exact sequences in mod C given in Lemma 1.2 and Lemma 1.3 with Proposition 1.5.

1.3. Cofinite ideals. Let C be a triangulated category and let A = mod C. Recall that an abelian category is a length category if every object has finite composition length.

Definition 1.7. Let ℑ be a cohomological ideal in C and S_{ℑ} be the corresponding Serre subcategory of A. Then ℑ is called cofinite if the quotient category A/S_{ℑ} is a length category.

In order to give an internal characterization of cofinite ideals in C we need another definition.

Definition 1.8. Let ℑ be a cohomological ideal in C.

(1) An ℑ-sequence is a sequence of maps X →^{ϕ_{0}} X_{0} →^{ϕ_{1}} … →^{ϕ_{n}} X_{n} in C such that for every i ∈ {1, …, n} the composition of ϕ_{i−1} … ϕ_{0} with a fibre of ϕ_{i} … ϕ_{0} does not belong to ℑ.

(2) The ℑ-length of an object X in C is

ℓ_{ℑ}(X) = sup{n ∈ ℕ | there exists an ℑ-sequence X →^{ϕ_{0}} X_{0} →^{ϕ_{1}} … →^{ϕ_{n}} X_{n}}.

We are now in a position to state a characterization of cofinite ideals in C.

Proposition 1.9. A cohomological ideal ℑ in C is cofinite if and only if the ℑ-length of every object in C is finite.

The proof of this proposition requires two lemmas.

Lemma 1.10. Let 0 = F_{0} ⊆ F_{1} ⊆ … ⊆ F_{n} = H_{X} be a chain of subobjects in A.

(1) There exists a sequence X →^{ϕ_{0}} X_{0} →^{ϕ_{1}} … →^{ϕ_{n}} X_{n} of maps in C such that F_{i} = Ker H_{ϕ_{i}...ϕ_{0}} for all i.

(2) Denote for every i by ψ_{i} a fibre of ϕ_{i} … ϕ_{0}. Then F_{i}/F_{i−1} ≅ Im H_{ϕ_{i−1}...ϕ_{0}ψ_{i}} for all i.

Proof. (1) Put X_{0} = X and ϕ_{0} = id_{X}. Choose for every i > 0 a copresentation 0 → F_{i} → H_{X} → H_{X_{i}}. The inclusion F_{i−1} → F_{i} induces a commutative diagram

0 ⟶ F_{i−1} ⟶ H_{X} ⟶ H_{X_{i−1}}
↓ ∥ ↓
0 ⟶ F_{i} ⟶ H_{X} ⟶ H_{X_{i}}

and the map H_{X_{i−1}} → H_{X_{i}} is induced by a map ϕ_{i}: X_{i−1} → X_{i}. This gives the desired

sequence X →^{ϕ_{0}} X_{0} →^{ϕ_{1}} … →^{ϕ_{n}} X_{n}.

(2) An application of the snake lemma shows that F_{i}/F_{i−1} is isomorphic to the kernel of the map Im H_{ϕ_{i−1}...ϕ_{0}} → Im H_{ϕ_{i}...ϕ_{0}} which is Im H_{ϕ_{i−1}...ϕ_{0}ψ_{i}} by Lemma 1.2.

Lemma 1.11. Let ℑ be a cohomological ideal and X ∈ C. Then ℓ_{ℑ}(X) is the length of H_{X} in A/S_{ℑ}.

Proof. Let Q: A → A/S_{ℑ} be the quotient functor. Any chain of subobjects of Q(H_{X}) is

the image of a chain 0 = F_{0} ⊆ F_{1} ⊆ … ⊆ F_{n} = H_{X} in A, and we denote by X →^{ϕ_{0}} X_{0} →^{ϕ_{1}} … →^{ϕ_{n}} X_{n} the sequence in C with F_{i} = Ker H_{ϕ_{i}...ϕ_{0}} which exists by Lemma 1.10. By construction, Q(F_{i})/Q(F_{i−1}) ≠ 0 if and only if F_{i}/F_{i−1} ∉ S_{ℑ}. Furthermore, F_{i}/F_{i−1} ∉ S_{ℑ} if and only if the composition of ϕ_{i−1} … ϕ_{0} with a fibre of ϕ_{i} … ϕ_{0} does not belong to ℑ by Lemma 1.10. Thus ℓ_{ℑ}(X) is the length of Q(H_{X}).

Proof of Proposition 1.9. Any object in A/S_{ℑ} is the quotient of H_{X} for some X in C. Therefore A/S_{ℑ} is a length category if and only if ℓ_{ℑ}(X) is finite for all X by Lemma 1.11.

Now suppose that ℑ is a cofinite cohomological ideal in C and denote by Q: A → A/S_{ℑ} the corresponding quotient functor. Then the assignment

X ↦ S_{X} = {A ∈ A | X does not occur as a composition factor of Q(A)}

induces a bijection between the isomorphism classes of simple objects in A/S_{ℑ} and the maximal Serre subcategories S ≠ A of A containing S_{ℑ}. We include the following consequence for later reference.

Lemma 1.12. Let ℑ be a cofinite cohomological ideal in C. The map X ↦ ℑ_{S_{X}} induces a bijection between the isomorphism classes of simple objects in A/S_{ℑ} and the maximal cohomological ideals 𝔍 ≠ C in C containing ℑ.

Proof. Apply the inclusion preserving bijection from Proposition 1.5 between Serre subcategories of A and cohomological ideals in C.

2. Annihilator ideals

Let C be a compactly generated triangulated category [17]. More precisely, C is a triangulated category and has arbitrary coproducts. An object X in C is called compact if for every family (Y_{i})_{i∈I} in C the canonical map ∐_{i} Hom(X, Y_{i}) → Hom(X, ∐_{i} Y_{i}) is an isomorphism. We denote by C_{0} the full subcategory of compact objects in C and observe that C_{0} is a triangulated subcategory of C. For C being compactly generated the isomorphism classes of objects in C_{0} need to form a set, and Hom(C, X) = 0 for all C in C_{0} implies X = 0 for every object X in C.

2.1. The restricted Yoneda functor. We denote by Mod C_{0} the category of C_{0}- modules, and our main tool for studying the category C is the restricted Yoneda functor

C ⟶ Mod C_{0}, X ↦ H_{X} = Hom(−, X)|_{C_{0}}.

We need to recall the concept of purity for a compactly generated triangulated category which has been introduced in [14]. A map X → Y in C is a pure monomorphism if the induced map Hom(C, X) → Hom(C, Y) is a monomorphism for all C in C_{0}. An object X in C is pure-injective if every pure monomorphism X → Y is a split monomorphism. The following lemma summarizes some essential properties of the restricted Yoneda functor.

Lemma 2.1. The restricted Yoneda functor C → Mod C_{0} identifies the pure-injectives in C with the injectives in Mod C_{0}. Moreover, an object X in C is pure-injective if and only if the map Hom(Y, X) → Hom(H_{Y} , H_{X}), ϕ ↦ H_{ϕ}, is bijective for every Y in C. A map ϕ in C is a pure monomorphism if and only if H_{ϕ} is a monomorphism in Mod C_{0}.

Proof. See Corollary 1.9 and Lemma 1.7 in [14].

We denote by Sp C the set of isomorphism classes of indecomposable pure-injectives in C, and Spec C_{0} denotes the set of isomorphism classes of indecomposable injectives in Mod C_{0}.

2.2. The annihilator. We define the annihilator of an object in C as follows:

Definition 2.2. The annihilator Ann X of an object X in C is the annihilator of the functor Hom(−, X)|_{C_{0}}.

Note that Ann X is a cohomological ideal in C_{0} since Hom(−, X)|_{C_{0}} is a cohomological functor. Given a cohomological ideal ℑ in C_{0}, we denote by E_{ℑ} the product of all X in Sp C with ℑ ⊆ Ann X. The following result shows that the indecomposable pureinjectives in C control the cohomological ideals in C_{0}.

Theorem 2.3. Let ℑ be a cohomological ideal in C_{0} and X be an object in C. (1) ℑ = Ann E_{ℑ}.

(2) ℑ ⊆ Ann X if and only if there exists a pure monomorphism X → ∏_{I} E_{ℑ} for some set I.

The theorem is a consequence of a result about the category of C_{0}-modules which we take from [12].

Lemma 2.4. Let S be a Serre subcategory of mod C_{0} and let E_{S} be the product of all Y ∈ Spec C_{0} with Hom(S, Y) = 0. Also, let X be an object in Mod C_{0}.

(1) If X ∈ mod C_{0}, then X ∈ S if and only if Hom(X, E_{S}) = 0.

(2) Hom(S, X) = 0 if and only if there exists a monomorphism X → ∏_{I} E_{S} for some set I.

Proof. See [12, Proposition 3.2].

Proof of Theorem 2.3. Let S = S_{ℑ} be the Serre subcategory of mod C_{0} corresponding to ℑ. Given a map ϕ in C_{0} and an object X in C, we have ϕ ∈ Ann X if and only if Hom(Im H_{ϕ}, H_{X}) = 0; in particular, ℑ ⊆ Ann X if and only if Hom(S, H_{X}) = 0. Therefore the restricted Yoneda functor C → Mod C_{0} identifies E_{ℑ} with E_{S} and induces a bijection Hom(X, E_{ℑ}) → Hom(H_{X}, E_{S}) for every X in C by Lemma 2.1. The assertion of the theorem now follows from these observations and the preceding lemma.

3. Endofiniteness

Let C be a compactly generated triangulated category. The notion of an endofinite object has been introduced by Crawley-Boevey for so-called locally finitely presented additive categories [4]. An analogue of this concept for compactly generated triangulated categories has been studied in [15]. Let us recall this definition.

Definition 3.1. An object X in C is called endofinite if for every compact object C in C the End(X)-module Hom(C, X) has finite composition length.

3.1. A characterization. We have the following characterization of endofinite objects in terms of cohomological ideals.

Theorem 3.2. An object X in C is endofinite if and only if the annihilator of X is a cofinite ideal in C_{0}.

Proof. Let Λ_{X} = End(X) and Γ_{X} = End(H_{X}). We define A = mod C_{0} and observe that the map Hom(−, C) ↦ Hom(C, −) induces an equivalence A^{op} → mod(C_{0}^{op}). Thus we can view the C_{0}-module H_{X}: C_{0}^{op} → Ab as an exact functor F_{X}: A^{op} → Ab by Freyd’s lemma. In fact, F_{X} is an exact functor A^{op} → Mod Γ_{X} in the obvious way, and the Serre subcategory S = S_{Ann X} of A corresponding to Ann X is the kernel of F_{X}. In [13, Theorem 10.2] it is shown that F_{X}(A) is a finite length Γ_{X}-module for every A in A if and only if A/S is a length category. Suppose now that X is endofinite. Then F_{X}(A) is for every A in A a finite length Λ_{X}-module, and therefore also a finite length Γ_{X}-module since the Λ_{X}-action is induced by the canonical map Λ_{X} → Γ_{X}. Thus A/S is a length category and Ann X is cofinite. Conversely, if Ann X is cofinite, then F_{X}(A) is a finite length Γ_{X}-module for every A in A. It follows from [13, Theorem 9.6] and the characterization of pure-injectives in [14, Theorem 1.8] that X is pure-injective. Thus the canonical map Λ_{X} → Γ_{X} is an isomorphism by Lemma 2.1, and therefore F_{X}(A) is of finite length over Λ_{X} for all A. We conclude that X is endofinite, and this finishes the proof.

3.2. Decomposing endofinite objects. Endofinite objects have nice decomposition properties. We recall the basic result which has been established in [15].

Proposition 3.3. An endofinite object X is pure-injective and has, up to isomorphism, a unique decomposition X ≅ ∐_{i∈I} X_{i} into indecomposable objects with local endomorphism ring.

Proof. See [15, Theorem 1.2].

The indecomposable endofinite objects can be described in terms of certain cohomological ideals. In order to state this result we denote by Max C_{0} the set of maximal elements among the cofinite cohomological ideals ℑ ≠ C_{0} in C_{0}.

Theorem 3.4. The assignments ℑ ↦ E_{ℑ} and X ↦ Ann X induce mutually inverse bijections between Max C_{0} and the set of isomorphism classes of indecomposable endofinite objects in C.

Proof. Suppose that ℑ is a cofinite cohomological ideal in C_{0} and let S = S_{ℑ} be the corresponding Serre subcategory of A = mod C_{0}. Then A/S is a length category. We denote by T the localizing subcategory of Mod C_{0} which is generated by S, i.e. T is the smallest Serre subcategory of Mod C_{0} containing S which is closed under taking coproducts. The quotient category Mod C_{0}/T is locally finite in the sense of [7, II.4] since the quotient functor Q: Mod C_{0} → Mod C_{0}/T sends A to a full subcategory of Mod C_{0}/T which is equivalent to A/S and generates Mod C_{0}/T ; see [12, Theorem 2.6] for details. In fact, A/S is equivalent to the full subcategory of finite length objects in Mod C_{0}/T . The quotient functor Q identifies the indecomposable injective objects X in Mod C_{0} satisfying Hom(S, X) = 0 with the indecomposable injective objects in Mod C_{0}/T [7, III.3]. Therefore the indecomposable pure-injective objects X in C satisfying ℑ ⊆ Ann X are identified via X ↦ Q(H_{X}) with the indecomposable injective objects in Mod C_{0}/T by Lemma 2.1. Every indecomposable injective object in a locally finite category is the injective envelope of a simple object [7, IV.1]. Therefore card{X ∈ Sp C | ℑ ⊆ Ann X} = 1 if and only if A/S has a unique simple object. The cofinite cohomological ideal ℑ is maximal if and only if A/S has a unique simple object by Lemma 1.12, and therefore E_{ℑ} is indecomposable if and only if ℑ is a maximal cofinite cohomological ideal in C_{0}. Moreover, E_{ℑ} is endofinite by Theorem 3.2 since ℑ = Ann E_{ℑ} which holds by Theorem 2.3.

Now suppose that X ∈ Sp C is endofinite and let ℑ = Ann X which is cofinite by Theorem 3.2. Let S be a simple object in A/S_{ℑ} which we view as a simple object in Mod C_{0}/T . By construction, Hom(S, Q(H_{X})) ≠ 0 and therefore Q(H_{X}) is isomorphic to an injective envelope of S. It follows that S is the unique simple object in A/S_{ℑ}. Thus ℑ is a maximal cofinite cohomological ideal in C_{0}; in particular X = E_{ℑ}. This finishes the proof.

We discuss now a number of consequences of Theorem 3.2 and Theorem 3.4. The following lemma will be useful.

Lemma 3.5. Let (X_{i})_{i∈I} be a family of objects in C. Then

Ann ∐_{i∈I} X_{i} = ⋂_{i∈I} Ann X_{i} = Ann ∏_{i∈I} X_{i}.

Proof. Use the isomorphisms ∐_{i} Hom(C, X_{i}) ≅ Hom(C, ∐_{i} X_{i}) and Hom(C, ∏_{i} X_{i}) ≅ ∏_{i} Hom(C, X_{i}) for C in C_{0}.

Corollary 3.6. Let X be an endofinite object in C with ℑ = Ann X. Then

ℑ = ⋂_{ℑ⊆𝔍∈Max C_{0}} 𝔍

and X has a unique decomposition

X ≅ ∐_{ℑ⊆𝔍∈Max C_{0}} X_{𝔍}

such that each X_{𝔍} is a non-empty coproduct of copies of E_{𝔍}.

Proof. Let X ≅ ∐_{i∈I} X_{i} be the decomposition into indecomposable objects which exists by Proposition 3.3. The annihilator Ann X_{i} contains Ann X and is therefore cofinite by Theorem 3.2. Thus X_{i} is endofinite for all i. Applying Theorem 3.4 and Lemma 3.5 we obtain

ℑ = Ann X = ⋂_{i∈I} Ann X_{i} = ⋂_{ℑ⊆𝔍∈Max C_{0}} 𝔍.

Now define for every 𝔍 ∈ Max C_{0} containing ℑ

X_{𝔍} = ∐_{Ann X_{i}=𝔍}X_{i}.

Clearly, this gives the desired decomposition

X ≅ ∐_{ℑ⊆𝔍∈Max C_{0}} X_{𝔍}.

It remains to show that X_{𝔍} ≠ 0 for every 𝔍 ∈ Max C_{0} containing ℑ. We apply the ideas from the proof of Theorem 3.4 and keep the same notation. We fix 𝔍. The ideal 𝔍 corresponds to a simple object S in A/S_{ℑ} by Lemma 1.12 which we view as a simple object in Mod C_{0}/T . The injective envelope of S is Q(H_{E_{𝔍}}) which is isomorphic to a direct factor of Q(H_{X}) since Hom(S, Q(H_{X})) ≠ 0. It follows that E_{𝔍} is isomorphic to a direct factor of X since Hom(E_{𝔍}, X) ≅ Hom(Q(H_{E_{𝔍}}), Q(H_{X})). The uniqueness of the decomposition X ≅ ∐_{i∈I} X_{i} implies that X_{i} ≅ E_{𝔍} for some i ∈ I, and therefore X_{𝔍} ≠ 0. This completes the proof.

Corollary 3.7. Let X and Y be objects in C and suppose that X is endofinite. Then Ann X ⊆ Ann Y if and only if Y is isomorphic to a direct factor of a (co)product of copies of X.

Proof. If Y is isomorphic to a direct factor of a (co)product of copies of X, then Ann X ⊆ Ann Y by Lemma 3.5. To prove the converse suppose that Ann X ⊆ Ann Y. Observe that Y is endofinite by Theorem 3.2 since Ann Y is cofinite. Let X ≅ ∐_{i∈I} X_{i} and Y ≅ ∐_{j∈J} Y_{j} be the decompositions into indecomposables which exist by Proposition 3.3. It follows from Corollary 3.6 that every factor Y_{j} is isomorphic to X_{i} for some i ∈ I since Y_{j} ≅ E_{𝔍} for some 𝔍 ∈ Max C_{0} containing Ann Y and therefore containing Ann X. The family of split monomorphisms Y_{j} → X induces a split monomorphism Y → ∐_{J} X and the composition with the canonical map ∐_{J} X → ∏_{J} X is again a split monomorphism since it is a pure monomorphism and Y is pure-injective by Proposition 3.3. We conclude that Y is isomorphic to a direct factor of a (co)product of copies of X.

Corollary 3.8. An indecomposable object X is endofinite if and only if every product of copies of X is a coproduct of copies of X.

Proof. Suppose first that X is indecomposable and endofinite. Any product ∏_{I} X is again endofinite by Theorem 3.2, since Ann ∏_{I} X = Ann X by Lemma 3.5. The decomposition of ∏_{I} X as a coproduct of copies of X then follows directly from Corollary 3.6. Now suppose that X is any indecomposable object such that every product of copies of X is a coproduct of copies of X. We consider the corresponding C_{0}-module M = H_{X}, and it is clear that every product of copies of M is a coproduct of copies of M. It follows from [8, Théorème] that M is a pure-injective C_{0}-module. In fact, M is injective since M is fp-injective by [14, Lemma 1.6] and therefore every monomorphism M → N in Mod C_{0} is a pure monomorphism, hence a split monomorphism. It follows from Lemma 2.1 that X is a pure-injective object in C. Viewing X as an exact functor F_{X}: A^{op} → Ab for A = mod C_{0} as in the proof of Theorem 3.2, it follows from the characterization of endofinite objects in [13, Corollary 10.5] that X is also an endofinite object in C.

4. Endofinite spectra

In the following we apply the results of the previous sections to the stable homotopy category C in the sense of [1]. The objects are CW-spectra, and the morphisms X → Y between two spectra X and Y are denoted by {X, Y }. We identify each CW-complex with its suspension spectrum. The stable homotopy category is a triangulated category where the suspension X ↦ ΣX = X ∧ S^{1} acts as a translation functor. In particular, S^{n} = Σ^{n}(S^{0}) is defined for all n ∈ ℤ, and for a spectrum X we denote by π_{n}^{s}(X) the n-th stable homotopy group {S^{n}, X}. Coproducts are given by one point unions called wedge, and arbitrary coproducts do exist in C. Recall that a spectrum X is finite if for some n ∈ ℕ the spectrum Σ^{n}X is the suspension spectrum of a finite CW-complex. Note that the finite spectra are precisely the compact objects in C. Therefore C is a compactly generated triangulated category since π_{n}^{s}(X) = 0 for all n ∈ ℤ implies X = 0. These facts are well known, see for instance [16].

4.1. 𝔭-localization. We denote for every prime ideal 𝔭 in ℤ by X → X_{𝔭} the 𝔭localization map for a spectrum X. The fibre of the rational localization X → X_{(0)} is denoted by X_{tor}.

Theorem 4.1. Let X be an endofinite spectrum.

(1) The 𝔭-localization map X → X_{𝔭} is a split epimorphism for every prime ideal 𝔭 in ℤ.

(2) X ≅ X_{tor} ∨ X_{(0)}.

(3) Let Y = X_{tor}. Then Y ≅ ⋁_{𝔭} Y_{𝔭} where 𝔭 runs through all non-zero prime ideals in ℤ.

Proof. (1) Fix a prime ideal 𝔭. The localization map X → X_{𝔭} induces, by definition, for every n ∈ ℤ an isomorphism π_{n}^{s}(X) ⊗_{ℤ} ℤ_{𝔭} ≅ π_{n}^{s}(X_{𝔭}) which extends to an isomorphism

{C, X} ⊗_{ℤ} ℤ_{𝔭} ≅ {C, X_{𝔭}}

for all C in C_{0}. Therefore Ann X ⊆ Ann X_{𝔭}. If X is indecomposable endofinite, then X_{𝔭} is a direct factor of a wedge of copies of X by Corollary 3.7. Thus X_{𝔭} = X or X_{𝔭} = 0, and therefore X → X_{𝔭} is a split epimorphism. An arbitrary endofinite spectrum X is a wedge of indecomposable endofinite spectra by Proposition 3.3, and therefore X → X_{𝔭} is a split epimorphism since 𝔭-localization preserves wedges; e.g. see [16, Corollary 8.3].

(2) The sequence 0 → X_{tor} → X → X_{(0)} → 0 is split exact by (1).

(3) Suppose first that Y is indecomposable. There exists a prime ideal 𝔮 ≠ 0 such that {C, Y } ⊗_{ℤ} ℤ_{𝔮} ≠ 0 for some C in C_{0}. Thus Ann Y_{𝔮} ≠ C_{0} and therefore Y = Y_{𝔮} by (1). Moreover, Y_{𝔭} = 0 for all 𝔭 ≠ 𝔮 since Y is torsion, and Y ≅ ⋁_{𝔭} Y_{𝔭} follows. Finally, the assertion holds for arbitrary Y since 𝔭-localization preserves wedges.

4.2. The endolength. Let X be any spectrum. We denote for every n ∈ ℤ by ℓ_{n}(X) the length of the stable homotopy group π_{n}^{s}(X) as {X, X}-module, and call

ℓ(X) = (ℓ_{n}(X))_{n∈ℤ}

the endolength of X. The endolength leads to a nice characterisation of endofinite spectra:

Proposition 4.2. A spectrum X is endofinite if and only if ℓ_{n}(X) is finite for all n ∈ ℤ.

Proof. Consider the full subcategory C_{X} of spectra Y such that {Σ^{n}Y, X} is of finite length over {X, X} for all n ∈ ℤ. This is easily seen to be closed under cofibrations and retracts, and therefore C_{X} is a thick subcategory of C. On the other hand, C_{0} is the smallest thick subcategory containing the sphere spectrum S^{0}; e.g. see [10, Theorem 2.1.3]. Consequently, if S^{0} is an object of C_{X} then C_{0} is contained in C_{X}. In particular, X is endofinite.

We continue with another interpretation of the length ℓ_{n}(X). Let ℑ = Ann X and denote by S = S_{ℑ} the corresponding Serre subcategory of A = mod C_{0}. The localizing subcategory of Mod C_{0} generated by S is denoted by T , and the quotient functor Q: Mod C_{0} → Mod C_{0}/T identifies A/S with a full subcategory of Mod C_{0}/T ; see [12, Theorem 2.6].

Lemma 4.3. Let X be a pure-injective spectrum. Then ℓ_{n}(X) equals the length of H_{S^{n}} in Mod C_{0}/T which is also the length of H_{S^{n}} in A/S; in particular ℓ_{n}(X) = ℓ_{ℑ}(S^{n}).

The proof of this lemma is essentially an application of the following easy lemma:

Lemma 4.4. Let M and N be objects in an abelian category and suppose that N is an injective cogenerator. Then the length of M equals the length of the End(N)-module Hom(M, N).

Proof. Straightforward.

Proof of Lemma 4.3. Put M = Q(H_{S^{n}}) and N = Q(H_{X}), and observe that

{Y, X} ≅ Hom(H_{Y} , H_{X}) ≅ Hom(Q(H_{Y}), Q(H_{X}))

for all Y in C by Lemma 2.1. Also, Q(H_{X}) is injective by the same lemma since X is pureinjective. The assertion now follows from the preceding lemma since Hom(M′, N) ≠ 0 for all M′ = Q(A) ≠ 0 with A ∈ A. Moreover, M′ is a simple object in Mod C_{0}/T if and only if M′ is simple when regarded as an object in A/S. Finally, ℓ_{n}(X) = ℓ_{ℑ}(S^{n}) follows from Lemma 1.11.

We are now in a position to compute the endolength of a spectrum X from a decomposition X ≅ ⋁_{i} X_{i}. We proceed in two steps.

Proposition 4.5. Let X be a spectrum and I be a non-empty set. Suppose that ⋁_{I} X is pure-injective. Then for all n ∈ ℤ

ℓ_{n}(⋁_{I} X) = ℓ_{n}(X) = ℓ_{n}(∏_{I} X).

Proof. The assertion follows from Lemma 4.3 since Ann ⋁_{I} X = Ann X = Ann ∏_{I} X by Lemma 3.5.

Proposition 4.6. Let X ≅ ⋁_{i∈I} X_{i} be a decomposition of a pure-injective spectrum. Suppose that any indecomposable spectrum arises as a wedge summand of X_{i} for at most one i ∈ I. Then for all n ∈ ℤ

ℓ_{n}(⋁_{i∈I} X_{i}) = ∑_{i∈I} ℓ_{n}(X_{i}).

Proof. We keep the above notation and denote for every i ∈ I by Q_{i}: Mod C_{0}/T → Mod C_{0}/T_{i} the quotient functor with respect to the localizing subcategory T_{i} which is generated by S_{Ann X_{i}}. Now suppose that A ≠ 0 is any subquotient of Q(H_{S^{n}}) in A/S. Then Q_{i}(A) ≠ 0 for some i ∈ I since

0 ≠ Hom(A, Q(H_{X})) ≅ ∐_{i} Hom(A, Q(H_{X_{i}})) ≅ ∐_{i} Hom(Q_{i}(A), Q_{i} ^{∘} Q(H_{X_{i}})).

If A is simple then Q_{i}(A) ≠ 0 for precisely one i ∈ I by our assumption on the family (X_{i})_{i∈I}, since the injective envelope of A in Mod C_{0}/T is isomorphic to Q(H_{Y}) for some indecomposable summand Y of X. Using the exactness of the Q_{i}, the formula for ℓ_{n}(⋁_{i∈I} X_{i}) follows from the fact that ℓ_{n} computes the length of Q(H_{S^{n}}) by Lemma 4.3.

Corollary 4.7. Let X be an endofinite spectrum with ℑ = Ann X. Then for all n ∈ ℤ

ℓ_{n}(X) = ∑_{ℑ⊆𝔍∈Max C_{0}} ℓ_{n}(E_{𝔍}).

Proof. Combine Proposition 4.5 and Proposition 4.6 with the decomposition X ≅ ⋁_{𝔍} X_{𝔍} from Corollary 3.6.

4.3. Subgroups of finite definition. Let X be any spectrum. Given a finite spectrum F and an element ϕ ∈ π_{n}^{s}(F) for some n ∈ ℤ, we denote by π_{n,ϕ}^{s}(X) the image of the induced map

{ϕ, X}: {F, X} ⟶ {S^{n}, X} = π_{n}^{s}(X).

A subgroup of π_{n}^{s}(X) which is of the form π_{n,ϕ}^{s}(X) for some ϕ ∈ π_{n}^{s}(F) is said to be of finite definition. Note that any subgroup of finite definition is automatically a {X, X}-submodule of π_{n}^{s}(X).

Lemma 4.8. Let ϕ ∈ π_{n}^{s}(F) for some finite spectrum F.

(1) The subgroups of finite definition form a lattice in π_{n}^{s}(X) for every spectrum X.

(2) If (X_{i})_{i∈I} is a family of spectra, then π_{n,ϕ}^{s}(⋁_{i} X_{i}) ≅ ∐_{i} π_{n,ϕ}^{s}(X_{i}) and π_{n,ϕ}^{s}(∏_{i} X_{i}) ≅ ∏_{i} π_{n,ϕ}^{s}(X_{i}).

Proof. (1) Let ϕ_{i} ∈ π_{n}^{s}(F_{i}) (i = 1, 2). Then π_{n,ϕ_{1}}^{s}(X) + π_{n,ϕ_{2}}^{s}(X) = π_{n,ϕ}^{s}(X) where ϕ: S^{n} → F_{1} ∨ F_{2} has components ϕ_{i}. Let ψ: F_{1} ∨ F_{2} → G be a cofibre of ϕ, with components ψ_{i}, and set χ = ψ_{1}ϕ_{1}. The sequence {G, X} → {F_{1} ∨ F_{2}, X} → {S^{n}, X} is exact, and it follows that π_{n,ϕ_{1}}^{s}(X) ∩ π_{n,ϕ_{2}}^{s}(X) = π_{n,χ}^{s}(X).

(2) Clear, since {ϕ, −} preserves wedges and products.

The endofiniteness of a spectrum can be characterized in terms of the lattice of subgroups of finite definition.

Proposition 4.9. A spectrum X is endofinite if and only if π_{n}^{s}(X) has the ascending and descending chain conditions on subgroups of finite definition for all n ∈ ℤ. If this holds, then the {X, X}-submodules of π_{n}^{s}(X) are precisely the subgroups of finite definition.

Proof. As for modules, see [3, Proposition 4.1].

For the sake of completeness we include the following characterization of the spectra having the descending chain condition on subgroups of finite definition:

Theorem 4.10. The following conditions are equivalent for a spectrum X: (1) X is Σ-pure-injective, i.e. ⋁_{I} X is pure-injective for every set I;

(2) the canonical map ⋁_{ℕ} X → ∏_{ℕ} X is a split monomorphism;

(3) the descending chain condition holds for the subgroups of finite definition of π_{n}^{s}(X) for every n ∈ ℤ;

(4) every product of copies of X is a wedge of indecomposable spectra with local endomorphism rings.

Proof. Adapt the argument for modules [11], using the restricted Yoneda functor C → Mod C_{0}.

Remark 4.11. Concepts and results from this section carry over to any compactly generated triangulated category C if one replaces the sphere spectrum S with any set of compact objects which generates C.

5. A criterion for endofiniteness

In this section we give a criterion for the suspension spectrum of a CW-complex to be endofinite in terms of its singular homology groups. Following the definition of an endofinite module [3], we say that a CW-complex is endofinite if the stable homotopy groups π_{n}^{s}(X) are modules of finite length over {X, X} for all n ∈ ℤ.

Theorem 5.1. Let X be a connected CW-complex such that the singular homology groups H_{n}(X; ℤ) are finite for all n > 0. Then X is an endofinite CW-complex, and therefore its suspension spectrum is also endofinite.

Proof. We prove that a CW-complex X satisfying the above assumptions has finite stable homotopy groups, i.e. π_{n}^{s}(X) is finite for all n ∈ ℤ. Clearly, this implies that X is an endofinite CW-complex, and therefore the suspension spectrum of X is endofinite by Proposition 4.2. To prove the finiteness of the stable homotopy groups, we apply the Atiyah-Hirzebruch-Whitehead spectral sequence for the (unreduced) homology theory h_{n}(−) = π_{n}^{s}(−_{+}). The subscript + means adding of a disjoint basepoint. The coefficient groups h_{n}(pt.) are the stable homotopy groups of S^{0} and are known to be trivial for n < 0, isomorphic to ℤ for n = 0, and finite for n > 0. Therefore, the E^{2}-term of the spectral sequence, E_{p,q}^{2} = H_{p}(X; h_{q}(pt.)), has non-trivial values only for p, q ≥ 0 and these are finite with the only exception E_{0,0}^{2} = ℤ. This form is preserved for all E^{k}-terms, k ≥ 2.

The spectral sequence converges to h_{∗}(X). This means that h_{n}(X) is the limit of groups F_{0,n} ⊆ F_{1,n−1} ⊆ … which are inductively calculated from F_{0,n} = E_{0,n}^{∞} and the extension problem 0 → F_{p−1,n−p+1} → F_{p,n−p} → E_{p,n−p}^{∞} → 0. From this inductive construction it follows easily that F_{p,n−p} = 0 for n < 0 and h_{n}(X) = F_{n,0} is finite for n > 0. For the respective reduced homology theory h̃_{n}(−) = π_{n}^{s}(−) we get

π_{n}^{s}(X) = π_{n}^{s}(X_{+})/π_{n}^{s}(S^{0}) = \begin{cases}0 & for n ≤ 0, \\ finite & for n > 0. \end{cases}

Important examples of CW-complexes with finite homology are the classifying spaces of finite groups.

Corollary 5.2. The classifying space BG of a finite group G is endofinite.

Proof. Since the homology groups H_{n}(BG; ℤ) are finite in all positive degrees we may apply the preceding theorem.

In Theorem 5.1, the condition on X to be connected cannot be omitted. For example, X = S^{0} has finite homology in all positive degrees but the {X, X}-module π_{0}^{s}(X) ≅ ℤ is not of finite length.

It has been pointed out by Stefan Schwede that Theorem 5.1 can be generalized as follows.

Theorem 5.3. A connected CW-spectrum X is endofinite if and only if the homology groups H_{n}(X; ℤ) with coefficients in ℤ are modules of finite length over {X, X} for all n ∈ ℤ.

Proof. The argument is essentialy that given in the proof of Theorem 5.1. To prove endofiniteness of X one uses the Atiyah-Hirzebruch-Whitehead spectral sequence

H_{∗}(X; π_{∗}^{s}) ⟹ π_{∗}^{s}(X)

as above. To prove finiteness of the homology of X one uses the less well-known spectral sequence

Tor^{π_{∗}^{s}}(π_{∗}^{s}(X), ℤ) ⟹ H_{∗}(X; ℤ).

Both spectral sequences are natural in X and can be found in [5].

Acknowledgement. Both authors would like to thank Stefan Bauer and Hans-Werner Henn for a number of helpful conversations concerning the material of this paper. The second named author was supported by the SFB 343.

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Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany

E-mail address: henning@mathematik.uni-bielefeld.de, reichenb@mathematik.uni-bielefeld.de