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of chain complexes and inverse limit of cohomology groups of corre- sponding cochain complexes. The result for non-free chain complexes is extended in the paper [Mdz-Sp]. On the other hand, to define an exact homology (Steenrod) theory [St], [Ed-Ha1], [Ed-Ha2], it is much convenient to obtain a chain complex from a cochain complex [Mas3], [Mas4], [Mil], [St]. In this case, a homology H∗ is generated by cochain complex C∗ and if the corresponding cochain complex is free, then there is an analogous Universal Coefficient Formula of a cohomology theory (not standard Universal Coefficient Formula of a homology theory) [Ber], [Ber-Mdz1], [Ber-Mdz2], [Bor-Mo], [Mas4], [Mil], [Skl]. In this paper the result is extended for non-free cochain complexes and using of it, the relation of homology groups of the direct limit of cochain complexes and the inverse limit of homology groups of cor- responding chain complexes is studied. As a corollary, the tautness property of a homology theory is obtained. Moreover, for the de- fined exact homology theory the continuous property (see Definition 1 [Mdz1]) is obtained on the category of compact pairs.

1 Introduction

α ′ β ′′ Let C∗ be a chain complex and 0 → G −→ G −→ G → 0 be an injective resolution of a R-module G over a principal ideal domain R. Let β# : ′ ′′ ′ ′′ Hom(C∗; G ) → Hom(C∗; G ) be the cochain map induced by β : G → G ∗ # n # and C (β )= {C (β#), δ} be the cone of the cochain map β , i.e.,

n # ′ ′′ (1.1) C (β ) ≃ Hom(Cn; G ) ⊕ Hom(Cn−1; G ),

′ ′′ ′ ′ ′′ ′ ′′ (1.2) δ(ϕ , ϕ ) = (ϕ ◦ ∂,β ◦ ϕ − ϕ ◦ ∂), ∀(ϕ , ϕ ) ∈ Cn(β#).

∗ # # It is natural that the cochain complex C (β ) = Hom(C∗; β ) is defined ∗ by the chain complex C∗ and R-module G. Consequently, let H¯ (C∗; G) be ∗ # a cohomology of the cochain complex C (β ). In the papers [Mdz3] (in the case of free chain complexes) and [Mdz-Sp] (in the general case) it is shown γ that for each direct system C∗ = {C∗ } of chain complexes there is a natural exact sequence: (1.3) (3) − (1) − n γ n (2) − ... lim H¯ n 2 lim H¯ n 1 H¯ limC∗ ; G limH¯ lim H¯ n 1 ... ←− γ ←− γ −→  ←− γ ←− γ

¯ ∗ ¯ ∗ γ ∗ ∗ where Hγ = H (C∗ ; G) = H (Cγ (β#)). Note that, using the methods de- veloped in [Mdz1] and [Mdz4], it is possible to show that if for a cohomology H∗ there exists a type (2.10) natural sequence, then there is an isomophism H¯ n(lim Cγ ; G) ≃ Hn(lim Cγ ; G). Therefore, it uniquely defines a cohomol- −→ ∗ −→ ∗ ∗ ogy H¯ generated by the chain complex C∗. On the other hand, our aim is 3 to develop tools which uniquely define a homology theory generated by the given cochain complex. Therefore, we will consider the dual case. ∗ ∗ ′ ∗ ′′ Let C be a cochain complex and β# : Hom(C ; G ) → Hom(C ; G ) ′ ′′ be the chain map induced by β : G → G . Consider the cone C∗(β#) = ∗ {Cn(β#), ∂} = {Hom(C , β#), ∂} of the chain map β#, i.e.,

n ′ n+1 ′′ (1.4) Cn(β#) ≃ Hom(C ; G ) ⊕ Hom(C ; G ),

′ ′′ ′ ′ ′′ ′ ′′ (1.5) ∂(ϕ , ϕ ) = (ϕ ◦ δ, β ◦ ϕ − ϕ ◦ δ), ∀(ϕ , ϕ ) ∈ Cn(β#).

∗ Consequently, the homology group H¯n is denoted by H¯n = H¯n(C ; G)

= Hn(C∗(β#)) and is called a homology with coefficient in G generated by the cochain complex C∗. Note that if f : C∗ −→ C′∗ is a cochain map, ′ then it induces the chain map f¯ : C∗(β#) −→ C∗(β#). In particular, for Z ¯ ′ each n ∈ the homomorphism fn : Cn(β#) −→ Cn(β#) is defined by the ¯ ′ ′′ ′ formula fn(ϕ ,ϕ )=(ϕ ◦fn,ϕ◦fn+1). Consequently, it induces a homomor- ′∗ ∗ phism of homology groups f¯ : H¯n(C ; G) −→ H¯n(C ; G). Therefore, H¯n is a naturally defined functor. C∗ ∗ Let = {Cγ } be a direct system of cochain complexes. Consider the C γ H ¯ ∗ corresponding inverse systems ∗ = {C∗ (β#)} and ∗ = {H∗(Cγ ; G)}. In this paper we have shown that there is a natural exact sequence: (1.6) (3) γ (1) γ ∗ γ (2) γ . . . lim H¯ lim H¯ H¯n limC ; G limH¯n lim H¯ . . . , ←− n+2 ←− n+1 −→ γ
←− ←− n+1

¯ γ ¯ ∗ γ where H∗ = H∗(Cγ ; G) = H∗(C∗ (β#)). On the other hand, to obtain se- quence (1.6), we have shown that for each cochain complex (it is not nec- essary to be free) C∗ and a R-module G over a principal ideal domain R, there exists a short exact sequence (Universal Coefficient Formula):

n+1 ∗ ∗ n ∗ (1.7) 0 −→ Ext(H (C ); G) −→ H¯n(C ; G) −→ Hom(H (C ); G) −→ 0.

At the end we have formulated and studied the tautness property for a ∗ homology theory H¯∗ generated by the Alexander-Spanier cochains C¯ on the category of paracompact Hausdorff spaces.

2 Universal Coefficient Formula

In this section we will prove the Universal Coefficient Formula for a homol- ∗ ogy theory H¯∗ generated by the given cochain complex C .

Theorem 2.1 (Universal Coefficient Formula). For each cochain complex C∗ and R-module G over a fixed principal ideal domain R, there exists a 4 A. Beridze and L. Mdzinarishvili short exact sequence:

¯ n+1 ∗ χ¯ ∗ ξ n ∗ (2.1) 0 −→ Ext(H (C ); G) −→ H¯n(C ; G) −→ Hom(H (C ); G) −→ 0.

n ∗ Proof. We will define a homomorphism ξ : Z¯n −→ Hom(H (C ); G), which ∗ n ∗ induces an epimorphism ξ¯ : H¯n(C ; G) −→ Hom(H (C ); G). On the other n+1 ∗ ′′ ∗ hand, we will define a homomorphism χ : Hom(H (C ); G ) −→ H¯n(C ; G) n+1 ∗ ∗ such that χ induces a monomorphism χ¯ : Ext(H (C ); G) −→ H¯n(C ; G) and the short sequence (2.1) is exact. n ∗ a. There is a homomorphism ξ : Z¯n → Hom(H (C ); G). Let ′ ′′ ′ n ′ ′′ n+1 ′′ (ϕ ,ϕ ) ∈ Z¯n be a cycle, i.e., ϕ : C → G and ϕ : C → G are homomorphisms such that ∂(ϕ′,ϕ′′)=(ϕ′ ◦ δ, β ◦ ϕ′ − ϕ′′ ◦ δ) = (0, 0)= 0 and therefore, the following diagram is commutative:

δ δ Cn−1 Cn Cn+1 0 ′ ′′ ϕ ϕ

′ β ′′ (2.2) G G where 0 is the zero map. Consider the groups of coboundaries Bn and cocy- cles Zn. Let i : Bn → Zn and j : Zn → Cn be natural monomorphisms and δ′ : Cn−1 → Bn be an epimophism induced by δ : Cn−1 → Cn. Therefore, we have the following sequence:

′ − δ i j δ (2.3) Cn 1 −→ Bn −→ Zn −→ Cn −→ Cn+1, where j ◦ i ◦ δ′ = δ and consequently δ ◦ j =0. ′ ′′ Since (ϕ ,ϕ ) ∈ Z¯n, we have the following commutative diagram:

j δ Zn Cn Cn+1

ϕ′ ϕ′′

α ′ β ′′ (2.4) 0 G G G 0 .

Hence, ϕ′′ ◦ δ ◦ j = β ◦ ϕ′ ◦ j, and by the equality δ ◦ j =0, we obtain that β ◦ ϕ′ ◦ j =0. So, Im(ϕ′ ◦ j) ⊂ Kerβ = Imα. Therefore, there is a uniquely defined map ϕ : Zn → G such that ϕ′ ◦ j = α ◦ ϕ (see the diagram (2.5).

j δ Zn Cn Cn+1

ϕ ϕ′ ϕ′′

α ′ β ′′ (2.5) 0 G G G 0 . 5

By the commutative diagram (2.2), we have ϕ′ ◦ δ = ϕ′ ◦ j ◦ i ◦ δ′ = 0. Hence, α ◦ ϕ ◦ i ◦ δ′ = ϕ′ ◦ j ◦ i ◦ δ′ = ϕ′ ◦ δ =0 (see the diagram (2.6)). α is a monomorphism and so ϕ ◦ i ◦ δ′ = 0. On the other hand, δ′ is an epi- morphism. Consequently, we have ϕ ◦ i =0. Therefore, the homomorphism ϕ : Zn → G induces a homomorphism ϕ¯ : Hn(C∗) → G which belongs to Hom(Hn(C∗); G). Hence, the following diagram is commutative:

δ′ i j δ Cn−1 Bn Zn Cn Cn+1 p n ′ ′′ 0 ϕ H ϕ ϕ ϕ¯ α ′ β ′′ (2.6) 0 G G G 0 ,

n n ∗ n where H = H (C ). Let ξ : Z¯n → Hom(H ; G) be the homomorphism defined by

′ ′′ ′ ′′ (2.7) ξ(ϕ , ϕ )=ϕ, ¯ ∀(ϕ , ϕ ) ∈ Z¯n.

n ∗ b. ξ : Z¯n → Hom(H (C ); G) is an epimorphism. Let ϕ¯ ∈ Hom(Hn(C∗); G) be a homomorphism and ϕ =ϕ ¯ ◦ p : Zn → G is the composition, where p : Zn → Hn(C∗) is a projection. Let ϕ′ : Cn → G′ be an extension of α ◦ ϕ : Zn → G′. In this case ϕ′ ◦ j = α ◦ ϕ and so β◦ϕ′◦j = β◦α◦ϕ =0. Therefore, β◦ϕ′ : Cn → G′′ vanishes on the subgroup Zn and so it induces a homomorphism ϕ˜′′ : Cn/Zn ≃ Bn+1 → G′′, which can be extended to a homomophism ϕ′′ : Cn+1 → G′′. Since ϕ˜′′ ◦ δ′ = β ◦ ϕ′, there is ∂(ϕ′,ϕ′′)=(ϕ′ ◦δ, β◦ϕ′ −ϕ′′ ◦δ)=(ϕ′ ◦j◦i◦δ′, β◦ϕ′ −ϕ′′ ◦j◦i◦δ′)= (α ◦ ϕ ◦ i ◦ δ′, β ◦ ϕ′ − ϕ˜′′ ◦ δ′)=(α ◦ ϕ¯ ◦ p ◦ i ◦ δ′, β ◦ ϕ′ − β ◦ ϕ′)=(0, 0)=0. ′ ′′ ′ ′′ Hence, (ϕ ,ϕ ) ∈ Z¯n and ξ¯(ϕ ,ϕ )=ϕ ¯ (see the diagram (2.8)).

′ j ′ j ◦ i − δ n i n n δ n n n+1 Cn 1 B Z p C C /Z ≃ B Cn+1 ′ ϕ n ′′ ′′ ϕ¯ H ϕ ϕ˜ ϕ α ′ β ′′ (2.8) 0 G G G 0.

n ∗ ∗ c. ξ : Z¯n → Hom(H (C ); G) induces a homomorphism ξ¯ : H¯n(C ; G) n ∗ → Hom(H (C ); G). We have to show that the homomorphism ξ : Z¯n → n ∗ ′ ′′ Hom(H (C ); G) vanishes on the subgroup B¯n. Indeed, let (ψ , ψ ) ∈ Cn+1(β#) ′ ′′ ′ ′ ′′ ′ ′′ be an element. For ∂(ψ , ψ )=(ψ ◦ δ, β ◦ ψ − ψ ◦ δ)=(ϕ ,ϕ ) ∈ B¯n ⊂ Z¯n we have ϕ′ ◦ j = 0. Indeed, ϕ′ ◦ j = ψ′ ◦ δ ◦ j = 0. Therefore, by the construction ξ, the homomorphism ϕ : Zn → G corresponding to the pair (ϕ′,ϕ′′) satisfies the equation α ◦ ϕ = ϕ′ ◦ j =0 and so ϕ =0, because α is 6 A. Beridze and L. Mdzinarishvili a monomorphism. Since ϕ =ϕ ¯ ◦ p and p is an epimorphism, we have ϕ¯ =0. Therefore, ξ∂(ψ′, ψ′′)= ξ(ϕ′,ϕ′′)=ϕ ¯ =0 (see the diagram (2.9)).

j n n δ δ Z p C Cn+1 Cn+2 ′ ϕ n ′ ′′ ′′ ϕ¯ H ϕ ψ ϕ ψ α ′ β ′′ (2.9) 0 G G G 0 .

∗ n ∗ n+1 ∗ d. The kernel of ξ¯ : H¯n(C ; G) → Hom(H (C ); G) is Ext(H (C ); G). If we apply the functor Hom(Hn+1(C∗); −) to the short exact sequence α β 0 → G −→ G′ −→ G′′ → 0, then we obtain:

α∗ β∗ 0 Hom(Hn+1(C∗); G) Hom(Hn+1(C∗); G′)

β∗ n+1 ∗ ′′ n+1 ∗ (2.10) Hom(H (C ); G ) Ext(H (C ); G) 0 .

Therefore, we have the following isomorphism:

n+1 ∗ n+1 ∗ ′′ (2.11) Ext(H (C ); G) ≃ Hom(H (C ); G )/Imβ∗.

Our aim is to define such a homomorpism χ : Hom(Hn+1(C∗); G′′) −→ ∗ H¯n(C ; G) that the following sequence is exact:

β∗ χ Hom(Hn+1(C∗); G′) Hom(Hn+1(C∗); G′′)

χ ξ¯ ¯ ∗ n ∗ (2.12) Hn(C ; G) Hom(H (C ); G) 0 .

Indeed, in this case, it is clear that for the homomorphisms ξ¯, χ and β∗, we have the following short exact sequences:

¯ ∗ ξ n ∗ (2.13) 0 −→ Kerξ¯ −→ H¯n(C ; G) −→ Hom(H (C ); G) −→ 0,

∗ ′′ χ (2.14) 0 −→ Kerχ −→ Hom(Hn+1(C ); G ) −→ Imχ −→ 0,

n+1 ∗ ′ β∗ (2.15) 0 −→ Kerβ∗ −→ Hom(H (C ); G ) −→ Imβ∗ −→ 0.

On the other hand, if we prove exactness of the sequence (2.12), then Kerξ¯ ≃ Imχ and Kerχ ≃ Imβ∗. Therefore, we have:

Kerξ¯ ≃ Imχ ≃ Hom(Hn+1(C∗); G′′)/Kerχ ≃

n+1 ∗ ′′ n+1 ∗ (2.16) ≃ Hom(H (C ); G )/Imβ∗ ≃ Ext(H (C ); G). 7

To define χ, consider an element ϕ¯′′ ∈ Hom(Hn+1(C∗); G′′). Let ϕ′′ : Cn+1 −→ G′′ be an extension of the composition ϕ¯′′ ◦ p : Zn+1 −→ G′′, where p : Zn+1 −→ Hn+1(C∗) is a natural projection. In this case, ϕ′′ ◦ δ = ϕ′′ ◦ j ◦ i ◦ δ′ =ϕ ˜′′ ◦ p ◦ i ◦ δ′ =0 and so, if we take ϕ′ =0, then the following diagram is commutative:

δ δ′ i j Cn−1 Cn Bn+1 Zn+1 Cn+1 p 0 ′ ′′ ϕ =0 Hn+1 ϕ ϕ¯′′ ′ β ′′ (2.17) G G .

′′ Therefore, (0,ϕ ) ∈ Z¯n and so, we can define χ in the following way:

′′ ′′ ′′ n+1 ∗ ′′ (2.18) χ(¯ϕ )=(0, −ϕ )+ B¯n, ∀ ϕ¯ ∈ Hom(H (C ); G ). ′′ ′′ Let check that χ is well defined. Consider two different extensions ϕ1 and ϕ2 ′′ n+1 ′′ ′′ ¯ ′′ ¯ of the map ϕ¯ ◦p : Z −→ G and show that (0, −ϕ1)+Bn = (0, −ϕ2)+Bn. ′′ ′′ ¯ For this, we have to show that (0,ϕ2 −ϕ1 ) ∈ Bn. Indeed, by the definition of ′′ ′′ ′′ ′′ ′′ ′′ ′′ ′′ ϕ1 and ϕ2, it is clear that (ϕ1 −ϕ2)◦j = ϕ1 ◦j−ϕ2 ◦j =ϕ ¯ ◦p−ϕ¯ ◦p =0 and ′′ ′′ n+1 n+1 ′′ so, ϕ1 − ϕ2 induces a homomorphism ψ : C /Z −→ G . On the other hand, Cn+1/Zn+1 ≃ Bn+2 and so, we have an extension ψ′′ : Cn+2 −→ G′′ of ψ (see the diagran (2.19)).

j δ′ j ◦ i Zn+1 Cn+1 Cn+1/Zn+1 ≃ Bn+2 Cn+2 ′′ ′ ′′ ψ ψ 0 ϕ1 − ϕ2

′′ (2.19) G In this case, it is easy to see that ′′ ′′ ′′ ′ ′ ′′ ′′ (2.20) ∂(0, ψ )=(0, −ψ ◦ δ)=(0, −ψ ◦ j ◦ i ◦ δ )=(0, −ψ ◦ δ )=(0, ϕ2 − ϕ1 ).

Therefore, it remains to show that Imχ ≃ Kerξ¯ and Imβ∗ = Kerχ. ′′ n+1 ∗ ′′ d1. Imχ ≃ Kerξ¯. Let ϕ¯ ∈ Hom(H (C ); G ) be an element, then ξ¯(χ(¯ϕ′′)) = ξ¯ (0, −ϕ′′)+ B¯ =ϕ ¯. On the other hand, by construction of ξ n
and the fact that the first coordinate of the pair (0, −ϕ′′) is zero, it is easy to check that ϕ¯ = 0. Therefore, Imχ ⊂ Kerξ.¯ Now consider an element ′ ′′ h¯ ∈ Kerξ¯ and any of its representatives (ϕ ,ϕ ) ∈ Z¯n. In this case, by the definition of ξ¯, there exists ϕ : Zn → G such that the following diagram is commutative: j δ Zn p Cn Cn+1 ′ ′′ ϕ ϕ¯Hn ϕ ϕ α ′ β ′′ (2.21) 0 G G G 0 . 8 A. Beridze and L. Mdzinarishvili

Moreover, h¯ ∈ Kerξ¯means that the homomorphism ϕ¯ : Hn → G induced by ϕ : Zn → G is zero. Therefore, ϕ =0 and so ϕ′ ◦j = α◦ϕ =0. Consequently, ϕ′ : Cn → G′ induces a homomorphism ψ˜′ : Cn/Zn ≃ Bn+1 → G′. Let ψ′ : Cn+1 → G′ be an extension of ψ˜′ ◦p : Zn+1 → G′ and ψ′′ = β◦ψ′ (see the diagram (2.22)). In this case, the homomorphism ψ = ψ′′ − ϕ′′ : Cn+1 → G′′ vanishes on the Bn+1. Indeed, ψ ◦ j ◦ i ◦ δ′ = ψ′′ ◦ j ◦ i ◦ δ′ − ϕ′′ ◦ j ◦ i ◦ δ′ = β ◦ψ˜′ ◦δ′ −β ◦ϕ′ = β ◦ϕ′ −β ◦ϕ′ =0. On the other hand, δ′ is an epimorphim and so ψ ◦ j ◦ i =0. Therefore, ψ ◦ j : Zn+1 → G′′ induces a homomorphism ψ¯ : Hn+1 → G′′ (see the diagram (2.22)). (2.22) j δ′ i p Zn Cn Bn+1 Zn+1 Hn+1 j ψ˜′ ′ n+1 ϕ ϕ ′ C ′′ ψ¯ ψ ψ ′′ α β ϕ 0 G G′ G′′ 0 .

′ ′′ Our aim is to show that χ(ψ¯)= h¯ =(ϕ ,ϕ )+ B¯n. Indeed, by the definition of χ, it is easy to see that χ(ψ¯)=(0, −ψ)+ B¯n. Therefore, we have to show ′ ′′ ′ ′′ ′ ′′ that (ϕ ,ϕ ) − (0, −ψ)=(ϕ ,ϕ + ψ)=(ϕ , ψ ) ∈ B¯n. Indeed,

′ ′ ′ ′ ′ ′ ′ ′ ′′ ′ ′′ (2.23) ∂(ψ , 0)=(ψ ◦ δ, β ◦ ψ ) = (ψ ◦ j ◦ i ◦ δ ,β ◦ ψ ) = (ψ˜ ◦ δ , ψ ) = (ϕ , ψ ).

′ n+1 ∗ ′ d2. Imβ∗ ≃ Kerχ. Let ϕ¯ ∈ Hom(H (C ); G ) be an element and ϕ′′ : Cn+1 → G′′ be an extension of the composition β ◦ ϕ¯′ ◦ p : Zn+1 → G′′. In this case we have ∂(0, −ϕ′′)=(0,ϕ′′ ◦ δ)=(0,ϕ′′ ◦ j ◦ i ◦ δ′)=(0, β ◦ ϕ¯′ ◦ ′ ′′ p ◦ i ◦ δ ) = (0, 0) = 0 (see the diagram (2.25)). Therefore, (0, −ϕ ) ∈ Z¯n and so we have

′ ′ ′ ′′ (2.24) (χ ◦ β∗)(¯ϕ )= χ (β∗ (¯ϕ )) = χ (β ◦ ϕ¯ )=(0, −ϕ )+ B¯n.

′′ ′ n+1 ′ Our aim is to show that (0, −ϕ ) ∈ B¯n. Indeed, let ϕ : C −→ G be an extension of the composition ϕ¯′ ◦ p : Zn+1 −→ G′. In this case (β ◦ ϕ′ − ϕ′′) ◦ j = β ◦ ϕ′ ◦ j − ϕ′′ ◦ j = β ◦ ϕ¯′ ◦ p − β ◦ ϕ¯′ ◦ p = 0 and so β ◦ ϕ′ − ϕ′′ : Cn+1 −→ G′′ induces a homomorphism ψ˜′′ : Cn+1/Zn+1 ≃ Bn+2 −→ G′′ such that β ◦ ϕ′ − ϕ′′ = ψ˜′′ ◦ δ′. Let ψ′′ : Cn+2 −→ G′′ be an extension of a homomorphism ψ˜′ : Bn+2 −→ G′′ (see the diagram (2.25) ). (2.25) δ′ i j δ′ j ◦ i Cn Bn+1 Zn+1 Cn+1 Cn+1/Zn+1 ≃ Bn+2 Cn+2 p ϕ′′ ˜′′ n+1 ′ ψ ′′ H ϕ¯′ ϕ ψ β G′ G′′ 9

In this case, we have (2.26) ∂(−ϕ′, −ψ′′) = (−ϕ′ ◦ δ, −β ◦ ϕ′ + ψ′′ ◦ δ) = (−ϕ′ ◦ j ◦ i ◦ δ′, −β ◦ ϕ′ + ψ˜′′ ◦ δ′)=

= (−ϕ¯′ ◦ p ◦ i ◦ δ′, −β ◦ ϕ′ + (β ◦ ϕ′ − ϕ′′)) = (0, −ϕ′′).

′′ Therefore, (0, −ϕ ) ∈ B¯n and so, χ ◦ β∗ = 0. Hence, Imβ∗ ⊂ Ker χ. Now consider an element ϕ¯′′ ∈ Kerχ. Let ϕ′′ : Cn+1 → G′′ be an extension of the composition ϕ¯′′ ◦ p : Zn+1 → G′′ (see the diagram (2.28)). Then, by ′′ ′′ ′ ′′ χ(¯ϕ )=(0, −ϕ )+ B¯n =0, there exists (ψ , ψ ) ∈ Cn+1(β#) such that ′ ′′ ′ ′ ′′ ′′ (2.27) ∂(ψ , ψ ) = (ψ ◦ δ, β ◦ ψ − ψ ◦ δ)=(0, −ϕ ).

Therefore, ψ′ ◦δ = ψ′ ◦j ◦i◦δ′ =0. Since δ′ : Cn → Bn+1 is an epimorphism, we have ψ′ ◦ j ◦ i = 0 and so ψ′ ◦ j : Zn+1 → G′ induces a homomorphism ψ¯′ : Hn+1(C∗) → G′. On the other hand, by β ◦ ψ′ − ψ′′ ◦ δ = −ϕ′′, we have −ϕ′′ ◦ j = β ◦ ψ′ ◦ j − ψ′′ ◦ δ ◦ j = β ◦ ψ′ ◦ j (see the diagram (2.28)). ′ ′′ Therefore, β∗(ψ¯ )=ϕ ¯ and so, Kerχ ⊂ Imβ∗ :

δ Cn+1 j Cn+2

Zn+1 ψ′ p ψ′′ n+1 ′′ ψ¯′ H ϕ¯ ′ β ′′ (2.28) G G

Since for each injective abelian group G, a group of extensions Ext(−; G) is trivial, by the exact sequence (2.1) we obtain the following corollary (cf.

Lemma VII.4.4 [Mas1]) Corollary 2.2. If G is an injective, then there is an isomorphism

∗ n ∗ (2.29) H¯n(C ; G) ≃ Hom(H (C ); G).

∗ n Let C∗ = Hom(C ; G) be a chain complex, where Cn = Hom(C ; G) and ∂ are defined by ∂(ϕ)= ϕ ◦ δ, for ϕ ∈ Hom(Cn; G). In this case, there is a ∗ ∗ map α∗ : Hom(C ; G) −→ Hom(C ; β#) defined by: n (2.30) α∗(ϕ) = (α ◦ ϕ, 0), ∀ϕ ∈ Hom(C ; G).

∗ ∗ Let Hn(C ; G) be a homology group of chain complex Hom(C ; G). Theorem 2.3. If a cochain complex C∗ is free, then the homomorphism ∗ ∗ α∗ : Hom(C ; G) −→ Hom(C ; β#) induces an isomoprhism ∗ ∗ (2.31) α¯∗ : Hn(C ; G) −→ H¯n(C ; G). 10 A. Beridze and L. Mdzinarishvili

Proof. Since C∗ is a free cochain complex, there is a short exact sequence:

˜ n+1 ∗ χ˜ ∗ ξ n ∗ (2.32) 0 −→ Ext(H (C ); G) −→ Hn(Hom(C ; G)) −→ Hom(H (C ); G) −→ 0.

Let review how the morphisms ξ˜and χ˜ are defined according to W. Massey’s

[Mas1] approach. Note that Massey has considered a free chain complex case and consequently, he has obtained Universal Coefficient Formula for cohomology theory and not homology theory. ∗ n ∗ a. For each ϕ¯ ∈ Hn(Hom(C ; G)) element let ξ˜(¯ϕ): H (C ) −→ G be homomorphism given by

∗ (2.33) ξ˜(¯ϕ)(¯c)= hϕ, ci = ϕ(c), ∀ c ∈ c,¯ c¯ ∈ Hn(C ), where ϕ is a representative of ϕ¯ [Mas4]. n+1 ∗ ∗ b. To define the homomorphism χ˜ : Ext(H (C ); G) −→ Hn(Hom(C ; G)), we need to use the isomorphism (2.11). Consequently, the homomorphism χ˜ n+1 ∗ ′′ ∗ is the homomorphism induced by χ0 : Hom(H (C ); G ) −→ Hn(Hom(C ; G)), ′′ n+1 ∗ ′′ where χ0 is defined in the following way. Let ϕ¯ ∈ Hom(H (C ); G ) be any element. Since G′′ is an injective, there is an extension ϕ′′ : Cn+1 −→ G′′ of the composition ϕ¯′′ ◦ p : Zn+1 −→ G′′. In this case ∂(ϕ′′) = ϕ′′ ◦ δ = ϕ′′ ◦ j ◦ i ◦ δ′ =ϕ ¯′′ ◦ p ◦ i ◦ δ′ =0 (see diagram (2.34)).

j δ′ i j Cn−1 Cn Bn+1 Zn+1 Cn+1 ψ′ p ′ n+1 ′′ ϕ ϕ H ϕ¯′′ ϕ α ′ β ′′ (2.34) 0 G G G 0 .

′′ ∗ ′′ Therefore, it defines homology class [ϕ ] ∈ Hn+1(Hom(C ; G )). Let E : ∗ ∗ Hn+1(Hom(C ; G)) −→ Hn(Hom(C ; G)) be a boundary homomorphism induced by the following exact sequence:

∗ α# ∗ ′ β# ∗ ′′ (2.35) 0 −→ Hom(C ; G) −→ Hom(C ; G ) −→ Hom(C ; G ) −→ 0.

Define a homomorphism χ0 by the formula

′′ ′′ ′′ n+1 ∗ ′′ (2.36) χ0 (¯ϕ )= E ([ϕ ]) , ∀ ϕ¯ ∈ Hom(H (C ); G ).

n+1 ∗ ∗ Note that the homomorphism χ0 : Hom(H (C ); G) −→ Hn(Hom(C ; G)) n+1 ∗ ≃ ∗ is a composition of the isomorphism Hom(H (C ); G) −→ Hn+1(Hom(C ; G)) ∗ ∗ and the homomorphism E : Hn+1(Hom(C ; G)) −→ Hn(Hom(C ; G)). To ′ n+1 ′ write the explicit formula for χ0, consider such a map ψ : C → G that β ◦ ψ′ = ϕ′′ (this is possible, because the cochain complex C∗ is free). Let ϕ′ = ψ′ ◦ δ : Cn → G′, then we have β ◦ ψ′ = ϕ′′ ◦ δ = 0. Therefore, ϕ′ ∈ Kerβ = Imα and so, there exists a unique map ϕ : Cn → G such that 11

α◦ϕ = ϕ′. In this case, we have α◦ϕ◦δ = ϕ′◦δ = ψ′◦δ◦δ =0 and so, ϕ◦δ =0 because α is a monomorphism. On the other hand, ∂(ϕ)= ϕ ◦ δ =0. Con- ∗ sequently, ϕ defines a homology class [ϕ] ∈ Hn((Hom(C ; G)) (see diagram (2.34)). Finally, by the formula (2.36) we have

′′ ′′ ′′ n+1 ∗ ′′ (2.37) χ0 (¯ϕ )= E ([ϕ ]) = [ϕ], ∀ ϕ¯ ∈ Hom(H (C ); G ).

In this case, the sequence (2.32) is induced by the following sequence

β∗ χ0 Hom(Hn+1(C∗); G′) Hom(Hn+1(C∗); G′′)

χ ξ¯ 0 ¯ ∗ n ∗ (2.38) Hn(C ; G) Hom(H (C ); G) 0 .

Therefore, by (2.12) and (2.38), it is sufficient to show that the following diagram is commutative: (2.39)

β∗ χ ξ˜ n+1 ∗ ′ n+1 ∗ ′′ 0 ∗ n ∗ Hom(H (C ); G ) Hom(H (C ); G ) Hn(Hom(C ; G)) Hom(H (C ); G) 0

1 1 α¯∗ 1 β∗ χ ξ¯ n+1 ∗ ′ n+1 ∗ ′′ ∗ n ∗ Hom(H (C ); G ) Hom(H (C ); G ) H¯n(Hom(C ; β#)) Hom(H (C ); G) 0.

Indeed, let ϕ¯′′ ∈ Hom(Hn+1(C∗); G′′) be an element and ϕ′′ : Cn+1 −→ G′′ be an extension of the composition ϕ¯′′ ◦ p : Zn+1 −→ G′′. Then, by ′ definition of χ0 and method of snake lemma, we must take an element ϕ ∈ n+1 ′ ′ ′′ Hom(C ; G ), such that β#(ϕ ) = ϕ . Note that this is possible because of exactness of the sequence (2.35). Then, there is a cycle ϕ ∈ Hom(Cn; G), ′ such that α#(ϕ)= ∂(ϕ ). Let [ϕ]= ϕ + Bn be the corresponding element in ∗ ′′ the homology group Hn(Hom(C ; G)), then χ0(¯ϕ ) = [ϕ]. By the definition ∗ ∗ of the map α˜ : Hn(Hom(C ); G) −→ H¯n(Hom(C ); β#), we have

′′ ′′ (2.40) (˜α∗ ◦ χ0)(¯ϕ )=˜α∗ (χ0 (¯ϕ ))=α ˜∗ ([ϕ]) = (α ◦ ϕ, 0)+ B¯n.

n+1 ∗ ′′ ∗ On the other hand, by the definition of χ : Hom(G (C ); G ) −→ H¯n(Hom(C ); β#), we have

′′ ′′ ′′ n+1 ′′ (2.41) χ(¯ϕ )=(0, −ϕ )+ B¯n, ∀ ϕ¯ ∈ Hom(H ; G ).

′′ ′′ Therefore, we have to show that (α ◦ ϕ, 0) − (0, −ϕ )=(α ◦ ϕ,ϕ ) ∈ B¯n. ′ ′ ′′ Indeed, by the equality α#(ϕ) = ∂(ϕ ) and β#(ϕ ) = ϕ , we have α ◦ ϕ = ϕ′ ◦ δ and β ◦ ϕ′ = ϕ′′. Therefore,

′ ′ ′ ′′ (2.42) ∂(ϕ , 0)=(ϕ ◦ δ, β ◦ ϕ ) = (α ◦ ϕ, ϕ ). 12 A. Beridze and L. Mdzinarishvili

By (2.40), (2.41), and (2.42), we obtain that α˜∗ ◦ χ0 = χ. So, it remains to show that ξ¯◦ α˜ = ξ˜. ∗ Let [ϕ] ∈ Hn(Hom(C ; G)) be an element and ϕ is its representative. Then, by the definitions of ξ¯ and α˜∗ we have

(2.43) ξ¯◦ α˜∗ ([ϕ]) = ξ¯(˜α∗ ([ϕ])) = ξ¯(α ◦ ϕ, 0)=(ϕ)+ B¯n.
Therefore, if we take an element c¯ ∈ Hn(C∗) and any of its representatives c ∈ c¯, then by (2.43) we have

(2.44) ξ¯◦ α˜ ([ϕ]) (¯c)= ϕ + B¯ (¯c)= ϕ(c).
n
Therefore, by (2.33) , (2.43) and (2.44), we obtain that

(2.45) ξ¯◦ α˜ = ξ.˜

Note that by the commutative diagram (2.39), we obtain the following commutative diagram:

χ˜ ξ˜ n+1 ∗ ∗ n ∗ 0 Ext(H (C ); G) Hn(Hom(C ; G)) Hom(H (C ); G) 0

1 α¯∗ 1 χ¯ ξ¯ n+1 ∗ ¯ ∗ n ∗ (2.46) 0 Ext(H (C ); G) Hn(C ; G) Hom(H (C ); G) 0.

Therefore, if a cochain complex C∗ is free, then the classical Univer- sal Coefficient Formula is isomorphic to the Universal Coefficient Formula defined in this paper.

3 Some property of Hom and inverse limit func- tors

As we have seen in the previous section, there exists an epimorphism ξ : n ∗ Z¯n −→ Hom(H (C ); G) which induces a homomorphism:

∗ n ∗ (3.1) ξ¯ : H¯n(C ; G) −→ Hom(H (C ); G) and the following diagram is commutative:

ξ n ∗ Z¯n Hom(H (C ); G) p¯ ξ¯

¯ ∗ (3.2) Hn(C ; G). 13

To investigate Kerξ we construct a homomorphism

′ ′′ (3.3) ω : Hom(Cn+1; G ) ⊕ Hom(Cn+1/Bn+1; G ) −→ Kerξ by ω(ψ′, ψ′′)=(ψ′ ◦ δ, β ◦ ψ′ − ψ′′ ◦ q), where q : Cn+1 −→ Cn+1/Bn+1 is the quotient map. Let show that ω(ψ′, ψ′′) ∈ Kerξ. Indeed, ∂ω(ψ′, ψ′′) = ∂(ψ′ ◦δ, β ◦ψ′ −ψ′′ ◦q)=(ψ ◦δ ◦δ, β ◦ψ′ ◦δ −(β ◦ψ′ −ψ′′ ◦q)◦δ)=(0, β ◦ψ′ ◦ δ − β ◦ ψ′ ◦ δ + ψ′′ ◦ q ◦ δ)=(0, ψ′′ ◦ q ◦ δ)=(0, 0), because q ◦ δ =0. Hence, ′ ′′ n ∗ ω(ψ , ψ ) ∈ Z¯n. By the definition of ξ : Z¯n −→ Hom(H (C ); G), there exists a uniquely defined map ϕ : Zn −→ G such that α ◦ ϕ = ψ′ ◦ δ ◦ j. On the other hand, δ ◦ j =0 and so, α ◦ ϕ =0, which induces a homomorphism ϕ¯ : Hn(C∗) −→ G. Note that α is a monomorphism and α ◦ ϕ = 0 implies that ϕ =0 and consequently (ω(ψ′, ψ′′))=(ψ′ ◦ δ, β ◦ ψ′ − ψ′′ ◦ q)=ϕ ¯ =0. Therefore, we obtain that ω(ψ′, ψ′′) ∈ Kerξ.

Lemma 3.1. For each integer n ∈ N, there exists the following short exact sequence (3.4) σ ω 0 Hom(Cn+1/Bn+1; G′) Hom(Cn+1; G′) ⊕ Hom(Cn+1/Bn+1; G′′) Kerξ 0, where σ : Hom(Cn+1/Bn+1; G′) −→ Hom(Cn+1; G′)⊕Hom(Cn+1/Bn+1; G′′) is defined by the formula

′ (3.5) σ(ϕ) = (ϕ ◦ q,β ◦ ϕ), ∀ϕ ∈ Hom(Cn+1/Bn+1; G ).

Proof. a. ω is an epimorphism. If (ϕ′,ϕ′′) ∈ Kerξ, then ξ(ϕ′,ϕ′′)=ϕ ¯ =0 and so ϕ : Zn −→ G is zero as well. On the other hand, α ◦ ϕ = ϕ′ ◦ j =0. Therefore, there is a unique homomorphism ϕ˜′ : Bn+1 −→ G′, such that ϕ′ =ϕ ˜′ ◦ δ′. Let ψ′ : Cn+1 −→ G′ be an extension of the map ϕ˜′ : Bn+1 −→ G′ (see the diagram (3.6)).

j δ′ j ◦ i Zn Cn Bn+1 Cn+1 ′ ϕ¯ ψ′ ϕ ϕ′ ϕ′′

α ′ β ′′ (3.6) 0 G G G 0.

′ ′′ n+1 ′′ ′ ′′ If we consider the map β ◦ ψ − ϕ : C −→ G , then by (ϕ ,ϕ ) ∈ Z¯n, we have (β ◦ ψ′ − ϕ′′) ◦ δ = β ◦ ψ′ ◦ δ − ϕ′′ ◦ δ = β ◦ ϕ′ − ϕ′′ ◦ δ =0. Since (β ◦ ψ′ − ϕ′′) ◦ δ = (β ◦ ψ′ − ϕ′′) ◦ j ◦ i ◦ δ′ = 0 and δ′ is an epimorphism, there is (β ◦ ψ′ − ψ′′) ◦ j ◦ i = 0. Therefore, there is a homomorphism ψ′′ : Cn+1/Bn+1 −→ G′′ such that β ◦ ψ′ − ϕ′′ = ψ′′ ◦ q and so ϕ′′ = β ◦ ψ′ − ψ′′ ◦ q (see the diagram (3.7)). 14 A. Beridze and L. Mdzinarishvili

j ◦ i q Bn+1 Cn+1 Cn+1/Bn+1 0 ψ′′ β ◦ ψ′ − ϕ′′

′′ (3.7) G

Hence, (ψ′, ψ′′) ∈ Hom(Cn+1; G′) ⊕ Hom(Cn+1/Bn+1; G′′) and ω(ψ′, ψ′′) = (ψ′ ◦ δ, β ◦ ψ′ − ψ′′ ◦ q)=(ϕ′,ϕ′′). So, ω is an epimorphism. b. There is an equality Imσ = Kerω. By the definition, we have (ω ◦ σ)(ϕ)= ω(σ(ϕ)) = ω(ϕ ◦ q, β ◦ ϕ)=(ϕ ◦ q ◦ δ, β ◦ ϕ ◦ q − β ◦ ϕ ◦ q)= (0, 0)=0, because q ◦ δ =0. Therefore, Imσ ⊂ Kerω. On the other hand, if (ψ′, ψ′′) ∈ Kerω, then ω(ψ′, ψ′′)=(ψ′ ◦ δ, β ◦ ψ′ − ψ′′ ◦ q)=0 and so, ψ′ ◦ δ =0 and β ◦ ψ′ = ψ′′ ◦ q. On the other hand, ψ′ ◦ δ = ψ′ ◦ j ◦ i ◦ δ′ =0. Therefore, we have ψ′ ◦j ◦i =0, because δ′ is an epimorphism. So, there is a unique homomorphism ϕ : Cn+1/Bn+1 −→ G′ such that ψ′ = ϕ ◦ q. In this case, β ◦ ϕ ◦ q = β ◦ ψ′ = ψ′′ ◦ q and since q is an epimorphism, β ◦ ϕ = ψ′′. Therefore, σ(ϕ)=(ψ′, ψ′′) and so, Kerω ⊂ Imσ. c. σ is a monomorphism. If σ(ϕ)=(ϕ ◦ q, β ◦ ϕ)=0, i.e. ϕ ◦ q = 0 and since q is an epimorphism, we have ϕ =0.

C∗ ∗ Let = {Cγ } be a direct system of cochain complexes. Consider the C γ # ∗ # corresponding inverse system ∗ = {C∗ (β )} = {Hom(Cγ ; β )} of chain complexes.

C∗ ∗ Lemma 3.2. For each direct system = {Cγ } of cochain complexes, there is an isomorphism

∗ ∗ (3.8) Hom(lim C ; β#) ≃ lim Hom(C ; β#). −→ γ ←− γ Proof. Consider a chain complex

∗ ′ ′′ (3.9) Hom(lim C ; β#)= {Hom(lim Cn; G ) ⊕ Hom(lim Cn+1; G ), ∂}, −→ γ −→ γ −→ γ where ∂(ϕ′,ϕ′′)=(ϕ′ ◦δ, β ◦ϕ′ −ϕ′′ ◦δ). Note that δ = lim δ : lim Cn−1 −→ −→ γ −→ γ lim Cn, where δ : Cn−1 −→ Cn is the coboundary map of the cochain −→ γ γ γ γ complex C∗. Since for any G there is an isomorphism Hom(lim C∗; G) ≃ γ −→ γ lim Hom(C∗; G), we have ←− γ ′ ′′ ′ ′′ (3.10) Hom(lim Cn; G )⊕Hom(lim Cn+1; G ) ≃ lim Hom(Cn; G )⊕lim Hom(Cn+1; G ). −→ γ −→ γ ←− γ ←− γ 15

Lemma 3.3. If f # : C∗ −→ C′∗ is a homomorphism of cochain complexes, then there is a commutative diagram: (3.11) ′ τ ′ µ 0 Hom(C′n+1/B′n+1; G′) Hom(C′n+1; G′) ⊕ Hom(C′n+1/B′n+1; G′′) Kerξ′ 0

f˜# (f#, f˜#) f˜ τ µ 0 Hom(Cn+1/Bn+1; G′) Hom(Cn+1; G′) ⊕ Hom(Cn+1/Bn+1; G′′) Kerξ 0.

′n+1 ′n+1 ′ n+1 n+1 ′ Proof. Note that homomorphisms f˜# : Hom(C /B ; G ) −→ Hom(C /B ; G ) ′n+1 ′ ′n+1 ′n+1 ′′ n+1 ′ and (f#, f˜#) : Hom(C ; G )⊕Hom(C /B ; G ) −→ Hom(C ; G )⊕ n+1 n+1 ′′ ˜ ′ ′ ˜ Hom(C /B ; G ) are naturally defined by f#(ϕ ) = ϕ ◦ fn+1 and ˜ ′ ′′ ′ ′′ ˜ ˜ n+1 n+1 (f#, f#)(ϕ ,ϕ ) = (ϕ ◦ fn+1,ϕ ◦ fn+1), where fn+1 : C /B −→ ′n+1 ′n+1 n+1 ′n+1 C /B is induced by fn+1 : C −→ C . ˜ ˜ ′ ˜ ′ a. τ ◦ f# = (f#, f#) ◦ τ . By the definition, we have τ ◦ f# (ϕ ) = ˜ ′ ′ ˜ ′ ˜ ′ ˜ ˜ ′ ′ τ f#(ϕ ) = τ ϕ ◦ fn+1 = ϕ ◦ fn+1 ◦ q, β ◦ ϕ ◦ fn+1 and (f#, f#) ◦ τ  (ϕ )= ˜ ′ ′ ˜ ′ ′ ′ ′ ′ ′ ˜ f#, f# (τ (ϕ )) = f#, f# (ϕ ◦ q , β ◦ ϕ )=(ϕ ◦ q ◦ fn+1, β ◦ ϕ ◦ fn+1). ˜ ′ ′ ˜ ′ ′ Since fn+1 ◦ q = q ◦ fn+1, we have ϕ ◦ fn+1 ◦ q = ϕ ◦ q ◦ fn+1. Hence, ′ τ ◦ f˜# =(f#, f˜#) ◦ τ . ˜ ˜ ′ ˜ ′ ′′ b. µ◦(f#, f#)= f◦µ . By the definition, we have µ ◦ (f#, f#) (ϕ ,ϕ )= ˜ ′ ′′ ′ ′′ ˜ ′ ′ µ (f#, f#)(ϕ ,ϕ ) = µ ϕ ◦ fn+1,ϕ ◦ fn+1 =(ϕ ◦fn+1 ◦δ, β ◦ϕ ◦fn+1 − ′′ ˜ ˜ ′ ′ ′′ ˜ ′ ′ ′′ ˜ ′ ′ ′ ′′ ′ ϕ ◦fn+1◦q) and f ◦ µ  (ϕ ,ϕ )= f (µ (ϕ ,ϕ )) = f (ϕ ◦ δ , β ◦ ϕ − ϕ ◦ q )= ′ ′ ′ ′′ ′ ′ (ϕ ◦ δ ◦ fn, β ◦ ϕ ◦ fn+1 − ϕ ◦ q ◦ fn+1). Since f˜n+1 ◦ q = q ◦ fn+1 and ′ ′ ′ ′ δ ◦ fn = fn+1 ◦ δ, there are qualities ϕ ◦ fn+1 ◦ δ = ϕ ◦ δ ◦ fn and ′′ ˜ ′′ ′ ˜ ˜ ′ ϕ ◦ fn+1 ◦ q = ϕ ◦ q ◦ fn+1. Hence, µ ◦ (f#, f#)= f ◦ µ .

n+1 ′ n+1 n+1 ′′ Let {Hom(Cγ ; G ) ⊕ Hom(Cγ /Bγ ; G )} be an inverse system gen- n+1 erated by the direct system {Cγ }. It is clear that for each γ there is an exact sequence (3.12)

n+1 ′ τ n+1 ′ n+1 n+1 ′′ µ n+1 n+1 ′′ 0 Hom(Cγ ; G ) Hom(Cγ ; G ) ⊕ Hom(Cγ /Bγ ; G ) Hom(Cγ /Bγ ; G ) 0.

Hence, by the main property of the derived functors lim(i) there is a long ←− exact sequence: (3.13) τ˜ µ˜ ... lim(i) Hom(Cn+1; G′) lim(i) Hom(Cn+1; G′) ⊕ Hom(Cn+1/Bn+1; G′′) ←− γ ←− γ γ γ

µ˜ lim(i) Hom(Cn+1/Bn+1; G′′) ... . ←− γ γ 16 A. Beridze and L. Mdzinarishvili

On the other hand, since for each injective abelian groups G, lim(i){Hom(Cn+1; G)} = ←− 0, i ≥ 1 (see Lemma 1.3 [Hub-Mei]), we obtain the following result. Corollary 3.4. For each injective abelian groups G′ and G′′, there is the following equality ′ ′′ (3.14) lim(i) Hom(Cn+1; G ) ⊕ Hom(Cn+1/Bn+1; G ) =0, i ≥ 1. ←− γ γ γ
Using the obtained result, we will prove the following lemma. Lemma 3.5. For each integer i ≥ 1, there is an equality (3.15) lim(i)Kerξ =0. ←− γ

Proof. By Lemma 1, for each ξγ, there is a short exact sequence (3.16) σγ ωγ n+1 n+1 ′ n+1 ′ n+1 n+1 ′′ 0 Hom(Cγ /Bγ ; G ) Hom(Cγ ; G ) ⊕ Hom(Cγ /Bγ ; G ) Kerξγ 0.

By the main property of a derived functor lim(i), there is a long exact ←− sequence

(3.17) (i) n+1 n+1 ′ (i) n+1 ′ n+1 n+1 ′′ (i) ... lim Hom(C /B ; G ) lim Hom(C ; G ) ⊕ Hom(C /B ; G ) lim Ker ξγ ... . ←− γ γ ←− γ γ γ
←−

By Lemma 1.3 [Hub-Mei] for each i ≥ 0, there is an equality lim(i) Hom(Cn+1/Bn+1; G′)= ←− γ γ 0 and by Corollary 2, for each i ≥ 1 we obtain ′ ′′ (3.18) lim(i) Hom(Cn+1; G ) ⊕ Hom(Cn+1/Bn+1; G ) =0. ←− γ γ γ
Hence, by the long exact sequence (3.17) we obtain that lim(i)Kerξ = 0, ←− γ i ≥ 1. Corollary 3.6. For each integer i ≥ 1, there is an isomorphism ∗ (3.19) lim(i)Z¯γ ≃ lim(i) Hom(Hn(C ); G). ←− n ←− γ ¯γ n ∗ Proof. By a. of Theorem 1, there is an epimorphism ξγ : Zn −→ Hom(H (Cγ ); G). Therefore, the following sequence is exact

ξγ Kerξ ¯γ Hom(Hn(C∗); G) (3.20) 0 γ Zn γ 0.

Consequently, it induces the following long exact sequence (3.21) ... lim(i)Kerξ lim(i)Z¯γ lim(i) Hom(Hn(C∗); G) lim(i+1)Kerξ ... . ←− γ ←− n ←− γ ←− γ

On the other hand, by Lemma 4, lim(i){Kerξ } =0, i ≥ 1. Therefore, for ←− γ i ≥ 1, we have an isomorphism lim(i)Z¯γ ≃ lim(i) Hom(Hn(C∗); G). ←− n ←− γ 17

Note that for each γ, there is a natural commutative triangle

ξγ ¯γ n ∗ Zn Hom(H (Cγ ); G) p¯γ ξ¯γ

H¯ (C∗; G). (3.22) n γ

Therefore, if we take lim(i) of this diagram, then by Corollary 2, we obtain ←− the following result.

Corollary 3.7. For each integer i ≥ 1, lim(i)Z¯γ is a direct summand of ←− n lim(i)H¯ (C∗; G) and the projection of lim(i)H¯ (C∗; G) onto lim(i)Z¯γ is nat- ←− n γ ←− n γ ←− n ural.

Finally, we obtain the following important property of lim(i) functor. ←− Theorem 3.8. For each integer i ≥ 0, there is a short exact sequence: (3.23)

ξγ lim(i) Ext(Hn+1(C∗); G) lim(i)H¯ (C∗; G) lim(i) Hom(Hn(C∗); G) 0, 0 ←− γ ←− n γ ←− γ and this sequence splits naturally for i ≥ 1.

Proof. Using the commutative diagram (3.22), for each γ we have a com- mutative diagram with exact rows:

ξγ ¯γ n ∗ 0 Kerξγ Zn Hom(H (Cγ ); G) 0

p¯γ 1 ξ¯γ Ext(Hn+1(C∗); G) H¯ (C∗; G) Hom(Hn(C∗); G) (3.24) 0 γ n γ γ 0.

This induces the following commutative diagram with exact rows: (3.25) (i) lim ξγ i i ←− i ∗ i ... lim( )Kerξ lim( )Z¯γ lim( ) Hom(Hn(C ); G) lim( +1)Kerξ ... ←− γ ←− n ←− γ ←− γ (i) lim p¯γ 1 ←− (i) lim ξ¯γ i ∗ i ∗ ←− i ∗ i ∗ ... lim( ) Ext(Hn+1(C ); G) lim( )H¯ (C ; G) lim( ) Hom(Hn(C ); G) lim( +1) Ext(Hn+1(C ); G) ... . ←− γ ←− n γ ←− γ ←− γ

By Lemma 4, lim(i)Kerξ = 0, for i ≥ 1, and so the beginning of the ←− γ diagram (3.25) of the following form: 18 A. Beridze and L. Mdzinarishvili

(3.26)

limξγ γ ←− n ∗ limKerξγ limZ¯ lim Hom(H (C ); G) 0 ←− ←− n ←− γ 0 lim(i)p¯ ←− γ 1 limξ¯γ lim Ext(Hn+1(C∗); G) limH¯ (C∗; G)←− lim Hom(Hn(C∗); G) ... . 0 ←− γ ←− n γ ←− γ

Therefore, the following sequence is exact: (3.27)

ξγ lim Ext(Hn+1(C∗); G) limH¯ (C∗; G) lim Hom(Hn(C∗); G) 0 ←− γ ←− n γ ←− γ 0 and the lim(1){Ext(Hn+1(C∗); G)} −→ lim(1){H¯ (C∗; G)} is a monomor- ←− γ ←− n γ phism. Therefore, we obtain the result for i = 0. On the other hand, for i ≥ 1, the result follows from the commutativity of the diagram (3.25) and Corollary 2 and 3.

4 The main theorem

Here we have formulated the main result of the paper. C∗ ∗ Theorem 4.1. Let = {Cγ } be a direct system of cochain complexes. Then, there is a natural exact sequence: (4.1) ... lim(3)H¯ γ lim(1)H¯ γ H¯ limC∗; G limH¯ γ lim(2)H¯ γ ... ←− n+2 ←− n+1 n −→ γ  ←− n ←− n+1

¯ γ ¯ ∗ where H∗ = H∗(Cγ ; G).

n+1 Proof. By Proposition 1.2 of [Hub-Mei], for the inverse system {Hγ } we have an exact sequence: (4.2) lim(1) Hom(Hn+1; G) Ext(lim Hn+1; G) lim Ext(Hn+1; G) lim(2) Hom(Hn+1; G) 0 ←− γ −→ γ ←− γ ←− γ 0 and (4.3) lim(i) Ext(Hn+1; G) ≃ lim(i+2) Hom(Hn+1; G), for i ≥ 1. ←− γ ←− γ Since cohomology commutes with direct limits, we have H∗(lim C∗; G) ≃ −→ γ lim H∗(C∗; G). Therefore, if C∗ ≃ lim C∗, then Hn+1(C∗) ≃ lim Hn+1, where −→ γ −→ γ −→ γ n+1 n+1 ∗ Hγ = H (Cγ ; G). So, we obtain an exact sequence: (4.4) lim(1) Hom(Hn+1; G) Ext(Hn+1(C∗); G) lim Ext(Hn+1; G) lim(2) Hom(Hn+1; G) 0 ←− γ ←− γ ←− γ 0. 19

∗ ∗ ¯ Note that, if iγ : Cγ −→ C is a natural map, then it induces πγ ; H∗(C; G) −→ ¯ ∗ Hn(Cγ ; G) map. On the other hand, by Theorem 1, the following diagram is commutative:

n+1 ∗ ∗ n ∗ 0 Ext(H (C ); G) H¯n(C ; G) Hom(H (C ); G) 0

π˜γ πγ π¯γ

Ext(Hn+1(C∗); G) H¯ (C∗; G) Hom(Hn(C∗); G) (4.5) 0 γ n γ γ 0.

The diagram (4.5) generates the following diagram:

n+1 ∗ ∗ n ∗ 0 Ext(H (C ); G) H¯n(C ; G) Hom(H (C ); G) 0

π˜ π ≃

lim Ext(Hn+1; G) lim H¯ γ lim Hom(Hn; G) (4.6) 0 ←− γ ←− n ←− γ 0.

Therefore, we have Kerπ˜ ≃ Kerπ and Cokerπ˜ ≃ Cokerπ and so, the following diagram is commutative:

0 0

≃ Kerπ˜ Kerπ

n+1 n 0 Ext(H (C); G) H¯n(C; G) Hom(H (C); G) 0

π˜ π ≃

lim Ext(Hn+1; G) lim H¯ γ lim Hom(Hn; G) 0 ←− γ ←− n ←− γ 0 .

≃ Cokerπ˜ Cokerπ

(4.7) 0 0

Using the exact sequence (4.4) and diagram (4.7), we obtain a four-term exact sequence: (4.8) lim(1) Hom(Hn+1; G) ¯ lim H¯ γ lim(2) Hom(Hn+1; G) 0 ←− γ Hn(C; G) ←− n ←− γ 0.

Using the exact sequence (4.8), Theorem 3 and isomorphism (4.3), we ob- tain the following diagram, which contains the long exact sequence of the 20 A. Beridze and L. Mdzinarishvili theorem: (4.9) . .

lim(3) Ext(Hn+3; G) lim(3)H¯ γ lim(3) Hom(Hn+2; G) 0 ←− γ ←− n+2 ←− γ 0 ≃

lim(1) Hom(Hn+1; G) lim(1)H¯ γ lim(1) Ext(Hn+2; G) 0 ←− γ ←− n+1 ←− γ 0 ≃

lim(1) Hom(Hn+1; G) H¯ (C; G) lim H¯ γ lim(2) Hom(Hn+1; G) 0 ←− γ n ←− n ←− γ 0. ≃

lim(2) Ext(Hn+2; G) lim(2)H¯ γ lim(2) Hom(Hn+1; G) 0 ←− γ ←− n+1 ←− γ 0 ≃

lim(4) Hom(Hn+2; G) lim(4)H¯ γ lim(4) Ext(Hn+3; G) 0 ←− γ ←− n+2 ←− γ 0

. .

C∗ ∗ Corollary 4.2. Let = {Cγ } be a direct system of cochain complexes. Then, for each injective abelian group G, there is an isomorphism ∗ ∗ (4.10) H¯ limC ; G ≃ limH¯∗(C ; G). n −→ γ  ←− γ

5 Applications

∗ 1. Let Cc (X,G) be the cochain complex of Massey [Mas2]. It is known that for each locally compact Hausdorff space X and each integer n the n Z cochain group Cc (X, ) with integer coefficient is a free abelian group (the- ∗ orem 4.1 [Mas2]). Using the cochain complex Cc (X,G), Massey defined an M exact homology H∗ , the so called Massey homology on the category of locally compact spaces and proper maps as a homology of the chain com- ∗ plex C∗(X,G) = Hom(Cc (X),G). Consequently, for the given category the Universal Coefficient Formula is obtained (see theorem 4.1, corollary 4.18

[Mas2] and theorem of universal coefficient 4.1 [Mac]): n+1 M n (5.1) 0 −→ Ext(Hc (X), G) −→ Hn (X, G) −→ Hom(Hc (X), G) −→ 0. ∗ ∗ Let Cc = Cc (X; G) be the cochain complex of Massey. Consider the chain M ∗ M complex C¯∗ (X; G) = Hom(C (X); β#). Let H¯∗ (X; G) be homology of the 21

M chain complex C¯∗ (X; G). In this case, by Theorem 1 we will obtain the Universal Coefficient Formula: n+1 ¯ M n (5.2) 0 −→ Ext(Hc (X), G) −→ Hn (X, G) −→ Hom(Hc (X), G) −→ 0.

Note that by Theorem 2, for the category of locally compact spaces the ¯ M M homologies Hn (X,G) and Hn (X,G) are isomorphic. M Note that for the Massey homology theory H¯∗ (−; G) our construction gives the following result:

Corollary 5.1. Let X be a locally compact Hausdorff space, then a) if {Nα} is the system of closed neighborhoods Nα of closed subspace A of X, directed by inclusion, then it induces the following exact sequence: (5.3) ... lim(2k+1)H¯ M (N ) ... lim(3)H¯ M (N ) lim(1)H¯ M (N ) ←− n++k+1 α ←− n+2 α ←− n+1 α

∗ M i M (2) M (2k) M ¯ limH¯ (Nα) lim H¯ (N ) ... lim H¯ (N ) .... Hn (A; G) ←− n ←− n+1 α ←− n+k α b) if {Uα} is the system of open subspaces of X, such that U¯α is compact and X = Uα directed by inclusion, then it induces the following exact S sequence: (5.4) ... lim(2k+1)H¯ M (U ) ... lim(3)H¯ M (U ) lim(1)H¯ M (U ) ←− n++k+1 α ←− n+2 α ←− n+1 α

∗ M i M (2) M (2k) M ¯ limH¯ (Uα) lim H¯ (U ) ... lim H¯ (U ) .... Hn (A; G) ←− n ←− n+1 α ←− n+k α

Note that the formula (??) is a generalization of Theorem 1.4.2 of [Mas2]. 2. Let G be an R-module over a principal ideal domain R and let X be a topological space. Denote by C¯∗(X; G) the cochain complex of Alexander- Spanier [Sp] and by H¯ ∗(X; G) the Alexander-Spanier cohomology. let A be a subspace of a topological space X and {Uα} be the family of all neigh- n borhoods of A in X directed downward by inclusion. Hence, {H¯ (Uα; G)} n n is a direct system. The restriction maps H¯ (Uα; G) −→ H¯ (A; G) define a natural homomorphism

(5.5) i : lim H¯ n(U ; G) −→ H¯ n(A; G). −→ α By Theorem 6.6.2 [Sp], if A is a closed subspace of a paracompact Hausdorff space X, then (5.5) is an isomorphism. In this case, A is called a taut subspace relative to the Alexander-Spanier cohomology theory. In the case of homology theory, we have a natural homomorphism

(5.6) i : H (A; G) −→ lim H (U ; G). n ←− n α 22 A. Beridze and L. Mdzinarishvili

The question whether the homomorphism (5.6) is an isomorphism or not was open. Let C¯∗ = C¯∗(X; G) be the cochain complex of Alexander-Spanier. Con- ∗ sider the chain complex C¯∗(X; G) = Hom(C¯ (X); β#). Let H¯∗(X; G) be the homology of the chain complex C¯∗(X; G). In this case, we will say that the homology H¯∗(X; G) is generated by the Alexander-Spanier cochains C¯∗(X; G). By Theorem 4 we have the long exact sequence, which contains the homomorphisms (5.6).

Corollary 5.2. If A is a closed subspace of a paracompact Hausdorff space

X and {Uα} is the family of all neighborhoods of A in X, then there is a long exact sequence: (5.7) ... lim(2k+1)H¯ (U ) ... lim(3)H¯ (U ) lim(1)H¯ (U ) ←− n++k+1 α ←− n+2 α ←− n+1 α

∗ i (2) (2k) H¯ (A; G) limH¯n(Uα) lim H¯ (U ) ... lim H¯ (U ) .... n ←− ←− n+1 α ←− n+k α

# ∗ ¯ 3. It is clear that there is a natural inclusion i : Cc (X; G) → C∗(X; G) from the Massey cochain complex to the Alexander-Spanier cochain com- ∗ plex, which induces the corresponding homomorphism i : H¯∗(−; G) → M M H¯∗ (−; G), where H¯∗(−; G) and H¯∗ (−; G) are homologies generated by the Alexander-Spanier and the Massey cochains, respectively. Therefore, M H¯∗(−; G) and H¯∗ (−; G) are homologies of the chain complexes C¯∗(−; G)= ¯∗ ¯M ∗ Hom(C (−); β#) and C∗ (−; G) = Hom(Cc (−); β#). On the other hand, on the category of compact Hausdorff spaces, the Alexsander-Spanier and the Massey cohomology are isomorphic and by the Universal Coefficient For- mula, we will obtain that for each compact Hausdorff space the there is an isomorphism:

∗ ≃ M (5.8) i : H¯∗(X; G) −→ H¯∗ (X; G).

On the other hand, since on the category of compact metric spaces the St Steenrod homology H∗ [St], [Ed-Ha1], [Ed-Ha2] and the Massey homology are isomorphic, using the isomorphism (5.8), we will obtain that

St (5.9) H¯∗(X; G) ≃ H∗ (X; G).

The same way, on the category of compact Hausdorff spaces, the Milnor Mil homology H∗ and the Massey homology are isomorphic and consequently, we have

Mil (5.10) H¯∗(X; G) ≃ H∗ (X; G). 23

BM If H∗ (−; G) is the Borel-Moore homology with coefficient in G, then by Theorem 3 [Kuz], we have the isomororphism

BM (5.11) H¯∗(X; G) ≃ H∗ (X; G).

4. Let KC be the category of compact pairs (X, A) and continuous maps and H∗ be an exact homology theory. Let {(Xα, Aα)} be an inverse system of compact pairs (X , A ) and (X, A) = lim(X , A ). The inverse system α α ←− α α {(Xα, Aα)} generates an inverse system {H∗(Xα, Aα)} and the projection

πα : (X, A) → (Xα, Aα) induces the homomorphism πα,∗ : H∗(X, A) →

H∗(Xα, Aα), which induces the homomorphism

(5.12) π∗ : H∗(X, A) → lim H∗(X , A ). ←− α α

Definition 5.3. An exact homology theory H∗ is said to be a continuous on the category KC , if for each inverse system {(Xα, Aα)} of the given category, there is an infinite exact sequence: (5.13) ... lim(2k+1)H¯ M (X , A ) ... lim(3)H¯ M (X , A ) lim(1)H¯ M (X , A ) ←− n++k+1 α α ←− n+2 α α ←− n+1 α α

∗ M i M (2) M (2k) M ¯ limH¯ (Xα, Aα) lim H¯ (X , A ) ... lim H¯ (X , A ) .... Hn (X, A; G) ←− n ←− n+1 α α ←− n+k α α

C∗ ∗ ∗ Definition 5.4. A direct system = {Cα} of the cochain complexes Cα is said to be associated with a cochain complex C∗, if there is a homomorphism C∗ → C∗ such that for each n ∈ Z the induced homomorphism ∗ ∗ ∗ ∗ (5.14) lim H (C ) → H (C ) −→ α is an isomorphism. C∗ ∗ ∗ Lemma 5.5. If a direct system = {Cα} of the cochain complexes Cα is associated with a cochain complex C∗, then there is an infinite exact sequence: (5.15) ... lim(2k+1)H¯ M (C∗ ; G) ... lim(3)H¯ M (C∗ ; G) lim(1)H¯ M (C∗ ; G) ←− n++k+1 α ←− n+2 α ←− n+1 α

i∗ ∗ H¯ M (C∗; G) limH¯ M (C ; G) lim(2)H¯ M (C∗ ; G) ... lim(2k)H¯ M (C∗; G) .... n ←− n α ←− n+1 α ←− n+k α

¯ ∗ ∗ ¯ ∗ ∗ where H∗(C )= H∗ ((Hom(C ; β#)) and H∗(Cα)= H∗ (Hom(Cα; β#)) .

Proof. By theorem 4, There is a natural exact sequence: (5.16) ... lim(3)H¯ α lim(1)H¯ α H¯ limC∗; G limH¯ α lim(2)H¯ α ... , ←− n+2 ←− n+1 n −→ α  ←− n ←− n+1 24 A. Beridze and L. Mdzinarishvili

¯ α ¯ ∗ C∗ ∗ where H∗ = H∗(Cα; G). Since the direct system = {Cα} of cochain com- ∗ ∗ plexes Cα is associated with a cochain complex C , there is an isomorphism

∗ ∗ ∗ ≃ ∗ (5.17) H¯∗(lim C ; G) ≃ lim H¯ (C ; G) −→ H¯∗(C ; G). −→ α −→ α On the other hand, by Universal Coefficient Formula, we have the following commutative diagram with exact rows: (5.18) n+1 ∗ ∗ n ∗ 0 Ext(H (C ); G) H¯n(C ; G) Hom(H (C ); G) 0

≃ π¯n ≃

Ext(Hn+1(lim C∗ ); G) H¯ (lim C∗ ; G) Hom(Hn(lim C∗ ); G) 0 −→ α n −→ α −→ α 0.

Hence, the homomorphism π¯n is an isomorphism for all n ∈ Z. Using the exact sequence (5.16) and the isomorphism π¯n, we obtain an infinite exact sequence (5.15).

Corollary 5.6. Let {(Xα, Aα)} be an inverse system of pairs of compact spaces (X , A ) and (X, A) = lim(X , A ). If H¯∗ is the homology theory α α ←− α α generated by the Alexander-Spanier cochains, then there is an infinite exact sequence: (5.19) ... lim(2k+1)H¯ (X , A ) ... lim(3)H¯ (X , A ) lim(1)H¯ (X , A ) ←− n+k+1 α α ←− n+2 α α ←− n+1 α α

∗ π (2) (2k) H¯ (X, A; G) limH¯n(Xα, Aα) lim H¯ (X , A ) ... lim H¯ (X , A ) .... n ←− ←− n+1 α α ←− n+k α α

Corollary 5.7. Let {(Xi, Ai)}i∈Z be an inverse sequnce of compact metric spaces (X , A ) and (X, A) = lim(X , A ). If H¯∗ is the homology theory gen- i i ←− i i erated by the Alexander-Spanier cochains, then there is an exact sequence:

π∗ (5.20) 0 −→ lim (1)H¯ (X , A ) −→ H¯ (X, A; G) −→ lim H¯ (X , A ) −→ 0. ←− n+1 i i n ←− n i i

Corollary 5.8. If H¯∗ is the homology theory generated by the Alexander- Spanier cochains, then there is an exact sequence:

π∗ (5.21) 0 −→ lim (1)H¯ (K ,L ) −→ H¯ (X, A; G) −→ lim H¯ (K ,L ) −→ 0, ←− n+1 α α n ←− n α α where (X, A) = lim(K , L ) and (K , L ) are finite polyhedral pairs. ←− α α α α ∗ 5. Let Cs (X; G) be the singular cochain complex of topological spaces ¯s ∗ ¯ s X and C∗ (X; G) = Hom(Cs (X); β#). Let H∗ (X; G) be the homology of s s the obtained chain complex C¯∗ (X; G). Therefore, H¯∗ (−; G) is the homol- ∗ ogy generated by the singular cochain complex Cs (−; G). It is known that 25

# ¯∗ ∗ there is a natural homomorphism j : C (X : G) → Cs (X; G) from the Alexander-Sapnier cochain complex to the singular cochain complex, which ∗ ¯ ∗ ¯ ∗ induces the isomorphism j : H (X; G) → Hs (X; G) on the category of manifolds. Therefore, by the Universal Coefficient Formula, we will obtain that if X is manifold, then there is an isomorphism:

s ≃ (5.22) j∗ : H¯∗ (X; G) −→ H¯∗(X; G).

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