# On the Universal Coefficient Formula and Derivative $\varprojlim^{(i)}$
Functor

Anzor Beridze
School of Mathematics
Kutaisi International University
Youth Avenue, 5th Lane, Kutaisi, 4600 Georgia
E-mail<EMAIL_ADDRESS>Leonard Mdzinarishvili
Department of Mathematics
Faculty of Informatics and Control Systems
Georgian Technical University
77, Kostava St., Tbilisi, Georgia
E-mail<EMAIL_ADDRESS>
###### Abstract

From the beginning of 1960, there were many approaches to define exact
homology theories using the methods of homological algebra (using an injective
resolution) [Bor-Mo], [Mas4], [Mas2], [Kuz], [Skl]. These approaches gave us
the unique homology theory on the category 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 homology theory, generated by the given cochain complex. If $H^{*}$ is
the cohomology of the cochain complex $C^{*}=\operatorname{Hom}(C_{*};G)$,
then the cohomology $H^{*}$ 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 of chain
complexes and inverse limit of cohomology groups of corresponding 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 corresponding chain
complexes is studied. As a corollary, the tautness property of a homology
theory is obtained. Moreover, for the defined exact homology theory the
continuous property (see Definition 1 [Mdz1]) is obtained on the category of
compact pairs.

††2020 _Mathematics Subject Classification_ : 55N10†† _Key words and phrases_
: Universal Coefficient Formula; inverse limit; derivative limit; tautness of
homology.

##  1 Introduction

Let $C_{*}$ be a chain complex and $0\to
G\overset{\alpha}{\longrightarrow}G^{\prime}\overset{\beta}{\longrightarrow}G^{\prime\prime}\to
0$ be an injective resolution of a $R$-module $G$ over a principal ideal
domain $R$. Let
$\beta^{\\#}:\operatorname{Hom}(C_{*};G^{\prime})\to\operatorname{Hom}(C_{*};G^{\prime\prime})$
be the cochain map induced by $\beta:G^{\prime}\to G^{\prime\prime}$ and
$C^{*}(\beta^{\\#})=\\{C^{n}(\beta_{\\#}),\delta\\}$ be the cone of the
cochain map $\beta^{\\#}$, i.e.,

(1.1) $\footnotesize
C^{n}(\beta^{\\#})\simeq\operatorname{Hom}(C_{n};G^{\prime})\oplus\operatorname{Hom}(C_{n-1};G^{\prime\prime}),$
(1.2)
$\footnotesize\delta(\varphi^{\prime},\varphi^{\prime\prime})=(\varphi^{\prime}\circ\partial,\beta\circ\varphi^{\prime}-\varphi^{\prime\prime}\circ\partial),~{}~{}~{}\forall(\varphi^{\prime},\varphi^{\prime\prime})\in
C^{n}(\beta^{\\#}).$

It is natural that the cochain complex
$C^{*}(\beta^{\\#})=\operatorname{Hom}(C_{*};\beta^{\\#})$ is defined by the
chain complex $C_{*}$ and $R$-module $G$. Consequently, let
$\bar{H}^{*}(C_{*};G)$ be a cohomology of the cochain complex
$C^{*}(\beta^{\\#})$. In the papers [Mdz3] (in the case of free chain
complexes) and [Mdz-Sp] (in the general case) it is shown that for each direct
system $\mathbf{C}_{*}=\\{C_{*}^{\gamma}\\}$ of chain complexes there is a
natural exact sequence:

(1.3)
$\dots$${\varprojlim}^{(3)}\bar{H}_{\gamma}^{n-2}$${\varprojlim}^{(1)}\bar{H}_{\gamma}^{n-1}$$\bar{H}^{n}\left({\varinjlim}C^{\gamma}_{*};G\right)$${\varprojlim}\bar{H}_{\gamma}^{n}$${\varprojlim}^{(2)}\bar{H}_{\gamma}^{n-1}$$\dots$

where
$\bar{H}^{*}_{\gamma}=\bar{H}^{*}(C_{*}^{\gamma};G)=H^{*}(C^{*}_{\gamma}(\beta_{\\#})).$
Note that, using the methods developed 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 $\bar{H}^{n}(\varinjlim
C_{*}^{\gamma};G)\simeq H^{n}(\varinjlim C_{*}^{\gamma};G)$. Therefore, it
uniquely defines a cohomology $\bar{H}^{*}$ generated by the chain complex
$C_{*}$. On the other hand, our aim is 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
$\beta_{\\#}:\operatorname{Hom}(C^{*};G^{\prime})\to\operatorname{Hom}(C^{*};G^{\prime\prime})$
be the chain map induced by $\beta:G^{\prime}\to G^{\prime\prime}$. Consider
the cone
$C_{*}(\beta_{\\#})=\\{C_{n}(\beta_{\\#}),\partial\\}=\\{\operatorname{Hom}(C^{*},\beta_{\\#}),\partial\\}$
of the chain map $\beta_{\\#}$, i.e.,

(1.4) $\footnotesize
C_{n}(\beta_{\\#})\simeq\operatorname{Hom}(C^{n};G^{\prime})\oplus\operatorname{Hom}(C^{n+1};G^{\prime\prime}),$
(1.5)
$\footnotesize\partial(\varphi^{\prime},\varphi^{\prime\prime})=(\varphi^{\prime}\circ\delta,\beta\circ\varphi^{\prime}-\varphi^{\prime\prime}\circ\delta),~{}~{}~{}\forall(\varphi^{\prime},\varphi^{\prime\prime})\in
C_{n}(\beta_{\\#}).$

Consequently, the homology group $\bar{H}_{n}$ is denoted by
$\bar{H}_{n}=\bar{H}_{n}(C^{*};G)$ $=H_{n}(C_{*}(\beta_{\\#}))$ and is called
a homology with coefficient in $G$ generated by the cochain complex $C^{*}$.
Note that if $f:C^{*}\longrightarrow C^{\prime*}$ is a cochain map, then it
induces the chain map $\bar{f}:C^{\prime}_{*}(\beta_{\\#})\longrightarrow
C_{*}(\beta_{\\#})$. In particular, for each $n\in\mathbb{Z}$ the homomorphism
$\bar{f}_{n}:C^{\prime}_{n}(\beta_{\\#})\longrightarrow C_{n}(\beta_{\\#})$ is
defined by the formula
$\bar{f}_{n}(\varphi^{\prime},\varphi^{\prime\prime})=(\varphi^{\prime}\circ
f_{n},\varphi\circ f_{n+1}).$ Consequently, it induces a homomorphism of
homology groups
$\bar{f}:\bar{H}_{n}(C^{\prime*};G)\longrightarrow\bar{H}_{n}(C^{*};G)$.
Therefore, $\bar{H}_{n}$ is a naturally defined functor.

Let $\mathbf{C}^{*}=\\{C^{*}_{\gamma}\\}$ be a direct system of cochain
complexes. Consider the corresponding inverse systems
$\mathbf{C}_{*}=\\{C_{*}^{\gamma}(\beta_{\\#})\\}$ and
$\mathbf{H}_{*}=\\{\bar{H}_{*}(C^{*}_{\gamma};G)\\}$. In this paper we have
shown that there is a natural exact sequence:

(1.6)
$\dots$${\varprojlim}^{(3)}\bar{H}^{\gamma}_{n+2}$${\varprojlim}^{(1)}\bar{H}^{\gamma}_{n+1}$$\bar{H}_{n}\left({\varinjlim}C^{*}_{\gamma};G\right)$${\varprojlim}\bar{H}^{\gamma}_{n}$${\varprojlim}^{(2)}\bar{H}^{\gamma}_{n+1}$$\dots$,

where
$\bar{H}^{\gamma}_{*}=\bar{H}_{*}(C^{*}_{\gamma};G)=H_{*}(C_{*}^{\gamma}(\beta_{\\#})).$
On the other hand, to obtain sequence (1.6), we have shown that for each
cochain complex (it is not necessary to be free) $C^{*}$ and a $R$-module $G$
over a principal ideal domain $R$, there exists a short exact sequence
(Universal Coefficient Formula):

(1.7) $\footnotesize
0\longrightarrow\operatorname{Ext}(H^{n+1}(C^{*});G)\overset{}{\longrightarrow}\bar{H}_{n}(C^{*};G)\overset{}{\longrightarrow}\operatorname{Hom}(H^{n}(C^{*});G)\longrightarrow
0.$

At the end we have formulated and studied the tautness property for a homology
theory $\bar{H}_{*}$ generated by the Alexander-Spanier cochains $\bar{C}^{*}$
on the category of paracompact Hausdorff spaces.

##  2 Universal Coefficient Formula

In this section we will prove the Universal Coefficient Formula for a homology
theory $\bar{H}_{*}$ 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 short exact sequence:

(2.1) $\footnotesize
0\longrightarrow\operatorname{Ext}(H^{n+1}(C^{*});G)\overset{\bar{\chi}}{\longrightarrow}{\bar{H}}_{n}(C^{*};G)\overset{\bar{\xi}}{\longrightarrow}\operatorname{Hom}(H^{n}(C^{*});G)\longrightarrow
0.$

###### Proof.

We will define a homomorphism
$\xi:\bar{Z}_{n}\longrightarrow\operatorname{Hom}(H^{n}(C^{*});G)$, which
induces an epimorphism
$\bar{\xi}:\bar{H}_{n}(C^{*};G)\longrightarrow\operatorname{Hom}(H^{n}(C^{*});G)$.
On the other hand, we will define a homomorphism
$\chi:\operatorname{Hom}(H^{n+1}(C^{*});G^{\prime\prime})\longrightarrow\bar{H}_{n}(C^{*};G)$
such that $\chi$ induces a monomorphism
$\bar{\chi}:\operatorname{Ext}(H^{n+1}(C^{*});G)\longrightarrow\bar{H}_{n}(C^{*};G)$
and the short sequence (2.1) is exact.

a. There is a homomorphism
$\xi:\bar{Z}_{n}\to\operatorname{Hom}(H^{n}(C^{*});G)$. Let
$(\varphi^{\prime},\varphi^{\prime\prime})\in\bar{Z}_{n}$ be a cycle, i.e.,
$\varphi^{\prime}:C^{n}\to G^{\prime}$ and $\varphi^{\prime\prime}:C^{n+1}\to
G^{\prime\prime}$ are homomorphisms such that
$\partial(\varphi^{\prime},\varphi^{\prime\prime})=(\varphi^{\prime}\circ\delta,\beta\circ\varphi^{\prime}-\varphi^{\prime\prime}\circ\delta)=(0,0)=0$
and therefore, the following diagram is commutative:

(2.2)
$C^{n-1}$$C^{n}$$C^{n+1}$$G^{\prime}$$G^{\prime\prime}$$\delta$$\delta$$\varphi^{\prime}$$\varphi^{\prime\prime}$$0$$\beta$

where $0$ is the zero map. Consider the groups of coboundaries $B^{n}$ and
cocycles $Z^{n}$. Let $i:B^{n}\to Z^{n}$ and $j:Z^{n}\to C^{n}$ be natural
monomorphisms and $\delta^{\prime}:C^{n-1}\to B^{n}$ be an epimophism induced
by $\delta:C^{n-1}\to C^{n}$. Therefore, we have the following sequence:

(2.3) $\footnotesize
C^{n-1}\overset{\delta^{\prime}}{\longrightarrow}B^{n}\overset{i}{\longrightarrow}Z^{n}\overset{j}{\longrightarrow}C^{n}\overset{\delta}{\longrightarrow}C^{n+1},$

where $j\circ i\circ\delta^{\prime}=\delta$ and consequently $\delta\circ
j=0$.

Since $(\varphi^{\prime},\varphi^{\prime\prime})\in\bar{Z}_{n}$, we have the
following commutative diagram:

(2.4)
$Z^{n}$$C^{n}$$C^{n+1}$$G$$G^{\prime}$$G^{\prime\prime}$$0~{}.$$0$$j$$\delta$$\varphi^{\prime}$$\varphi^{\prime\prime}$$\alpha$$\beta$

Hence, $\varphi^{\prime\prime}\circ\delta\circ
j=\beta\circ\varphi^{\prime}\circ j$, and by the equality $\delta\circ j=0$,
we obtain that $\beta\circ\varphi^{\prime}\circ j=0$. So,
$Im(\varphi^{\prime}\circ j)\subset Ker\beta=Im\alpha$. Therefore, there is a
uniquely defined map $\varphi:Z^{n}\to G$ such that $\varphi^{\prime}\circ
j=\alpha\circ\varphi$ (see the diagram (2.5).

(2.5)
$Z^{n}$$C^{n}$$C^{n+1}$$G$$G^{\prime}$$G^{\prime\prime}$$0~{}.$$0$$j$$\delta$$\varphi$$\varphi^{\prime}$$\varphi^{\prime\prime}$$\alpha$$\beta$

By the commutative diagram (2.2), we have
$\varphi^{\prime}\circ\delta=\varphi^{\prime}\circ j\circ
i\circ\delta^{\prime}=0$. Hence, $\alpha\circ\varphi\circ
i\circ\delta^{\prime}=\varphi^{\prime}\circ j\circ
i\circ\delta^{\prime}=\varphi^{\prime}\circ\delta=0$ (see the diagram (2.6)).
$\alpha$ is a monomorphism and so $\varphi\circ i\circ\delta^{\prime}=0$. On
the other hand, $\delta^{\prime}$ is an epimorphism. Consequently, we have
$\varphi\circ i=0$. Therefore, the homomorphism $\varphi:Z^{n}\to G$ induces a
homomorphism $\bar{\varphi}:H^{n}(C^{*})\to G$ which belongs to
$\operatorname{Hom}(H^{n}(C^{*});G)$. Hence, the following diagram is
commutative:

(2.6)
$Z^{n}$$B^{n}$$C^{n-1}$$C^{n}$$C^{n+1}$$G$$H^{n}$$G^{\prime}$$G^{\prime\prime}$$0~{},$$0$$\delta^{\prime}$$i$$j$$\delta$$\varphi$$\varphi^{\prime}$$\varphi^{\prime\prime}$$0$$\alpha$$\beta$$p$$\bar{\varphi}$

where $H^{n}=H^{n}(C^{*}).$ Let
$\xi:\bar{Z}_{n}\to\operatorname{Hom}(H^{n};G)$ be the homomorphism defined by

(2.7)
$\footnotesize\xi(\varphi^{\prime},\varphi^{\prime\prime})=\bar{\varphi},~{}~{}\forall(\varphi^{\prime},\varphi^{\prime\prime})\in\bar{Z}_{n}.$

b. $\xi:\bar{Z}_{n}\to\operatorname{Hom}(H^{n}(C^{*});G)$ is an epimorphism.
Let $\bar{\varphi}\in\operatorname{Hom}(H^{n}(C^{*});G)$ be a homomorphism and
$\varphi=\bar{\varphi}\circ p:Z^{n}\to G$ is the composition, where
$p:Z^{n}\to H^{n}(C^{*})$ is a projection. Let $\varphi^{\prime}:C^{n}\to
G^{\prime}$ be an extension of $\alpha\circ\varphi:Z^{n}\to G^{\prime}$. In
this case $\varphi^{\prime}\circ j=\alpha\circ\varphi$ and so
$\beta\circ\varphi^{\prime}\circ j=\beta\circ\alpha\circ\varphi=0.$ Therefore,
$\beta\circ\varphi^{\prime}:C^{n}\to G^{\prime\prime}$ vanishes on the
subgroup $Z^{n}$ and so it induces a homomorphism
$\tilde{\varphi}^{\prime\prime}:C^{n}/Z^{n}\simeq B^{n+1}\to
G^{\prime\prime}$, which can be extended to a homomophism
$\varphi^{\prime\prime}:C^{n+1}\to G^{\prime\prime}$. Since
$\tilde{\varphi}^{\prime\prime}\circ\delta^{\prime}=\beta\circ\varphi^{\prime}$,
there is
$\partial(\varphi^{\prime},\varphi^{\prime\prime})=(\varphi^{\prime}\circ\delta,\beta\circ\varphi^{\prime}-\varphi^{\prime\prime}\circ\delta)=(\varphi^{\prime}\circ
j\circ
i\circ\delta^{\prime},\beta\circ\varphi^{\prime}-\varphi^{\prime\prime}\circ
j\circ i\circ\delta^{\prime})=(\alpha\circ\varphi\circ
i\circ\delta^{\prime},\beta\circ\varphi^{\prime}-\tilde{\varphi}^{\prime\prime}\circ\delta^{\prime})=$
$(\alpha\circ\bar{\varphi}\circ p\circ
i\circ\delta^{\prime},\beta\circ\varphi^{\prime}-\beta\circ\varphi^{\prime})=(0,0)=0$.
Hence, $(\varphi^{\prime},\varphi^{\prime\prime})\in\bar{Z}_{n}$ and
$\bar{\xi}(\varphi^{\prime},\varphi^{\prime\prime})=\bar{\varphi}$ (see the
diagram (2.8)).

(2.8) $B^{n}$$C^{n-1}$$Z^{n}$$C^{n}$$C^{n}/Z^{n}\simeq
B^{n+1}$$C^{n+1}$$H^{n}$$0$$G$$G^{\prime}$$G^{\prime\prime}$$0.$$\bar{\varphi}$$p$$\delta^{\prime}$$i$$j$$\delta^{\prime}$$j\circ
i$$\alpha$$\beta$$\varphi$$\varphi^{\prime}$$\tilde{\varphi}^{\prime\prime}$$\varphi^{\prime\prime}$

c. $\xi:\bar{Z}_{n}\to\operatorname{Hom}(H^{n}(C^{*});G)$ induces a
homomorphism $\bar{\xi}:\bar{H}_{n}(C^{*};G)$
$\to\operatorname{Hom}(H^{n}(C^{*});G)$. We have to show that the homomorphism
$\xi:\bar{Z}_{n}\to\operatorname{Hom}(H^{n}(C^{*});G)$ vanishes on the
subgroup $\bar{B}_{n}$. Indeed, let $(\psi^{\prime},\psi^{\prime\prime})\in
C_{n+1}(\beta_{\\#})$ be an element. For
$\partial(\psi^{\prime},\psi^{\prime\prime})=(\psi^{\prime}\circ\delta,\beta\circ\psi^{\prime}-\psi^{\prime\prime}\circ\delta)=(\varphi^{\prime},\varphi^{\prime\prime})\in\bar{B}_{n}\subset\bar{Z}_{n}$
we have $\varphi^{\prime}\circ j=0$. Indeed, $\varphi^{\prime}\circ
j=\psi^{\prime}\circ\delta\circ j=0$. Therefore, by the construction $\xi$,
the homomorphism $\varphi:Z^{n}\to G$ corresponding to the pair
$(\varphi^{\prime},\varphi^{\prime\prime})$ satisfies the equation
$\alpha\circ\varphi=\varphi^{\prime}\circ j=0$ and so $\varphi=0$, because
$\alpha$ is a monomorphism. Since $\varphi=\bar{\varphi}\circ p$ and $p$ is an
epimorphism, we have $\bar{\varphi}=0$. Therefore,
$\xi\partial(\psi^{\prime},\psi^{\prime\prime})=\xi(\varphi^{\prime},\varphi^{\prime\prime})=\bar{\varphi}=0$
(see the diagram (2.9)).

(2.9)
$Z^{n}$$C^{n}$$C^{n+1}$$C^{n+2}$$G$$H^{n}$$G^{\prime}$$G^{\prime\prime}$$0~{}.$$0$$p$$\bar{\varphi}$$j$$\delta$$\delta$$\psi^{\prime}$$\psi^{\prime\prime}$$\varphi$$\varphi^{\prime}$$\varphi^{\prime\prime}$$\alpha$$\beta$

d. The kernel of
$\bar{\xi}:\bar{H}_{n}(C^{*};G)\to\operatorname{Hom}(H^{n}(C^{*});G)$ is
$\operatorname{Ext}(H^{n+1}(C^{*});G)$. If we apply the functor
$\operatorname{Hom}(H^{n+1}(C^{*});-)$ to the short exact sequence $0\to
G\overset{\alpha}{\longrightarrow}G^{\prime}\overset{\beta}{\longrightarrow}G^{\prime\prime}\to
0$, then we obtain:

(2.10)
$0$$\operatorname{Hom}(H^{n+1}(C^{*});G)$$\operatorname{Hom}(H^{n+1}(C^{*});G^{\prime})$$\operatorname{Hom}(H^{n+1}(C^{*});G^{\prime\prime})$$\operatorname{Ext}(H^{n+1}(C^{*});G)$$0~{}.$$\alpha_{*}$$\beta_{*}$$\beta_{*}$

Therefore, we have the following isomorphism:

(2.11)
$\footnotesize\operatorname{Ext}(H^{n+1}(C^{*});G)\simeq\operatorname{Hom}(H^{n+1}(C^{*});G^{\prime\prime})/Im\beta_{*}.$

Our aim is to define such a homomorpism
$\chi:\operatorname{Hom}(H^{n+1}(C^{*});G^{\prime\prime})\longrightarrow\bar{H}_{n}(C^{*};G)$
that the following sequence is exact:

(2.12)
$\operatorname{Hom}(H^{n+1}(C^{*});G^{\prime})$$\operatorname{Hom}(H^{n+1}(C^{*});G^{\prime\prime})$$\bar{H}_{n}(C^{*};G)$$\operatorname{Hom}(H^{n}(C^{*});G)$$0~{}.$$\beta_{*}$$\chi$$\chi$$\bar{\xi}$

Indeed, in this case, it is clear that for the homomorphisms $\bar{\xi}$,
$\chi$ and $\beta_{*}$, we have the following short exact sequences:

(2.13) $\footnotesize 0\longrightarrow
Ker\bar{\xi}\overset{}{\longrightarrow}\bar{H}_{n}(C^{*};G)\overset{\bar{\xi}}{\longrightarrow}\operatorname{Hom}(H^{n}(C^{*});G)\overset{}{\longrightarrow}0,$
(2.14) $\footnotesize 0\longrightarrow
Ker\chi\overset{}{\longrightarrow}\operatorname{Hom}(H^{n+1}(C^{*});G^{\prime\prime})\overset{\chi}{\longrightarrow}Im\chi\overset{}{\longrightarrow}0,$
(2.15) $\footnotesize 0\longrightarrow
Ker\beta_{*}\overset{}{\longrightarrow}\operatorname{Hom}(H^{n+1}(C^{*});G^{\prime})\overset{\beta_{*}}{\longrightarrow}Im\beta_{*}\overset{}{\longrightarrow}0.$

On the other hand, if we prove exactness of the sequence (2.12), then
$Ker\bar{\xi}\simeq Im\chi$ and $Ker\chi\simeq Im\beta_{*}$. Therefore, we
have:

(2.16) $Ker\bar{\xi}\simeq
Im\chi\simeq\operatorname{Hom}(H^{n+1}(C^{*});G^{\prime\prime})/Ker\chi\simeq$$\simeq\operatorname{Hom}(H^{n+1}(C^{*});G^{\prime\prime})/Im\beta_{*}\simeq\operatorname{Ext}(H^{n+1}(C^{*});G).$

To define $\chi$, consider an element
$\bar{\varphi}^{\prime\prime}\in\operatorname{Hom}(H^{n+1}(C^{*});G^{\prime\prime})$.
Let $\varphi^{\prime\prime}:C^{n+1}\longrightarrow G^{\prime\prime}$ be an
extension of the composition $\bar{\varphi}^{\prime\prime}\circ
p:Z^{n+1}\longrightarrow G^{\prime\prime}$, where $p:Z^{n+1}\longrightarrow
H^{n+1}(C^{*})$ is a natural projection. In this case,
$\varphi^{\prime\prime}\circ\delta=\varphi^{\prime\prime}\circ j\circ
i\circ\delta^{\prime}=\tilde{\varphi}^{\prime\prime}\circ p\circ
i\circ\delta^{\prime}=0$ and so, if we take $\varphi^{\prime}=0$, then the
following diagram is commutative:

(2.17)
$C^{n-1}$$C^{n}$$B^{n+1}$$Z^{n+1}$$C^{n+1}$$G^{\prime}$$H^{n+1}$$G^{\prime\prime}$
.$p$$\bar{\varphi}^{\prime\prime}$$\delta$$\delta^{\prime}$$i$$j$$\varphi^{\prime}=0$$\varphi^{\prime\prime}$$0$$\beta$

Therefore, $(0,\varphi^{\prime\prime})\in\bar{Z}_{n}$ and so, we can define
$\chi$ in the following way:

(2.18)
$\chi(\bar{\varphi}^{\prime\prime})=(0,-\varphi^{\prime\prime})+\bar{B}_{n},~{}~{}~{}\forall~{}\bar{\varphi}^{\prime\prime}\in\operatorname{Hom}(H^{n+1}(C^{*});G^{\prime\prime}).$

Let check that $\chi$ is well defined. Consider two different extensions
$\varphi_{1}^{\prime\prime}$ and $\varphi_{2}^{\prime\prime}$ of the map
$\bar{\varphi}^{\prime\prime}\circ p:Z^{n+1}\longrightarrow G^{\prime\prime}$
and show that
$(0,-\varphi_{1}^{\prime\prime})+\bar{B}_{n}=(0,-\varphi_{2}^{\prime\prime})+\bar{B}_{n}$.
For this, we have to show that
$(0,\varphi_{2}^{\prime\prime}-\varphi_{1}^{\prime\prime})\in\bar{B}_{n}$.
Indeed, by the definition of $\varphi_{1}^{\prime\prime}$ and
$\varphi_{2}^{\prime\prime}$, it is clear that
$(\varphi_{1}^{\prime\prime}-\varphi_{2}^{\prime\prime})\circ
j=\varphi_{1}^{\prime\prime}\circ j-\varphi_{2}^{\prime\prime}\circ
j=\bar{\varphi}^{\prime\prime}\circ p-\bar{\varphi}^{\prime\prime}\circ p=0$
and so, $\varphi_{1}^{\prime\prime}-\varphi_{2}^{\prime\prime}$ induces a
homomorphism $\psi:C^{n+1}/Z^{n+1}\longrightarrow G^{\prime\prime}$. On the
other hand, $C^{n+1}/Z^{n+1}\simeq B^{n+2}$ and so, we have an extension
$\psi^{\prime\prime}:C^{n+2}\longrightarrow G^{\prime\prime}$ of $\psi$ (see
the diagran (2.19)).

(2.19) $Z^{n+1}$$C^{n+1}$$C^{n+1}/Z^{n+1}\simeq
B^{n+2}$$C^{n+2}$$G^{\prime\prime}$$j$$\delta^{\prime}$$j\circ
i$$0$$\varphi_{1}^{\prime}-\varphi_{2}^{\prime\prime}$$\psi$$\psi^{\prime\prime}$

In this case, it is easy to see that

(2.20)
$\footnotesize\partial(0,\psi^{\prime\prime})=(0,-\psi^{\prime\prime}\circ\delta)=(0,-\psi^{\prime\prime}\circ
j\circ
i\circ\delta^{\prime})=(0,-\psi\circ\delta^{\prime})=(0,\varphi_{2}^{\prime\prime}-\varphi_{1}^{\prime\prime}).$

Therefore, it remains to show that $Im\chi\simeq Ker\bar{\xi}$ and
$Im\beta_{*}=Ker\chi$.

d1. $Im\chi\simeq Ker\bar{\xi}$. Let
$\bar{\varphi}^{\prime\prime}\in\operatorname{Hom}(H^{n+1}(C^{*});G^{\prime\prime})$
be an element, then
$\bar{\xi}\left(\chi(\bar{\varphi}^{\prime\prime})\right)=\bar{\xi}\left((0,-\varphi^{\prime\prime})+\bar{B}_{n}\right)=\bar{\varphi}$.
On the other hand, by construction of $\xi$ and the fact that the first
coordinate of the pair $(0,-\varphi^{\prime\prime})$ is zero, it is easy to
check that $\bar{\varphi}=0$. Therefore, $Im\chi\subset Ker\bar{\xi}.$ Now
consider an element $\bar{h}\in Ker\bar{\xi}$ and any of its representatives
$(\varphi^{\prime},\varphi^{\prime\prime})\in\bar{Z_{n}}$. In this case, by
the definition of $\bar{\xi}$, there exists $\varphi:Z^{n}\to G$ such that the
following diagram is commutative:

(2.21)
$Z^{n}$$C^{n}$$C^{n+1}$$G$$H^{n}$$G^{\prime}$$G^{\prime\prime}$$0~{}.$$0$$p$$\bar{\varphi}$$j$$\delta$$\varphi$$\varphi^{\prime}$$\varphi^{\prime\prime}$$\alpha$$\beta$

Moreover, $\bar{h}\in Ker\bar{\xi}$ means that the homomorphism
$\bar{\varphi}:H^{n}\to G$ induced by $\varphi:Z^{n}\to G$ is zero. Therefore,
$\varphi=0$ and so $\varphi^{\prime}\circ j=\alpha\circ\varphi=0$.
Consequently, $\varphi^{\prime}:C^{n}\to G^{\prime}$ induces a homomorphism
$\tilde{\psi}^{\prime}:C^{n}/Z^{n}\simeq B^{n+1}\to G^{\prime}$. Let
$\psi^{\prime}:C^{n+1}\to G^{\prime}$ be an extension of
$\tilde{\psi}^{\prime}\circ p:Z^{n+1}\to G^{\prime}$ and
$\psi^{\prime\prime}=\beta\circ\psi^{\prime}$ (see the diagram (2.22)). In
this case, the homomorphism
$\psi=\psi^{\prime\prime}-\varphi^{\prime\prime}:C^{n+1}\to G^{\prime\prime}$
vanishes on the $B^{n+1}$. Indeed, $\psi\circ j\circ
i\circ\delta^{\prime}=\psi^{\prime\prime}\circ j\circ
i\circ\delta^{\prime}-\varphi^{\prime\prime}\circ j\circ
i\circ\delta^{\prime}=\beta\circ\tilde{\psi}^{\prime}\circ\delta^{\prime}-\beta\circ\varphi^{\prime}=\beta\circ\varphi^{\prime}-\beta\circ\varphi^{\prime}=0$.
On the other hand, $\delta^{\prime}$ is an epimorphim and so $\psi\circ j\circ
i=0$. Therefore, $\psi\circ j:Z^{n+1}\to G^{\prime\prime}$ induces a
homomorphism $\bar{\psi}:H^{n+1}\to G^{\prime\prime}$ (see the diagram
(2.22)).

(2.22)
$Z^{n}$$C^{n}$$B^{n+1}$$Z^{n+1}$$H^{n+1}$$C^{n+1}$$G$$G^{\prime}$$G^{\prime\prime}$$0~{}.$$0$$j$$\psi^{\prime\prime}$$\varphi^{\prime\prime}$$j$$\delta^{\prime}$$i$$p$$\varphi$$\varphi^{\prime}$$\bar{\psi}$$\tilde{\psi}^{\prime}$${\psi}^{\prime}$$\alpha$$\beta$

Our aim is to show that
$\chi(\bar{\psi})=\bar{h}=(\varphi^{\prime},\varphi^{\prime\prime})+\bar{B}_{n}$.
Indeed, by the definition of $\chi$, it is easy to see that
$\chi(\bar{\psi})=(0,-\psi)+\bar{B}_{n}$. Therefore, we have to show that
$(\varphi^{\prime},\varphi^{\prime\prime})-(0,-\psi)=(\varphi^{\prime},\varphi^{\prime\prime}+\psi)=(\varphi^{\prime},\psi^{\prime\prime})\in\bar{B}_{n}$.
Indeed,

(2.23)
$\footnotesize\partial(\psi^{\prime},0)=(\psi^{\prime}\circ\delta,\beta\circ\psi^{\prime})=(\psi^{\prime}\circ
j\circ
i\circ\delta^{\prime},\beta\circ\psi^{\prime})=(\tilde{\psi}^{\prime}\circ\delta^{\prime},\psi^{\prime\prime})=(\varphi^{\prime},\psi^{\prime\prime}).$

d2. $Im\beta_{*}\simeq Ker\chi$. Let
$\bar{\varphi}^{\prime}\in\operatorname{Hom}(H^{n+1}(C^{*});G^{\prime})$ be an
element and $\varphi^{\prime\prime}:C^{n+1}\to G^{\prime\prime}$ be an
extension of the composition $\beta\circ\bar{\varphi}^{\prime}\circ
p:Z^{n+1}\to G^{\prime\prime}$. In this case we have
$\partial(0,-\varphi^{\prime\prime})=(0,\varphi^{\prime\prime}\circ\delta)=(0,\varphi^{\prime\prime}\circ
j\circ i\circ\delta^{\prime})=(0,\beta\circ\bar{\varphi}^{\prime}\circ p\circ
i\circ\delta^{\prime})=(0,0)=0$ (see the diagram (2.25)). Therefore,
$(0,-\varphi^{\prime\prime})\in\bar{Z}_{n}$ and so we have

(2.24)
$\footnotesize\left(\chi\circ\beta_{*}\right)\left(\bar{\varphi}^{\prime}\right)=\chi\left(\beta_{*}\left(\bar{\varphi}^{\prime}\right)\right)=\chi\left(\beta\circ\bar{\varphi}^{\prime}\right)=\left(0,-\varphi^{\prime\prime}\right)+\bar{B}_{n}.$

Our aim is to show that
$\left(0,-\varphi^{\prime\prime}\right)\in\bar{B}_{n}$. Indeed, let
$\varphi^{\prime}:C^{n+1}\longrightarrow G^{\prime}$ be an extension of the
composition $\bar{\varphi}^{\prime}\circ p:Z^{n+1}\longrightarrow G^{\prime}$.
In this case $(\beta\circ\varphi^{\prime}-\varphi^{\prime\prime})\circ
j=\beta\circ\varphi^{\prime}\circ j-\varphi^{\prime\prime}\circ
j=\beta\circ\bar{\varphi}^{\prime}\circ
p-\beta\circ\bar{\varphi}^{\prime}\circ p=0$ and so
$\beta\circ\varphi^{\prime}-\varphi^{\prime\prime}:C^{n+1}\longrightarrow
G^{\prime\prime}$ induces a homomorphism
$\tilde{\psi}^{\prime\prime}:C^{n+1}/Z^{n+1}\simeq B^{n+2}\longrightarrow
G^{\prime\prime}$ such that
$\beta\circ\varphi^{\prime}-\varphi^{\prime\prime}=\tilde{\psi}^{\prime\prime}\circ\delta^{\prime}$.
Let $\psi^{\prime\prime}:C^{n+2}\longrightarrow G^{\prime\prime}$ be an
extension of a homomorphism $\tilde{\psi}^{\prime}:B^{n+2}\longrightarrow
G^{\prime\prime}$ (see the diagram (2.25) ).

(2.25) $Z^{n+1}$$B^{n+1}$$C^{n}$$C^{n+1}$$C^{n+1}/Z^{n+1}\simeq
B^{n+2}$$C^{n+2}$$H^{n+1}$$G^{\prime}$$G^{\prime\prime}$$\delta^{\prime}$$i$$j$$\delta^{\prime}$$j\circ
i$$p$$\bar{\varphi}^{\prime}$$\varphi^{\prime}$$\psi^{\prime\prime}$$\varphi^{\prime\prime}$$\tilde{\psi}^{\prime\prime}$$\beta$

In this case, we have

(2.26)
$\partial(-\varphi^{\prime},-\psi^{\prime\prime})=(-\varphi^{\prime}\circ\delta,-\beta\circ\varphi^{\prime}+\psi^{\prime\prime}\circ\delta)=(-\varphi^{\prime}\circ
j\circ
i\circ\delta^{\prime},-\beta\circ\varphi^{\prime}+\tilde{\psi}^{\prime\prime}\circ\delta^{\prime})=$$=(-\bar{\varphi}^{\prime}\circ
p\circ
i\circ\delta^{\prime},-\beta\circ\varphi^{\prime}+(\beta\circ\varphi^{\prime}-\varphi^{\prime\prime}))=(0,-\varphi^{\prime\prime}).$

Therefore, $(0,-\varphi^{\prime\prime})\in\bar{B}_{n}$ and so,
$\chi\circ\beta_{*}=0$. Hence, $Im\beta_{*}\subset\operatorname{Ker}\chi$. Now
consider an element $\bar{\varphi}^{\prime\prime}\in Ker\chi$. Let
$\varphi^{\prime\prime}:C^{n+1}\to G^{\prime\prime}$ be an extension of the
composition $\bar{\varphi}^{\prime\prime}\circ p:Z^{n+1}\to G^{\prime\prime}$
(see the diagram (2.28)). Then, by
$\chi(\bar{\varphi}^{\prime\prime})=(0,-\varphi^{\prime\prime})+\bar{B}_{n}=0$,
there exists $(\psi^{\prime},\psi^{\prime\prime})\in C_{n+1}(\beta_{\\#})$
such that

(2.27)
$\footnotesize\partial(\psi^{\prime},\psi^{\prime\prime})=(\psi^{\prime}\circ\delta,\beta\circ\psi^{\prime}-\psi^{\prime\prime}\circ\delta)=(0,-\varphi^{\prime\prime}).$

Therefore, $\psi^{\prime}\circ\delta=\psi^{\prime}\circ j\circ
i\circ\delta^{\prime}=0$. Since $\delta^{\prime}:C^{n}\to B^{n+1}$ is an
epimorphism, we have $\psi^{\prime}\circ j\circ i=0$ and so
$\psi^{\prime}\circ j:Z^{n+1}\to G^{\prime}$ induces a homomorphism
$\bar{\psi}^{\prime}:H^{n+1}(C^{*})\to G^{\prime}$. On the other hand, by
$\beta\circ\psi^{\prime}-\psi^{\prime\prime}\circ\delta=-\varphi^{\prime\prime}$,
we have $-\varphi^{\prime\prime}\circ j=\beta\circ\psi^{\prime}\circ
j-\psi^{\prime\prime}\circ\delta\circ j=\beta\circ\psi^{\prime}\circ j$ (see
the diagram (2.28)). Therefore,
$\beta_{*}(\bar{\psi}^{\prime})=\bar{\varphi}^{\prime\prime}$ and so,
$Ker\chi\subset Im\beta_{*}:$

(2.28)
$C^{n+1}$$C^{n+2}$$Z^{n+1}$$H^{n+1}$$G^{\prime}$$G^{\prime\prime}$$\delta$$\beta$$j$$p$$\bar{\psi}^{\prime}$$\psi^{\prime}$$\psi^{\prime\prime}$$\bar{\varphi}^{\prime\prime}$

∎

Since for each injective abelian group $G$, a group of extensions
$\operatorname{Ext}(-;G)$ 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

(2.29)
$\footnotesize\bar{H}_{n}(C^{*};G)\simeq\operatorname{Hom}(H^{n}(C^{*});G).$

Let $C_{*}=\operatorname{Hom}(C^{*};G)$ be a chain complex, where
$C_{n}=\operatorname{Hom}(C^{n};G)$ and $\partial$ are defined by
$\partial(\varphi)=\varphi\circ\delta,$ for
$\varphi\in\operatorname{Hom}(C^{n};G)$. In this case, there is a map
$\alpha_{*}:\operatorname{Hom}(C^{*};G)\longrightarrow\operatorname{Hom}(C^{*};\beta_{\\#})$
defined by:

(2.30)
$\footnotesize\alpha_{*}(\varphi)=(\alpha\circ\varphi,0),~{}~{}~{}\forall\varphi\in\operatorname{Hom}(C^{n};G).$

Let $H_{n}(C^{*};G)$ be a homology group of chain complex
$\operatorname{Hom}(C^{*};G)$.

######  Theorem 2.3.

If a cochain complex $C^{*}$ is free, then the homomorphism
$\alpha_{*}:\operatorname{Hom}(C^{*};G)\longrightarrow\operatorname{Hom}(C^{*};\beta_{\\#})$
induces an isomoprhism

(2.31)
$\footnotesize\bar{\alpha}_{*}:H_{n}(C^{*};G)\longrightarrow\bar{H}_{n}(C^{*};G).$

###### Proof.

Since $C^{*}$ is a free cochain complex, there is a short exact sequence:

(2.32) $\footnotesize
0\longrightarrow\operatorname{Ext}(H^{n+1}(C^{*});G)\overset{\tilde{\chi}}{\longrightarrow}{H}_{n}(\operatorname{Hom}(C^{*};G))\overset{\tilde{\xi}}{\longrightarrow}\operatorname{Hom}(H^{n}(C^{*});G)\longrightarrow
0.$

Let review how the morphisms $\tilde{\xi}$ and $\tilde{\chi}$ 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.
a. For each $\bar{\varphi}\in H_{n}(\operatorname{Hom}(C^{*};G))$ element let
$\tilde{\xi}(\bar{\varphi}):H^{n}(C^{*})\longrightarrow G$ be homomorphism
given by

(2.33)
$\footnotesize\tilde{\xi}(\bar{\varphi})(\bar{c})=\langle\varphi,c\rangle=\varphi(c),~{}~{}~{}~{}\forall~{}c\in\bar{c},~{}\bar{c}\in
H^{n}(C^{*}),$

where $\varphi$ is a representative of $\bar{\varphi}$ [Mas4].
b. To define the homomorphism
$\tilde{\chi}:\operatorname{Ext}(H^{n+1}(C^{*});G)\longrightarrow
H_{n}(\operatorname{Hom}(C^{*};G))$, we need to use the isomorphism (2.11).
Consequently, the homomorphism $\tilde{\chi}$ is the homomorphism induced by
$\chi_{0}:\operatorname{Hom}(H^{n+1}(C^{*});G^{\prime\prime})\longrightarrow
H_{n}(\operatorname{Hom}(C^{*};G)),$ where $\chi_{0}$ is defined in the
following way. Let
$\bar{\varphi}^{\prime\prime}\in\operatorname{Hom}(H^{n+1}(C^{*});G^{\prime\prime})$
be any element. Since $G^{\prime\prime}$ is an injective, there is an
extension $\varphi^{\prime\prime}:C^{n+1}\longrightarrow G^{\prime\prime}$ of
the composition $\bar{\varphi}^{\prime\prime}\circ p:Z^{n+1}\longrightarrow
G^{\prime\prime}.$ In this case
$\partial(\varphi^{\prime\prime})=\varphi^{\prime\prime}\circ\delta=\varphi^{\prime\prime}\circ
j\circ i\circ\delta^{\prime}=\bar{\varphi}^{\prime\prime}\circ p\circ
i\circ\delta^{\prime}=0$ (see diagram (2.34)).

(2.34)
$C^{n-1}$$C^{n}$$B^{n+1}$$Z^{n+1}$$C^{n+1}$$H^{n+1}$$G$$G^{\prime}$$G^{\prime\prime}$$0~{}.$$0$$p$$\bar{\varphi}^{\prime\prime}$$j$$\delta^{\prime}$$i$$j$$\varphi$$\varphi^{\prime}$$\varphi^{\prime\prime}$${\psi}^{\prime}$$\alpha$$\beta$

Therefore, it defines homology class $[\varphi^{\prime\prime}]\in
H_{n+1}(\operatorname{Hom}(C^{*};G^{\prime\prime}))$. Let
$E:H_{n+1}(\operatorname{Hom}(C^{*};G))\longrightarrow
H_{n}(\operatorname{Hom}(C^{*};G))$ be a boundary homomorphism induced by the
following exact sequence:

(2.35) $\footnotesize
0\longrightarrow\operatorname{Hom}(C^{*};G)\overset{\alpha_{\\#}}{\longrightarrow}\operatorname{Hom}(C^{*};G^{\prime})\overset{\beta_{\\#}}{\longrightarrow}\operatorname{Hom}(C^{*};G^{\prime\prime})\longrightarrow
0.$

Define a homomorphism $\chi_{0}$ by the formula

(2.36)
$\footnotesize\chi_{0}\left(\bar{\varphi}^{\prime\prime}\right)=E\left([{\varphi}^{\prime\prime}]\right),~{}~{}~{}\forall~{}\bar{\varphi}^{\prime\prime}\in\operatorname{Hom}(H^{n+1}(C^{*});G^{\prime\prime}).$

Note that the homomorphism
$\chi_{0}:\operatorname{Hom}(H^{n+1}(C^{*});G)\longrightarrow
H_{n}(\operatorname{Hom}(C^{*};G))$ is a composition of the isomorphism
$\operatorname{Hom}(H^{n+1}(C^{*});G)\overset{\simeq}{\longrightarrow}H_{n+1}(\operatorname{Hom}(C^{*};G))$
and the homomorphism $E:H_{n+1}(\operatorname{Hom}(C^{*};G))\longrightarrow
H_{n}(\operatorname{Hom}(C^{*};G))$. To write the explicit formula for
$\chi_{0}$, consider such a map $\psi^{\prime}:C^{n+1}\to G^{\prime}$ that
$\beta\circ\psi^{\prime}=\varphi^{\prime\prime}$ (this is possible, because
the cochain complex $C^{*}$ is free). Let
$\varphi^{\prime}=\psi^{\prime}\circ\delta:C^{n}\to G^{\prime}$, then we have
$\beta\circ\psi^{\prime}=\varphi^{\prime\prime}\circ\delta=0$. Therefore,
$\varphi^{\prime}\in Ker\beta=Im\alpha$ and so, there exists a unique map
$\varphi:C^{n}\to G$ such that $\alpha\circ\varphi=\varphi^{\prime}$. In this
case, we have
$\alpha\circ\varphi\circ\delta=\varphi^{\prime}\circ\delta=\psi^{\prime}\circ\delta\circ\delta=0$
and so, $\varphi\circ\delta=0$ because $\alpha$ is a monomorphism. On the
other hand, $\partial(\varphi)=\varphi\circ\delta=0.$ Consequently, $\varphi$
defines a homology class $[\varphi]\in H_{n}((\operatorname{Hom}(C^{*};G))$
(see diagram (2.34)). Finally, by the formula (2.36) we have

(2.37)
$\footnotesize\chi_{0}\left(\bar{\varphi}^{\prime\prime}\right)=E\left([{\varphi}^{\prime\prime}]\right)=[\varphi],~{}~{}~{}\forall~{}\bar{\varphi}^{\prime\prime}\in\operatorname{Hom}(H^{n+1}(C^{*});G^{\prime\prime}).$

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

(2.38)
$\operatorname{Hom}(H^{n+1}(C^{*});G^{\prime})$$\operatorname{Hom}(H^{n+1}(C^{*});G^{\prime\prime})$$\bar{H}_{n}(C^{*};G)$$\operatorname{Hom}(H^{n}(C^{*});G)$$0~{}.$$\beta_{*}$$\chi_{0}$$\chi_{0}$$\bar{\xi}$

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

(2.39)
$\operatorname{Hom}(H^{n+1}(C^{*});G^{\prime})$$\operatorname{Hom}(H^{n+1}(C^{*});G^{\prime\prime})$$H_{n}(\operatorname{Hom}(C^{*};G))$$\operatorname{Hom}(H^{n}(C^{*});G)$$0$$\operatorname{Hom}(H^{n+1}(C^{*});G^{\prime})$$\operatorname{Hom}(H^{n+1}(C^{*});G^{\prime\prime})$$\bar{H}_{n}(\operatorname{Hom}(C^{*};\beta_{\\#}))$$\operatorname{Hom}(H^{n}(C^{*});G)$$0$.$\beta_{*}$$\chi_{0}$$\tilde{\xi}$$\beta_{*}$$\chi$$\bar{\xi}$$1$$1$$\bar{\alpha}_{*}$$1$

Indeed, let
$\bar{\varphi}^{\prime\prime}\in\operatorname{Hom}(H^{n+1}(C^{*});G^{\prime\prime})$
be an element and $\varphi^{\prime\prime}:C^{n+1}\longrightarrow
G^{\prime\prime}$ be an extension of the composition
$\bar{\varphi}^{\prime\prime}\circ p:Z^{n+1}\longrightarrow G^{\prime\prime}.$
Then, by definition of $\chi_{0}$ and method of snake lemma, we must take an
element $\varphi^{\prime}\in\operatorname{Hom}(C^{n+1};G^{\prime})$, such that
$\beta_{\\#}(\varphi^{\prime})=\varphi^{\prime\prime}$. Note that this is
possible because of exactness of the sequence (2.35). Then, there is a cycle
$\varphi\in\operatorname{Hom}(C^{n};G)$, such that
$\alpha_{\\#}(\varphi)=\partial(\varphi^{\prime}).$ Let
$[\varphi]=\varphi+B_{n}$ be the corresponding element in the homology group
$H_{n}(\operatorname{Hom}(C^{*};G)),$ then
$\chi_{0}(\bar{\varphi}^{\prime\prime})=[\varphi].$ By the definition of the
map
$\tilde{\alpha}:H_{n}(\operatorname{Hom}(C^{*});G)\longrightarrow\bar{H}_{n}(\operatorname{Hom}(C^{*});\beta_{\\#})$,
we have

(2.40)
$\footnotesize\left(\tilde{\alpha}_{*}\circ\chi_{0}\right)\left(\bar{\varphi}^{\prime\prime}\right)=\tilde{\alpha}_{*}\left(\chi_{0}\left(\bar{\varphi}^{\prime\prime}\right)\right)=\tilde{\alpha}_{*}\left([\varphi]\right)=\left(\alpha\circ\varphi,0\right)+\bar{B}_{n}.$

On the other hand, by the definition of
$\chi:\operatorname{Hom}(G^{n+1}(C^{*});G^{\prime\prime})\longrightarrow\bar{H}_{n}(\operatorname{Hom}(C^{*});\beta_{\\#}),$
we have

(2.41)
$\footnotesize\chi(\bar{\varphi}^{\prime\prime})=(0,-\varphi^{\prime\prime})+\bar{B}_{n},~{}~{}~{}\forall~{}\bar{\varphi}^{\prime\prime}\in\operatorname{Hom}(H^{n+1};G^{\prime\prime}).$

Therefore, we have to show that
$(\alpha\circ\varphi,0)-(0,-\varphi^{\prime\prime})=(\alpha\circ\varphi,\varphi^{\prime\prime})\in\bar{B_{n}}.$
Indeed, by the equality $\alpha_{\\#}(\varphi)=\partial(\varphi^{\prime})$ and
$\beta_{\\#}(\varphi^{\prime})=\varphi^{\prime\prime}$, we have
$\alpha\circ\varphi=\varphi^{\prime}\circ\delta$ and
$\beta\circ\varphi^{\prime}=\varphi^{\prime\prime}$. Therefore,

(2.42)
$\footnotesize\partial(\varphi^{\prime},0)=(\varphi^{\prime}\circ\delta,\beta\circ\varphi^{\prime})=(\alpha\circ\varphi,\varphi^{\prime\prime}).$

By (2.40), (2.41), and (2.42), we obtain that
$\tilde{\alpha}_{*}\circ\chi_{0}=\chi$. So, it remains to show that
$\bar{\xi}\circ\tilde{\alpha}=\tilde{\xi}$.

Let $[\varphi]\in H_{n}(\operatorname{Hom}(C^{*};G))$ be an element and
$\varphi$ is its representative. Then, by the definitions of $\bar{\xi}$ and
$\tilde{\alpha}_{*}$ we have

(2.43)
$\footnotesize\left(\bar{\xi}\circ\tilde{\alpha}_{*}\right)\left([\varphi]\right)=\bar{\xi}\left(\tilde{\alpha}_{*}\left([\varphi]\right)\right)=\bar{\xi}\left(\alpha\circ\varphi,0\right)=\left(\varphi\right)+\bar{B}_{n}.$

Therefore, if we take an element $\bar{c}\in H^{n}(C^{*})$ and any of its
representatives $c\in\bar{c}$, then by (2.43) we have

(2.44)
$\footnotesize\left(\bar{\xi}\circ\tilde{\alpha}\right)\left([\varphi]\right)\left(\bar{c}\right)=\left(\varphi+\bar{B}_{n}\right)\left(\bar{c}\right)=\varphi(c).$

Therefore, by (2.33) , (2.43) and (2.44), we obtain that

(2.45) $\footnotesize\bar{\xi}\circ\tilde{\alpha}=\tilde{\xi}.$

∎

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

(2.46)
$0$$\operatorname{Ext}(H^{n+1}(C^{*});G)$$H_{n}(\operatorname{Hom}(C^{*};G))$$\operatorname{Hom}(H^{n}(C^{*});G)$$0$$0$$\operatorname{Ext}(H^{n+1}(C^{*});G)$$\bar{H}_{n}(C^{*};G)$$\operatorname{Hom}(H^{n}(C^{*});G)$$0$.$\tilde{\chi}$$\tilde{\xi}$$\bar{\chi}$$\bar{\xi}$$1$$\bar{\alpha}_{*}$$1$

Therefore, if a cochain complex $C^{*}$ is free, then the classical Universal
Coefficient Formula is isomorphic to the Universal Coefficient Formula defined
in this paper.

##  3 Some property of $\operatorname{Hom}$ and inverse limit functors

As we have seen in the previous section, there exists an epimorphism
$\xi:\bar{Z}_{n}\longrightarrow\operatorname{Hom}(H^{n}(C^{*});G)$ which
induces a homomorphism:

(3.1)
$\footnotesize\bar{\xi}:\bar{H}_{n}(C^{*};G)\longrightarrow\operatorname{Hom}(H^{n}(C^{*});G)$

and the following diagram is commutative:

(3.2)
$\bar{Z}_{n}$$\operatorname{Hom}(H^{n}(C^{*});G)$$\bar{H}_{n}(C^{*};G)$.$\xi$$\bar{p}$$\bar{\xi}$

To investigate $Ker\xi$ we construct a homomorphism

(3.3)
$\footnotesize\omega:\operatorname{Hom}(C^{n+1};G^{\prime})\oplus\operatorname{Hom}(C^{n+1}/B^{n+1};G^{\prime\prime})\longrightarrow
Ker\xi$

by
$\omega(\psi^{\prime},\psi^{\prime\prime})=(\psi^{\prime}\circ\delta,\beta\circ\psi^{\prime}-\psi^{\prime\prime}\circ
q),$ where $q:C^{n+1}\longrightarrow C^{n+1}/B^{n+1}$ is the quotient map. Let
show that $\omega(\psi^{\prime},\psi^{\prime\prime})\in Ker\xi$. Indeed,
$\partial\omega(\psi^{\prime},\psi^{\prime\prime})=\partial(\psi^{\prime}\circ\delta,\beta\circ\psi^{\prime}-\psi^{\prime\prime}\circ
q)=(\psi\circ\delta\circ\delta,\beta\circ\psi^{\prime}\circ\delta-(\beta\circ\psi^{\prime}-\psi^{\prime\prime}\circ
q)\circ\delta)=(0,\beta\circ\psi^{\prime}\circ\delta-\beta\circ\psi^{\prime}\circ\delta+\psi^{\prime\prime}\circ
q\circ\delta)=(0,\psi^{\prime\prime}\circ q\circ\delta)=(0,0),$ because
$q\circ\delta=0.$ Hence,
$\omega(\psi^{\prime},\psi^{\prime\prime})\in\bar{Z}_{n}$. By the definition
of $\xi:\bar{Z}_{n}\longrightarrow\operatorname{Hom}(H^{n}(C^{*});G)$, there
exists a uniquely defined map $\varphi:Z^{n}\longrightarrow G$ such that
$\alpha\circ\varphi=\psi^{\prime}\circ\delta\circ j$. On the other hand,
$\delta\circ j=0$ and so, $\alpha\circ\varphi=0$, which induces a homomorphism
$\bar{\varphi}:H^{n}(C^{*})\longrightarrow G$. Note that $\alpha$ is a
monomorphism and $\alpha\circ\varphi=0$ implies that $\varphi=0$ and
consequently
$\left(\omega(\psi^{\prime},\psi^{\prime\prime})\right)=(\psi^{\prime}\circ\delta,\beta\circ\psi^{\prime}-\psi^{\prime\prime}\circ
q)=\bar{\varphi}=0$. Therefore, we obtain that
$\omega(\psi^{\prime},\psi^{\prime\prime})\in Ker\xi.$

######  Lemma 3.1.

For each integer $n\in\mathbb{N}$, there exists the following short exact
sequence

(3.4)
$0$$\operatorname{Hom}(C^{n+1}/B^{n+1};G^{\prime})$$\operatorname{Hom}(C^{n+1};G^{\prime})\oplus\operatorname{Hom}(C^{n+1}/B^{n+1};G^{\prime\prime})$$Ker\xi$$0,$$\sigma$$\omega$

where
$\sigma:\operatorname{Hom}(C^{n+1}/B^{n+1};G^{\prime})\longrightarrow\operatorname{Hom}(C^{n+1};G^{\prime})\oplus\operatorname{Hom}(C^{n+1}/B^{n+1};G^{\prime\prime})$
is defined by the formula

(3.5) $\footnotesize\sigma(\varphi)=(\varphi\circ
q,\beta\circ\varphi),~{}~{}~{}\forall\varphi\in\operatorname{Hom}(C^{n+1}/B^{n+1};G^{\prime}).$

###### Proof.

a. $\omega$ is an epimorphism. If
$(\varphi^{\prime},\varphi^{\prime\prime})\in Ker\xi,$ then
$\xi(\varphi^{\prime},\varphi^{\prime\prime})=\bar{\varphi}=0$ and so
$\varphi:Z^{n}\longrightarrow G$ is zero as well. On the other hand,
$\alpha\circ\varphi=\varphi^{\prime}\circ j=0$. Therefore, there is a unique
homomorphism $\tilde{\varphi}^{\prime}:B^{n+1}\longrightarrow G^{\prime}$,
such that $\varphi^{\prime}=\tilde{\varphi}^{\prime}\circ\delta^{\prime}$. Let
$\psi^{\prime}:C^{n+1}\longrightarrow G^{\prime}$ be an extension of the map
$\tilde{\varphi}^{\prime}:B^{n+1}\longrightarrow G^{\prime}$ (see the diagram
(3.6)).

(3.6) $Z^{n}$$C^{n}$$B^{n+1}$$C^{n+1}$$j$$\delta^{\prime}$$j\circ
i$$0$$G$$G^{\prime}$$G^{\prime\prime}$$0$.$\alpha$$\beta$$\varphi$$\varphi^{\prime}$$\bar{\varphi}^{\prime}$$\psi^{\prime}$$\varphi^{\prime\prime}$

If we consider the map
$\beta\circ\psi^{\prime}-\varphi^{\prime\prime}:C^{n+1}\longrightarrow
G^{\prime\prime},$ then by
$(\varphi^{\prime},\varphi^{\prime\prime})\in\bar{Z}_{n},$ we have
$(\beta\circ\psi^{\prime}-\varphi^{\prime\prime})\circ\delta=\beta\circ\psi^{\prime}\circ\delta-\varphi^{\prime\prime}\circ\delta=\beta\circ\varphi^{\prime}-\varphi^{\prime\prime}\circ\delta=0$.
Since
$(\beta\circ\psi^{\prime}-\varphi^{\prime\prime})\circ\delta=(\beta\circ\psi^{\prime}-\varphi^{\prime\prime})\circ
j\circ i\circ\delta^{\prime}=0$ and $\delta^{\prime}$ is an epimorphism, there
is $(\beta\circ\psi^{\prime}-\psi^{\prime\prime})\circ j\circ i=0.$ Therefore,
there is a homomorphism $\psi^{\prime\prime}:C^{n+1}/B^{n+1}\longrightarrow
G^{\prime\prime}$ such that
$\beta\circ\psi^{\prime}-\varphi^{\prime\prime}=\psi^{\prime\prime}\circ q$
and so
$\varphi^{\prime\prime}=\beta\circ\psi^{\prime}-\psi^{\prime\prime}\circ q$
(see the diagram (3.7)).

(3.7) $B^{n+1}$$C^{n+1}$$C^{n+1}/B^{n+1}$$0$$j\circ
i$$q$$G^{\prime\prime}$$\beta\circ\psi^{\prime}-\varphi^{\prime\prime}$$\psi^{\prime\prime}$

Hence,
$(\psi^{\prime},\psi^{\prime\prime})\in\operatorname{Hom}(C^{n+1};G^{\prime})\oplus\operatorname{Hom}(C^{n+1}/B^{n+1};G^{\prime\prime})$
and
$\omega(\psi^{\prime},\psi^{\prime\prime})=(\psi^{\prime}\circ\delta,\beta\circ\psi^{\prime}-\psi^{\prime\prime}\circ
q)=(\varphi^{\prime},\varphi^{\prime\prime})$. So, $\omega$ is an epimorphism.

b. There is an equality $Im\sigma=Ker\omega$. By the definition, we have
$(\omega\circ\sigma)(\varphi)=\omega(\sigma(\varphi))=\omega(\varphi\circ
q,\beta\circ\varphi)=(\varphi\circ q\circ\delta,\beta\circ\varphi\circ
q-\beta\circ\varphi\circ q)=(0,0)=0,$ because $q\circ\delta=0$. Therefore,
$Im\sigma\subset Ker\omega.$ On the other hand, if
$(\psi^{\prime},\psi^{\prime\prime})\in Ker\omega,$ then
$\omega(\psi^{\prime},\psi^{\prime\prime})=(\psi^{\prime}\circ\delta,\beta\circ\psi^{\prime}-\psi^{\prime\prime}\circ
q)=0$ and so, $\psi^{\prime}\circ\delta=0$ and
$\beta\circ\psi^{\prime}=\psi^{\prime\prime}\circ q.$ On the other hand,
$\psi^{\prime}\circ\delta=\psi^{\prime}\circ j\circ i\circ\delta^{\prime}=0$.
Therefore, we have $\psi^{\prime}\circ j\circ i=0,$ because $\delta^{\prime}$
is an epimorphism. So, there is a unique homomorphism
$\varphi:C^{n+1}/B^{n+1}\longrightarrow G^{\prime}$ such that
$\psi^{\prime}=\varphi\circ q.$ In this case, $\beta\circ\varphi\circ
q=\beta\circ\psi^{\prime}=\psi^{\prime\prime}\circ q$ and since $q$ is an
epimorphism, $\beta\circ\varphi=\psi^{\prime\prime}.$ Therefore,
$\sigma(\varphi)=(\psi^{\prime},\psi^{\prime\prime})$ and so,
$Ker\omega\subset Im\sigma.$

c. $\sigma$ is a monomorphism. If $\sigma(\varphi)=(\varphi\circ
q,\beta\circ\varphi)=0,$ i.e. $\varphi\circ q=0$ and since $q$ is an
epimorphism, we have $\varphi=0$.

∎

Let $\mathbf{C}^{*}=\\{C^{*}_{\gamma}\\}$ be a direct system of cochain
complexes. Consider the corresponding inverse system
$\mathbf{C}_{*}=\\{C_{*}^{\gamma}(\beta^{\\#})\\}=\\{\operatorname{Hom}(C^{*}_{\gamma};\beta^{\\#})\\}$
of chain complexes.

######  Lemma 3.2.

For each direct system $\mathbf{C}^{*}=\\{C^{*}_{\gamma}\\}$ of cochain
complexes, there is an isomorphism

(3.8) $\footnotesize\operatorname{Hom}(\varinjlim
C^{*}_{\gamma};\beta^{\\#})\simeq\varprojlim\operatorname{Hom}(C^{*}_{\gamma};\beta^{\\#}).$

###### Proof.

Consider a chain complex

(3.9) $\footnotesize\operatorname{Hom}(\varinjlim
C^{*}_{\gamma};\beta^{\\#})=\\{\operatorname{Hom}(\varinjlim
C^{n}_{\gamma};G^{\prime})\oplus\operatorname{Hom}(\varinjlim
C^{n+1}_{\gamma};G^{\prime\prime}),\partial\\},$

where
$\partial(\varphi^{\prime},\varphi^{\prime\prime})=(\varphi^{\prime}\circ\delta,\beta\circ\varphi^{\prime}-\varphi^{\prime\prime}\circ\delta).$
Note that $\delta=\varinjlim\delta_{\gamma}:\varinjlim
C_{\gamma}^{n-1}\longrightarrow\varinjlim C_{\gamma}^{n},$ where
$\delta_{\gamma}:C_{\gamma}^{n-1}\longrightarrow C_{\gamma}^{n}$ is the
coboundary map of the cochain complex $C^{*}_{\gamma}.$ Since for any $G$
there is an isomorphism $\operatorname{Hom}(\varinjlim
C^{*}_{\gamma};G)\simeq\varprojlim\operatorname{Hom}(C^{*}_{\gamma};G)$, we
have

(3.10) $\footnotesize\operatorname{Hom}(\varinjlim
C_{\gamma}^{n};G^{\prime})\oplus\operatorname{Hom}(\varinjlim
C_{\gamma}^{n+1};G^{\prime\prime})\simeq\varprojlim\operatorname{Hom}(C_{\gamma}^{n};G^{\prime})\oplus\varprojlim\operatorname{Hom}(C_{\gamma}^{n+1};G^{\prime\prime}).$

∎

######  Lemma 3.3.

If $f^{\\#}:C^{*}\longrightarrow C^{\prime*}$ is a homomorphism of cochain
complexes, then there is a commutative diagram:

(3.11) $0$$\operatorname{Hom}(C^{\prime n+1}/B^{\prime
n+1};G^{\prime})$$\operatorname{Hom}(C^{\prime
n+1};G^{\prime})\oplus\operatorname{Hom}(C^{\prime n+1}/B^{\prime
n+1};G^{\prime\prime})$$Ker\xi^{\prime}$$0$$\tau^{\prime}$$\mu^{\prime}$$0$$\operatorname{Hom}(C^{n+1}/B^{n+1};G^{\prime})$$\operatorname{Hom}(C^{n+1};G^{\prime})\oplus\operatorname{Hom}(C^{n+1}/B^{n+1};G^{\prime\prime})$$Ker\xi$$0.$$\tau$$\mu$$\tilde{f}_{\\#}$$(f_{\\#},\tilde{f}_{\\#})$$\tilde{f}$

###### Proof.

Note that homomorphisms $\tilde{f}_{\\#}:\operatorname{Hom}(C^{\prime
n+1}/B^{\prime
n+1};G^{\prime})\longrightarrow\operatorname{Hom}(C^{n+1}/B^{n+1};G^{\prime})$
and $(f_{\\#},\tilde{f}_{\\#}):\operatorname{Hom}(C^{\prime
n+1};G^{\prime})\oplus\operatorname{Hom}(C^{\prime n+1}/B^{\prime
n+1};G^{\prime\prime})\longrightarrow\operatorname{Hom}(C^{n+1};G^{\prime})\oplus\operatorname{Hom}(C^{n+1}/B^{n+1};G^{\prime\prime})$
are naturally defined by
$\tilde{f}_{\\#}(\varphi^{\prime})=\varphi^{\prime}\circ\tilde{f}_{n+1}$ and
$(f_{\\#},\tilde{f}_{\\#})(\varphi^{\prime},\varphi^{\prime\prime})=(\varphi^{\prime}\circ
f_{n+1},\varphi^{\prime\prime}\circ\tilde{f}_{n+1}),$ where
$\tilde{f}_{n+1}:C^{n+1}/B^{n+1}\longrightarrow C^{\prime n+1}/B^{\prime n+1}$
is induced by $f_{n+1}:C^{n+1}\longrightarrow C^{\prime n+1}.$

a. $\tau\circ\tilde{f}_{\\#}=(f_{\\#},\tilde{f}_{\\#})\circ\tau^{\prime}$. By
the definition, we have
$\left(\tau\circ\tilde{f}_{\\#}\right)(\varphi^{\prime})=\tau\left(\tilde{f}_{\\#}(\varphi^{\prime})\right)=\tau\left(\varphi^{\prime}\circ\tilde{f}_{n+1}\right)=\left(\varphi^{\prime}\circ\tilde{f}_{n+1}\circ
q,\beta\circ\varphi^{\prime}\circ\tilde{f}_{n+1}\right)$ and
$\left((f_{\\#},\tilde{f}_{\\#})\circ\tau^{\prime}\right)\left(\varphi^{\prime}\right)=\left(f_{\\#},\tilde{f}_{\\#}\right)\left(\tau^{\prime}\left(\varphi^{\prime}\right)\right)=\left(f_{\\#},\tilde{f}_{\\#}\right)\left(\varphi^{\prime}\circ
q^{\prime},\beta\circ\varphi^{\prime}\right)=(\varphi^{\prime}\circ
q^{\prime}\circ f_{n+1},\beta\circ\varphi^{\prime}\circ\tilde{f}_{n+1}).$
Since $\tilde{f}_{n+1}\circ q=q^{\prime}\circ f_{n+1},$ we have
$\varphi^{\prime}\circ\tilde{f}_{n+1}\circ q=\varphi^{\prime}\circ
q^{\prime}\circ f_{n+1}.$ Hence,
$\tau\circ\tilde{f}_{\\#}=(f_{\\#},\tilde{f}_{\\#})\circ\tau^{\prime}$.

b. $\mu\circ(f_{\\#},\tilde{f}_{\\#})=\tilde{f}\circ\mu^{\prime}$. By the
definition, we have
$\left(\mu\circ(f_{\\#},\tilde{f}_{\\#})\right)(\varphi^{\prime},\varphi^{\prime\prime})=\mu\left((f_{\\#},\tilde{f}_{\\#})(\varphi^{\prime},\varphi^{\prime\prime})\right)=\mu\left(\varphi^{\prime}\circ
f_{n+1},\varphi^{\prime\prime}\circ\tilde{f}_{n+1}\right)=(\varphi^{\prime}\circ
f_{n+1}\circ\delta,\beta\circ\varphi^{\prime}\circ
f_{n+1}-\varphi^{\prime\prime}\circ\tilde{f}_{n+1}\circ q)$ and
$\left(\tilde{f}\circ\mu^{\prime}\right)(\varphi^{\prime},\varphi^{\prime\prime})=\tilde{f}\left(\mu^{\prime}(\varphi^{\prime},\varphi^{\prime\prime})\right)=\tilde{f}\left(\varphi^{\prime}\circ\delta^{\prime},\beta\circ\varphi^{\prime}-\varphi^{\prime\prime}\circ
q^{\prime}\right)=(\varphi^{\prime}\circ\delta^{\prime}\circ
f_{n},\beta\circ\varphi^{\prime}\circ f_{n+1}-\varphi^{\prime\prime}\circ
q^{\prime}\circ f_{n+1}).$ Since $\tilde{f}_{n+1}\circ q=q^{\prime}\circ
f_{n+1}$ and $\delta^{\prime}\circ f_{n}=f_{n+1}\circ\delta,$ there are
qualities $\varphi^{\prime}\circ
f_{n+1}\circ\delta=\varphi^{\prime}\circ\delta^{\prime}\circ f_{n}$ and
$\varphi^{\prime\prime}\circ\tilde{f}_{n+1}\circ q=\varphi^{\prime\prime}\circ
q^{\prime}\circ f_{n+1}.$ Hence,
$\mu\circ(f_{\\#},\tilde{f}_{\\#})=\tilde{f}\circ\mu^{\prime}$. ∎

Let
$\\{\operatorname{Hom}(C_{\gamma}^{n+1};G^{\prime})\oplus\operatorname{Hom}(C_{\gamma}^{n+1}/B_{\gamma}^{n+1};G^{\prime\prime})\\}$
be an inverse system generated by the direct system $\\{C^{n+1}_{\gamma}\\}$.
It is clear that for each $\gamma$ there is an exact sequence

(3.12)
$0$$\operatorname{Hom}(C^{n+1}_{\gamma};G^{\prime})$$\operatorname{Hom}(C^{n+1}_{\gamma};G^{\prime})\oplus\operatorname{Hom}(C^{n+1}_{\gamma}/B^{n+1}_{\gamma};G^{\prime\prime})$$\operatorname{Hom}(C^{n+1}_{\gamma}/B^{n+1}_{\gamma};G^{\prime\prime})$$0$.$\tau$$\mu$

Hence, by the main property of the derived functors $\varprojlim^{(i)}$ there
is a long exact sequence:

(3.13)
$\dots$$\varprojlim^{(i)}\operatorname{Hom}(C^{n+1}_{\gamma};G^{\prime})$$\varprojlim^{(i)}\left(\operatorname{Hom}(C^{n+1}_{\gamma};G^{\prime})\oplus\operatorname{Hom}(C^{n+1}_{\gamma}/B^{n+1}_{\gamma};G^{\prime\prime})\right)$$\varprojlim^{(i)}\operatorname{Hom}(C^{n+1}_{\gamma}/B^{n+1}_{\gamma};G^{\prime\prime})$$\dots$.$\tilde{\tau}$$\tilde{\mu}$$\tilde{\mu}$

On the other hand, since for each injective abelian groups $G$,
$\varprojlim^{(i)}\\{\operatorname{Hom}(C^{n+1};G)\\}=0,$ $i\geq 1$ (see Lemma
1.3 [Hub-Mei]), we obtain the following result.

######  Corollary 3.4.

For each injective abelian groups $G^{\prime}$ and $G^{\prime\prime}$, there
is the following equality

(3.14)
$\footnotesize{\varprojlim}^{(i)}\left(\operatorname{Hom}(C^{n+1}_{\gamma};G^{\prime})\oplus\operatorname{Hom}(C^{n+1}_{\gamma}/B^{n+1}_{\gamma};G^{\prime\prime})\right)=0,~{}~{}i\geq
1.$

Using the obtained result, we will prove the following lemma.

######  Lemma 3.5.

For each integer $i\geq 1$, there is an equality

(3.15) $\footnotesize{\varprojlim}^{(i)}Ker\xi_{\gamma}=0.$

###### Proof.

By Lemma 1, for each $\xi_{\gamma}$, there is a short exact sequence

(3.16)
$0$$\operatorname{Hom}(C^{n+1}_{\gamma}/B^{n+1}_{\gamma};G^{\prime})$$\operatorname{Hom}(C^{n+1}_{\gamma};G^{\prime})\oplus\operatorname{Hom}(C^{n+1}_{\gamma}/B^{n+1}_{\gamma};G^{\prime\prime})$$Ker\xi_{\gamma}$$0$.$\sigma_{\gamma}$$\omega_{\gamma}$

By the main property of a derived functor $\varprojlim^{(i)}$, there is a long
exact sequence

(3.17)
$\dots$$\varprojlim^{(i)}\operatorname{Hom}(C^{n+1}_{\gamma}/B^{n+1}_{\gamma};G^{\prime})$$\varprojlim^{(i)}\left(\operatorname{Hom}(C^{n+1}_{\gamma};G^{\prime})\oplus\operatorname{Hom}(C^{n+1}_{\gamma}/B^{n+1}_{\gamma};G^{\prime\prime})\right)$$\varprojlim^{(i)}\operatorname{Ker}\xi_{\gamma}$$\dots$.

By Lemma 1.3 [Hub-Mei] for each $i\geq 0,$ there is an equality
${\varprojlim}^{(i)}\operatorname{Hom}(C_{\gamma}^{n+1}/B_{\gamma}^{n+1};G^{\prime})=0$
and by Corollary 2, for each $i\geq 1$ we obtain

(3.18)
$\footnotesize{\varprojlim}^{(i)}\left(\operatorname{Hom}(C_{\gamma}^{n+1};G^{\prime})\oplus\operatorname{Hom}(C_{\gamma}^{n+1}/B_{\gamma}^{n+1};G^{\prime\prime})\right)=0.$

Hence, by the long exact sequence (3.17) we obtain that
${\varprojlim}^{(i)}Ker\xi_{\gamma}=0$, $i\geq 1$. ∎

######  Corollary 3.6.

For each integer $i\geq 1,$ there is an isomorphism

(3.19)
$\footnotesize{\varprojlim}^{(i)}\bar{Z}_{n}^{\gamma}\simeq{\varprojlim}^{(i)}\operatorname{Hom}(H^{n}(C^{*}_{\gamma});G).$

###### Proof.

By a. of Theorem 1, there is an epimorphism
$\xi_{\gamma}:\bar{Z}_{n}^{\gamma}\longrightarrow\operatorname{Hom}(H^{n}(C^{*}_{\gamma});G).$
Therefore, the following sequence is exact

(3.20)
$0$$Ker\xi_{\gamma}$$\bar{Z}_{n}^{\gamma}$$\operatorname{Hom}(H^{n}(C^{*}_{\gamma});G)$$0$.$\xi_{\gamma}$

Consequently, it induces the following long exact sequence

(3.21)
$\dots$${\varprojlim}^{(i)}Ker\xi_{\gamma}$${\varprojlim}^{(i)}\bar{Z}_{n}^{\gamma}$${\varprojlim}^{(i)}\operatorname{Hom}(H^{n}(C^{*}_{\gamma});G)$${\varprojlim}^{(i+1)}Ker\xi_{\gamma}$$\dots$.

On the other hand, by Lemma 4,
${\varprojlim}^{(i)}\\{Ker\xi_{\gamma}\\}=0,~{}~{}i\geq 1.$ Therefore, for
$i\geq 1,$ we have an isomorphism
${\varprojlim}^{(i)}\bar{Z}_{n}^{\gamma}\simeq{\varprojlim}^{(i)}\operatorname{Hom}(H^{n}(C^{*}_{\gamma});G).$
∎

Note that for each $\gamma$, there is a natural commutative triangle

(3.22)
$\bar{Z}_{n}^{\gamma}$$\operatorname{Hom}(H^{n}(C^{*}_{\gamma});G)$$\bar{H}_{n}(C^{*}_{\gamma};G)$.$\xi_{\gamma}$$\bar{p}_{\gamma}$$\bar{\xi}_{\gamma}$

Therefore, if we take ${\varprojlim}^{(i)}$ of this diagram, then by Corollary
2, we obtain the following result.

######  Corollary 3.7.

For each integer $i\geq 1$, ${\varprojlim}^{(i)}\bar{Z}_{n}^{\gamma}$ is a
direct summand of ${\varprojlim}^{(i)}\bar{H}_{n}(C^{*}_{\gamma};G)$ and the
projection of ${\varprojlim}^{(i)}\bar{H}_{n}(C^{*}_{\gamma};G)$ onto
${\varprojlim}^{(i)}\bar{Z}_{n}^{\gamma}$ is natural.

Finally, we obtain the following important property of $\varprojlim^{(i)}$
functor.

######  Theorem 3.8.

For each integer $i\geq 0$, there is a short exact sequence:

(3.23)
$0$${\varprojlim}^{(i)}\operatorname{Ext}(H^{n+1}(C^{*}_{\gamma});G)$${\varprojlim}^{(i)}\bar{H}_{n}(C^{*}_{\gamma};G)$${\varprojlim}^{(i)}\operatorname{Hom}(H^{n}(C^{*}_{\gamma});G)$$0,$$\xi_{\gamma}$

and this sequence splits naturally for $i\geq 1$.

###### Proof.

Using the commutative diagram (3.22), for each $\gamma$ we have a commutative
diagram with exact rows:

(3.24)
$0$$Ker\xi_{\gamma}$$\bar{Z}_{n}^{\gamma}$$\operatorname{Hom}(H^{n}(C^{*}_{\gamma});G)$$0$$\xi_{\gamma}$$0$$\operatorname{Ext}(H^{n+1}(C^{*}_{\gamma});G)$$\bar{H}_{n}(C^{*}_{\gamma};G)$$\operatorname{Hom}(H^{n}(C^{*}_{\gamma});G)$$0$.$\bar{\xi}_{\gamma}$$\bar{p}_{\gamma}$$1$

This induces the following commutative diagram with exact rows:

(3.25)
$\dots$${\varprojlim}^{(i)}Ker\xi_{\gamma}$${\varprojlim}^{(i)}\bar{Z}_{n}^{\gamma}$${\varprojlim}^{(i)}\operatorname{Hom}(H^{n}(C^{*}_{\gamma});G)$${\varprojlim}^{(i+1)}Ker\xi_{\gamma}$$\dots$${\varprojlim}^{(i)}\xi_{\gamma}$$\dots$${\varprojlim}^{(i)}\operatorname{Ext}(H^{n+1}(C^{*}_{\gamma});G)$${\varprojlim}^{(i)}\bar{H}_{n}(C^{*}_{\gamma};G)$${\varprojlim}^{(i)}\operatorname{Hom}(H^{n}(C^{*}_{\gamma});G)$${\varprojlim}^{(i+1)}\operatorname{Ext}(H^{n+1}(C^{*}_{\gamma});G)$$\dots$
.${\varprojlim}^{(i)}\bar{\xi}_{\gamma}$${\varprojlim}^{(i)}\bar{p}_{\gamma}$$1$

By Lemma 4, ${\varprojlim}^{(i)}Ker\xi_{\gamma}=0,$ for $i\geq 1,$ and so the
beginning of the diagram (3.25) of the following form:

(3.26)
$0$${\varprojlim}Ker\xi_{\gamma}$${\varprojlim}\bar{Z}_{n}^{\gamma}$${\varprojlim}\operatorname{Hom}(H^{n}(C^{*}_{\gamma});G)$$0$${\varprojlim}\xi_{\gamma}$$0$${\varprojlim}\operatorname{Ext}(H^{n+1}(C^{*}_{\gamma});G)$${\varprojlim}\bar{H}_{n}(C^{*}_{\gamma};G)$${\varprojlim}\operatorname{Hom}(H^{n}(C^{*}_{\gamma});G)$$\dots$
.${\varprojlim}\bar{\xi}_{\gamma}$${\varprojlim}^{(i)}\bar{p}_{\gamma}$$1$

Therefore, the following sequence is exact:

(3.27)
$0$${\varprojlim}\operatorname{Ext}(H^{n+1}(C^{*}_{\gamma});G)$${\varprojlim}\bar{H}_{n}(C^{*}_{\gamma};G)$${\varprojlim}\operatorname{Hom}(H^{n}(C^{*}_{\gamma});G)$$0$$\xi_{\gamma}$

and the
${\varprojlim}^{(1)}\\{\operatorname{Ext}(H^{n+1}(C^{*}_{\gamma});G)\\}\longrightarrow{\varprojlim}^{(1)}\\{\bar{H}_{n}(C^{*}_{\gamma};G)\\}$
is a monomorphism. Therefore, we obtain the result for $i=0$. On the other
hand, for $i\geq 1,$ 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.

######  Theorem 4.1.

Let ${\mathbf{C}}^{*}=\\{C^{*}_{\gamma}\\}$ be a direct system of cochain
complexes. Then, there is a natural exact sequence:

(4.1)
$\dots$${\varprojlim}^{(3)}\bar{H}^{\gamma}_{n+2}$${\varprojlim}^{(1)}\bar{H}^{\gamma}_{n+1}$$\bar{H}_{n}\left({\varinjlim}C^{*}_{\gamma};G\right)$${\varprojlim}\bar{H}^{\gamma}_{n}$${\varprojlim}^{(2)}\bar{H}^{\gamma}_{n+1}$$\dots$

where $\bar{H}_{*}^{\gamma}=\bar{H}_{*}(C^{*}_{\gamma};G).$

###### Proof.

By Proposition 1.2 of [Hub-Mei], for the inverse system
$\\{H_{\gamma}^{n+1}\\}$ we have an exact sequence:

(4.2)
$0$${\varprojlim}^{(1)}\operatorname{Hom}(H_{\gamma}^{n+1};G)$$\operatorname{Ext}(\varinjlim
H_{\gamma}^{n+1};G)$$\varprojlim\operatorname{Ext}(H_{\gamma}^{n+1};G)$${\varprojlim}^{(2)}\operatorname{Hom}(H_{\gamma}^{n+1};G)$$0$

and

(4.3)
$\footnotesize{\varprojlim}^{(i)}\operatorname{Ext}(H^{n+1}_{\gamma};G)\simeq{\varprojlim}^{(i+2)}\operatorname{Hom}(H^{n+1}_{\gamma};G),\text{
for }i\geq 1.$

Since cohomology commutes with direct limits, we have $H^{*}(\varinjlim
C^{*}_{\gamma};G)\simeq\varinjlim H^{*}(C^{*}_{\gamma};G)$. Therefore, if
$C^{*}\simeq\varinjlim C^{*}_{\gamma}$, then $H^{n+1}(C^{*})\simeq\varinjlim
H^{n+1}_{\gamma},$ where $H^{n+1}_{\gamma}=H^{n+1}(C^{*}_{\gamma};G).$ So, we
obtain an exact sequence:

(4.4)
$0$${\varprojlim}^{(1)}\operatorname{Hom}(H_{\gamma}^{n+1};G)$$\operatorname{Ext}(H^{n+1}(C^{*});G)$$\varprojlim\operatorname{Ext}(H_{\gamma}^{n+1};G)$${\varprojlim}^{(2)}\operatorname{Hom}(H_{\gamma}^{n+1};G)$$0$.

Note that, if $i_{\gamma}:C^{*}_{\gamma}\longrightarrow C^{*}$ is a natural
map, then it induces
$\pi_{\gamma};\bar{H}_{*}(C;G)\longrightarrow\bar{H}_{n}(C^{*}_{\gamma};G)$
map. On the other hand, by Theorem 1, the following diagram is commutative:

(4.5)
$0$$\operatorname{Ext}(H^{n+1}(C^{*});G)$$\bar{H}_{n}(C^{*};G)$$\operatorname{Hom}(H^{n}(C^{*});G)$$0$$0$$\operatorname{Ext}(H^{n+1}(C^{*}_{\gamma});G)$$\bar{H}_{n}(C^{*}_{\gamma};G)$$\operatorname{Hom}(H^{n}(C^{*}_{\gamma});G)$$0$.$\tilde{\pi}_{\gamma}$$\pi_{\gamma}$$\bar{\pi}_{\gamma}$

The diagram (4.5) generates the following diagram:

(4.6)
$0$$\operatorname{Ext}(H^{n+1}(C^{*});G)$$\bar{H}_{n}(C^{*};G)$$\operatorname{Hom}(H^{n}(C^{*});G)$$0$$0$$\varprojlim\operatorname{Ext}(H^{n+1}_{\gamma};G)$$\varprojlim\bar{H}_{n}^{\gamma}$$\varprojlim\operatorname{Hom}(H^{n}_{\gamma};G)$$0$.$\tilde{\pi}$$\pi$$\simeq$

Therefore, we have $Ker\tilde{\pi}\simeq Ker\pi$ and $Coker\tilde{\pi}\simeq
Coker\pi$ and so, the following diagram is commutative:

(4.7)
$0$$\operatorname{Ext}(H^{n+1}(C);G)$$\bar{H}_{n}(C;G)$$\operatorname{Hom}(H^{n}(C);G)$$0$$Ker\tilde{\pi}$$Ker\pi$$0$$0$$\simeq$$0$$\varprojlim\operatorname{Ext}(H^{n+1}_{\gamma};G)$$\varprojlim\bar{H}_{n}^{\gamma}$$\varprojlim\operatorname{Hom}(H^{n}_{\gamma};G)$$0$
.$\tilde{\pi}$$\pi$$\simeq$$Coker\tilde{\pi}$$Coker\pi$$0$$0$$\simeq$

Using the exact sequence (4.4) and diagram (4.7), we obtain a four-term exact
sequence:

(4.8)
$0$${\varprojlim}^{(1)}\operatorname{Hom}(H_{\gamma}^{n+1};G)$$\bar{H}_{n}(C;G)$$\varprojlim\bar{H}_{n}^{\gamma}$${\varprojlim}^{(2)}\operatorname{Hom}(H_{\gamma}^{n+1};G)$$0$.

Using the exact sequence (4.8), Theorem 3 and isomorphism (4.3), we obtain the
following diagram, which contains the long exact sequence of the theorem:

(4.9)
$0$${\varprojlim}^{(1)}\operatorname{Hom}(H_{\gamma}^{n+1};G)$$\bar{H}_{n}(C;G)$$\varprojlim\bar{H}_{n}^{\gamma}$${\varprojlim}^{(2)}\operatorname{Hom}(H_{\gamma}^{n+1};G)$$0$.$0$${\varprojlim}^{(2)}\operatorname{Ext}(H_{\gamma}^{n+2};G)$${\varprojlim}^{(2)}\bar{H}_{n+1}^{\gamma}$${\varprojlim}^{(2)}\operatorname{Hom}(H_{\gamma}^{n+1};G)$$0$$0$${\varprojlim}^{(4)}\operatorname{Hom}(H_{\gamma}^{n+2};G)$${\varprojlim}^{(4)}\bar{H}_{n+2}^{\gamma}$${\varprojlim}^{(4)}\operatorname{Ext}(H_{\gamma}^{n+3};G)$$0$$\vdots$$0$${\varprojlim}^{(1)}\operatorname{Hom}(H_{\gamma}^{n+1};G)$${\varprojlim}^{(1)}\bar{H}_{n+1}^{\gamma}$${\varprojlim}^{(1)}\operatorname{Ext}(H_{\gamma}^{n+2};G)$$0$$0$${\varprojlim}^{(3)}\operatorname{Ext}(H_{\gamma}^{n+3};G)$${\varprojlim}^{(3)}\bar{H}_{n+2}^{\gamma}$${\varprojlim}^{(3)}\operatorname{Hom}(H_{\gamma}^{n+2};G)$$0$$\vdots$$\simeq$$\simeq$$\simeq$$\simeq$

∎

######  Corollary 4.2.

Let ${\mathbf{C}}^{*}=\\{C^{*}_{\gamma}\\}$ be a direct system of cochain
complexes. Then, for each injective abelian group $G,$ there is an isomorphism

(4.10)
$\footnotesize\bar{H}_{n}\left({\varinjlim}C^{*}_{\gamma};G\right)\simeq{\varprojlim}\bar{H}_{*}(C^{*}_{\gamma};G).$

##  5 Applications

1\. Let $C_{c}^{*}(X,G)$ be the cochain complex of Massey [Mas2]. It is known
that for each locally compact Hausdorff space $X$ and each integer $n$ the
cochain group $C_{c}^{n}(X,\mathbb{Z})$ with integer coefficient is a free
abelian group (theorem 4.1 [Mas2]). Using the cochain complex
$C_{c}^{*}(X,G)$, Massey defined an exact homology $H_{*}^{M}$, the so called
Massey homology on the category of locally compact spaces and proper maps as a
homology of the chain complex $C_{*}(X,G)=\operatorname{Hom}(C_{c}^{*}(X),G)$.
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]):

(5.1) $\footnotesize
0\longrightarrow\operatorname{Ext}(H_{c}^{n+1}(X),G)\longrightarrow{H}^{M}_{n}(X,G)\longrightarrow\operatorname{Hom}(H_{c}^{n}(X),G)\longrightarrow
0.$

Let $C^{*}_{c}=C^{*}_{c}(X;G)$ be the cochain complex of Massey. Consider the
chain complex $\bar{C}^{M}_{*}(X;G)=\operatorname{Hom}(C^{*}(X);\beta_{\\#})$.
Let $\bar{H}^{M}_{*}(X;G)$ be homology of the chain complex
$\bar{C}^{M}_{*}(X;G)$. In this case, by Theorem 1 we will obtain the
Universal Coefficient Formula:

(5.2) $\footnotesize
0\longrightarrow\operatorname{Ext}(H_{c}^{n+1}(X),G)\longrightarrow\bar{H}^{M}_{n}(X,G)\longrightarrow\operatorname{Hom}(H_{c}^{n}(X),G)\longrightarrow
0.$

Note that by Theorem 2, for the category of locally compact spaces the
homologies $\bar{H}^{M}_{n}(X,G)$ and $H^{M}_{n}(X,G)$ are isomorphic.

Note that for the Massey homology theory $\bar{H}^{M}_{*}(-;G)$ our
construction gives the following result:

######  Corollary 5.1.

Let $X$ be a locally compact Hausdorff space, then
a) if $\\{N_{\alpha}\\}$ is the system of closed neighborhoods $N_{\alpha}$ of
closed subspace $A$ of $X$, directed by inclusion, then it induces the
following exact sequence:

(5.3)
$\dots$${\varprojlim}^{(2k+1)}\bar{H}^{M}_{n++k+1}(N_{\alpha})$$\dots$${\varprojlim}^{(3)}\bar{H}^{M}_{n+2}(N_{\alpha})$${\varprojlim}^{(1)}\bar{H}^{M}_{n+1}(N_{\alpha})$$\bar{H}^{M}_{n}(A;G)$${\varprojlim}\bar{H}^{M}_{n}(N_{\alpha})$${\varprojlim}^{(2)}\bar{H}^{M}_{n+1}(N_{\alpha})$$\dots$${\varprojlim}^{(2k)}\bar{H}^{M}_{n+k}(N_{\alpha})$$\dots~{}~{}.$$i_{*}$

b) if $\\{U_{\alpha}\\}$ is the system of open subspaces of $X$, such that
$\bar{U}_{\alpha}$ is compact and $X=\bigcup U_{\alpha}$ directed by
inclusion, then it induces the following exact sequence:

(5.4)
$\dots$${\varprojlim}^{(2k+1)}\bar{H}^{M}_{n++k+1}(U_{\alpha})$$\dots$${\varprojlim}^{(3)}\bar{H}^{M}_{n+2}(U_{\alpha})$${\varprojlim}^{(1)}\bar{H}^{M}_{n+1}(U_{\alpha})$$\bar{H}^{M}_{n}(A;G)$${\varprojlim}\bar{H}^{M}_{n}(U_{\alpha})$${\varprojlim}^{(2)}\bar{H}^{M}_{n+1}(U_{\alpha})$$\dots$${\varprojlim}^{(2k)}\bar{H}^{M}_{n+k}(U_{\alpha})$$\dots~{}~{}.$$i_{*}$

Note that the formula (LABEL:eq89.1) 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 $\bar{C}^{*}(X;G)$ the cochain complex of Alexander-Spanier [Sp] and by $\bar{H}^{*}(X;G)$ the Alexander-Spanier cohomology. let $A$ be a subspace of a topological space $X$ and $\\{U_{\alpha}\\}$ be the family of all neighborhoods of $A$ in $X$ directed downward by inclusion. Hence, $\\{\bar{H}^{n}(U_{\alpha};G)\\}$ is a direct system. The restriction maps $\bar{H}^{n}(U_{\alpha};G)\longrightarrow\bar{H}^{n}(A;G)$ define a natural homomorphism

(5.5) $\footnotesize
i:\varinjlim\bar{H}^{n}(U_{\alpha};G)\longrightarrow\bar{H}^{n}(A;G).$

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) $\footnotesize
i:{H}_{n}(A;G)\longrightarrow\varprojlim{H}_{n}(U_{\alpha};G).$

The question whether the homomorphism (5.6) is an isomorphism or not was open.

Let $\bar{C}^{*}=\bar{C}^{*}(X;G)$ be the cochain complex of Alexander-
Spanier. Consider the chain complex
$\bar{C}_{*}(X;G)=\operatorname{Hom}(\bar{C}^{*}(X);\beta_{\\#})$. Let
$\bar{H}_{*}(X;G)$ be the homology of the chain complex $\bar{C}_{*}(X;G)$. In
this case, we will say that the homology $\bar{H}_{*}(X;G)$ is generated by
the Alexander-Spanier cochains $\bar{C}^{*}(X;G)$. 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
$\\{U_{\alpha}\\}$ is the family of all neighborhoods of $A$ in $X$, then
there is a long exact sequence:

(5.7)
$\dots$${\varprojlim}^{(2k+1)}\bar{H}_{n++k+1}(U_{\alpha})$$\dots$${\varprojlim}^{(3)}\bar{H}_{n+2}(U_{\alpha})$${\varprojlim}^{(1)}\bar{H}_{n+1}(U_{\alpha})$$\bar{H}_{n}(A;G)$${\varprojlim}\bar{H}_{n}(U_{\alpha})$${\varprojlim}^{(2)}\bar{H}_{n+1}(U_{\alpha})$$\dots$${\varprojlim}^{(2k)}\bar{H}_{n+k}(U_{\alpha})$$\dots~{}~{}.$$i_{*}$

3. It is clear that there is a natural inclusion $i^{\\#}:C^{*}_{c}(X;G)\to\bar{C}_{*}(X;G)$ from the Massey cochain complex to the Alexander-Spanier cochain complex, which induces the corresponding homomorphism $i^{*}:\bar{H}_{*}(-;G)\to\bar{H}^{M}_{*}(-;G),$ where $\bar{H}_{*}(-;G)$ and $\bar{H}^{M}_{*}(-;G)$ are homologies generated by the Alexander-Spanier and the Massey cochains, respectively. Therefore, $\bar{H}_{*}(-;G)$ and $\bar{H}^{M}_{*}(-;G)$ are homologies of the chain complexes $\bar{C}_{*}(-;G)=\operatorname{Hom}(\bar{C}^{*}(-);\beta_{\\#})$ and $\bar{C}_{*}^{M}(-;G)=\operatorname{Hom}(C^{*}_{c}(-);\beta_{\\#}).$ 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 Formula, we will obtain that for each compact Hausdorff space the there is an isomorphism:

(5.8) $\footnotesize
i^{*}:\bar{H}_{*}(X;G)\overset{\simeq}{\longrightarrow}\bar{H}^{M}_{*}(X;G).$

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

(5.9) $\footnotesize\bar{H}_{*}(X;G)\simeq{H}^{St}_{*}(X;G).$

The same way, on the category of compact Hausdorff spaces, the Milnor homology
$H^{Mil}_{*}$ and the Massey homology are isomorphic and consequently, we have

(5.10) $\footnotesize\bar{H}_{*}(X;G)\simeq{H}^{Mil}_{*}(X;G).$

If $H_{*}^{BM}(-;G)$ is the Borel-Moore homology with coefficient in $G$, then
by Theorem 3 [Kuz], we have the isomororphism

(5.11) $\footnotesize\bar{H}_{*}(X;G)\simeq{H}^{BM}_{*}(X;G).$

4. Let $K_{C}$ be the category of compact pairs $(X,A)$ and continuous maps and $H_{*}$ be an exact homology theory. Let $\\{(X_{\alpha},A_{\alpha})\\}$ be an inverse system of compact pairs $(X_{\alpha},A_{\alpha})$ and $(X,A)=\varprojlim(X_{\alpha},A_{\alpha}).$ The inverse system $\\{(X_{\alpha},A_{\alpha})\\}$ generates an inverse system $\\{H_{*}(X_{\alpha},A_{\alpha})\\}$ and the projection $\pi_{\alpha}:(X,A)\to(X_{\alpha},A_{\alpha})$ induces the homomorphism $\pi_{\alpha,*}:H_{*}(X,A)\to H_{*}(X_{\alpha},A_{\alpha}),$ which induces the homomorphism

(5.12) $\footnotesize\pi_{*}:H_{*}(X,A)\to\varprojlim
H_{*}(X_{\alpha},A_{\alpha}).$

######  Definition 5.3.

An exact homology theory $H_{*}$ is said to be a continuous on the category
$K_{C}$, if for each inverse system $\\{(X_{\alpha},A_{\alpha})\\}$ of the
given category, there is an infinite exact sequence:

(5.13)
$\dots$${\varprojlim}^{(2k+1)}\bar{H}^{M}_{n++k+1}(X_{\alpha},A_{\alpha})$$\dots$${\varprojlim}^{(3)}\bar{H}^{M}_{n+2}(X_{\alpha},A_{\alpha})$${\varprojlim}^{(1)}\bar{H}^{M}_{n+1}(X_{\alpha},A_{\alpha})$$\bar{H}^{M}_{n}(X,A;G)$${\varprojlim}\bar{H}^{M}_{n}(X_{\alpha},A_{\alpha})$${\varprojlim}^{(2)}\bar{H}^{M}_{n+1}(X_{\alpha},A_{\alpha})$$\dots$${\varprojlim}^{(2k)}\bar{H}^{M}_{n+k}(X_{\alpha},A_{\alpha})$$\dots~{}~{}.$$i_{*}$

######  Definition 5.4.

A direct system $\mathbf{C}^{*}=\\{C^{*}_{\alpha}\\}$ of the cochain complexes
$C^{*}_{\alpha}$ is said to be associated with a cochain complex $C^{*},$ if
there is a homomorphism $\mathbf{C}^{*}\to C^{*}$ such that for each
$n\in\mathbb{Z}$ the induced homomorphism

(5.14) $\footnotesize\varinjlim H^{*}(C^{*}_{\alpha})\to H^{*}(C^{*})$

is an isomorphism.

######  Lemma 5.5.

If a direct system $\mathbf{C}^{*}=\\{C^{*}_{\alpha}\\}$ of the cochain
complexes $C^{*}_{\alpha}$ is associated with a cochain complex $C^{*}$, then
there is an infinite exact sequence:

(5.15)
$\dots$${\varprojlim}^{(2k+1)}\bar{H}^{M}_{n++k+1}(C^{*}_{\alpha};G)$$\dots$${\varprojlim}^{(3)}\bar{H}^{M}_{n+2}(C^{*}_{\alpha};G)$${\varprojlim}^{(1)}\bar{H}^{M}_{n+1}(C^{*}_{\alpha};G)$$\bar{H}^{M}_{n}(C^{*};G)$${\varprojlim}\bar{H}^{M}_{n}(C^{*}_{\alpha};G)$${\varprojlim}^{(2)}\bar{H}^{M}_{n+1}(C^{*}_{\alpha};G)$$\dots$${\varprojlim}^{(2k)}\bar{H}^{M}_{n+k}(C^{*}_{\alpha};G)$$\dots~{}~{}.$$i_{*}$

where
$\bar{H}_{*}(C^{*})=H_{*}\left((\operatorname{Hom}(C^{*};\beta_{\\#})\right)$
and
$\bar{H}_{*}(C^{*}_{\alpha})=H_{*}\left(\operatorname{Hom}(C^{*}_{\alpha};\beta_{\\#})\right).$

###### Proof.

By theorem 4, There is a natural exact sequence:

(5.16)
$\dots$${\varprojlim}^{(3)}\bar{H}^{\alpha}_{n+2}$${\varprojlim}^{(1)}\bar{H}^{\alpha}_{n+1}$$\bar{H}_{n}\left({\varinjlim}C^{*}_{\alpha};G\right)$${\varprojlim}\bar{H}^{\alpha}_{n}$${\varprojlim}^{(2)}\bar{H}^{\alpha}_{n+1}$$\dots$
,

where $\bar{H}_{*}^{\alpha}=\bar{H}_{*}(C^{*}_{\alpha};G).$ Since the direct
system ${\bf C}^{*}=\\{C^{*}_{\alpha}\\}$ of cochain complexes
$C^{*}_{\alpha}$ is associated with a cochain complex $C^{*}$, there is an
isomorphism

(5.17) $\footnotesize\bar{H}_{*}(\varinjlim
C^{*}_{\alpha};G)\simeq\varinjlim\bar{H}^{*}(C^{*}_{\alpha};G)\overset{\simeq}{\longrightarrow}\bar{H}_{*}(C^{*};G).$

On the other hand, by Universal Coefficient Formula, we have the following
commutative diagram with exact rows:

(5.18)
$0$$\operatorname{Ext}(H^{n+1}(C^{*});G)$$\bar{H}_{n}(C^{*};G)$$\operatorname{Hom}(H^{n}(C^{*});G)$$0$$0$$\operatorname{Ext}(H^{n+1}(\varinjlim
C^{*}_{\alpha});G)$$\bar{H}_{n}(\varinjlim
C^{*}_{\alpha};G)$$\operatorname{Hom}(H^{n}(\varinjlim
C^{*}_{\alpha});G)$$0$.$\simeq$$\bar{\pi}_{n}$$\simeq$

Hence, the homomorphism $\bar{\pi}_{n}$ is an isomorphism for all
$n\in\mathbb{Z}.$ Using the exact sequence (5.16) and the isomorphism
$\bar{\pi}_{n},$ we obtain an infinite exact sequence (5.15). ∎

######  Corollary 5.6.

Let $\\{(X_{\alpha},A_{\alpha})\\}$ be an inverse system of pairs of compact
spaces $(X_{\alpha},A_{\alpha})$ and
$(X,A)=\varprojlim(X_{\alpha},A_{\alpha})$. If $\bar{H}_{*}$ is the homology
theory generated by the Alexander-Spanier cochains, then there is an infinite
exact sequence:

(5.19)
$\dots$${\varprojlim}^{(2k+1)}\bar{H}_{n+k+1}(X_{\alpha},A_{\alpha})$$\dots$${\varprojlim}^{(3)}\bar{H}_{n+2}(X_{\alpha},A_{\alpha})$${\varprojlim}^{(1)}\bar{H}_{n+1}(X_{\alpha},A_{\alpha})$$\bar{H}_{n}(X,A;G)$${\varprojlim}\bar{H}_{n}(X_{\alpha},A_{\alpha})$${\varprojlim}^{(2)}\bar{H}_{n+1}(X_{\alpha},A_{\alpha})$$\dots$${\varprojlim}^{(2k)}\bar{H}_{n+k}(X_{\alpha},A_{\alpha})$$\dots~{}~{}.$$\pi_{*}$

######  Corollary 5.7.

Let $\\{(X_{i},A_{i})\\}_{i\in\mathbb{Z}}$ be an inverse sequnce of compact
metric spaces $(X_{i},A_{i})$ and $(X,A)=\varprojlim(X_{i},A_{i})$. If
$\bar{H}_{*}$ is the homology theory generated by the Alexander-Spanier
cochains, then there is an exact sequence:

(5.20) $\footnotesize
0\longrightarrow\underset{\longleftarrow}{\lim}\,^{(1)}\bar{H}_{n+1}(X_{i},A_{i})\longrightarrow\bar{H}_{n}(X,A;G)\overset{\pi_{*}}{\longrightarrow}\underset{\longleftarrow}{\lim}\,\bar{H}_{n}(X_{i},A_{i})\longrightarrow
0.$

######  Corollary 5.8.

If $\bar{H}_{*}$ is the homology theory generated by the Alexander-Spanier
cochains, then there is an exact sequence:

(5.21) $\footnotesize
0\longrightarrow\underset{\longleftarrow}{\lim}\,^{(1)}\bar{H}_{n+1}(K_{\alpha},L_{\alpha})\longrightarrow\bar{H}_{n}(X,A;G)\overset{\pi_{*}}{\longrightarrow}\underset{\longleftarrow}{\lim}\,\bar{H}_{n}(K_{\alpha},L_{\alpha})\longrightarrow
0,$

where $(X,A)=\varprojlim(K_{\alpha},L_{\alpha})$ and $(K_{\alpha},L_{\alpha})$
are finite polyhedral pairs.

5. Let $C^{*}_{s}(X;G)$ be the singular cochain complex of topological spaces $X$ and $\bar{C}^{s}_{*}(X;G)=\operatorname{Hom}(C^{*}_{s}(X);\beta_{\\#}).$ Let $\bar{H}_{*}^{s}(X;G)$ be the homology of the obtained chain complex $\bar{C}^{s}_{*}(X;G).$ Therefore, $\bar{H}_{*}^{s}(-;G)$ is the homology generated by the singular cochain complex ${C}^{*}_{s}(-;G).$ It is known that there is a natural homomorphism $j^{\\#}:\bar{C}^{*}(X:G)\to C^{*}_{s}(X;G)$ from the Alexander-Sapnier cochain complex to the singular cochain complex, which induces the isomorphism $j^{*}:\bar{H}^{*}(X;G)\to\bar{H}^{*}_{s}(X;G)$ on the category of manifolds. Therefore, by the Universal Coefficient Formula, we will obtain that if $X$ is manifold, then there is an isomorphism:

(5.22) $\footnotesize
j_{*}:\bar{H}^{s}_{*}(X;G)\overset{\simeq}{\longrightarrow}\bar{H}_{*}(X;G).$

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