Homology, Homotopy and Applications, vol.3, No.3, 2001, pp.55–85

TENSOR PRODUCTS OF SPECTRA AND LOCALIZATIONS

FRIEDRICH W. BAUER

(communicated by Walter Tholen)

Abstract 1) The tensor product of two spectra, different from the ∧– product, is introduced in such a way that a Künneth theorem holds. 2) Localizations of spectra are treated by using the more algebraic category of chain functors (instead of the category of CW-spectra). 3) The localization of a given chain functor K_{∗} can up to chain homotopy be expressed by tensoring K_{∗} with the localization of a fixed chain functor ℤ_{∗}.

0. Introduction

Concerning details about chain functors we refer to [5] §4, resp. §5 for the motivation. A chain functor K K K_{∗} = {K_{∗}, K′_{∗}, i′, l, φ, κ} consists of a functor

K_{∗} : 𝔎 ⟶ ch (= category of chain complexes),  (1)

𝔎 a suitable category of pairs of topological spaces or of spectra, a subfunctor l : K′_{∗} ⊂ K_{∗}, a natural inclusion i′ : K_{∗}(A) ⊂ K′_{∗}(X, A) together with some other structural (non-necessary natural) mappings φ : K′_{∗}(X, A) ⟶ K_{∗}(X), κ : K_{∗}(X) ⟶ K′_{∗}(X, A), satisfying certain properties. Every homology theory h_{∗}( ) = {h_{n}( ), ∂, n ∈ ℤ} admits a chain functor such that the derived homology homology H_{∗}(K_{∗}(X, A)) is naturally isomorphic to h_{∗}(X, A). In addition the boundary ∂ : h_{∗}(X, A) ⟶ h_{n−1}(A) is determined by the chain functor K K K. This last property distinguishes chain functors from simple functors (1) admitting natural sequences

0 ⟶ K_{∗}(A) ⟶^{i_{∗}} K_{∗}(X) ⟶^{j_{∗}} K_{∗}(X, A) ⟶ 0  (2)

giving rise to a boundary operator

∂ : H_{∗}(K_{∗}(X, A)) ⟶ H_{n−1}(K_{∗}(A))

(called chain theories, which are only available for ordinary, but not necessarily for generalized homology theories). In the present paper we are dealing with the following three objectives:

1) Definition of tensor products of spectra in such a way that, unlike for ∧– products, a Künneth–theorem holds (§1, 2).

Received February 24, 2000, revised November 23, 2000; published on February 3, 2001. 2000 Mathematics Subject Classification: 55P60, 55P42, 18G15, 18G35. Key words and phrases: tensor products; homology theories associated with a chain functor; localizations of chain functors; regular and irregular–chain–functors. \begin{matrix}c \\ ◯ \end{matrix} 2001, Friedrich W. Bauer. Permission to copy for private use granted.

2) Establishing localizations of chain functors and verification of the Bousfield exact sequence of a localization but now for chain functors instead of spectra (§5). Finally:

3) Expressing localizations of chain functors by tensoring them with the localization of a specific chain functor (cf. §3 resp. §6). Therefore localizations can, at least up to a chain homotopy, be described, similar to the algebraic case of group–localization, by means of a tensor product (cf. [11] appendix). Although our main objective lies in the localization of CW–spectra (i.e. of elements of the Boardman category), we deal mainly with chain functors, recalling that there is a close relationship between CW–spectra A, the associated homology theory A_{∗} and the chain functors C_{∗} whose related homology theory H_{∗}(C_{∗}) is isomorphic to A_{∗} (cf. §4 in ). More precisely: Every generalized homology theory h_{∗} (even on Top^{2} or a suitable subcategory) originates from a chain functor C_{∗} such that h_{∗} ≈ H_{∗}(C_{∗}) (cf. theorem 8.1. in [2]) while on the other hand, each chain functor C_{∗} with compact carrier on CW^{2} = 𝔓^{2}, the category of CW–pairs defines a CW– spectrum |C_{∗}| such that |C_{∗}|_{∗} ≈ H_{∗}(C_{∗}) (theorem 1.1. in [3]). Together with the first result this leads to E. H. Brown’s representation theorem for homology theories. In §1 we introduce the tensor product of chain functors K_{∗} ⊗ L_{∗} which is defined in analogy to the classical ⊗–product of chain complexes (cf. [8]). Due to this fact there is a Künneth theorem (theorem 1.7.) for tensor products of chain functors. The tensor–product of spectra A, B is defined in §2 by assigning to A_{∗}, B_{∗}, the associated homology theories, the chain functors ^{A}C_{∗}, ^{B}C_{∗} and setting

A ⊗ B = |^{A}C_{∗} ⊗ ^{B}C_{∗}|.

The independence of A ⊗ B of the choices of ^{A}C_{∗}, ^{B}C_{∗} can be confirmed by using the main result of [5], where among other things the concept of a chain functor, a mapping between chain functors as well as some other relevant details are recorded. In particular, we call chain functors, whose associated homology satisfies an excision axiom (hence becoming in this way a generalized homology theory) regular. Moreover there is need for structures which do not enjoy all properties of a chain functor (in particular not of a regular one). We call, by an abuse of notation, these structures irregular–chain–functors (cf. [5] definition 4.1.2). Unlike in [5] we do not insist, that a regular chain functor has automatically compact carriers. In §3 we present an example of such an irregular–chain–functor ℤ_{∗} ℤ_{∗} ℤ_{∗} (which is not associated with any spectrum, because the realization theorem does not apply) having the property that for any chain functor K_{∗} one obtains an isomorphism of irregular chain functors

K_{∗} ⊗ ℤ ℤ ℤ_{∗} ≈ K_{∗}

(theorem 3.1.). In §5 we describe the process of 𝔏–localization with respect to a subcategory 𝔏 ⊂ 𝔎^{2} (= the category on which the chain functors are defined). Theorem 5.5. asserts the existence of a Bousfield exact sequence for the localizations of chain functors:

𝔏A_{∗} ⟶ A_{∗} ⟶ A_{𝔏∗}

with 𝔏–acyclic _{𝔏}A_{∗} and 𝔏– local A_{𝔏∗}. (cf. definition 4.4. resp. 5.1.). We may regard as a special case the category 𝔏 consisting of one single object E ∈ 𝔎 (with the identity as single morphism) (corollary 5.6.). If 𝔎^{2} is a subcategory of Top^{2} and E ∈ 𝔅 (=Boardman category) a spectrum, then we can look upon E as a specific subcategory of 𝔎 and define A_{E ∗} by using this category (corollary 5.8.). In §6 we prove the main theorem (theorem 6.5.) asserting that the 𝔏–localization sequence (4) in §5) can be achieved, up to a chain homotopy equivalence, by tensoring A_{∗} with the specific 𝔏–localization of ℤ ℤ ℤ_{∗}. This constitutes the topological analogue of a well-known algebraic result about localizations of a group at a set of primes. Recently the authors of [9] developed another algebraic approach to stable homotopy theory together with an appropriate concept of a ∧- product. The proofs in §5, §6 are preceeded by some general results in §4. All notations and definitions concerning chain functors are recorded in [5] §4 to which we freely refer without further mentioning. The results of the present paper are reported in the expository article [4].

1. Tensor products of chain functors

The concept of a chain functor and the associated homology is recorded in §4. Let K_{∗}, L_{∗} be regular or irregular–chain–functors as defined in [5] §4, definition 4.1., then we define their tensor product (K ⊗ L)_{∗} :

Suppose (X, A) ∈ 𝔎^{2}, then we set

(K ⊗ L)_{n}(X, A) = ⨁_{p+q=n} K_{p}(X, A) ⊗ L_{q}(X, A) ⊕  (1)

⊕ ⨁_{p+q=n} K_{p}(X, A) ⊗ L_{q}(X) ⊕ ⨁_{p+q=n} K_{p}(X) ⊗ L_{q}(X, A)

where the boundary

d_{n} : (K ⊗ L)_{n}(X, A) ⟶ (K ⊗ L)_{n−1}(X, A)

is defined as usual for tensor products of chain complexes. These direct sums are to be understood as amalgamated sums whenever inclusions like j : X ⊂ (X, A) which induce inclusions (e.g. for regular chain complexes) are involved: So, for example,

x ⊗ y ∈ K_{p}(X, A) ⊗ L_{q}(X)

is identified with

x ⊗ j_{#}(y) ∈ K_{p}(X, A) ⊗ L_{q}(X, A)

As a result for regular chain functors this reduces to:

(1’) (K ⊗ L)_{n}(X, A) = ⨁_{p+q=n} K_{p}(X, A) ⊗ L_{q}(X, A)

which is the ordinary tensor product of chain complexes (cf. [8]). We define:

(K ⊗ L)′_{n}(X, A) = ⨁_{p+q=n} K′_{p}(X, A) ⊗ L′_{q}(X, A) ⊕  (2)

⊕ ⨁_{p+q=n} K′_{p}(X, A) ⊗ L_{q}(A, A) ⊕ ⨁_{p+q=n} K_{p}(A, A) ⊗ L′_{q}(X, A).

Since K_{∗}(·) and L_{∗}(·) are free, we have for regular chain functors an inclusion

l : (K ⊗ L)′_{∗}(X, A) ⊂ (K ⊗ L)_{∗}(X, A)

For irregular–chain–functors there is still a natural l available, which is not monic. The mappings i′^{K} : K_{∗}(A) ⟶ K′_{∗}(X, A) resp. for L induce a chain mapping i′ : (K ⊗ L)_{∗} (A) ⟶ (K ⊗ L)′_{∗} (X, A), which for regular K_{∗}, L_{∗} is an inclusion. The chain mapping φ : (K ⊗ L)′_{∗} (X, A) ⟶ (K ⊗ L)_{∗} (X) is φ^{K} ⊗ φ^{L} on the first summand of (2) (φ^{K}, φ^{L} denoting the corresponding maps for K_{∗}, L_{∗}) and it is trivial on the remaining two summands ⨁_{p+q=n} · · ·.

κ : (K ⊗ L)_{∗} (X) ⟶ (K ⊗ L)′_{∗} (X, A)

maps the first summand of (1) by κ^{K} ⊗ κ^{L}. We are now going to verify the properties of a chain functor as they were recorded in [5], §4: Since we have chain homotopies φ^{K} κ^{K} ≃ 1 resp. for L, the existence of a chain homotopy (as required in D1) [5], §4)

φ κ ≃ 1

follows. We need

1.1. Lemma: Let C_{∗} be any chain functor, d c = k z, in C_{∗} (X, A), k ∈ ℤ, then there exist: z′, c′ ∈ C′_{∗} (X, A), a_{i} ∈ C_{∗} (A, A), i = 1, 2, a_{3} ∈ C_{∗} (A) such that

\{ \begin{matrix}l z′ + q_{#} a_{1} ∼ z, dz′ ∈ im i′ \\ l c′ + q_{#} a_{2} = c + dw, w ∈ C_{∗} (X, A) \\ k z′ + i′ a_{3} = dc′. \end{matrix}  (3)

Proof: There exists z ∼ l z′ +q_{#} a_{1} so that without loss of generality we are allowed to assume that z = l z′ + q_{#} a_{1}. Because of [5] §4 there exist non-natural chain homotopies j_{#} φ ≃ l, φ κ ≃ 1, hence a chain homotopy (writing simply j for j_{#})

l κ φ ≃ j φ κ φ ≃ j φ ≃ l.

More precisely there are due to [5], §4 chains D(z′), D(dz′) such that

l κ φ z′ − l z′ = d D(z′) + D (dz′).

Since d z′ ∈ im i′, we can, because of [5] §4, D1”), assume that D(dz′) = 0 so that

z = l z′ + q_{#} a_{1} ∼ l κ φ z′ + q_{#} a_{1}.

By substituting z we assume without loss of generality that

z = l κ φ z′ + q_{#} a_{1}.

Since kz = dc, D3) (in [5] §4) implies the existence of ā ∈ C_{∗}(A) such that

dk κ φ z′ = −i′ dā,
kz = kl κ φ z′ + l i′ ā + (q_{#} a_{1} − l i′ ā)
= kl κ φ z′ + l i′ ā + dv

taking into account that q_{#} k a_{1} − l i′ā ∈ C_{∗} (A, A) is a cycle, hence bounding (since C_{∗} (A, A) is acyclic). So

kz = l (kκ φ z′ + i′ ā) + dv, v ∈ C_{∗} (A, A).  (4)

However, since kz, hence l(kκ φ z′ + i′ ā) is bounding, D2) in [5] §4 ensures the existence of a c′_{1} ∈ C′_{∗+1} (X, A), b ∈ C_{∗} (A) such that

kκ φ κ φ z′ + κ φ i′ ā = dc′_{1} + i′(b).

Due to [5] §4 D1’) we can assume

κ φ i′ = κ i = i′

hence

kκ φ κ φ z′ + κ φ i′ ā ∼ kκ φ z′ + i′ ā.

As a result we find a c′_{2} ∈ C′_{∗} (X, A) satisfying

\{ \begin{matrix}dc′_{2} = kκ φ z′ + i′ ā − i′ b \\ = kκ φ z′ + i′ ã_{3}, ã_{3} ∈ C_{∗} (A), \end{matrix}  (5)
ã_{3} = ā − b.

Due to (4) and (5) we realize that

d l c′_{2} − d c = d l c′_{2} − kz ∈ C_{∗} (A, A)

hence we detect a b_{1} ∈ C_{∗} (A, A) such that

d(l c′_{2} − c − q_{#} b_{1}) = 0.

Because l c′_{2} − c − q_{#} b_{1} is a cycle, there exist a u′ ∈ C′_{∗} (X, A), b_{2} ∈ C_{∗} (A, A) satisfying

l u′ + q_{#} b_{2} = l c′_{2} − c + q_{#} b_{1} − dw
w ∈ C_{∗} (X, A), du′ ∈ im i′.

Defining c′ = c′_{2} − u′ we conclude with a_{2} = b_{1} − b_{2}

l c′ + q_{#} a_{2} = c + dw  (6)
dc′ = dc′_{2} − du′ = kκφz′ + i′ ã_{3} − i′ a_{4},
du′ = i′ a_{4} ∈ im i′.

With

a_{3} = ã_{3} − a_{4}

we obtain

dc′ = kκ φ z′ + i′a_{3}.  (7)

Implementing the chain homotopy of the beginning of this proof allows us to replace κφz′ by z′. This completes the proof of the lemma.-

1.2. Lemma ψ : H_{∗} ((K ⊗ L)′′_{∗})(X, A) ⟶ H_{∗} (K ⊗ L)(X, A) is epic (cf. [5] §4, D2)). Proof: Suppose z^{K}, z^{L} are cycles in K_{p} (X, A) resp. L_{q} (X, A), then we have relative cycles z′^{K}, z′^{L} in K′_{p} resp. L′_{q} as well as chains a^{K}, a^{L} in K_{p} (A, A) resp. L_{q} (A, A) such that

\{ \begin{matrix}z^{K} ∼ l^{K} z′^{K} + q_{#} a^{K} \\ z^{L} ∼ l^{L} z′^{L} + q_{#} a^{L}. \end{matrix}  (8)

Let us assume that we have identities (rather than homologies) in (8), then we deduce

z^{K} ⊗ z^{L} = l(z′^{K} ⊗ z′^{L}) + q_{#}(a^{K}) ⊗ l^{L} z′^{L} + l^{K} z′^{K} ⊗ q_{#} (a^{L})+
+q_{#}(a^{K} ⊗ a^{L}) = l z′^{K⊗L} + q_{#}(a^{K⊗L}),

with z′^{K⊗L} ∈ (K ⊗ L)′_{∗} (X, A), a^{K⊗L} ∈ (K ⊗ L)_{∗} (A, A). Hence the homology class {z^{K} ⊗ z^{L}} is contained in im ψ. Now it is easy to realize that the same conclusion holds in the general case (8). Suppose that

z^{K⊗L} = ∑ c_{i}^{K} ⊗ c_{i}^{L}

is any cycle in (K ⊗ L)_{∗} (X, A), then c_{i}^{K}, c_{i}^{L} are either cycles in K_{∗} resp. L_{∗} or the summands are of the form z_{i}^{K} ⊗ c_{i}^{L}, d c_{i}^{L} = mz_{i}^{L}, mz_{i}^{K} = d c_{i}^{K}, for m ∈ ℤ. In this case we have

d(z_{i}^{K} ⊗ c_{i}^{L} + (−1)^{p−1} c_{i}^{K} ⊗ z_{i}^{L}) = 0.

Now we have due to lemma 1.1. (eventually replacing a cycle by a homologous one)

z_{i}^{K} = l^{K} z′_{i}^{K} + q_{#} b_{i}^{K},
z_{i}^{L} = l^{L} z′_{i}^{L} + q_{#} b_{i}^{L},
c_{i}^{K} = l^{K} c′_{i}^{K} + q_{#} a_{i}^{K},

Homology, Homotopy and Applications, vol. 3, No. 3, 2001 61

c_{i}^{L} = l^{L} c′_{i}^{L} + q_{#} a_{i}^{L}

and therefore z = z_{i}^{K} ⊗ c_{i}^{L} + (−1)^{p−1} c_{i}^{K} ⊗ z_{i}^{L} = l(z′_{i}^{K} ⊗ c′_{i}^{L}) + (−1)^{p−1} l(c′_{i}^{K} ⊗ z′_{i}^{L}) +

+ l(z′_{i}^{K} ⊗ a_{i}^{L} + (−1)^{p−1} a_{i}^{K} ⊗ z′_{i}^{L} + (−1)^{p−1} c′_{i}^{K} ⊗ b_{i}^{L}) +

+ q_{#}(b_{i}^{K} ⊗ a_{i}^{L}) + (−1)^{p−1} q_{#} (a_{i}^{K} ⊗ b_{i}^{L}).

as a result

z = l z′^{K⊗L} + q_{#} a^{K⊗L}, a^{K⊗L} ∈ (K ⊗ L)_{∗} (A, A).

This completes the proof of lemma 1.2.- 1.3. Lemma: One has (cf. [5] §4 D3))

ker ψ ⊂ ker ¯∂.

Proof: Suppose we have

d ∑ c_{i}^{K} ⊗ c_{i}^{L} = z^{K⊗L} ∈ Z_{n} ((K ⊗ L)_{∗} (X, A)),

then we obtain

z^{K⊗L} = ∑ d c_{i}^{K} ⊗ c_{i}^{L} + (−1)^{p} c_{i}^{K} ⊗ d c_{i}^{L}.

Since d c_{i}^{K}, d c_{i}^{L} are per definitionem bounding, application of lemma 1.1. yields the same situation as in the proof of the preceding lemma, but with all b_{i}^{K} = b_{i}^{L} = 0. As a result the q_{#}(b_{i}^{K} ⊗ a_{i}^{L} + (−1)^{p} a_{i}^{K} ⊗ b_{i}^{L}) terms vanish and we establish

z^{K⊗L} = l z′^{K⊗L}.

This is equivalent to the assertion of the lemma.- 1.4. Lemma: (K ⊗ L)_{∗} satisfies [5], §4] D2), hence there exists a

ρ_{#} : im j_{#} ⟶ (K ⊗ L)′′_{∗} (X, A)

such that ψ ρ_{#∗} = 1 : im j_{∗} ⟶ im j_{∗} and

p κ ≃ ρ_{#} j_{#}.

Proof: We use the fact that 1) ρ_{#}^{K}, ρ_{#}^{L} with the corresponding properties exist and 2) that D2) turns out to be equivalent to

ker j_{∗} ⊂ ker p_{∗} κ.  (9)

We form ρ = ρ_{#}^{K} ⊗ ρ_{#}^{L} : im j_{#} ⟶ (K ⊗ L)′′_{∗} (X, A) which implies obviously (9).- We summarize: 1.5. Proposition: The tensor product ((K ⊗ L)_{∗}, l, κ, φ, i′) of two (irregular) chain functors is an irregular–chain–functor. Proof: The crucial points of the proofs are the subject of the preceding lemmas, the remaining points (as [5] §4 D2), D4) to the extent in which they have not already been treated, cf. also definition 4.1) are trivial or immediate.

1.6. Proposition: Let λ^{K} : K_{∗} ⟶ K̃_{∗}, λ^{L} : L_{∗} ⟶ L̃_{∗} be mappings of chain functors, then

λ^{K} ⊗ λ^{L} : (K ⊗ L)_{∗} ⟶ (K̃ ⊗ L̃)_{∗}

furnishes a mapping of chain functors in a canonical way.

Proof: Follows from the definition of a mapping of chain functors (cf. [5] §4).-

1.7. Theorem: Let K_{∗}, L_{∗} be two regular chain functors, then there exists for any (X, A) ∈ 𝔎^{2} a Künneth formula which is natural in (X, A) as well as in K_{∗} and L_{∗}:

0 ⟶ (H_{∗}(K_{∗})(X, A) ⊗ H_{∗}(L_{∗})(X, A))_{n} ⟶ H_{n} ((K ⊗ L)_{∗}(X, A) ⟶
⟶ T or(H_{∗}(K_{∗})(X, A), H_{∗}(L)(X, A))_{n−1} ⟶ 0.

Proof: Since according to (1’) (K ⊗ L)_{∗} (·) ≈ K_{∗}(·) ⊗ L_{∗}(·), this is a repetition of the proof of the Künneth formula for chain complexes (cf. [8]).

1.8. Proposition: The tensor product of two regular chain functors (resp. with compact carriers) is regular (resp. has compact carriers).

Proof: The fact that (K ⊗ L)_{∗}( ) has compact carrier is an immediate consequence of the same property for K_{∗}( ), L_{∗}( ). In particular are (K ⊗ L)_{∗} (·) free with canonical base. Because K_{∗} (·), L_{∗} (·) are free chain complexes, inclusions as well as i′, l (for non- irregular–chain–functors) induce monomorphisms onto direct summands.- Let f : (X, A) ⟶ (Y, B) be any mapping such that H_{∗}(K_{∗})(f) = f_{∗K} as well as H_{∗}(L_{∗})(f) = f_{∗L} is an isomorphism. Then we have a mapping between the two Künneth formulae for (X, A) and (Y, B). Since

f_{∗K} ⊗ f_{∗L} : (H_{∗}(K_{∗})(X, A) ⊗ H_{∗}(L_{∗})(X, A))_{n} ⟶ (H_{∗}(K_{∗})(Y, B) ⊗ H_{∗}(L_{∗})(Y, B))_{n}

as well as the same expression for the torsion product are isomorphisms, and the Künneth formula is natural, we conclude that

f_{∗} = H_{n}((K ⊗ L)_{∗})(f) : H_{n}((K ⊗ L)_{∗})(X, A) ⟶ H_{n}((K ⊗ L)_{∗})(Y, B)

is an isomorphism. Applying this to an excision map (X, A) ⟶ (X/A, ⋆) assures us that the tensor product of two chain functors satisfies an excision property, whenever both factors do. Since due to 1.5. (K ⊗ L)_{∗} are chain functors, 1.8. follows.-

1.9. Lemma: The tensor product of chain functors is up to an isomorphism associative and commutative:

(K ⊗ L)_{∗} ⊗ M_{∗} ≈ K_{∗} ⊗ (L ⊗ M)_{∗}

(K ⊗ L)_{∗} ≈ (L ⊗ K)_{∗}

Proof: This is an immediate consequence of the corresponding well-known fact for chain complexes.-

2. Tensor products of spectra

The (Boardman) category of spectra is defined in [1] or [13]. To each such spectrum A there exists a homology theory A_{∗}, defined on the subcategory 𝔓^{2} of CW pairs. According to [2] theorem 8.1. we obtain a (regular) chain functor ^{A}C_{∗} such that the associated homology theory H_{∗}(^{A}C_{∗}) is isomorphic to A_{∗}. This is true even if A_{∗} is defined on Top^{2} and A not necessarily a CW spectrum. On the other hand there exists a functorial relationship between regular chain functors C_{∗} and spectra |C_{∗}|, such that H_{∗}(C_{∗}) ≈ |C_{∗}|_{∗} (cf. [3] theorem 1.1.). Let A, B be two spectra, then we define their tensor product by

A ⊗ B = |^{A}C_{∗} ⊗ ^{B}C_{∗}|.  (1)

In order to ensure the independence of A ⊗ B of the choices of ^{A}C_{∗}, ^{B}C_{∗} we need:

2.1. Lemma: Suppose ^{A}K_{∗}, ^{A}L_{∗} (^{B}K_{∗}, ^{B}L_{∗}) are two regular chain functors such that there exists a mapping of chain functors

Aλ : ^{A}K_{∗} ⟶ ^{A}L_{∗} (^{B}λ : ^{B}K_{∗} ⟶ ^{B}L_{∗})

inducing isomorphisms ^{A}λ_{∗} : H_{∗} (^{A}K_{∗}) ≈ H_{∗} (^{A}L_{∗}) (resp. for ^{B}λ_{∗}). Then

(^{A}λ ⊗ ^{B}λ)_{∗} : H_{∗} (^{A}K ⊗ ^{B}K) ⟶^{≈} H_{∗} (^{A}L ⊗ ^{B}L)

is an isomorphism of homology theories. Proof: Observe that (^{A}λ ⊗ ^{B}λ)_{∗} is a natural transformation of homology theories. We have short exact Künneth sequences

[figure: 0 → (H_{∗}(^{A}K)(·) ⊗ H_{∗}(^{B}K)(·))_{n} → H_{n}(^{A}K ⊗^{B}K)(·) → T or(H_{∗}(^{A}K)(·), H_{∗}(^{B}K)(·))_{n−1}) → 0; 0 → (H_{∗}(^{A}L)(·) ⊗ H_{∗}(^{B}L)(·))_{n} → H_{n}(^{A}L ⊗^{B}L)(·) → T or(H_{∗}(^{A}L)(·), H_{∗}(^{B}L)(·))_{n−1} → 0; ^{A}λ_{∗} ⊗^{B}λ_{∗}; (^{A}λ ⊗^{B}λ)_{∗}; T or(^{A}λ_{∗},^{B}λ_{∗})]

fitting into the commutative diagram with isomorphisms^{A}λ_{∗} ⊗ ^{B}λ_{∗}, T or(^{A}λ_{∗}, ^{B}λ_{∗}). Hence the third morphism (^{A}λ ⊗ ^{B}λ)_{∗} is an isomorphism for any (·) ∈ 𝔎^{2}. This completes the proof of the lemma.-

2.2. Lemma: Suppose ^{A}K_{∗}, ^{A}L_{∗} ^{B}K_{∗}, ^{B}L_{∗} are as in 2.1. and assume that there exist chain transformations

\{ \begin{matrix}^{A}K_{∗} ⟶ · ⟵ · ⟶ · · · · · ⟶ ^{A}L_{∗} \\ ^{B}K_{∗} ⟶ · ⟵ · ⟶ ⋯ ⟶ ^{B}L_{∗} \end{matrix}  (2)

(not necessarily of the same length) such that each transformation induces an isomorphism of the associated homology. Then there exists a chain of transformations

(^{A}K ⊗ ^{B}K)_{∗} ⟶ · ⟵ ⋯ ⟶ (^{A}L ⊗ ^{B}L)_{∗}

such that each transformation induces an isomorphism on the homology level. Hence there exists a natural isomorphism

(^{A}K ⊗ ^{B}K)_{∗} ≈ (^{A}L ⊗ ^{B}L)_{∗}.  (3)

Proof: We can without loss of generality assume that the chains in (2) are of equal length (by eventually inserting identities in one or the other direction). Application of lemma 2.1. to each seperate step yields the result.-

2.3. Lemma: Let ^{A}K_{∗}, ^{B}K_{∗}, ^{A}L_{∗}, ^{B}L_{∗} be chain functors such that ^{A}K_{∗} and ^{A}L_{∗} (resp. for B) yield isomorphic homology theories. Then there exists an isomorphism (3), which is induced by a chain of transformations of chain functors, each inducing an isomorphism of homology theories. Proof: According to theorem 1.1. in [5] there exist two pairs of transformations of chain functors

AK_{∗} ⟶ · ⟵ ^{A}L_{∗}
BK_{∗} ⟶ · ⟵ ^{B}L_{∗}

satisfying the assumption of 2.2. So we accomplish an isomorphism

(^{A}K ⊗ ^{B}K)_{∗} ≈ (^{A}L ⊗ ^{B}L)_{∗}

which is induced by a chain of transformations of chain functors.-

2.4. Corollary: The tensor product A ⊗ B of two spectra A and B is determined by (1) up to an isomorphism of spectra in the Boardman homotopy category.

Proof: Since | · | is a functor such that H_{∗} (K_{∗})(·) ≈ |K_{∗}|_{∗}, every transformation of chain functors λ : K_{∗} ⟶ L_{∗}, with λ_{∗} = isomorphism induces an isomorphism |K_{∗}| ≈ |L_{∗}| in the Boardman homotopy category (cf. [1] or [13]).) The assertion follows therefore from 2.3.-

Remark: Corollary 2.4. guarantees that, due to the main result of [5], the isomorphism class of A ⊗ B depends only on the isomorphism classes of the spectra A and B, it does in particular not depend on the choices of the chain functors ^{A}C_{∗} resp. ^{B}C_{∗}. Since there are many different ways to detect for a given spectrum A such a chain functor ^{A}C_{∗} (cf. for example the remarks in [5] §5) this facts provides us with some freedom for the individual construction of a tensor product of two spectra.

As an immediate corollary we deduce from 1.7., (1) and 2.4.:

2.5. Corollary: Let A, B be two spectra, then there exists a Kuenneth formula

0 ⟶ (A_{∗}(·) ⊗ B_{∗}(·))_{n} ⟶ (A ⊗ B_{∗})(·) ⟶ Tor(A_{∗}(·), B_{∗}(·))_{n−1} ⟶ 0 (4)

which is natural in (·), as well as in A, B.

As a first application of this tensor product let G_{1}, G_{2} ∈ Ab, be abelian groups, K(G_{i}) the associated Eilenberg-MacLane spectra, then we have:

2.6. Theorem: Suppose T or(G_{1}, G_{2}) = 0, then there exists an isomorphism in the Boardman homotopy category

K(G_{1} ⊗ G_{2}) ≈ K(G_{1}) ⊗ K(G_{2}).  (5)

Proof: We deduce from 2.5. (T or(⋯, ⋯) = ⋯ ∗ ⋯)

0 ⟶ (K(G_{1})_{∗}(·) ⊗ K(G_{2})_{∗}(·))_{n} ⟶ (K(G_{1}) ⊗ K(G_{2})))_{n}(·) ⟶
(K(G_{1})_{∗}(·) ∗ K(G_{2})_{∗}(·))_{n−1} ⟶ 0

that the derived homology theory of K(G_{1})⊗K(G_{2}) is an ordinary homology theory with G_{1} ⊗ G_{2} as coefficient group. Hence (1) follows.-

3. An example

We pointed out in §1 already that for irregular–chain–functors K_{∗} (X, A) ⊗ L_{∗} (X, A) is not necessarily isomorphic to K_{∗} (X, A) ⊗ L_{∗} (X, A) ⊕ K_{∗} (X, A) ⊗ L_{∗}(X) ⊕ K_{∗} (X) ⊗ L_{∗} (X, A). We display an irregular–chain–functor ℤ ℤ ℤ_{∗} such that

(K ⊗ ℤ ℤ ℤ)_{∗} ≈ K_{∗},

for any chain functor K_{∗} (cf. [5] §4 concerning the definition of equivalences of irregular–chain–functors). We set

ℤ_{n} (X) = \{ \begin{matrix}⟨z_{X}⟩ | dz_{X} = 0 ⋯ n = 0, X ≠ ∅ \\ 0 ⋯ elsewhere \end{matrix}  (1)

denoting by ⟨a⟩ the free abelian group generated by a, and

ℤ_{n} (X, A) = 0 ⋯ A ≠ ∅.

Let f : (X, A) ⟶ (Y, B) be a mapping, then we define either f_{#}(z_{X}) = z_{Y} or f_{#} = 0. Moreover we set ℤ′_{n} (X, A) = ℤ_{n} (X), φ(z_{X}) = z_{X}, κ(z_{X}) = z_{X}, whenever X ≠ ∅,

l = \{ \begin{matrix}id ⋯ A = ∅ \\ 0 ⋯ A ≠ ∅ \end{matrix}

This furnishes the structure of an irregular chain functor. We have (for X ≠ ∅, the remaining case is trivial)

kerψ = \{ \begin{matrix}0 ⋯ A = ∅ \\ 0 ⋯ A ≠ ∅ \end{matrix} ⊂ ker ¯∂ = \{ \begin{matrix}⟨z_{X}⟩ ⋯ A = ∅ \\ 0 ⋯ A ≠ ∅ \end{matrix}

ℤ_{∗} (X, X) is always acyclic, and i′(z_{A}) = z_{X}, so that

κ i_{#} = i′
l = j_{#} φ

follows. In the same way we get ker j_{∗} ⊂ ker p_{∗}κ_{∗}. On the other hand j_{#}, l are not inclusions. This furnishes an irregular–chain–functor ℤ_{∗}. We have for any chain functor K_{∗}

(K ⊗ ℤ)_{∗} (X, A) = K_{∗} (X, A) ⊗ ℤ_{∗} (X, A) ⊕ K_{∗} (X, A) ⊗ ℤ_{∗} (X) ⊕
⊕ K_{∗} (X) ⊗ ℤ_{∗} (X, A) = K_{∗} (X, A),

and

\{ \begin{matrix}(K ⊗ ℤ)′_{n}(X, A) = ⨁_{p+q=n} K′_{p} (X, A) ⊗ ℤ′_{q} (X, A) ⊕ \\ ⊕ ⨁_{p+q=n} K′_{p} (X, A) ⊗ ℤ_{q} (A, A) ⊕ ⨁_{p+q=n} K_{p} (A, A) ⊗ ℤ′_{q} (X, A) \\ ≈ K′_{n} (X, A) ⊕ K_{n} (A, A). \end{matrix}  (2)

The inclusion λ : K_{∗}(·) ⟶ (K ⊗ ℤ)_{∗}(·) which is defined by this procedure, is a mapping of (irregular) chain functors, i.e. compatible with the structure maps i′, l and, since K_{∗}(A, A) is acyclic and (K ⊗ ℤ)′_{∗} ⊃ K′_{∗}, an equivalence of irregular– chain–functors in the sense of definition 4.2. in [5]. To this end we have to observe that the sum (2) is an amalgamated sum in the sense of the definition of the ⊗- product of chain functors (cf. §1). Moreover ℤ_{∗}, ℤ′_{∗} have obviously compact carriers and satisfy a homotopy axiom ([5] §4 D4)). We have

H_{n} (ℤ_{∗})(X) = \{ \begin{matrix}ℤ ⋯ X ≠ ∅, n = 0 \\ 0 ⋯ elsewhere \end{matrix}

H_{n} (ℤ_{∗})(X, A) = 0 ⋯ A ≠ ∅, n ∈ ℤ

and summarize: 3.1. Theorem: There exists an irregular–chain–functor with compact carriers ℤ ℤ ℤ_{∗} such that for any chain functor K_{∗} there exists an equivalence of irregular–chain– functors (cf. [5] definition 4.2)

(K ⊗ ℤ ℤ ℤ)_{∗} ≈ K_{∗}

. Remarks: 1) The chain functor ℤ ℤ ℤ_{∗} a) does not induce monomorphisms for inclusions, b) does not admit a l : ℤ′_{∗} (X, A) ⟶ ℤ_{∗} (X, A) which is monic and c) does not turn all excision maps into isomorphisms (e.g. (X, ∅) ⟶ (X^{+}, ⋆), X ≠ ∅).

So ℤ_{∗} is not a regular chain functor in the sense of [5] definition 4.1.-

2) There can be no regular chain functor C_{∗} such that

(K ⊗ C)_{∗} ≈ K_{∗}

for any regular chain functor K_{∗}: Assume to the contrary that such a C_{∗} exists, then we would have a Künneth formula

0 ⟶ (H_{∗}(K_{∗})(·) ⊗ H_{∗}(C_{∗})(·))_{n} ⟶ H_{n}(K_{∗}(·)) ⟶
T or(H_{∗}(K_{∗})(·), H_{∗}(C_{∗}(·))_{n−1} ⟶ 0

implying that always

H_{n} (C_{∗})(·) = \{ \begin{matrix}ℤ ⋯ n = 0 \\ 0 ⋯ n ≠ 0. \end{matrix}  (3)

Assume to this end that H_{q}(C_{∗}) ≠ 0, q ≠ 0, then we detect a K_{∗} such that H_{0}(K_{∗})(·) = 0, H_{−q}(K_{∗})(·) = ℤ. Such a generalized homology theory clearly exists. This leads to a contradiction. Take on the other hand a K_{∗} satisfying H_{0}(K_{∗})(·) = ℤ, then H_{0}(C_{∗})(·) = ℤ follows from the Künneth formula. However (3) is impossible for a homology theory.

3) In particular there does not exist a spectrum B B B such that

B B B ⊗ E E E ≈ E E E

for every spectrum E E E.

4. Preparation of the localization process

The localization process requires some general constructions which we display in this section: Suppose 𝔏 ⊂ 𝔎^{2} is a subcategory, C_{∗} a (regular or irregular) chain functor on 𝔎^{2}, K_{∗} a (regular or irregular) chain functor on 𝔏. Since we do not know anything about 𝔏 (it may not contain any compact objects at all) it does not make sense to assume that K_{∗} has compact carriers. Let α : K_{∗} ⟶ C_{∗}|𝔏 be a g-transformation of chain functors (cf. [5] definition 4.3.).

4.1. Theorem There exists a chain functor K̄_{∗} on 𝔎^{2}, mappings of chain functors τ : K_{∗} ⟶ K̄_{∗} | 𝔏, ᾱ : K̄_{∗} ⟶ C_{∗} such that

1) (ᾱ|𝔏)τ = α, and τ : K_{∗} ⊂ K̄_{∗} | 𝔏 is a strong deformation retract.

2) For any g-transformation β : L_{∗} ⟶ C_{∗}, L_{∗} any chain functor on 𝔎^{2}, γ : L_{∗}|𝔏 ⟶ K_{∗}, satisfying (β | 𝔏) = α γ, there exists a up to chain homotopy unique r : L_{∗} ⟶ K̄_{∗} such that ᾱ r = β, r | 𝔏 = τ γ

3) If K_{∗}, C_{∗} are regular, then l, i′, ((X, A) ⊂ (Y, B))_{#} for K_{∗} are inclusions onto direct summands (cf. [5] §4 D3) ⋆)); if all involved chain functors have compact carriers, then K̄_{∗} can be assumed to have compact carriers.

Proof: The proof is a slight adaptation of the classical way to accomplish Kan extensions.

I) Assume we are dealing not with chain functors C_{∗}, K_{∗} but merely with functors C_{∗} : 𝔎^{2} ⟶ ch (=category of free chain complexes with natural bases), K_{∗} : 𝔎^{2} ⟶ ch (without the additional structure of chain functors). Suppose a ∈ C_{∗}(X, A), (X, A) ∈ 𝔎^{2}, then we consider collections

a = {a; a_{f} ∈ K_{∗}(E, F)| f : (X, A) ⟶ (E, F) ∈ 𝔏}  (1)

satisfying

a_{gf} = g_{#} a_{f}, g ∈ 𝔏((E, F), (E′, F′))  (2)

and

f_{#} (a) = α(a_{f}).  (3)

We set

K̂_{n} (X, A) = ⟨{a| a ∈ C_{n} (X, A)}⟩

⟨⋯ ⟩ denoting the abelian group generated by · · ·. If h ∈ 𝔎^{2}((X, A), (Y, B)), we define

h_{#}(a) = {h_{#}(a); a_{fh}| f ∈ 𝔎^{2}( (Y, B), (E, F)), (E, F) ∈ 𝔏},  (4)

and

da = {da; da_{f}}.

Assuming that C_{∗} has compact carrier, then, by a well-known technique we equip K̂_{∗} : 𝔎^{2} ⟶ ch with compact carriers: We perform the previous construction only for compact (X, A) and form the direct limit. So K̂_{∗} : 𝔎^{2} ⟶ ch becomes a functor with compact carrier, whenever K_{∗} and C_{∗} have this property. Setting ᾱ(a) = a, yields a natural transformation ᾱ : K̂_{∗} ⟶ C_{∗}. We set

τ : K_{∗} ⟶ K̂_{∗}|𝔏
c ⟼ {α(c); f_{#}(c)}

c in a base of K_{∗}(X, A), (X, A) ∈ 𝔏 and for given β : L_{∗} ⟶ K_{∗} as in the theorem:

r : L_{∗} ⟶ K̂_{∗}
c ⟼ {β(c); γf_{#}(c)}.

Since K_{∗}(X, A) has a natural base, τ, β are natural. We deduce

ᾱτ(c) = ᾱ{α(c); f_{#}(c)} = α(c)

and

ᾱr(c) = β(c), c ∈ L_{∗}(X, A),
τγ(c) = {αγ(c); f_{#}γ(c)} = {β(c); γf_{#}(c)}

c ∈ L_{∗}(E, F), f : (E, F) ⟶ (E′, F′) in 𝔏. Moreover for (E′, F′), (X, A) ∈ 𝔏, f : (X, A) ⟶ (E′, F′), we have a

K̂_{∗}(X, A) ⟶ K_{∗}(X, A)
{a; a_{f}} ⟼  a_{1},  1 = 1_{(X,A)}

and observe that in this case τ|𝔏 is an isomorphism. We have a subcomplex K_{n}^{0}(X, A) ⊂ K̂_{n}(X, A) consisting of all those a with a = ᾱ(a) = 0. Now we define

K̄_{∗}(X, A) = K̂_{∗}(X, A) ∪ cone K_{∗}^{0}(X, A).

The mappings ᾱ, r an be extended over K̄_{n}. We will deal with τ : K_{∗} ⟶ K̄_{∗} | 𝔏 at the end of II) of the present proof. Unlike for K̂_{n} we have a uniqueness assertion for r: Suppose r, r̃: L_{∗} ⟶ K̄_{∗} be such that ᾱr = ᾱr̃, then we deduce r ≃ r̃. Proof: If r̃(c) = {a; a_{f}}, then ᾱr̃(c) = a = β(c) implies r̃(c) = {β(c); a_{f}}, hence r̃(c) − r(c) = {0; a_{f} − f_{#}(c)} ∈ K_{n}^{0}, so that the existence of a chain homotopy r ≃ r̃ follows. II) We are now obliged to endow K̄_{∗} with the structure of a chain functor. This is in a first step accomplished for K̂_{∗}, followed by attaching the cone over K_{∗}^{0} afterwards. In detail we have

1) to establish non-natural mappings φ, κ together with chain homotopies D_{1} : φ κ ≃ 1, D_{2} : j_{#} φ ≃ l, such that the following additional conditions hold for C_{∗} = K̂_{∗}:

2) Every cycle z ∈ Z_{n}(C_{∗} (X, A)) is associated with a x ∈ C_{n+1}(X, A), z′ ∈ C′_{n} (X, A), d z′ ∈ im i′, a ∈ C_{n} (A, A) such that

l z′ + q_{#} a = z + dx;

3) ker ψ ⊂ ker ¯∂ (cf. [5] §4, ψ : H_{∗}(C′′_{∗}(X, A)) ⟶ H_{∗}(C_{∗}(X, A)), C′′_{∗}(X, A) = C′_{∗}(X, A)/im i′);

4) ψ is epic; ker j_{∗} ⊂ ker p_{∗} κ, p : C′_{∗}(X, A) ⟶ C′′_{∗}(X, A);

5) every cycle z ∈ Z_{n}(C_{∗}(X, X)) is bounding: dx = z.

6) Let H : f_{0} ≃ f_{1} : (X, A) ⟶ (Y, B) be a homotopy, then we have a chain homotopy χ : C_{n}(X, A) ⟶ C_{n+1}(Y, B), satisfying dχ(a)+χda = f_{0#}(a)−f_{1#}(a), (Property D4) in the definition of a chain functor [5] §4.) All these objectives are achieved by a modification of the procedure in I), which we explain in the case of 1): To that end assume that C_{∗}, K_{∗} are endowed with the structure of chain functors and that α is a mapping of chain functors. We change the definition of (1) in the following way: Instead of using continuous mappings f ∈ 𝔎^{2} we introduce formal arrows (X, A) ⟶^{φ} X, X ⟶^{κ} (X, A) and substitute for f words

(X, A) ⟶^{k_{1}} · ⟶^{k_{2}} ⋯ ⟶^{k_{n}} (E, F)  (5)

where each k_{i} is either of the form φ or κ or a continuous mapping, whereas φ can only appear whenever the image of a under the word (X, A) ⟶ · · · · · ⟶ (X_{1}, A_{1}) lies in C′_{n}(X_{1}, A_{1}). Condition (2) in I) is replaced by the same one, but now words (5) replacing continuous mappings f, while g is still assumed to be continuous. There are no relations between the letters φ, κ or between one of these and continuous mappings. So (2) takes the form (2’) a_{gw} = g_{#} a_{w}, where w is any suitable word and g is continuous. Condition (3) reads now:

(3′) w_{#}(a) = α(a_{w}).

We set

K̂′_{n} (X, A) = ⟨{a ∈ K̂_{n} (X, A)| a ∈ C′_{n} (X, A)}⟩

resp. for K̄_{∗}, and detect a natural inclusion induced by l whenever C_{∗}, K_{∗} are regular,

l : K̄′_{∗} (X, A) ⊂ K̄_{∗} (X, A).

We define

φ(a) = {φ(a); a_{wφ}}
κ(a) = {κ(a); a_{wκ}},

for suitable words w : X ⟶ (E, F) resp. w : (X, A) ⟶ (E, F) of the form (5), realizing that this provides us with elements in K̂_{∗}. Observe that we still have

d{a; a_{w}} = {da; da_{w}}

Finally setting

i′ = κ i_{#}

ensures that ᾱ is compatible with φ, κ, i′, l. In a next step we have to establish chain homotopies D_{1} : φ κ ≃ 1, D_{2} : j_{#} φ ≃ l and other relations 2)-5) which are required of a chain functor. This is in all cases accomplished by translating the specific problem into that of finding suitable (with the exception of χ, non- natural) mappings δ_{1}, δ_{2}, η, γ_{1}, γ_{2}, δ, χ, which in turn is finally solved in the same way as we established κ and φ before. Let us start with 5): By choosing a basis for the free abelian group Z_{n}(C_{∗}(X, X)) of all cycles in C_{∗}(X, X) we find a homomorphism δ : Z_{n}(C_{∗}(X, X)) ⟶ C_{n+1}(X, X) satisfying dδ(z) = z. Again we form words as in (5) but now also incorporating formal arrows of the form δ, finally obtaining a δ : Z_{n}(K̂_{∗}(X, X)) ⟶ K̂_{n+1}(X, X) guaranteeing that K̂_{∗}(X, X) and K̄_{∗}(X, X) are acyclic. We have for any a = {a; a_{w}} ∈ K̂_{∗}(X, X)

δa = {δa; a_{wδ}}
dδa = a.

Here for the first time we encounter a case where d{a; a_{w}} = {da; ã_{w}} ≠ {da; da_{w}}. The homotopy D_{1} : φκ ≃ 1 is associated with a mapping δ_{1} : C_{n}(X) ⟶ C_{n+1}(X) which again is established by using the fact that all occuring abelian groups are free, satisfying dδ_{1}(a) + δ_{1}(da) = φκ(a) − a. We introduce a new arrow δ_{1} : X ⟶ X and define for a = {a, a_{w}}

δ_{1}(a) = {δ_{1}(a); a_{wδ_{1}}},

confirming that δ_{1}a is well-defined in K̂_{∗}. Suppose dâ = {da; ã_{w}}, then we set

dδ_{1}a = {δ_{1}a; a_{wφκ} − a_{w} − ã_{wδ_{1}}}.

Because

δ_{1}da = {δ_{1}da; â_{wδ_{1}}},

we have

dδ_{1}a + δ_{1}da = φκa − a.

The chain homotopy D_{2} : jφ ≃ l is treated in the same way Dealing with 2) we have to translate this into a mapping

η : Z_{n}(C_{∗}(X, A)) ⟶ C′_{n}(X, A) ⊕ C_{n+1}(X, A) ⊕ C_{n−1}(A)
z ⟼ (η′(z), η′′(z), η′′′(z))

satisfying

dη′′(z) + z = lη′(z) + q_{#}δs_{#}η′′′(z)
dη′(z) = i′η′′′(z).

So we introduce three arrows η′, η′′, η′′′:

η^{(k)}a = {η^{(k)}(a); a_{wη^{(k)}}},

with the following boundary relations:

dη′a = {dη′a; a_{wη′′′κi}}
dη′′a = {dη′′a; a_{wη′} + a_{wη′′′sδq} − a_{w}}
dη′′′a = 0.

As a result we obtain for a ∈ Z_{n}(K̂_{∗}(X, A))

dη′′a + a = η′a + q_{#}δs_{#}η′′′a.

Condition 3) is easily translated into a mapping

γ_{1} : B_{n}(C_{∗}(X, A)) ⟶ C_{n}(A)

(B_{n} = bounding cycles) such that dγ_{1}(z) = dη′(z). By the same procedure we associate 4) with a mapping and ultimately with a suitable arrow γ_{2}. We arrive at 6):

For any chain functor, a homotopy H : i_{0} ≃ i_{1} : (X, A) ⊂ (X, A) × I determines a chain homotopy

χ : C_{n}(X, A) ⟶ C_{n+1}((X, A) × I)

satisfying

dχ(a) + χ(da) = i_{0#}(a) − i_{1#}(a).

We invent an arrow χ and set as before χa = {χa; a_{wχ}. If da = {da; ã_{w}}, we define

dχa = {dχa; w_{wi_{0}} − a_{wi_{1}} − ã_{wχ}}

accomplishing a chain homotopy

dχa + χda = i_{0#}a − i_{1#}a.

Since every homotopy H̃ : f_{0} ≃ f_{1} : (X, A) ⟶ (Y, B), H̃ : (X, A) × I ⟶ (Y, B) is associated with a

χ̃ : C_{n}(X, A) ⟶ C_{n+1}(Y, B),
χ̃ = H̃χ,

we obtain a word χ̃ = H̃χ furnishing a chain homotopy

dχ̃a + χ̃da = f_{0#}a − f_{1#}a.

The final K̃_{∗} is now constructed by the devices of I) but with words involving all arrows φ, κ, δ, δ_{1}, δ_{2}, η, γ_{1}, γ_{2}, δ, χ together with continuous mappings as letters. Since the induced chain homotopies are natural, cf. [5] D4), we have to assume that χ̃ commutes appropriately with continuous mappings. By construction this yields a chain functor, such that ᾱ becomes a mapping of chain functors. We form K_{∗}^{0}(X, A) and K̄_{∗}(X, A) = K̂_{∗}(X, A) ∪ cone(K_{∗}^{0}(X, A) as in I). This yields a chain functor; ᾱ is immediately extended over K̄_{∗}. Since α and β are supposed to be g-transformations, we can assume without loss of generality, i.e. up to isomorphisms (cf. [5] 4.4.), that they commute stricly with all words w_{#}. Let β : L_{∗} ⟶ C_{∗} be as in the theorem, and set r̃(c) = {β(c); γw_{#} (c)} ∈ K̂_{∗}(X, A) where c ∈ L_{∗}(X, A) observing that

r̃(dc) − dr̃(c) ∈ K_{∗}^{0}(X, A)

and

ᾱr̃ = β

so that there exists a D(c) ∈ K̄_{∗}(X, A) with

dD(c) + D(dc) = r̃(dc) − dr̃(c).

Hence r(c) = r̃(c) + D(c) is a chain mapping, satisfying ᾱr = β. We define on the subcategory 𝔏:

τ̂ : K_{∗} ⟶ K̂_{∗}|𝔏
b ⟼ {α(b); w_{#}(b)}

and

τ̂_{0} : K̂_{∗}|𝔏 ⟶ K_{∗}
{a; a_{w}} ⟼ a_{1}.

We calculate

w_{#}α(b) = α w_{#}(b)
g_{#}w_{#}(b) = (gw)_{#}(b)
τ̂_{0} τ̂(b) = b,
τ̂ τ̂_{0}{a; a_{w}} = τ̂(a_{1}) = {α(a_{1}); w_{#} a_{1}} = {a; w_{#} a_{1}}.

Since τ̂ τ̂_{0}{a; a_{w}} − {a; a_{w}} ∈ K̂_{∗}^{0}, the existence of a homotopy τ τ_{0} ≃ 1 follows for the corresponding

τ : K_{∗} ⟶ K̄_{∗}|𝔏
τ_{0} : K̄_{∗}|𝔏 ⟶ K_{∗},

which are easily established, turning K̂_{∗} into a strong deformation retract of K_{∗}. This can easily be extended to a transformation τ : K_{∗} ⟶ K̄_{∗}, displaying the same property. One has r | 𝔏 = τ γ. All transformations ᾱ, r, τ commute with l and i′, so turning out to be mappings of chain functors. All K̂_{∗}( ) and of course all K̄_{∗}( ) are free chain complexes with natural bases. Suppose that K_{∗}, C_{∗} are regular: Since l, i′ and ((X, A) ⊂ (Y, B))_{#} are monomorphisms onto direct summands for K_{∗}, the same holds clearly for K̂_{∗} and this carries immediately over to K̄_{∗}. As in the course of I), we can assume that K̂_{∗}, hence K̄_{∗} have compact carriers, whenever K_{∗} has. This settles the assertion 3) of the theorem. The universality of K̄_{∗}, ᾱ follows as in I). This completes the proof of theorem 4.1.-

4.2. Corollary: Under the same assumptions as in 4.1. let ᾱ^{i} : K̄_{∗}^{(i)} ⟶ C_{∗}, r^{i} : L_{∗} ⟶ K̄_{∗}^{(i)}, i = 1, 2 be two solutions, then there exists a natural chain homotopy K̄_{∗}^{(1)} ≃ K̄_{∗}^{(2)} commuting with ᾱ^{i}, r^{i} up to homotopy. Proof: Follows immediately from the uniqueness assertion (up to homotopy) in 4.1.-

One important application of the preceding assertion deals with the case that 𝔏 consists of a single object E ∈ 𝔎 ( with respect to which one is going to localize a chain functor resp. a spectrum), with the identity as the only morphism in 𝔏.

In addition we need some simple facts about chain complexes, which are presumably well-known:

4.3. Lemma: Let C_{∗} be any chain complex, then there exists an acyclic chain complex U_{∗} and a chain mapping α : U_{∗} ⟶ C_{∗} such that for any chain mapping β : V_{∗} ⟶ C_{∗}, V_{∗} acyclic, there exists a chain mapping ρ : V_{∗} ⟶ U_{∗}, such that α ρ = β. Proof: Let D_{∗} be the chain complex

D_{0} = ⟨x⟩, D_{−1} = ⟨dx⟩, D_{n} = 0, n ≠ 0, 1

(⟨⋯ ⟩ = free abelian group generated by ⋯). We can take for U_{∗} the chain complex D_{∗} ⊗ C_{∗} and observe that:

1) D_{∗} ⊗ C_{∗} is acyclic.

2) α : D_{∗} ⊗ C_{∗} ⟶ C_{∗}, α(x ⊗ c) = c, α(dx ⊗ c) = 0 is a chain mapping

3) Let β : V_{∗} ⟶ C_{∗} be a chain mapping, then we have 1_{D_{∗}} ⊗ β : D_{∗} ⊗ V_{∗} ⟶ D_{∗} ⊗ C_{∗} and since V_{∗} is acyclic, there exists a chain mapping γ : V_{∗} ⟶ D_{∗} ⊗ V_{∗} such that

[figure: (6); V_{∗}; D_{∗} ⊗ V_{∗}; C_{∗}; D_{∗} ⊗ C_{∗}; γ; α; β; 1_{D_{∗}} ⊗ β]

is commutative, so that we are allowed to set

ρ = (1_{D_{∗}} ⊗ β)γ

Only 3) needs proof: We establish γ at first on the cycles z ∈ V_{n}: Since V_{∗} is acyclic there exists a z̄ ∈ V_{n+1} such that dz̄ = z. We set

γ(z) = z ⊗ x + (−1)^{n+1} z̄ ⊗ dx

so that dγ(z) = 0. Let c ∈ V_{n} be any chain, then we find \overline{dc} and c̄ ∈ V_{n+1} such that dc̄ = c − \overline{dc}, and set γ(c) = c ⊗ x + (−1)^{n+1} c̄ ⊗ dx, so that

dγ(c) = dc ⊗ x + (−1)^{n} c ⊗ dx + (−1)^{n+1} dc̄ ⊗ dx
= dc ⊗ x + (−1)^{n} c ⊗ dx + (−1)^{n+1} c ⊗ dx + (−1)^{n} \overline{dc} ⊗ dx = γ(dc).

The commutativity of (6) is immediate.-

Suppose we fix a subcategory 𝔏 ⊂ 𝔎^{2} and to each (K, L) ∈ 𝔏 a U_{∗}(K, L) = U_{∗}(A_{∗}(K, L)) as in 4.3. The fact that U_{∗}(C_{∗}) = D_{∗} ⊗ C_{∗} ensures that the construction of U_{∗} as well as that of α is functorial and inherits the structure of a chain functor from the given chain functor A_{∗}. Moreover the mapping α is a gtransformation of chain functors (cf. [5] 4.3.). In order to transfer 4.3. from chain complexes to chain functors, we need:

4.4. Definition: Suppose a chain functor A_{∗} is defined on a category 𝔎^{2} and 𝔏 ⊂ 𝔎^{2} is a given subcategory. A_{∗} is 𝔏 -acyclic whenever there exists a natural chain homotopy D between the identity 1 : A_{∗}|𝔏 ⟶ A_{∗}|𝔏 and the trivial map, i.e. one has

d D(c) + D(dc) = c, c ∈ A_{∗}(K, L), (K, L) ∈ 𝔏,

and

D(c) ∈ A′_{∗}

whenever c ∈ A′_{∗}.

This makes sense also if 𝔏 = 𝔎^{2}. In general 𝔏 -acyclicity is stronger than the mere requirement that H_{∗}(A_{∗}|𝔏) = 0. however ist is easy to realize that in some cases both concepts coincide:

1) Suppose 𝔏 consists of one single object E with no morphisms but the identity.

2) 𝔏 is a tower L_{1} ⟶ L_{2} ⟶ ⋯

3) 𝔏 is any category and A_{∗} = cone B_{∗} for some chain functor B_{∗}. This allows us to assert:

4.5. Lemma: Let A_{∗} be a chain functor on 𝔎^{2}, 𝔏 ⊂ 𝔎^{2} a subcategory, then there exists an 𝔏 -acyclic chain functor U_{∗} on 𝔏, a g-transformation of chain functors α : U_{∗} ⟶ A_{∗}|𝔏, such that for any 𝔏 -acyclic chain functor V_{∗} on 𝔏 and transformation of chain functors β : V_{∗} ⟶ A_{∗}|𝔏 there exists a transformation of chain functors ρ : V_{∗} ⟶ U_{∗} such that

α ρ = β.

Regularity of A_{∗}, V_{∗} implies that of U_{∗}.

Proof: U_{∗} as defined before is clearly 𝔏 -acyclic (not just acyclic) by construction. Moreover α is a g-transformation. Since the assignment determined by c ⟼ c̄ in the proof of 4.3. is functorial and commutes with l : V′_{∗}(X, A) ⟶ V_{∗}(X, A), i′ : V_{∗}(A) ⟶ V′_{∗}(X, A) and the chain homotopies D(H) of [5] D4), γ turns out to be a transformation of chain functors, so that this part of 4.5. follows.- The last assertion is immediate.-

Combining this with 4.1. allows us to talk about Ū_{∗} and we find a morphism

cone Ū_{∗} ⟶ A_{∗} ∪_{ζ} cone Ū_{∗}

with ζ = ᾱ : Ū_{∗} ⟶ A_{∗}.

4.6. Lemma: There are g-transformations U_{∗} ⟶ A_{∗}, hence

cone U_{∗} ⟶ A_{∗} ∪_{ζ} cone Ū_{∗} and cone Ū_{∗} ⟶ A_{∗} ∪_{ζ} cone Ū_{∗}.

One has

cone Ū_{∗} = \overline{cone U_{∗}}.

Proof. The formation of the cone is functorial so that the assertion follows from the construction.-

On the subcategory 𝔏 ⊂ 𝔎^{2} we have the chain functor U_{∗}(A_{∗} ∪_{ζ} cone Ū_{∗}) which belongs to A_{∗} ∪_{ζ} cone Ū_{∗}|𝔏 in the sense of 4.3. (for each object in 𝔏 seperately). On the other hand we have cone U_{∗}|𝔏 and claim:

4.7. Lemma: cone Ū_{∗}|𝔏 satisfies the universality property of U_{∗}(A_{∗} ∪_{ζ} cone Ū_{∗}) in 4.5.

Proof: The proof is accomplished in three steps:

1) Suppose K_{∗} in theorem 4.1. is 𝔏- acyclic, then K̄_{∗} is 𝔏-acyclic. Proof: According to 4.1. τ : K_{∗} ⊂ K̄_{∗}|𝔏 is a strong deformation retract. So 1) follows.- 2) Let A_{∗} ∪_{ζ} cone Ū_{∗}(K, L) (K, L) ∈ 𝔏 be given, then there exists a retraction

r : A_{∗} ∪_{ζ} cone Ū_{∗}(K, L) ⟶ A_{∗}(K, L).

Proof: Since Ū_{∗}(K, L) is acyclic, hence (as a free chain complex) contractible, we can deform cone Ū_{∗}(K, L) onto Ū_{∗}(K, L). This implies the assertion.-

3) Let (K, L) ∈ 𝔏 be any object, V_{∗} an acyclic chain complex and

β : V_{∗} ⟶ (A_{∗} ∪_{ζ} cone Ū_{∗})(K, L)

a given chain mapping. We are going to detect a

ρ : V_{∗} ⟶ (cone Ū_{∗})(K, L)

such that η ρ = β , where η : cone Ū_{∗}(K, L) ⟶ (A_{∗} ∪_{ζ} cone Ū_{∗})(K, L) is induced by ζ. Proof: According to 2) we have a retraction r : A_{∗} ∪_{ζ} cone Ū_{∗} ⟶ A_{∗} and therefore a ρ′ : V_{∗} ⟶ U_{∗}(K, L) such that r β = α ρ′. Suppose β(v) = a + c, a ∈ A_{∗}, c ∈ cone Ū_{∗}, then α ρ′(v) = r β(v) = a, so that a ∈ im α ⊂ U_{∗} ⊂ Ū_{∗} ⊂ cone Ū_{∗}, hence β(v) ∈ im η. As a result we detect a ρ : V_{∗} ⟶ cone Ū_{∗} such that ηρ = β as required. Hence cone Ū_{∗}(K, L) satisfies the universality condition of 4.3. and cone Ū_{∗}|𝔏 that of 4.5.-

5. Localizations

The kind of localization which we are now developing for chain functors has been originally discovered by A. K. Bousfield [6] for spectra as well as for mappings between spectra, and, more recently, extended for axiomatically defined stable homotopy categories by the authors of [10]. Let A_{∗} be any (regular or irregular) chain functor on a category 𝔎^{2} of pairs of topological spaces (or of Boardman spectra) and 𝔏 ⊂ 𝔎^{2} any subcategory. We are going to localize A_{∗} at 𝔏. There are localizations in the world of chain functors with as well as in the world of chain functors without compact carriers. They can be treated by the same procedure. In the preceding section we mentioned two interesting categories 𝔏:

1) 𝔏 consists of one single object, or of one single map.

2) Let Φ be a homology theory, defined on the category of all chain functors. Associate with Φ the category 𝔏 = 𝔏_{Φ} = {(K, L)| ∀A_{∗}; Φ(A_{∗}) = 0 ⇒ H_{∗}(A_{∗})(K, L) = 0}.

5.1. Definition: 1) A morphism of chain functors (cf. [5] §4) η : B_{∗} ⟶ C_{∗} is an 𝔏 --isomorphism whenever (η|𝔏)_{∗} : H_{∗}(B_{∗})|𝔏 ⟶^{≈} H_{∗}(C_{∗})|𝔏 is an isomorphism.

2) C_{∗} is 𝔏 -local whenever one has [V_{∗}, C_{∗}] = 0 for any 𝔏 -acyclic chain functor V_{∗} (cf. definition 4.4.) ([V_{∗}, C_{∗}] denoting the set of all chain homotopy classes of morphisms V_{∗} ⟶ C_{∗}, cf. [5] §4).

Alternatively C_{∗} is 𝔏 -local, whenever any 𝔏 -isomorphism f : V_{∗} ⟶ Ṽ_{∗} induces an isomorphism

f^{∗} : [Ṽ_{∗}, C_{∗}] ⟶^{≈} [V_{∗}, C_{∗}].

Observe that if the chain functor B_{∗} is associated with a spectrum B, then for any spectrum E one has

B_{∗}(E) = H_{∗}(^{E}C_{∗})(B),

so that the preceding definition agrees with the usual one ([1], [11], [13]). According to 4.5. we have an acyclic U_{∗}|𝔏 = U_{∗} and a mapping α : U_{∗} ⟶ A_{∗}|𝔏. We set

𝔏A_{∗} = Ū_{∗},  (1)

where the ¯(·) -construction stems from 4.1.

5.2. Lemma: _{𝔏}A_{∗} is 𝔏 -acyclic. Regularity (resp. compact carriers) of A_{∗} implies the same for _{𝔏}A_{∗}.

Proof: We have

𝔏A_{∗}(L) = Ū_{∗}(L) ≈ U_{∗}(L),

which is acyclic for any L ∈ 𝔏, so that Ū_{∗} is in view of 4.5. 𝔏 -acyclic. The second assertion will be settled in §6 (theorem 6.8. which will be independently verified). -

According to theorem 4.1. we have

ζ = ᾱ : Ū_{∗} ⟶ A_{∗}

and define

A_{𝔏}_{∗} = A_{∗} ∪_{ζ} cone (_{𝔏}A_{∗})  (2)

denoting by

η : A_{∗} ⟶ A_{𝔏}_{∗}  (3)

the mapping induced by ζ. As in the preceding section, the formation of the cone over a chain functor as well as that of the amalgamated sum carries immediately over from chain complexes ([8]) to chain functors.

5.3. Lemma: Let V_{∗} be 𝔏 -acyclic, then

[V_{∗}, A_{𝔏}_{∗}] = 0,

hence A_{𝔏}_{∗} is 𝔏 - local.

Proof. Let f : V_{∗} ⟶ A_{∗} ∪_{ζ} cone Ū_{∗} be any mapping of chain functors, V_{∗} 𝔏 -acyclic. According to [5] 4.5. we can assume that f is a g-transformation. Due to 4.5. we find a factorization

[figure: V_{∗}|𝔏; A_{𝔏}_{∗}|𝔏; U_{∗}(A_{𝔏}_{∗}|𝔏) = Ũ_{∗}; f|𝔏]

hence according to theorem 4.1. a factorization

[figure: V_{∗}; A_{𝔏}_{∗}; ^{¯Ũ^{∗}}; f]

However according to 4.7. we are allowed to use cone U_{∗} instead of Ũ_{∗} and therefore, because of 4.6.,

¯Ũ_{∗} = cone Ū_{∗}

with

U_{∗} = U_{∗}(A_{∗})|𝔏.

So we have a factorization

[figure: V_{∗}; A_{𝔏}_{∗}; cone Ū_{∗}; f]

implying that f ≃ 0, since the cone over everything is always acyclic.-

5.4. Lemma: η is an 𝔏 -isomorphism.

Proof: 1) To any cycle z ∈ A_{𝔏}_{∗}(L), L ∈ 𝔏 there exists a z′ ∼ z, z′ ∈ im η, hence H_{∗}(η(L)) is epic. Proof. Suppose v ∈ cone U_{∗}(L), dv ∈ ζ(U_{∗}), then, since U_{∗} is acyclic, dv is bounding in U_{∗}(L), ζ^{−1}(dv) = du, so that dv = dζ(u). If z = u_{A} + v, u_{A} ∈ im η, v as before, then we have

z = u_{A} + (v − u) + u.

Since v − u ∼ 0 (as a cycle in cone U_{∗}) the assertion follows.

2) Let z ∈ A_{∗}(L), L ∈ 𝔏 be a cycle, such that z bounds, then the same argument ensures that z bounds already in A_{∗}(L). Hence H_{∗}(η(L)) is monic.- So H_{∗}(η(L)) is an isomorphism.-

We summarize:

5.5. Theorem: Let A_{∗} be any chain functor, 𝔏 ⊂ 𝔎^{2} any subcategory, then there exists a (with respect to A_{∗}) natural exact sequence of chain functors

𝔏A_{∗} ⟶^{ζ} A_{∗} ⟶^{η} A_{𝔏∗}  (4)

where _{𝔏}A_{∗} is 𝔏 -acyclic, A_{𝔏}_{∗} 𝔏 -local and η an 𝔏 -isomorphism. A_{𝔏}, _{𝔏}A_{∗} are regular (resp. have compact carriers), whenever we start with a regular A_{∗} (resp. with one having compact carriers).

Proof: The existence of (4) is a consequence of the preceding lemmas, while the exactness (to be understood in an obvious sense) follows from (1)-(3). The second assertion will be treated by theorem 6.8. (cf. 5.2.) -

5.6. Corollary: Let A_{∗} be as in 5.5., E ∈ 𝔎 any object, then there exists an exact sequence

EA_{∗} ⟶^{ζ} A_{∗} ⟶^{η} A_{E}_{∗}  (5)

with E -acyclic _{E}A_{∗} and E -local A_{E}_{∗}.

In [5] we mentioned the possibility to extend the definition of a chain functor to a category of spectra (i.e. to the Boardman category 𝔅 resp. to the category 𝔅^{2} of pairs of spectra). We can immediately rephrase 5.6. for this case, accomplishing

5.7. Corollary: Let A_{∗} be a chain functor on 𝔅^{2}, E ∈ 𝔅, then there exists an exact sequence (5).

This covers the localization process described in [6] by A. Bousfield. We return to example 2) at the beginning of this section: If the category on which our chain functors are defined is the Boardman category 𝔅, any homology theory Φ which is defined on 𝔅 admits an extension over the category of all chain functors by setting Φ(A_{∗}) = Φ(|A_{∗}|), where |A_{∗}| is the realization of the chain functor in the sense of theorem 1.1. in [3]. Moreover Φ has a classifying spectrum E so that E_{∗} = Φ. Now it turns out easily that 𝔏_{Φ} -localization is the same as E -localization.

Suppose now that Φ is a homology theory on 𝔎^{2} ⊂ Top^{2}, then we define 𝔏_{Φ} = {(K, L)|∀X ∈ 𝔎; Φ(X) = 0 ⇒ X_{∗}(K, L) = 0}, where X_{∗} denotes the homology theory associated with X. This provides us with a second access to the localization of a chain functor A_{∗} defined on a category 𝔎^{2} (⊂ Top^{2}) with respect to a spectrum E ∈ 𝔅 (not being an object of 𝔎^{2}!) by using the homology theory E_{∗} = Φ and localizing A_{∗} with respect to 𝔏_{Φ} in the sense of 5.5. As a result we get:

5.8. Corollary: Let A_{∗} be as in 5.5., E ∈ 𝔅 a spectrum, then there exists an exact localization sequence (5).

6. Localizations and tensor products

In §3 we introduced a chain functor ℤ ℤ ℤ_{∗} to which the localization process can be applied. Hence for given subcategory 𝔏 ⊂ 𝔎^{2} we obtain _{𝔏}ℤ ℤ ℤ_{∗} and ℤ ℤ ℤ_{𝔏}_{∗}. It turns out that for any chain functor A_{∗} the exact sequence (4) in §5 originates from the sequence

𝔏ℤ ℤ ℤ_{∗} ⟶^{ζ} ℤ ℤ ℤ_{∗} ⟶^{η} ℤ ℤ ℤ_{𝔏}_{∗}  (1)

by tensoring with A_{∗}. Recall that the chain functor U_{∗} of 4.5. is isomorphic to A_{∗}|𝔏 ⊗ D_{∗} (cf. proof of 4.3.).

6.1. Lemma: For any chain functor A_{∗} we have a natural homotopy equivalence

A_{∗} ⊗ _{𝔏}ℤ ℤ ℤ_{∗} ⟶^{≃} _{𝔏}A_{∗}.

Proof: At first we introduce and recall some notations which are mostly taken from the proof of 4.1.: There are elements {z_{X}; z_{K} ⊗x}, {0; z_{K} ⊗dx} in _{𝔏}ℤ_{n}(X, A), n = 0, −1 as well as a = {a; f_{#}(a)⊗x}, a = {a; a_{f} ⊗dx}, f_{#}(a) = 0 for all f : (X, A) ⟶ (E, F) ∈ 𝔏 , which are continuous mappings or words in the sense of the proof of 4.1. (II), a ∈ _{𝔏}A_{n}(X, A). Moreover in cone K_{∗}^{0}(X, A) ⊂ _{𝔏}A_{∗}(X, A) we have elements Δ({0; a_{f} ⊗ dx}), satisfying dΔ({0; a_{f} ⊗ dx}) = {0; a_{f} ⊗ dx} − Δ({0; da_{f} ⊗ dx}). In _{𝔏}ℤ ℤ ℤ_{∗} we encounter chains Δ({0; z_{K} ⊗ dx}) = Δ ∈ K_{∗}^{0}(X, A) and {z_{X}; z_{K} ⊗ x} with the same boundary. We denote the cycle {z_{X}; z_{K} ⊗ x} − Δ by ∇ ∈ _{𝔏}ℤ_{0}(X, A). Using these notations we define

η : A A A_{∗} ⊗_{𝔏}ℤ ℤ ℤ_{∗} ⟶ _{𝔏}A_{∗}

by setting

η(a ⊗ {z_{X}; z_{K} ⊗ x}) = {a; f_{#}(a) ⊗ x}
η(a ⊗ {0; z_{K} ⊗ dx}) = {0; f_{#}(a) ⊗ dx})
η(a ⊗ Δ) = Δ{0; f_{#}(a) ⊗ dx}.

One checks immediately that η is a natural chain mapping. The inverse

ζ : _{𝔏}A_{∗} ⟶ A_{∗} ⊗ _{𝔏}ℤ ℤ ℤ_{∗}

is defined by setting

ζ({a; f_{#}(a) ⊗ x}) = a ⊗ ∇

while on all elements in cone K_{∗}^{0} we set ζ = 0. This establishes again a natural chain mapping. We have

ζη(a ⊗ {z_{X}; z_{K} ⊗ x}) = a ⊗ ∇ = a ⊗ {z_{X}; z_{K} ⊗ x} − a ⊗ Δ,
ηζ({a; f_{#}(a) ⊗ x}) = {a; f_{#}(a) ⊗ x} − Δ({0; f_{#}(a) ⊗ dx}).

Here, as in all remaining cases, we observe that

ζη( ) − ( )

resp.

ηζ( ) − ( )

are elements of A_{∗} ⊗ cone K_{∗}^{0} (K_{∗}^{0} understood for _{𝔏}ℤ_{∗}) resp. of cone K_{∗}^{0} (now for A_{∗}). Since cone K_{∗}^{0} (contained in _{𝔏}A_{∗}) as well as A_{∗} ⊗ cone K_{∗}^{0} are acyclic (the latter e.g. due to the Künneth-formula), we conclude that

ζη ≃ 1
ηζ ≃ 1.

This chain homotopy can of course be realized in detail, which we will accomplish for ζη(a ⊗ {z_{X}; z_{K} ⊗ x}), The remaining cases are settled by a similar procedure. We need a Γ(u) for u = a ⊗ {z_{X}; z_{K} ⊗ x} such that ζη(u) − u = dΓ(u) + Γdu. Observe that ζη(du) = da⊗{z; z_{K} ⊗x}−(−1)^{n−1}da⊗{0; z_{K} ⊗dx}+(−1)^{n−1}da⊗ {0; z_{K} ⊗ dx} − da ⊗ Δ. Since −(−1)^{n−1}da ⊗ {0; z_{K} ⊗ dx} − da ⊗ Δ = w is a cycle in A_{∗}⊗cone K_{∗}^{0}, hence bounding, there exists a Γ(du) in A_{∗}⊗cone K_{∗}^{0}, such that

w = dΓ(du) = ζη(du) − du

Now we have

ζη(u) − u − Γ(du) = −a ⊗ Δ − Γ(du) = w′

with w′ ∈ A ⊗ cone K_{∗}^{0}. Since w′ is a cycle, there exists a dΓ(u) satisfying dΓ(u) = w′ = ζη(u) − u − Γ(du). This can be performed in a natural way, establishing a Γ(u) such that

dΓ(u) + Γ(du) = ζη(u) − u.

Up to this point η as well as ζ are mappings of chain complexes (rather than of chain functors). However endowing them with the structure of chain functor transformations is immediate. This completes the proof of 6.1.-

6.2. Lemma: For any chain functor A_{∗} we have a natural isomorphism

A_{∗} ⊗ cone _{𝔏}ℤ ℤ ℤ_{∗} ≈ cone (A_{∗} ⊗ _{𝔏}ℤ ℤ ℤ_{∗})

Proof: We have for any chain functor C_{∗} an isomorphism

(cone C_{∗})(X) ≈ C_{∗}(X) ⊗ B_{∗}

where B_{∗} = ∑ D_{∗} (denoting by ∑ the suspension), i.e. B_{1} = ⟨x⟩, B_{0} = ⟨dx⟩, B_{n} = 0 elsewhere (§4). Hence we detect a sequence of isomorphisms

A_{∗} ⊗ cone _{𝔏}ℤ ℤ ℤ_{∗} ≈ A_{∗} ⊗ (_{𝔏}ℤ_{∗} ⊗ B_{∗}) ≈
≈ (A_{∗} ⊗ _{𝔏}ℤ ℤ ℤ_{∗}) ⊗ B_{∗} ≈ cone (A_{∗} ⊗ _{𝔏}ℤ ℤ ℤ_{∗}).−

6.3. Lemma We have a natural homotopy equivalence

A_{𝔏}_{∗} ≃ A_{∗} ⊗ ℤ ℤ ℤ_{𝔏}_{∗}.

Proof: We have

A_{𝔏}_{∗} ≈ A_{∗} ∪_{ζ} cone (_{𝔏}A_{∗}) ≃
A_{∗} ∪_{ζ} cone (A_{∗} ⊗ _{𝔏}ℤ ℤ ℤ_{∗}) ≈
A_{∗} ∪_{ζ} (A_{∗} ⊗ cone _{𝔏}ℤ ℤ ℤ_{∗}) ≈ A_{∗} ⊗ (ℤ ℤ ℤ_{∗} ∪_{ζ} cone (_{𝔏}ℤ ℤ ℤ_{∗})) ≈
≈ A_{∗} ⊗ ℤ ℤ ℤ_{𝔏}_{∗}.−

6.4. Lemma: 1_{A_{∗}} ⊗ η : A_{∗} ⊗ ℤ ℤ ℤ_{∗} ⟶ A_{∗} ⊗ ℤ ℤ ℤ_{𝔏}_{∗} is an 𝔏 - isomorphism. Proof: Follows immediately.-

We summarize this in the following main theorem:

6.5. Theorem: Let A_{∗} be any chain functor, 𝔏 ⊂ 𝔎^{2} any subcategory, then the exact 𝔏 -localization sequence of A_{∗} is given by:

[figure: A_{∗} ⊗ _{𝔏}ℤ ℤ ℤ_{∗}; 1 ⊗ ζ; A_{∗} ⊗ ℤ ℤ ℤ_{∗}; 1 ⊗ η; A_{∗} ⊗ ℤ ℤ ℤ_{𝔏}_{∗}; _{𝔏}A_{∗}; A_{∗}; A_{𝔏}_{∗}; ≃; ≃; ≃]

Proof: The only remaining fact which needs proof is that 1 ⊗ ζ resp. 1 ⊗ η are in fact the mappings ζ resp. η of §5(2) for A_{∗}, but this is due to the construction straightforward.-

6.6. Corollary: Let E ∈ 𝔎 be any object, then the exact E -localization sequence of a chain functor takes the form

A_{∗} ⊗ _{E}ℤ ℤ ℤ_{∗} ⟶ A_{∗} ⟶ A_{∗} ⊗ ℤ ℤ ℤ_{E}_{∗}.

6.7. Corollary: There exist natural homotopy equivalences

A_{𝔏∗} ⊗ B_{𝔏∗} ≃ (A_{∗} ⊗ B_{∗})_{𝔏∗} ≃ A_{∗} ⊗ B_{𝔏∗}.  (2)

Proof: One has:

(A_{∗} ⊗ B)_{𝔏∗} ≃ (A_{∗} ⊗ B_{∗}) ⊗ ℤ ℤ ℤ_{𝔏∗} ≃ A_{∗} ⊗ (B_{∗} ⊗ ℤ ℤ ℤ_{𝔏∗}).

However since (A_{𝔏∗})_{𝔏∗} ≃ A_{𝔏∗} as an immediate consequence of the definition of an 𝔏-local object in definition 5.1., in particular ℤ ℤ ℤ_{𝔏∗} ⊗ ℤ ℤ ℤ_{𝔏∗} ≃ (ℤ ℤ ℤ_{𝔏∗})_{𝔏∗}, we have

A_{∗} ⊗ B_{∗} ⊗ ℤ ℤ ℤ_{𝔏∗} ≃ A_{∗} ⊗ B_{∗} ⊗ ℤ ℤ ℤ_{𝔏∗} ⊗ ℤ ℤ ℤ_{𝔏∗}
≃ (A_{∗} ⊗ ℤ ℤ ℤ_{𝔏∗}) ⊗ (B_{∗} ⊗ ℤ ℤ ℤ_{𝔏∗}) ≃ A_{𝔏∗} ⊗ B_{𝔏∗}.

Remarks: 1) Since there is no spectrum |ℤ ℤ ℤ_{∗}| associated with the irregular–chain– functor ℤ ℤ ℤ_{∗}, theorem 6.5. has no immediate counterpart for spectra, unless one agrees to define tensor products |B ⊗ C_{∗}| between spectra B and chain functors C_{∗}, which is of course possible.-

2) Corollary 6.7. ensures that, unlike the ∧-product, the ⊗- product turns out to be compatible with localizations.

6.8. Theorem If A_{∗} is regular (resp. has compact carriers), then _{𝔏}A_{∗} and A_{𝔏∗} are regular chain functors (resp. they have compact carriers), i.e. the derived homology is a homology theory (with, resp. without compact carriers).

Proof: If the chain functors K_{∗} and C_{∗} in 4.1. are regular, we have that \overline{K}_{∗} satisfies the requirement that l, i′ and ((X, A) ⊂ (Y, B)_{#}) are inclusions onto direct summands (cf. 4.1. 3)). Hence _{𝔏}A_{∗} has this property. Moreover it is immediate that this property carries over to A_{𝔏∗} (§5 (2)). So it remains to deal with excision. To this end we retain the notation of the proof of 6.1. and consider A_{∗} ⊗_{𝔏}ℤ ℤ ℤ_{∗}. Suppose that h : (X, A) ⟶ (X/A, ⋆) is an excision map, then the second component in the mapping of tensor products

h_{#} : (A_{∗} ⊗_{𝔏}ℤ_{∗})(X, A) ⟶ (A_{∗} ⊗_{𝔏}ℤ_{∗})(X/A, ⋆)

does not change: Due to the definition of tensorproducts in §1 (1) as well as that of ℤ ℤ ℤ_{∗} in §3, only h : X ⟶ X/A enters, inducing an isomorphism even on the chain level. So one has to deal merely with h_{#} on A_{∗}, which, by assumption induces an isomorphism of the derived homology. Consequently h_{#} on A_{∗} ⊗ _{𝔏}ℤ ℤ ℤ_{∗} induces an isomorphism of the derived homology of A_{∗} ⊗_{𝔏}ℤ ℤ ℤ_{∗} and therefore, because of 6.1., of 𝔏A_{∗}. By a similar argument we ensure, employing 6.3., that A_{𝔏∗} satisfies the same excision property as A_{∗}.

According to 4.1. _{𝔏}A_{∗} = Ū_{∗} can be assumed to have compact carriers; the same holds for cone Ū_{∗} and consequently A_{∗} ∪_{ζ} cone Ū_{∗} inherit this property from that of A_{∗} and of cone Ū_{∗}.-

Although tensor products do not appear explicitely in the formulation of 6.8., it nevertheless displays itself as an implication of 6.1., 6.3.. We can apply 6.8. to regain the existence of Bousfield-localizations of spectra (rather than of chain functors):

6.9. Corollary: Let 𝔏 ⊂ 𝔎^{2}, 𝔎 = 𝔅 be as before, A a spectrum, then there exists an exact sequence of spectra

𝔏A ⟶^{ζ} A ⟶^{η} A_{𝔏}  (3)

with 𝔏- acyclic _{𝔏}A, 𝔏- local A_{𝔏} and 𝔏-isomosphic η. Proof: The spectrum A is associated with a regular chain functor C_{∗} = ^{A}C_{∗}. Theorem 6.8. in combination with [3] theorem 1.1. ensure that | _{𝔏}C_{∗} |= _{𝔏}A as well as | C_{𝔏∗} | exists. The remaining points are settled by standard arguments.-

6.9. Corollary: (A. Bousfield [6]) Let A, B ∈ 𝔅 be spectra, then there exists a localization sequence (2), with E replacing 𝔏.

We are presently not treating the unstable localization theory of [7].

References

[1] F. Adams, ”Stable homotopy and generalized homology”. Univ. of Chicago Press, Chicago 1974

[2] F. W. Bauer, Generalized homology theories and chain functors. Annali di Matematica pura ed appl.(IV) Vol.CLV (1989) 143-191

[3] —”—, Classifying spectra for generalized homology theories. Annali di Matematica pura ed appl. (IV) Vol. CLXIV (1993) 365-399

[4] —”—, The Boardman category of spectra, chain complexes and (co-)localizations. Homology, Homotopy and Applications Vol. 1 No. 3 (1999) 95 - 116

[5] —”—, Chain functors with isomorphic homology

[6] A.K. Bousfield, Localization of a spectrum with respect to homology. Topology 18 (1979) 257-281

[7] —”—, Homotopical localizations of spaces. Am. Journ. of Math. 119 (1997) 1321 - 1354

[8] A. Dold, ”Lectures on Algebraic Topology”. Springer, Berlin-Heidelberg-NewYork (1972)

[9] A. D. Elmendorf, I. Kriz, M. A. Mandell, J. P. May ” Rings, modules and algebras in stable homotopy theory”. Amer. Math. Soc. Surveys and Monographs Vol. 47 (1996)

[10] M. Hovey, J. H. Palmieri, N. P. Strickland Axiomatic stable homotopy theory. Vol. 128, Mem. Am. Math. Soc. no. 610 AMS (1997)

[11] H. R. Margolis, ”Spectra and the Steenrod Algebra”. North- Holland Mathem. Library, Amsterdam-NewYork-Oxford (1983)

[12] D. C. Ravenel, Nilpotence and periodicity in stable homotopy theory. Ann. Math. Stud. no. 128, Princeton Univ. Press, Princeton NJ, (1992)

[13] R. Switzer, ”Algebraic Topology-Homotopy and Homology”. Springer, Berlin-Heidelberg-NewYork (1975)

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Friedrich W. Bauer f.w.bauer@mathematik.uni-frankfurt.de

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