On the Universal Coefficient Formula and Derivative lim(i) Functor ←− Anzor Beridze School of Mathematics Kutaisi International University Youth Avenue, 5th Lane, Kutaisi, 4600 Georgia E-mail: [email protected] Leonard Mdzinarishvili Department of Mathematics Faculty of Informatics and Control Systems Georgian Technical University 77, Kostava St., Tbilisi, Georgia E-mail: [email protected] Abstract From the beginning of 1960, there were many approaches to define exact homology theories using the methods of homological algebra arXiv:2102.00468v2 [math.AT] 8 Feb 2021 (using an injective resolution) [Bor-Mo], [Mas4], [Mas2], [Kuz], [Skl]. These approaches gave us the unique homology theory on the cate- gory of compact Hausdorff spaces [Kuz], [Skl]. Our aim is to develop a method of homological algebra which gives opportunity to define on the category of general topological spaces a unique exact homol- ogy theory, generated by the given cochain complex. If H∗ is the ∗ cohomology of the cochain complex C = Hom(C∗; G), then the co- ∗ homology H is said to be generated by chain complex C∗. If a chain complex C∗ is free, then there is a Universal Coefficient Formula of a cohomology theory [Eil-St], [Mas1], [Sp]. In the paper [Mdz3], using this formula and derivatives of inverse limit, a long exact sequence is written, which shows a relation of a cohomology of direct limit 2020 Mathematics Subject Classification: 55N10 Key words and phrases: Universal Coefficient Formula; inverse limit; derivative limit; tautness of homology. 1 2 A. Beridze and L. Mdzinarishvili 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.