\section{Introduction}

Let $C_*$ be a chain complex  and $0 \to G \overset{\alpha}{\longrightarrow} G' \overset{\beta}{\longrightarrow} G'' \to 0 $ be an injective resolution of a $R$-module $G$ over a principal ideal domain $R$. Let $\beta^{\#}:\operatorname{Hom}(C_*;G') \to \operatorname{Hom}(C_*;G'')$ be the cochain map induced by $\beta: G' \to G''$ and  $C^*(\beta
^{\#})=\{ C^n(\beta_{\#}), \delta \}$ be the cone of the cochain map $\beta^{\#}$, i.e., 
\begin{equation}\label{eq01}
\footnotesize
C^n(\beta^{\#}) \simeq \operatorname{Hom}(C_{n};G') \oplus \operatorname{Hom}(C_{n-1};G''),
\end{equation}
\begin{equation}\label{eq02}
\footnotesize
\delta(\varphi ',\varphi '')=(\varphi ' \circ \partial, \beta \circ \varphi ' -\varphi '' \circ \partial), ~~~ \forall (\varphi ',\varphi '') \in C^n(\beta^{\#}).
\end{equation}
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 \cite{10} (in the case of free chain complexes) and \cite{12} (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:
\begin{equation}\label{eq03}
\begin{tikzpicture}
\footnotesize

\node (A) {$\dots$};
\node (B) [node distance=1.8cm, right of=A] {$ {\varprojlim} ^ {(3)}   \bar{H}_\gamma ^{n-2} $};
\node (C) [node distance=2.3cm, right of=B] {${\varprojlim} ^ {(1)}  \bar{H}_\gamma ^{n-1} $};
\node (D) [node distance=2.7cm, right of=C] {$ \bar{H}^n \left( {\varinjlim}   C^\gamma_*;G \right) $};
\node (E) [node distance=2.3cm, right of=D] {${\varprojlim}  \bar{H}_\gamma ^{n} $};
\node (F) [node distance=2cm, right of=E] {${\varprojlim} ^ {(2)}  \bar{H}_\gamma ^{n-1} $};
\node (H) [node distance=1.7cm, right of=F] {$\dots$};

\draw[->] (A) to node [above]{}(B);
\draw[->] (B) to node [above]{}(C);
\draw[->] (C) to node [above]{}(D);
\draw[->] (D) to node [above]{}(E);
\draw[->] (E) to node [above]{}(F);
\draw[->] (F) to node [above]{}(H);

\end{tikzpicture}
\end{equation}
where $\bar{H}^*_\gamma=\bar{H}^* (C_*^\gamma ;G)=H^* (C^*_\gamma (\beta_{\#})).$ Note that, using the methods developed in \cite{91} and \cite{11}, it is possible to show that if for a cohomology $H^*$ there exists a type \eqref{eq03} 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  $\beta_{\#}:\operatorname{Hom}(C^*;G') \to \operatorname{Hom}(C^*;G'')$ be the chain map induced by $\beta: G' \to G''$. Consider the cone  $C_*(\beta_{\#})=\{ C_n(\beta_{\#}), \partial \}=\{\operatorname{Hom}( C^*,\beta_{\#}), \partial \}$ of the chain map $\beta_{\#}$, i.e.,
\begin{equation}\label{eq04}
\footnotesize
C_n(\beta_{\#}) \simeq \operatorname{Hom}(C^{n};G') \oplus \operatorname{Hom}(C^{n+1};G''),
\end{equation}
\begin{equation}\label{eq05}
\footnotesize
\partial(\varphi ',\varphi '')=(\varphi ' \circ \delta, \beta \circ \varphi ' -\varphi '' \circ \delta), ~~~ \forall (\varphi ',\varphi '') \in C_n(\beta_{\#}).
\end{equation}
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 $f:C^* \longrightarrow C'^*$ is a cochain  map, then it induces the chain map $\bar{f}:C'_* (\beta_{\#}) \longrightarrow C_*(\beta_{\#})$. In particular, for each $n \in \mathbb{Z}$ the homomorphism $\bar{f}_n: C'_n (\beta_{\#}) \longrightarrow C_n (\beta_{\#})$ is defined by the formula
$\bar{f}_n(\varphi ', \varphi '')=(\varphi ' \circ f_n , \varphi \circ f_{n+1}).$  Consequently, it induces a homomorphism of homology groups $\bar{f}:\bar{H}_n(C'^*;G) \longrightarrow \bar{H}_n(C^*;G)$. 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:
\begin{equation}\label{eq06}
\footnotesize
\begin{tikzpicture}\small

\node (A) {$\dots$};
\node (B) [node distance=1.8cm, right of=A] {$ {\varprojlim} ^ {(3)}   \bar{H}^\gamma_{n+2} $};
\node (C) [node distance=2.4cm, right of=B] {${\varprojlim} ^ {(1)}  \bar{H}^\gamma_{n+1} $};
\node (D) [node distance=2.7cm, right of=C] {$ \bar{H}_n \left( {\varinjlim}   C^*_\gamma ;G \right) $};
\node (E) [node distance=2.4cm, right of=D] {${\varprojlim}  \bar{H}^\gamma_{n} $};
\node (F) [node distance=2cm, right of=E] {${\varprojlim} ^ {(2)}  \bar{H}^\gamma_{n+1} $};
\node (H) [node distance=1.8cm, right of=F] {$\dots$,};

\draw[->] (A) to node [above]{}(B);
\draw[->] (B) to node [above]{}(C);
\draw[->] (C) to node [above]{}(D);
\draw[->] (D) to node [above]{}(E);
\draw[->] (E) to node [above]{}(F);
\draw[->] (F) to node [above]{}(H);
\end{tikzpicture}
\end{equation}
where $ \bar{H}^\gamma_*=\bar{H}_* (C^*_\gamma ;G)=H_* (C_*^\gamma (\beta_{\#})).$ On the other hand, to obtain sequence \eqref{eq06}, 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):
\begin{equation}\label{eq07}
\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.
\end{equation}
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.  

\section{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 $C^*$.

\begin{theorem} [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:
	\begin{equation}\label{eq3}
	\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.
	\end{equation}
\end{theorem}
\begin{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  $\chi:\operatorname{Hom}(H^{n+1}(C^*);G'') \longrightarrow \bar{H}_n(C^*;G)$ 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 \eqref{eq3} is exact. 
	
	
	{\bf a. There is a homomorphism $\xi:\bar{Z}_n \to \operatorname{Hom}(H^n(C^*);G)$.} Let $(\varphi ' , \varphi '') \in \bar{Z}_n$ be a cycle, i.e., $\varphi' :C^n \to G'$ and $\varphi '': C^{n+1} \to G''$ are homomorphisms such that $\partial(\varphi ',\varphi '')=(\varphi ' \circ \delta, \beta \circ \varphi ' -\varphi '' \circ \delta)=(0,0)=0$ and therefore, the following diagram is commutative:
\begin{equation}\label{eq4}
\begin{tikzpicture}\footnotesize

\node (A) {$C^{n-1}$};
\node (B) [node distance=2.5cm, right of=A] {$C^n$};
\node (C) [node distance=2.5cm, right of=B] {$C^{n+1}$};

\node (B1) [node distance=1.5cm, below of=B] {$G'$};
\node (C1) [node distance=1.5cm, below of=C] {$G''$};


\draw[->] (A) to node [above]{$\delta$}(B);
\draw[->] (B) to node [above]{$\delta$}(C);

\draw[->] (B) to node [right]{$\varphi '$}(B1);
\draw[->] (C) to node [right]{$\varphi ''$}(C1);

\draw[->] (A) to node [above]{$0$}(B1);

\draw[->] (B1) to node [above]{$\beta$}(C1);

\end{tikzpicture}
\end{equation}
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 $\delta ' : C^{n-1} \to B^n$ be an epimophism induced by $\delta:C^{n-1} \to C^n$. Therefore, we have the following sequence:
\begin{equation}\label{eq5}
\footnotesize
C^{n-1} \overset{\delta '}{\longrightarrow} B^n \overset{i}{\longrightarrow} Z^n \overset{j}{\longrightarrow} C^n \overset{\delta}{\longrightarrow} C^{n+1},
\end{equation}
where $j \circ i \circ \delta ' = \delta$ and consequently $\delta \circ j =0$.  


Since $(\varphi',\varphi '') \in \bar{Z}_n$, we have the following commutative diagram:

\begin{equation}\label{eq6}
\begin{tikzpicture}
\footnotesize

\node (A) {$Z^n$};
\node (B) [node distance=2.5cm, right of=A] {$C^n$};
\node (C) [node distance=2.5cm, right of=B] {$C^{n+1}$};

\node (A1) [node distance=1.5cm, below of=A] {$G$};
\node (B1) [node distance=1.5cm, below of=B] {$G'$};
\node (C1) [node distance=1.5cm, below of=C] {$G''$};
\node (C0) [node distance=1.5cm, right of=C1] {$0~.$};
\node (A0) [node distance=1.5cm, left of=A1] {$0$};

\draw[->] (A) to node [above]{$j$}(B);
\draw[->] (B) to node [above]{$\delta$}(C);

\draw[->] (B) to node [right]{$\varphi '$}(B1);
\draw[->] (C) to node [right]{$\varphi ''$}(C1);

\draw[->] (A1) to node [above]{$\alpha$}(B1);
\draw[->] (B1) to node [above]{$\beta$}(C1);

\draw[->] (A0) to node [above]{}(A1);
\draw[->] (C1) to node [above]{}(C0);
\end{tikzpicture}
\end{equation}
Hence, $\varphi '' \circ \delta \circ j = \beta \circ \varphi ' \circ j$, and by the equality $\delta \circ j=0$, we obtain that $\beta \circ \varphi ' \circ j=0$.  So, $Im (\varphi ' \circ j) \subset Ker \beta = Im \alpha$. Therefore, there is a uniquely defined map $\varphi :Z^n \to G$ such that $\varphi ' \circ j =\alpha \circ \varphi$ (see the diagram \eqref{eq7}. 
\begin{equation}\label{eq7}
\begin{tikzpicture}
\footnotesize

\node (A) {$Z^n$};
\node (B) [node distance=2.5cm, right of=A] {$C^n$};
\node (C) [node distance=2.5cm, right of=B] {$C^{n+1}$};

\node (A1) [node distance=1.5cm, below of=A] {$G$};
\node (B1) [node distance=1.5cm, below of=B] {$G'$};
\node (C1) [node distance=1.5cm, below of=C] {$G''$};
\node (C0) [node distance=1.5cm, right of=C1] {$0~.$};
\node (A0) [node distance=1.5cm, left of=A1] {$0$};

\draw[->] (A) to node [above]{$j$}(B);
\draw[->] (B) to node [above]{$\delta$}(C);

\draw[dashed,->] (A) to node [right]{$\varphi $}(A1);
\draw[->] (B) to node [right]{$\varphi '$}(B1);
\draw[->] (C) to node [right]{$\varphi ''$}(C1);

\draw[->] (A1) to node [above]{$\alpha$}(B1);
\draw[->] (B1) to node [above]{$\beta$}(C1);

\draw[->] (A0) to node [above]{}(A1);
\draw[->] (C1) to node [above]{}(C0);
\end{tikzpicture}
\end{equation}
By the commutative diagram \eqref{eq4}, we have $\varphi ' \circ \delta=\varphi ' \circ j \circ i \circ \delta '=0$. Hence,  $ \alpha  \circ \varphi \circ i \circ \delta '   = \varphi '  \circ j \circ i \circ \delta' = \varphi ' \circ \delta=0$ (see the diagram \eqref{eq8}).  $\alpha$ is a monomorphism and so $ \varphi \circ i \circ \delta ' =0$. On the other hand, $\delta '$ 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:
\begin{equation}\label{eq8}
\begin{tikzpicture}
\footnotesize

\node (A) {$Z^n$};
\node (ABC) [node distance=1cm, right of=A] {};
\node (AA) [node distance=2cm, left of=A] {$B^n$};
\node (AB) [node distance=2cm, left of=AA] {$C^{n-1}$};
\node (B) [node distance=2.5cm, right of=A] {$C^n$};
\node (C) [node distance=2.5cm, right of=B] {$C^{n+1}$};

\node (A1) [node distance=1.5cm, below of=A] {$G$};
\node (ABC1) [node distance=0.7cm, below of=ABC] {$H^n$};
\node (B1) [node distance=1.5cm, below of=B] {$G'$};
\node (C1) [node distance=1.5cm, below of=C] {$G''$};
\node (C0) [node distance=1.5cm, right of=C1] {$0~,$};
\node (A0) [node distance=1.5cm, left of=A1] {$0$};

\draw[->] (AB) to node [above]{$\delta '$}(AA);
\draw[->] (AA) to node [above]{$i$}(A);
\draw[->] (A) to node [above]{$j$}(B);
\draw[->] (B) to node [above]{$\delta$}(C);

\draw[->] (A) to node [right]{$\varphi $}(A1);
\draw[->] (B) to node [right]{$\varphi '$}(B1);
\draw[->] (C) to node [right]{$\varphi ''$}(C1);
\draw[dashed,->] (AA) to node [right]{$0$}(A1);

\draw[->] (A1) to node [above]{$\alpha$}(B1);
\draw[->] (B1) to node [above]{$\beta$}(C1);

\draw[->] (A) to node [right]{$p$}(ABC1);
\draw[dashed,->] (ABC1) to node [right]{$\bar{\varphi}$}(A1);

\draw[->] (A0) to node [above]{}(A1);
\draw[->] (C1) to node [above]{}(C0);
\end{tikzpicture}
\end{equation}
where $H^n = H^n(C^*).$ Let $\xi :\bar{Z}_n \to \operatorname{Hom}(H^n;G)$ be the homomorphism defined by
\begin{equation}\label{eq9}
\footnotesize
\xi(\varphi ', \varphi '')=\bar{\varphi}, ~~\forall (\varphi ', \varphi '') \in \bar{Z}_n.
\end{equation}

{\bf 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 $\varphi ' :C^n \to G'$  be an extension of  $\alpha \circ \varphi  : Z^n \to G'$. In this case $\varphi ' \circ j = \alpha \circ \varphi$ and so $\beta \circ \varphi ' \circ j  =\beta \circ  \alpha \circ \varphi =0.$ Therefore, $\beta  \circ \varphi ' : C^n \to G''$ vanishes on the subgroup $Z^n$ and so it induces a homomorphism $\tilde{\varphi} '' :C^n/Z^n \simeq B^{n+1} \to G''$, which can be extended to a homomophism $\varphi '' : C^{n+1} \to G''$.  Since $\tilde{\varphi}'' \circ \delta '= \beta \circ \varphi '$, there is $\partial (\varphi ', \varphi '') = (\varphi ' \circ \delta ,\beta \circ \varphi ' - \varphi '' \circ \delta)=(\varphi ' \circ j \circ i \circ \delta' ,\beta \circ \varphi ' - \varphi '' \circ j \circ i \circ \delta')= (\alpha \circ \varphi \circ i \circ \delta' ,\beta \circ \varphi ' -  \tilde{\varphi}'' \circ \delta ' )=$ $(\alpha \circ \bar{\varphi} \circ p \circ i \circ \delta' , \beta \circ \varphi ' - \beta \circ \varphi ')=(0,0)=0$. Hence,  $(\varphi ', \varphi '') \in \bar{Z}_n$ and $\bar{\xi}(\varphi ', \varphi '')=\bar{\varphi}$ (see the diagram \eqref{eq10}).

\begin{equation}\label{eq10}
\begin{tikzpicture}
\footnotesize

\node (A) {$B^n$};
\node (1A) [node distance=1.5cm, left of=A] {$C^{n-1}$};
\node (AA) [node distance=2.6cm, right of=A] {};
\node (B) [node distance=1.5cm, right of=A] {$Z^n$};
\node (C) [node distance=2cm, right of=B] {$C^n$};
\node (D) [node distance=2.2cm, right of=C] {$C^n/Z^n \simeq B^{n+1}$};
\node (E) [node distance=2.5cm, right of=D] {$C^{n+1}$};

\node (AA1) [node distance=0.8cm, below of=AA] {$H^n$};


\node (A1) [node distance=1.5cm, below of=A] {$0$};
\node (B1) [node distance=1.5cm, below of=B] {$G$};
\node (C1) [node distance=1.5cm, below of=C] {$G'$};
\node (D1) [node distance=1.5cm, below of=D] {$G''$};
\node (E1) [node distance=1.5cm, below of=E] {$0.$};

\draw[->] (AA1) to node [above]{$\bar{\varphi}$}(B1);
\draw[->] (B) to node [above]{$p$}(AA1);
\draw[->] (1A) to node [above]{$\delta '$}(A);
\draw[->] (A) to node [above]{$i$}(B);

\draw[->] (B) to node [above]{$j $}(C);
\draw[->] (C) to node [above]{$\delta' $}(D);
\draw[->] (D) to node [above]{$j \circ i$}(E);

\draw[->] (A1) to node [above]{$ $}(B1);
\draw[->] (B1) to node [above]{$\alpha$}(C1);
\draw[->] (C1) to node [above]{$\beta $}(D1);
\draw[->] (D1) to node [above]{$ $}(E1);

\draw[->] (B) to node [right]{$\varphi $}(B1);
\draw[dashed, ->] (C) to node [right]{$\varphi '$}(C1);
\draw[dashed, ->] (D) to node [right]{$\tilde{\varphi} ''$}(D1);
\draw[dashed, ->] (E) to node [right]{$\varphi ''$}(D1);

\end{tikzpicture}
\end{equation}
{\bf 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 $(\psi ' , \psi '' ) \in C_{n+1}(\beta_\#)$ be an element. For $\partial (\psi ' , \psi '')=(\psi ' \circ \delta , \beta \circ \psi ' - \psi '' \circ \delta)=(\varphi ', \varphi '') \in \bar{B}_n \subset \bar{Z}_n$ we have $\varphi ' \circ j =0$. Indeed, $\varphi '  \circ j=\psi ' \circ  \delta \circ j=0$. Therefore, by the construction $\xi$, the homomorphism $\varphi : Z^n \to G$ corresponding to the pair $(\varphi ', \varphi '')$ satisfies the equation $\alpha \circ \varphi = \varphi ' \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 ' , \psi '') = \xi (\varphi ' , \varphi '') = \bar{\varphi}=0 $  (see the diagram \eqref{eq11}).
\begin{equation}\label{eq11}
\begin{tikzpicture}
\footnotesize
\node (A) {$Z^n$};
\node (AA) [node distance=1cm, right of=A] {};
\node (B) [node distance=2.5cm, right of=A] {$C^n$};
\node (C) [node distance=2.5cm, right of=B] {$C^{n+1}$};
\node (D) [node distance=2.5cm, right of=C] {$C^{n+2}$};

\node (A1) [node distance=1.5cm, below of=A] {$G$};
\node (AA1) [node distance=0.8cm, below of=AA] {$H^n$};
\node (B1) [node distance=1.5cm, below of=B] {$G'$};
\node (C1) [node distance=1.5cm, below of=C] {$G''$};
\node (C0) [node distance=1.5cm, right of=C1] {$0~.$};
\node (A0) [node distance=1.5cm, left of=A1] {$0$};

\draw[->] (A) to node [above]{$p$}(AA1);
\draw[->] (AA1) to node [above]{$\bar{\varphi}$}(A1);
\draw[->] (A) to node [above]{$j$}(B);
\draw[->] (B) to node [above]{$\delta$}(C);
\draw[->] (C) to node [above]{$\delta$}(D);

\draw[->] (C) to node [right]{$\psi ' $}(B1);
\draw[->] (D) to node [right]{$\psi '' $}(C1);

\draw[->] (A) to node [right]{$\varphi $}(A1);
\draw[->] (B) to node [right]{$\varphi '$}(B1);
\draw[->] (C) to node [right]{$\varphi ''$}(C1);

\draw[->] (A1) to node [above]{$\alpha$}(B1);
\draw[->] (B1) to node [above]{$\beta$}(C1);

\draw[->] (A0) to node [above]{}(A1);
\draw[->] (C1) to node [above]{}(C0);
\end{tikzpicture}
\end{equation}

{\bf 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 $0 \to G \overset{\alpha}{\longrightarrow} G' \overset{\beta}{\longrightarrow}  G'' \to 0$, then we obtain:
\begin{equation}\label{eq03}
\begin{tikzpicture}
\footnotesize

\node (A) {$0$};
\node (B) [node distance=2.5cm, right of=A] {$ \operatorname{Hom}(H^{n+1}(C^*);G)$};
\node (C) [node distance=4cm, right of=B] {$\operatorname{Hom}(H^{n+1}(C^*);G')$};
\node (D) [node distance=2.7cm, right of=C] {};
\node (A1) [node distance=1.5cm, below of=A] {};
\node (B1) [node distance=1.5cm, below of=B] {$\operatorname{Hom}(H^{n+1}(C^*);G'')$};
\node (C1) [node distance=1.5cm, below of=C] {$\operatorname{Ext}(H^{n+1}(C^*);G)$};
\node (D1) [node distance=1.5cm, below of=D] {$0~.$};

\draw[->] (A) to node [above]{}(B);
\draw[->] (B) to node [above]{$\alpha_*$}(C);
\draw[->] (C) to node [above]{$\beta_*$}(D);
\draw[->] (A1) to node [above]{$\beta_*$}(B1);
\draw[->] (B1) to node [above]{}(C1);
\draw[->] (C1) to node [above]{}(D1);
\end{tikzpicture}
\end{equation}

Therefore, we have the following isomorphism:
\begin{equation}\label{eq13}
\footnotesize
\operatorname{Ext}(H^{n+1}(C^*);G) \simeq \operatorname{Hom}(H^{n+1}(C^*);G'')/Im \beta_*.
\end{equation}
Our aim is to define such a homomorpism $\chi : \operatorname{Hom}(H^{n+1}(C^*);G'') \longrightarrow \bar{H}_n(C^*;G) $ that the following sequence is exact:
\begin{equation}\label{eq14}
\begin{tikzpicture}
\footnotesize

\node (A) {};
\node (B) [node distance=2cm, right of=A] {$ \operatorname{Hom}(H^{n+1}(C^*);G')$};
\node (C) [node distance=4cm, right of=B] {$ \operatorname{Hom}(H^{n+1}(C^*);G'')$};
\node (D) [node distance=2.5cm, right of=C] {};

\node (A1) [node distance=1.5cm, below of=A] {};
\node (B1) [node distance=1.5cm, below of=B] {$\bar{H}_n(C^*;G)$};
\node (C1) [node distance=1.5cm, below of=C] {$ \operatorname{Hom}(H^{n}(C^*);G)$};
\node (D1) [node distance=1.5cm, below of=D] {$0~.$};


\draw[->] (B) to node [above]{$\beta_*$}(C);
\draw[->] (C) to node [above]{$\chi$}(D);
\draw[->] (A1) to node [above]{$\chi$}(B1);
\draw[->] (B1) to node [above]{$\bar{\xi}$}(C1);
\draw[->] (C1) to node [above]{}(D1);
\end{tikzpicture}
\end{equation}
Indeed, in this case, it is clear that for the homomorphisms $\bar{\xi}$, $\chi$ and $\beta_*$, we have the following short exact sequences:
\begin{equation}\label{eq15}
\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,
\end{equation}
\begin{equation}\label{eq16}
\footnotesize
0 \longrightarrow Ker \chi \overset{}{\longrightarrow} \operatorname{Hom}(H^{n+1}(C^*);G'') \overset{\chi}{\longrightarrow} Im \chi \overset{}{\longrightarrow}  0,
\end{equation}
\begin{equation}\label{eq17}
\footnotesize
0 \longrightarrow Ker \beta_* \overset{}{\longrightarrow} \operatorname{Hom}(H^{n+1}(C^*);G') \overset{\beta_*}{\longrightarrow} Im \beta_* \overset{}{\longrightarrow}  0.
\end{equation}
On the other hand, if we prove exactness of the sequence  \eqref{eq14}, then $Ker \bar{\xi} \simeq Im \chi$ and $Ker \chi \simeq Im \beta_*$. Therefore, we have:
\begin{equation}\label{eq18}
\begin{tikzpicture}
\footnotesize

\node (A) {$Ker \bar{\xi} \simeq Im \chi \simeq  \operatorname{Hom} (H^{n+1}(C^*);G'')/Ker \chi \simeq$};


\node (A1) [node distance=0.8cm, below of=A] {$\simeq   \operatorname{Hom} (H^{n+1}(C^*);G'')/Im \beta_* \simeq \operatorname{Ext}(H^{n+1}(C^*);G).$};

\end{tikzpicture}
\end{equation}
To define $\chi$, consider an element $\bar{\varphi}'' \in \operatorname{Hom}(H^{n+1}(C^*);G'')$. Let $\varphi '' :C^{n+1} \longrightarrow G''$ be an extension of the composition $\bar{\varphi}'' \circ p : Z^{n+1} \longrightarrow G''$, where $p:Z^{n+1} \longrightarrow H^{n+1}(C^*)$ is a natural projection. In this case,  $\varphi'' \circ \delta =\varphi'' \circ    j \circ i \circ \delta ' = \tilde{\varphi}'' \circ p \circ i \circ \delta '  =0$ and so, if we take $\varphi ' =0$, then the following diagram is commutative:
\begin{equation}\label{eq19}
\begin{tikzpicture}
\footnotesize

\node (A) {$C^{n-1}$};
\node (B) [node distance=2.5cm, right of=A] {$C^n$};
\node (C) [node distance=1.5cm, right of=B] {$B^{n+1}$};
\node (D) [node distance=1.5cm, right of=C] {$Z^{n+1}$};
\node (E) [node distance=1.5cm, right of=D] {$C^{n+1}$};

\node (B1) [node distance=1.5cm, below of=B] {$G'$};
\node (D1) [node distance=0.8cm, below of=D] {$H^{n+1}$};
\node (E1) [node distance=1.5cm, below of=E] {$G''$ .};


\draw[->] (D) to node [left]{$p$}(D1);
\draw[->] (D1) to node [right]{$\bar{\varphi}''$}(E1);

\draw[->] (A) to node [above]{$\delta$}(B);
\draw[->] (B) to node [above]{$\delta '$}(C);
\draw[->] (C) to node [above]{$i$}(D);
\draw[->] (D) to node [above]{$j$}(E);


\draw[->] (B) to node [right]{$\varphi '=0$}(B1);
\draw[->] (E) to node [right]{$\varphi ''$}(E1);

\draw[->] (A) to node [above]{$0$}(B1);

\draw[->] (B1) to node [above]{$\beta$}(E1);

\end{tikzpicture}
\end{equation}
Therefore, $(0,\varphi '') \in \bar{Z}_n$ and so, we can define $\chi$  in the following way:
\begin{equation}\label{eq20}
\chi(\bar{\varphi}'')= (0,- \varphi '')+\bar{B}_n,  ~~~\forall~ \bar{\varphi}'' \in \operatorname{Hom}(H^{n+1}(C^*);G'').
\end{equation}
Let check that $\chi$ is well defined. Consider two different extensions $\varphi _1 ''$ and $\varphi _2 ''$ of the map $\bar{\varphi}'' \circ p :Z^{n+1} \longrightarrow G''$ and show that $(0, -\varphi_1 '')+\bar{B}_n=(0, -\varphi_2 '')+\bar{B}_n$. For this, we have to show that $(0, \varphi _2 '' - \varphi _1 '') \in \bar{B}_n$. Indeed, by the definition of $\varphi _1 ''$ and $\varphi _2 ''$, it is clear that $(\varphi _1 '' - \varphi _2 '') \circ j=\varphi _1 '' \circ j - \varphi _2 '' \circ j = \bar{\varphi}'' \circ p -\bar{\varphi}'' \circ p=0$ and so, $\varphi _1 '' - \varphi _2 ''$ induces a homomorphism $\psi : C^{n+1}/Z^{n+1} \longrightarrow G''$. On the other hand, $C^{n+1}/Z^{n+1} \simeq B^{n+2}$ and so, we have an extension $\psi '': C^{n+2} \longrightarrow G''$ of $\psi$ (see the diagran \eqref{eq261}). 
\begin{equation}\label{eq261}
\begin{tikzpicture}
\footnotesize

\node (A) {$Z^{n+1}$};
\node (B) [node distance=2.5cm, right of=A] {$C^{n+1}$};
\node (C) [node distance=4cm, right of=B] {$C^{n+1}/Z^{n+1} \simeq B^{n+2}$};
\node (D) [node distance=3.5cm, right of=C] {$C^{n+2}$};


\node (B1) [node distance=1.5cm, below of=B] {$G''$};


\draw[->] (A) to node [above]{$j$}(B);
\draw[->] (B) to node [above]{$\delta '$}(C);
\draw[->] (C) to node [above]{$j \circ i$}(D);

\draw[->] (A) to node [right]{$0$}(B1);
\draw[->] (B) to node [right]{$\varphi_1 ' - \varphi_2 ''$}(B1);
\draw[->] (C) to node [above]{$\psi $}(B1);
\draw[->] (D) to node [above]{$\psi ''$}(B1);

\end{tikzpicture}
\end{equation}
In this case, it is easy to see that
\begin{equation}\label{eq18}
\footnotesize
\partial(0, \psi '')=(0, -\psi '' \circ \delta ) =(0, - \psi '' \circ j \circ i \circ \delta ' )=(0, -\psi  \circ \delta ' )=(0, \varphi _2 '' -\varphi _1 '').
\end{equation}
Therefore, it remains to show that $Im \chi \simeq Ker \bar{\xi}$ and $Im \beta _* = Ker \chi$.

 {\bf d$_1$. $Im \chi \simeq Ker \bar{\xi}$.} Let $\bar{\varphi}'' \in \operatorname{Hom}(H^{n+1}(C^*);G'')$ be an element, then $\bar{\xi} \left(\chi (\bar{\varphi}'')\right)= \bar{\xi} \left((0, -\varphi '')+\bar{B}_{n}\right)=\bar{\varphi}$. On the other hand, by construction of $\xi$ and the fact that the first coordinate of the pair $(0,-\varphi '')$ 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 $(\varphi ', \varphi '') \in \bar{Z_n}$. In this case, by the definition of $\bar{\xi}$, there exists $\varphi :Z^n \to G$ such that the following diagram is commutative:
\begin{equation}\label{eq22}
\begin{tikzpicture}
\footnotesize
\node (A) {$Z^n$};
\node (AA) [node distance=0.75cm, right of=A] {};
\node (B) [node distance=2.5cm, right of=A] {$C^n$};
\node (C) [node distance=2.5cm, right of=B] {$C^{n+1}$};

\node (A1) [node distance=1.5cm, below of=A] {$G$};
\node (AA1) [node distance=0.75cm, below of=AA] {$H^n$};
\node (B1) [node distance=1.5cm, below of=B] {$G'$};
\node (C1) [node distance=1.5cm, below of=C] {$G''$};
\node (C0) [node distance=1.5cm, right of=C1] {$0~.$};
\node (A0) [node distance=1.5cm, left of=A1] {$0$};

\draw[->] (A) to node [above]{$p$}(AA1);
\draw[dashed,->] (AA1) to node [above]{$\bar{\varphi}$}(A1);

\draw[->] (A) to node [above]{$j$}(B);
\draw[->] (B) to node [above]{$\delta$}(C);

\draw[dashed,->] (A) to node [left]{$\varphi $}(A1);
\draw[->] (B) to node [right]{$\varphi '$}(B1);
\draw[->] (C) to node [right]{$\varphi ''$}(C1);

\draw[->] (A1) to node [above]{$\alpha$}(B1);
\draw[->] (B1) to node [above]{$\beta$}(C1);

\draw[->] (A0) to node [above]{}(A1);
\draw[->] (C1) to node [above]{}(C0);
\end{tikzpicture}
\end{equation}
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 $\varphi ' \circ j = \alpha \circ \varphi =0$. Consequently, $\varphi ' :C^n \to G'$ induces a homomorphism $ \tilde{\psi} ' : C^n / Z^n \simeq B^{n+1} \to G'$. Let $\psi ' : C^{n+1} \to G'$ be an extension of $ \tilde{\psi}' \circ p :Z^{n+1} \to G'$ and $\psi '' = \beta \circ \psi '$ (see the diagram \eqref{eq23}).  In this case, the homomorphism $\psi = \psi '' - \varphi ''  : C^{n+1} \to G''$ vanishes on the $B^{n+1}$. Indeed,  $\psi \circ j \circ i \circ \delta ' = \psi '' \circ j \circ i \circ \delta ' -\varphi '' \circ j \circ i \circ \delta '  =\beta \circ \tilde{\psi}' \circ \delta ' -  \beta \circ \varphi ' = \beta \circ \varphi '-\beta \circ \varphi '=0$. On the other hand, $\delta '$ is an epimorphim and so $\psi \circ j \circ i =0$. Therefore,  $\psi \circ j : Z ^{n+1} \to G''$ induces a homomorphism $\bar{\psi}: H^{n+1} \to G''$ (see the diagram \eqref{eq23}).
 \begin{equation}\label{eq23}
\begin{tikzpicture}
\footnotesize

\node (A) {$Z^n$};
\node (B) [node distance=3cm, right of=A] {$C^n$};
\node (C) [node distance=2.5cm, right of=B] {$ B^{n+1}$};
\node (D) [node distance=2cm, right of=C] {$Z^{n+1}$};
\node (E) [node distance=1.5cm, right of=D] {$H^{n+1}$};
\node (F) [node distance=1cm, right of=E] {};
\node (F1) [node distance=1cm, below of=D] {$C^{n+1}$};

\node (A1) [node distance=2cm, below of=A] {$G$};
\node (B1) [node distance=2cm, below of=B] {$G'$};
\node (E1) [node distance=2cm, below of=E] {$G''$};
\node (E0) [node distance=2cm, right of=E1] {$0~.$};
\node (A0) [node distance=2cm, left of=A1] {$0$};

\draw[->] (D) to node [left]{$j$}(F1);
\draw[->] (F1) to node [above]{$\psi ''$}(E1);
\draw[<-] (9+0.3-0.75,-2.2-0.2+0.5) to node [left]{$\varphi ''$} (7.5+0.3,-1.2-0.2);

\draw[->] (A) to node [above]{$j$}(B);
\draw[->] (B) to node [above]{$\delta '$}(C);
\draw[->] (C) to node [above]{$i$}(D);
\draw[->] (D) to node [above]{$p$}(E);

\draw[dashed,->] (A) to node [right]{$\varphi $}(A1);
\draw[->] (B) to node [right]{$\varphi '$}(B1);
\draw[->] (E) to node [right]{$\bar{\psi}$}(E1);

\draw[->] (C) to node [above]{$\tilde {\psi} '$}(B1);
\draw[->] (F1) to node [above]{$ {\psi} '$}(B1);

\draw[->] (A1) to node [above]{$\alpha$}(B1);
\draw[->] (B1) to node [above]{$\beta$}(E1);

\draw[->] (A0) to node [above]{}(A1);
\draw[->] (E1) to node [above]{}(E0);
\end{tikzpicture}
\end{equation}
Our aim is to show that $\chi(\bar{\psi})=\bar{h}=(\varphi ', \varphi '')+\bar{B}_n$. 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 $(\varphi ', \varphi '')-(0, -\psi)=(\varphi ', \varphi '' +\psi)=(\varphi ', \psi '') \in \bar{B}_n$. Indeed,
\begin{equation}\label{eq24}
\footnotesize
\partial(\psi ', 0)=(\psi ' \circ \delta, \beta \circ \psi ')=(\psi ' \circ j \circ i \circ \delta ', \beta \circ \psi ')=(\tilde{\psi}' \circ \delta ', \psi '')=(\varphi ', \psi '').  
\end{equation}

{\bf d$_2$. $Im \beta_* \simeq Ker \chi$.} Let $\bar{\varphi}' \in \operatorname{Hom}(H^{n+1}(C^*);G')$ be an element and $\varphi '' :C^{n+1} \to G''$ be an extension of the composition $\beta \circ \bar{\varphi}' \circ p :Z^{n+1} \to G''$. In this case we have $\partial(0,-\varphi '')=(0,\varphi '' \circ \delta)=(0,\varphi '' \circ j \circ i \circ \delta')= (0,\beta \circ  \bar{\varphi} ' \circ p \circ i \circ \delta')=(0,0)=0$ (see the diagram \eqref{eq26}). Therefore, $(0, -\varphi '') \in \bar{Z}_n$ and so we have
\begin{equation}\label{eq25}
\footnotesize
\left(\chi \circ \beta_* \right) \left(\bar{\varphi}'\right) =\chi \left( \beta_* \left(\bar{\varphi}'\right) \right) =\chi \left( \beta \circ  \bar{\varphi }' \right) =
\left(0, -\varphi'' \right)+\bar{B}_n.  
\end{equation}
Our aim is to show that $\left(0, -\varphi'' \right) \in \bar{B}_n$. Indeed, let $\varphi ' :C^{n+1} \longrightarrow G'$ be an extension of the composition $\bar{\varphi}' \circ p : Z^{n+1} \longrightarrow G'$. In this case $( \beta \circ \varphi ' -\varphi '') \circ j= \beta \circ \varphi ' \circ j - \varphi '' \circ j =\beta \circ \bar{\varphi}' \circ p - \beta \circ \bar{\varphi}' \circ p =0$ and so $ \beta \circ \varphi ' -\varphi '' :C^{n+1} \longrightarrow G'' $  induces a homomorphism $\tilde{\psi}'': C^{n+1}/Z^{n+1}\simeq B^{n+2} \longrightarrow G''$  such that $\beta \circ \varphi ' - \varphi '' =\tilde{\psi}'' \circ \delta '$. Let $\psi '' :C^{n+2} \longrightarrow G''$ be an extension of a homomorphism $\tilde{\psi}':B^{n+2} \longrightarrow G''$ (see the diagram \eqref{eq26} ). 
\begin{equation}\label{eq26}
\begin{tikzpicture}
\footnotesize

\node (A) {$Z^{n+1}$};
\node (AA) [node distance=1.5cm, left of=A] {$B^{n+1}$};
\node (AAA) [node distance=1.5cm, left of=AA] {$C^n$};
\node (B) [node distance=2cm, right of=A] {$C^{n+1}$};
\node (CC) [node distance=3.5cm, right of=B] {$C^{n+1}/Z^{n+1} \simeq B^{n+2}$};
\node (C) [node distance=3.5cm, right of=CC] {$C^{n+2}$};

\node (AA1) [node distance=0.75cm, below of=A] {$H^{n+1}$};


\node (B1) [node distance=1.5cm, below of=B] {$G'$};
\node (C1) [node distance=1.5cm, below of=C] {$G''$};

\draw[->] (AAA) to node [above]{$\delta '$}(AA);
\draw[->] (AA) to node [above]{$i$}(A);
\draw[->] (A) to node [above]{$j$}(B);
\draw[->] (B) to node [above]{$\delta '$}(CC);
\draw[->] (CC) to node [above]{$j \circ i$}(C);

\draw[->] (A) to node [right]{$p$}(AA1);
\draw[->] (AA1) to node [above]{$\bar{\varphi}'$}(B1);
\draw[dashed, ->] (B) to node [right]{$\varphi '$}(B1);
\draw[dashed, ->] (C) to node [right]{$\psi ''$}(C1);

\draw[dashed, ->] (B) to node [above]{$ \varphi '' $}(C1);
\draw[dashed,->] (CC) to node [above]{$\tilde {\psi} ''$}(C1);

\draw[->] (B1) to node [above]{$\beta$}(C1);

\end{tikzpicture}
\end{equation}
In this case, we have
\begin{equation}\label{eq27}
\begin{tikzpicture}
\footnotesize

\node (A) {$\partial (-\varphi',-\psi'')=(-\varphi' \circ \delta,  - \beta \circ \varphi' + \psi '' \circ \delta )=(-\varphi' \circ j \circ i \circ \delta',   - \beta \circ \varphi' + \tilde{\psi} '' \circ \delta' )=$};


\node (A1) [node distance=0.8cm, below of=A] {$=(-\bar{\varphi}' \circ p \circ i \circ \delta',  - \beta \circ \varphi' +(\beta \circ \varphi' -\varphi '') )=(0, - \varphi '').$};

\end{tikzpicture}
\end{equation}
Therefore, $(0,-\varphi'')\in \bar{B}_n$ and so, $\chi \circ \beta_*=0$. Hence, $Im \beta_* \subset \operatorname{Ker} \chi$. Now consider an element $\bar{\varphi}'' \in Ker \chi$. Let $\varphi '' :C^{n+1} \to G''$ be an extension of the composition $\bar{\varphi}'' \circ p :Z^{n+1} \to G''$ (see the diagram \eqref{eq28}). Then, by $\chi(\bar{\varphi}'')=(0, -\varphi '')+\bar{B}_n=0$, there exists $(\psi',\psi'') \in C_{n+1}(\beta_\#)$ such that 
\begin{equation}\label{eq27}
\footnotesize
\partial (\psi',\psi'')=(\psi' \circ \delta,  \beta \circ \psi' - \psi '' \circ \delta )=(0, - \varphi ''). 
\end{equation}
Therefore, $\psi ' \circ \delta =\psi ' \circ j \circ i \circ \delta ' = 0$. Since $\delta ':C^n \to B^{n+1}$ is an epimorphism, we have $\psi ' \circ j \circ i=0$ and  so $\psi ' \circ j :Z^{n+1} \to G'$ induces a homomorphism $\bar{\psi}' : H^{n+1}(C^*) \to G'$. On the other hand, by $  \beta \circ \psi' - \psi '' \circ \delta =  -\varphi ''$, we have $ - \varphi '' \circ j  = \beta \circ \psi' \circ j - \psi '' \circ \delta \circ j =\beta \circ \psi' \circ j$ (see the diagram \eqref{eq28}). Therefore, $\beta_*(\bar{\psi}')=\bar{\varphi}''$ and so, $ Ker \chi \subset Im \beta_*:$ 
\begin{equation}\label{eq28}
\begin{tikzpicture}
\footnotesize
\node (A) {$C^{n+1}$};
\node (B) [node distance=1.5cm, right of=A] {};
\node (C) [node distance=2.5cm, right of=B] {$C^{n+2}$};

\node (B1) [node distance=0.75cm, below of=B] {$Z^{n+1}$};
\node (B2) [node distance=1.5cm, below of=B] {$H^{n+1}$};

\node (A1) [node distance=2.25cm, below of=A] {$G'$};
\node (C1) [node distance=2.25cm, below of=C] {$G''$};


\draw[->] (A) to node [above]{$\delta $}(C);
\draw[->] (A1) to node [above]{$\beta $}(C1);

\draw[->] (B1) to node [above]{$j$}(A);
\draw[->] (B1) to node [right]{$p$}(B2);
\draw[->] (B2) to node [above]{$\bar{\psi}'$}(A1);
\draw[->] (A) to node [right]{$\psi'$}(A1);
\draw[->] (C) to node [right]{$\psi''$}(C1);
\draw[->] (B2) to node [above]{$\bar{\varphi} ''$}(C1);

\end{tikzpicture}
\end{equation}
\end{proof}


Since for each injective abelian group $G$, a group of extensions $\operatorname{Ext}(-;G)$ is trivial, by the exact sequence \eqref{eq3} we obtain the following corollary (cf. Lemma VII.4.4 \cite{71})

\begin{corollary}
	If $G$ is an injective, then there is an isomorphism
	\begin{equation}\label{eq2.0}
	\footnotesize
	\bar{H}_n(C^*;G) \simeq \operatorname{Hom}(H^n(C^*);G).
	\end{equation}
\end{corollary}

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:
\begin{equation}\label{eq2.1}
\footnotesize
\alpha_*(\varphi)=(\alpha \circ \varphi, 0), ~~~ \forall \varphi \in \operatorname{Hom}(C^n;G).
\end{equation}
Let $H_n(C^*;G)$ be a homology group of chain complex $\operatorname{Hom}(C^*;G)$.
\begin{theorem}
	If a cochain complex $C^*$ is free, then the homomorphism $\alpha_*:\operatorname{Hom}(C^*;G) \longrightarrow \operatorname{Hom}(C^*;\beta_\#)$ induces an isomoprhism
	\begin{equation}\label{eq2.2}
	\footnotesize
	\bar{\alpha}_*:H_n(C^*;G) \longrightarrow \bar{H}_n(C^*;G).
	\end{equation}
\end{theorem}  
\begin{proof}
	Since $C^*$ is a free cochain complex, there is a short exact sequence:
	\begin{equation}\label{eq2.3} \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.
	\end{equation}
	Let review how the morphisms $\tilde{\xi}$ and $\tilde{\chi}$ are defined according to W. Massey's \cite{71} 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.\\
	{\bf 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 
	\begin{equation}\label{eq2.4} \footnotesize
	\tilde{\xi}(\bar{\varphi})(\bar{c})= \langle \varphi, c \rangle =\varphi(c), ~~~~\forall~c \in \bar{c}, ~\bar{c} \in H^n(C^*),
	\end{equation}
	where $\varphi$ is a representative of $\bar{\varphi}$ \cite{8}.\\ 
	{\bf 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 \eqref{eq13}. Consequently, the homomorphism $\tilde{\chi}$ is the homomorphism induced by  $\chi_0: \operatorname{Hom}(H^{n+1}(C^*);G'') \longrightarrow H_n(\operatorname{Hom}(C^*;G)),$ where $\chi_0$ is defined in the following way.  Let $\bar{\varphi}'' \in \operatorname{Hom}(H^{n+1}(C^*);G'') $ be any element. Since $G''$ is an injective, there is  an extension $\varphi '' :C^{n+1} \longrightarrow G''$ of the composition $\bar{\varphi}'' \circ p :Z^{n+1} \longrightarrow G''.$  In this case $\partial(\varphi '')= \varphi '' \circ \delta= \varphi '' \circ j \circ i \circ \delta ' =\bar{\varphi}'' \circ p \circ i \circ \delta ' =0$  (see diagram \eqref{eq41.1}). 
	\begin{equation}\label{eq41.1} \footnotesize
	\begin{tikzpicture}
	\footnotesize
	
	\node (A) {$C^{n-1}$};
	\node (B) [node distance=1.8cm, right of=A] {$C^n$};
	\node (C) [node distance=1.8cm, right of=B] {$B^{n+1}$};
	\node (D) [node distance=1.8cm, right of=C] {$Z^{n+1}$};
	\node (E) [node distance=1.5cm, right of=D] {$C^{n+1}$};
	\node (F) [node distance=1cm, right of=E] {};
	\node (F1) [node distance=0.75cm, below of=D] {$H^{n+1}$};
	
	\node (A1) [node distance=1.5cm, below of=A] {$G$};
	\node (B1) [node distance=1.5cm, below of=B] {$G'$};
	\node (E1) [node distance=1.5cm, below of=E] {$G''$};
	\node (E0) [node distance=1.5cm, right of=E1] {$0~.$};
	\node (A0) [node distance=1.5cm, left of=A1] {$0$};
	
	\draw[->] (D) to node [left]{$p$}(F1);
	\draw[->] (F1) to node [above]{$\bar{\varphi}''$}(E1);
	
	\draw[->] (A) to node [above]{$j$}(B);
	\draw[->] (B) to node [above]{$\delta '$}(C);
	\draw[->] (C) to node [above]{$i$}(D);
	\draw[->] (D) to node [above]{$j$}(E);
	
	\draw[dashed,->] (B) to node [right]{$\varphi $}(A1);
	\draw[dashed, ->] (B) to node [right]{$\varphi '$}(B1);
	\draw[->] (E) to node [right]{$\varphi ''$}(E1);
	
	\draw[dashed, ->] (E) to node [above]{$ {\psi} '$}(B1);
	
	\draw[->] (A1) to node [above]{$\alpha$}(B1);
	\draw[->] (B1) to node [above]{$\beta$}(E1);
	
	\draw[->] (A0) to node [above]{}(A1);
	\draw[->] (E1) to node [above]{}(E0);
	\end{tikzpicture}
	\end{equation}	
	Therefore, it defines homology class $[\varphi''] \in H_{n+1}(\operatorname{Hom}(C^*;G''))$. 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:
	\begin{equation}\label{eq2.5} \footnotesize
	0 \longrightarrow \operatorname{Hom}(C^*;G) \overset{\alpha_\#}{\longrightarrow} \operatorname{Hom}(C^*;G') \overset{\beta_{\#}}{\longrightarrow} \operatorname{Hom}(C^*;G'') \longrightarrow 0.
	\end{equation}
	Define a homomorphism $\chi_0 $ by the formula
	\begin{equation}\label{eq2.6} \footnotesize
	\chi_0\left(\bar{\varphi}''\right)=E\left([{\varphi}'']\right),~~~\forall~\bar{\varphi}'' \in \operatorname{Hom}(H^{n+1}(C^*);G'').
	\end{equation}
	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 $\psi':C^{n+1} \to G'$ that $\beta \circ \psi ' =\varphi ''$ (this is possible, because the cochain complex $C^*$ is free). Let $\varphi ' = \psi ' \circ \delta :C^n \to G'$, then we have $\beta \circ \psi ' = \varphi '' \circ \delta = 0$. Therefore, $\varphi ' \in Ker \beta = Im \alpha$ and so, there exists a unique map $\varphi :C^n \to G $ such that $\alpha \circ \varphi = \varphi '$. In this case, we have $ \alpha \circ \varphi  \circ \delta = \varphi ' \circ \delta = \psi ' \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 \eqref{eq41.1}). Finally, by the formula \eqref{eq2.6} we have
	\begin{equation}\label{eq2.61} \footnotesize
	\chi_0\left(\bar{\varphi}''\right)=E\left([{\varphi}'']\right)=[\varphi],~~~\forall~\bar{\varphi}'' \in \operatorname{Hom}(H^{n+1}(C^*);G'').
	\end{equation}  
	

	In this case, the sequence \eqref{eq2.3} is induced by the following sequence
	\begin{equation}\label{eq2.7}
	\begin{tikzpicture}
	\footnotesize
	
	\node (A) {};
	\node (B) [node distance=2cm, right of=A] {$ \operatorname{Hom}(H^{n+1}(C^*);G')$};
	\node (C) [node distance=4cm, right of=B] {$ \operatorname{Hom}(H^{n+1}(C^*);G'')$};
	\node (D) [node distance=2.5cm, right of=C] {};
	
	\node (A1) [node distance=1.5cm, below of=A] {};
	\node (B1) [node distance=1.5cm, below of=B] {$\bar{H}_n(C^*;G)$};
	\node (C1) [node distance=1.5cm, below of=C] {$ \operatorname{Hom}(H^{n}(C^*);G)$};
	\node (D1) [node distance=1.5cm, below of=D] {$0~.$};
	
	
	\draw[->] (B) to node [above]{$\beta_*$}(C);
	\draw[->] (C) to node [above]{$\chi _0$}(D);
	\draw[->] (A1) to node [above]{$\chi _0$}(B1);
	\draw[->] (B1) to node [above]{$\bar{\xi}$}(C1);
	\draw[->] (C1) to node [above]{}(D1);
	\end{tikzpicture}
	\end{equation}
	Therefore, by \eqref{eq14} and \eqref{eq2.7}, it is sufficient to show that the following diagram is commutative:
	\begin{equation}\label{eq2.8}
	\begin{tikzpicture}\footnotesize
	
	\node (A) {$\operatorname{Hom}(H^{n+1}(C^*);G')$};
	\node (B) [node distance=3.8cm, right of=A] {$\operatorname{Hom}(H^{n+1}(C^*);G'')$};
	c\node (C) [node distance=3.7cm, right of=B] {$ H_n(\operatorname{Hom}(C^*;G))$};
	\node (D) [node distance=3.4cm, right of=C] {$\operatorname{Hom}(H^n(C^*);G)$};
	\node (E) [node distance=2cm, right of=D] {$0$};
	
	\node (A1) [node distance=1.5cm, below of=A] {$\operatorname{Hom}(H^{n+1}(C^*);G')$};
	\node (B1) [node distance=1.5cm, below of=B] {$\operatorname{Hom}(H^{n+1}(C^*);G'')$};
	\node (C1) [node distance=1.5cm, below of=C] {$ \bar{H}_n(\operatorname{Hom}(C^*;\beta_{\#}))$};
	\node (D1) [node distance=1.5cm, below of=D] {$\operatorname{Hom}(H^n(C^*);G)$};
	\node (E1) [node distance=1.5cm, below of=E] {$0$.};
	
	
	\draw[->] (A) to node [above]{$\beta_*$}(B);
	\draw[->] (B) to node [above]{$\chi_0$}(C);
	\draw[->] (C) to node [above]{$\tilde{\xi}$}(D);
	\draw[->] (D) to node [above]{}(E);
	
	\draw[->] (A1) to node [above]{$\beta_*$}(B1);
	\draw[->] (B1) to node [above]{$\chi$}(C1);
	\draw[->] (C1) to node [above]{$\bar{\xi}$}(D1);
	\draw[->] (D1) to node [above]{}(E1);
	
	\draw[->] (A) to node [right]{$1$}(A1);
	\draw[->] (B) to node [right]{$1$}(B1);
	\draw[->] (C) to node [right]{$\bar{\alpha}_*$}(C1);
	\draw[->] (D) to node [right]{$1$}(D1);
	\end{tikzpicture}
	\end{equation}
	Indeed, let $\bar{\varphi}'' \in \operatorname{Hom}(H^{n+1}(C^*);G'')$ be an element and $\varphi '' :C^{n+1} \longrightarrow G''$ be an extension of the composition $\bar{\varphi}'' \circ p :Z^{n+1} \longrightarrow G''.$ Then, by definition of $\chi_0$ and method of snake lemma, we must take an element $\varphi' \in \operatorname{Hom}(C^{n+1};G')$, such that $\beta_{\#}(\varphi')=\varphi ''$. Note that this is possible because of exactness of the sequence \eqref{eq2.5}. Then, there is a cycle $\varphi \in \operatorname{Hom}(C^n;G)$, such that $\alpha_\#(\varphi)=\partial(\varphi').$ Let $[\varphi]=\varphi+B_n$ be the corresponding element in the homology group $H_n(\operatorname{Hom}(C^*;G)),$ then $\chi_0(\bar{\varphi}'')=[\varphi].$ By the definition of the map $\tilde{\alpha}:H_n(\operatorname{Hom}(C^*);G) \longrightarrow \bar{H}_n(\operatorname{Hom}(C^*);\beta_{\#})$, we have
	\begin{equation}\label{eq2.10} \footnotesize
	\left(\tilde{\alpha}_* \circ \chi_0 \right) \left(\bar{\varphi}'' \right)=\tilde{\alpha}_* \left( \chi_0  \left(\bar{\varphi}'' \right) \right)=\tilde{\alpha}_* \left([\varphi] \right)=\left(\alpha \circ \varphi, 0 \right)+\bar{B}_n.
	\end{equation}
	On the other hand, by the definition of $\chi: \operatorname{Hom}(G^{n+1}(C^*);G'') \longrightarrow \bar{H}_n(\operatorname{Hom}(C^*);\beta_{\#}),$ we have
	\begin{equation}\label{eq2.11} \footnotesize
	\chi(\bar{\varphi}'')=(0, - \varphi '')+\bar{B}_n,  ~~~\forall~ \bar{\varphi}'' \in \operatorname{Hom}(H^{n+1};G'').
	\end{equation}
	Therefore, we have to show that $(\alpha \circ \varphi , 0)-(0, - \varphi'')=(\alpha \circ \varphi,  \varphi '') \in \bar{B_n}.$ Indeed, by the equality $\alpha_\#(\varphi)=\partial(\varphi ')$ and $\beta_{\#}(\varphi')=\varphi''$, we have $\alpha \circ \varphi=\varphi ' \circ \delta$ and $\beta \circ \varphi'=\varphi''$. Therefore,
	\begin{equation}\label{eq2.12} \footnotesize
	\partial (\varphi ', 0)=(\varphi ' \circ \delta , \beta \circ \varphi ')=(\alpha \circ \varphi, \varphi '').
	\end{equation}
	By \eqref{eq2.10}, \eqref{eq2.11}, and \eqref{eq2.12}, 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
	\begin{equation}\label{eq2.13} \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.
	\end{equation}
	Therefore, if we take an element $\bar{c} \in H^n(C^*)$ and  any of its representatives $c \in \bar{c}$, then  by \eqref{eq2.13} we have	
	\begin{equation}\label{eq2.14} \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).
	\end{equation}
	Therefore, by \eqref{eq2.4} , \eqref{eq2.13} and \eqref{eq2.14}, we obtain that
	\begin{equation}\label{eq2.15} \footnotesize
	\bar{\xi} \circ \tilde{\alpha}= \tilde{\xi}.
	\end{equation}
\end{proof}

Note that by the commutative diagram \eqref{eq2.8}, we obtain the following commutative diagram:
\begin{equation}\label{eq2.16}
\begin{tikzpicture}
\footnotesize
\node (A) {$0$};
\node (B) [node distance=2cm, right of=A] {$\operatorname{Ext}(H^{n+1}(C^*);G)$};
c\node (C) [node distance=3.5cm, right of=B] {$ H_n(\operatorname{Hom}(C^*;G))$};
\node (D) [node distance=3.5cm, right of=C] {$\operatorname{Hom}(H^n(C^*);G)$};
\node (E) [node distance=2cm, right of=D] {$0$};

\node (A1) [node distance=1.5cm, below of=A] {$0$};
\node (B1) [node distance=1.5cm, below of=B] {$\operatorname{Ext}(H^{n+1}(C^*);G)$};
\node (C1) [node distance=1.5cm, below of=C] {$ \bar{H}_n(C^*;G)$};
\node (D1) [node distance=1.5cm, below of=D] {$\operatorname{Hom}(H^n(C^*);G)$};
\node (E1) [node distance=1.5cm, below of=E] {$0$.};

\draw[->] (A) to node [above]{}(B);
\draw[->] (B) to node [above]{$\tilde{\chi}$}(C);
\draw[->] (C) to node [above]{$\tilde{\xi}$}(D);
\draw[->] (D) to node [above]{}(E);

\draw[->] (A1) to node [above]{}(B1);
\draw[->] (B1) to node [above]{$\bar{\chi}$}(C1);
\draw[->] (C1) to node [above]{$\bar{\xi}$}(D1);
\draw[->] (D1) to node [above]{}(E1);


\draw[->] (B) to node [right]{$1$}(B1);
\draw[->] (C) to node [right]{$\bar{\alpha}_*$}(C1);
\draw[->] (D) to node [right]{$1$}(D1);
\end{tikzpicture}
\end{equation}

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.  


\section{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:
\begin{equation}\label{eq32}
\footnotesize
\bar{\xi}:\bar{H}_n(C^*;G) \longrightarrow \operatorname{Hom} (H^n(C^*);G)
\end{equation}
and the following diagram is commutative:
\begin{equation}\label{eq33}
\begin{tikzpicture}
\footnotesize
\node (A) {$\bar{Z}_n$};
\node (B) [node distance=2cm, right of=A] {};
\node (C) [node distance=2cm, right of=B] {$\operatorname{Hom}(H^n(C^*);G)$};
\node (B') [node distance=1.5cm, below of=B] {$\bar{H}_n(C^*;G)$.};


\draw[->] (A) to node [above]{$\xi$}(C);
\draw[->] (A) to node [above]{$\bar{p}$}(B');
\draw[->] (B') to node [above]{$\bar{\xi}$}(C);
\end{tikzpicture}
\end{equation}
To investigate $Ker \xi$ we construct a homomorphism
\begin{equation}\label{eq34}
\footnotesize
\omega:\operatorname{Hom}(C^{n+1};G') \oplus \operatorname{Hom}(C^{n+1}/B^{n+1};G'') \longrightarrow Ker \xi
\end{equation}
by $\omega(\psi', \psi '')=(\psi' \circ \delta, \beta \circ \psi' -\psi '' \circ q),$ where $q:C^{n+1} \longrightarrow C^{n+1}/B^{n+1}$ is the quotient  map. Let show that $\omega(\psi',\psi'') \in Ker \xi$. Indeed,  $\partial \omega(\psi',\psi'')= \partial (\psi' \circ \delta, \beta \circ \psi' -\psi '' \circ q)=(\psi \circ \delta \circ \delta, \beta \circ \psi' \circ \delta -(\beta \circ \psi ' - \psi'' \circ  q)\circ \delta)=(0,\beta \circ \psi' \circ \delta - \beta \circ \psi ' \circ \delta + \psi'' \circ  q \circ \delta )=(0,\psi'' \circ  q  \circ \delta )=(0,0),$ because $q \circ \delta=0.$ Hence, $\omega (\psi ', \psi '') \in \bar{Z}_n$. 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  $\alpha \circ \varphi = \psi' \circ \delta \circ j$. 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 $\left(\omega(\psi ', \psi '')\right)=(\psi' \circ \delta, \beta \circ \psi' -\psi '' \circ q)=\bar{\varphi}=0$. Therefore, we obtain that $\omega(\psi ', \psi '') \in Ker \xi.$

\begin{lemma} For each integer $n \in \mathbb{N}$, there exists the following short exact sequence 
	\begin{equation}\label{eq36} \footnotesize
	\begin{tikzpicture}\footnotesize
	
	\node (A) {$0$};
	\node (B) [node distance=2.4cm, right of=A] {$\operatorname{Hom}(C^{n+1}/B^{n+1};G')$};
	\node (C) [node distance=5.5cm, right of=B] {$\operatorname{Hom}(C^{n+1};G') \oplus \operatorname{Hom}(C^{n+1}/B^{n+1};G'')$};
	\node (D) [node distance=4.2cm, right of=C] {$Ker \xi$};
	\node (E) [node distance=1.3cm, right of=D] {$0,$};
	
	
	\draw[->] (A) to node [above]{}(B);
	\draw[->] (B) to node [above]{$\sigma$}(C);
	\draw[->] (C) to node [above]{$\omega$}(D);
	\draw[->] (D) to node [above]{}(E);
	
	\end{tikzpicture}
	\end{equation}
	where $\sigma: \operatorname{Hom}(C^{n+1}/B^{n+1};G') \longrightarrow \operatorname{Hom}(C^{n+1};G') \oplus \operatorname{Hom}(C^{n+1}/B^{n+1};G'')$ is defined by the formula
	\begin{equation}\label{eq35} \footnotesize
	\sigma(\varphi)=(\varphi \circ q, \beta \circ \varphi), ~~~\forall \varphi \in \operatorname{Hom}(C^{n+1}/B^{n+1};G').
	\end{equation} 
\end{lemma}

\begin{proof}
	{\bf a. $\omega$ is an epimorphism.} If $(\varphi ' , \varphi '') \in Ker \xi,$ then $\xi(\varphi ' , \varphi '')=\bar{\varphi}=0$ and so $\varphi:Z^n \longrightarrow G$ is zero as well. On the other hand, $\alpha \circ \varphi =\varphi ' \circ j=0$. Therefore, there is a unique homomorphism $\tilde{\varphi}':B^{n+1} \longrightarrow G'$, such that $\varphi ' = \tilde{\varphi}' \circ \delta '$. Let $\psi ' : C^{n+1} \longrightarrow G'$ be an extension of the map $\tilde{\varphi} ' : B^{n+1} \longrightarrow G'$ (see the diagram \eqref{eq37}).
	\begin{equation}\label{eq37}
	\begin{tikzpicture}
	\footnotesize
	\node (A) {};
	\node (B) [node distance=1.5cm, right of=A] {$Z^n$};
	\node (C) [node distance=2cm, right of=B] {$C^n$};
	\node (D) [node distance=2cm, right of=C] {$B^{n+1}$};
	\node (E) [node distance=2cm, right of=D] {$C^{n+1}$};
	\node (F) [node distance=1.5cm, right of=E] {};
	
	\draw[->] (B) to node [above]{$j$}(C);
	\draw[->] (C) to node [above]{$\delta '$}(D);
	\draw[->] (D) to node [above]{$j \circ i$}(E);
	
	\node (A1) [node distance=1.5cm, below of=A] {$0$};
	\node (B1) [node distance=1.5cm, below of=B] {$G$};
	\node (C1) [node distance=1.5cm, below of=C] {$G'$};

	\node (E1) [node distance=1.5cm, below of=E] {$G''$};
	\node (F1) [node distance=1.5cm, below of=F] {$0$.};
	
	\draw[->] (A1) to node [above]{}(B1);
	\draw[->] (B1) to node [above]{$\alpha$}(C1);
	\draw[->] (C1) to node [above]{$\beta$}(E1);
	\draw[->] (E1) to node [above]{}(F1);
	
	
	\draw[->] (B) to node [left]{$\varphi$}(B1);
	\draw[->] (C) to node [left]{$\varphi '$}(C1);
	\draw[dashed, ->] (D) to node [above]{$\bar{\varphi}'$}(C1);
	\draw[dashed,->] (E) to node [above]{$\psi '$}(C1);
	\draw[->] (E) to node [right]{$\varphi ''$}(E1);
	\end{tikzpicture}
	\end{equation}
	If we consider the map $\beta \circ \psi ' - \varphi '': C^{n+1} \longrightarrow G'',$ then by $(\varphi ', \varphi '') \in \bar{Z}_n,$ we have $(\beta \circ \psi ' - \varphi'') \circ \delta = \beta \circ \psi' \circ \delta - \varphi '' \circ \delta = \beta \circ \varphi ' - \varphi '' \circ \delta=0$.  Since $(\beta \circ \psi ' - \varphi'') \circ \delta = (\beta \circ \psi ' - \varphi '') \circ j \circ i \circ \delta ' =0$ and $\delta '$ is an epimorphism, there is $(\beta \circ \psi ' - \psi'') \circ j \circ i =0.$ Therefore, there is a homomorphism $\psi '' :C^{n+1}/B^{n+1} \longrightarrow G''$ such that $\beta \circ \psi ' -\varphi '' =\psi '' \circ q$ and so $\varphi '' = \beta \circ \psi ' - \psi '' \circ q$ (see the diagram \eqref{eq38}).
	
	\begin{equation}\label{eq38}
	\begin{tikzpicture}
	\footnotesize
	\node (A) {};
	\node (B) [node distance=2cm, right of=A] {$B^{n+1}$};
	\node (C) [node distance=2cm, right of=B] {$C^{n+1}$};
	\node (D) [node distance=2cm, right of=C] {$C^{n+1}/B^{n+1}$};
	\node (E) [node distance=2cm, right of=D] {$0$};
	
	
	\draw[->] (B) to node [above]{$j \circ i$}(C);
	\draw[->] (C) to node [above]{$q$}(D);
	\draw[->] (D) to node [above]{}(E);
	
	\node (C1) [node distance=1.5cm, below of=C] {$G''$};
	
	\draw[->] (C) to node [left]{$\beta \circ \psi '-\varphi ''$}(C1);
	\draw[dashed, ->] (D) to node [above]{$\psi''$}(C1);
	\end{tikzpicture}
	\end{equation}
	Hence, $(\psi ' , \psi'') \in \operatorname{Hom}(C^{n+1};G') \oplus \operatorname{Hom}(C^{n+1}/B^{n+1};G'')$ and $\omega(\psi ', \psi'')=(\psi ' \circ \delta, \beta \circ \psi ' - \psi '' \circ q)=(\varphi ', \varphi '')$. So,  $\omega$ is an epimorphism.
	
	{\bf 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 $(\psi', \psi'')  \in Ker \omega,$ then  $\omega(\psi ', \psi '')=(\psi ' \circ \delta, \beta \circ \psi' - \psi '' \circ q)=0$ and so, $\psi ' \circ \delta =0$ and $\beta \circ \psi' =\psi '' \circ q.$ On the other hand, $\psi ' \circ \delta = \psi ' \circ j \circ i \circ \delta '=0$. Therefore,  we have $\psi' \circ j \circ i =0,$ because $\delta'$ is an epimorphism. So, there is a unique homomorphism $\varphi:C^{n+1}/B^{n+1} \longrightarrow G'$ such that $\psi'=\varphi \circ q.$ In this case, $\beta \circ \varphi \circ q=\beta \circ \psi' = \psi'' \circ q$ and since $q$ is an epimorphism, $\beta \circ \varphi=\psi''.$ Therefore, $\sigma(\varphi)=(\psi', \psi'')$ and so, $Ker \omega \subset  Im \sigma.$
	
	{\bf 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 $\varphi=0$.
	
	
\end{proof}



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. 


\begin{lemma}
	For each direct system $\mathbf{C}^*=\{C^*_\gamma \}$  of cochain complexes, there is an isomorphism
	\begin{equation}\label{eq39} \footnotesize
	\operatorname{Hom}(\varinjlim C^*_\gamma  ; \beta^\#) \simeq \varprojlim \operatorname{Hom}(C^*_\gamma;\beta^\#).
	\end{equation}
\end{lemma} 

\begin{proof}
	Consider a chain complex
	\begin{equation}\label{eq40} \footnotesize
	\operatorname{Hom}(\varinjlim C^*_\gamma;\beta^\#)=\{\operatorname{Hom}(\varinjlim C^n_\gamma; G') \oplus \operatorname{Hom}(\varinjlim C^{n+1}_\gamma;G''), \partial \},
	\end{equation} 
	where $\partial (\varphi ', \varphi '')= (\varphi ' \circ \delta , \beta \circ \varphi ' - \varphi '' \circ \delta ).$ 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
	\begin{equation}\label{eq41}	\footnotesize
	\operatorname{Hom}(\varinjlim C_\gamma^n;G') \oplus \operatorname{Hom}(\varinjlim C_\gamma^{n+1};G'') \simeq \varprojlim \operatorname{Hom}(C_\gamma^n;G') \oplus \varprojlim \operatorname{Hom}(C_\gamma^{n+1};G'').
	\end{equation}
\end{proof}

\begin{lemma}
	If $f^\#:C^* \longrightarrow C'^*$ is a homomorphism of cochain complexes, then there is a commutative diagram:
	\begin{equation}\label{eq42}	
	\begin{tikzpicture}\footnotesize
	
	\node (A) {$0$};
	\node (B) [node distance=2.5cm, right of=A] {$\operatorname{Hom}(C'^{n+1}/B'^{n+1};G')$};
	\node (C) [node distance=5.7cm, right of=B] {$\operatorname{Hom}(C'^{n+1};G') \oplus \operatorname{Hom}(C'^{n+1}/B'^{n+1};G'')$};
	\node (D) [node distance=4.5cm, right of=C] {$Ker \xi'$};
	\node (E) [node distance=1.3cm, right of=D] {$0$};
	
	\draw[->] (A) to node [left]{}(B);
	\draw[->] (B) to node [above]{$\tau '$}(C);
	\draw[->] (C) to node [above]{$\mu '$}(D);
	\draw[->] (D) to node [left]{}(E);
	
	\node (A1) [node distance=1.5cm, below of=A] {$0$};
	\node (B1) [node distance=1.5cm, below of=B] {$\operatorname{Hom}(C^{n+1}/B^{n+1};G')$};
	\node (C1) [node distance=1.5cm, below of=C] {$\operatorname{Hom}(C^{n+1};G') \oplus \operatorname{Hom}(C^{n+1}/B^{n+1};G'')$};
	\node (D1) [node distance=1.5cm, below of=D] {$Ker \xi$};
	\node (E1) [node distance=1.5cm, below of=E] {$0.$};
	
	\draw[->] (A1) to node [left]{}(B1);
	\draw[->] (B1) to node [above]{$\tau$}(C1);
	\draw[->] (C1) to node [above]{$\mu$}(D1);
	\draw[->] (D1) to node [left]{}(E1);
	
	\draw[->] (B) to node [left]{$\tilde{f}_\#$}(B1);
	\draw[->] (C) to node [left]{$(f_\#, \tilde{f}_\#)$}(C1);
	\draw[->] (D) to node [left]{$\tilde{f}$}(D1);
	
	\end{tikzpicture}
	\end{equation}
\end{lemma}
\begin{proof} Note that homomorphisms $\tilde{f}_{\#}:\operatorname{Hom}(C'^{n+1}/B'^{n+1};G' ) \longrightarrow \operatorname{Hom}(C^{n+1}/B^{n+1};G')$	and $(f_{\#},\tilde{f}_{\#}):\operatorname{Hom}(C'^{n+1};G') \oplus \operatorname{Hom}(C'^{n+1}/B'^{n+1};G'') \longrightarrow \operatorname{Hom}(C^{n+1};G') \oplus \operatorname{Hom}(C^{n+1}/B^{n+1};G'')$ are naturally defined by $\tilde{f}_\#(\varphi') = \varphi' \circ \tilde{f}_{n+1}$ and $(f_\#, \tilde{f}_\#)(\varphi ',\varphi '')=(\varphi ' \circ f_{n+1}, \varphi '' \circ \tilde{f}_{n+1}),$ where $\tilde{f}_{n+1}:C^{n+1}/B^{n+1} \longrightarrow C'^{n+1}/B'^{n+1}$ is induced by $f_{n+1}:C^{n+1} \longrightarrow C'^{n+1}.$
	
	{\bf a. $\tau \circ \tilde{f}_\# = (f_\#, \tilde{f}_\#) \circ \tau '$.} By the definition, we have $\left(\tau \circ \tilde{f}_\# \right)(\varphi ')=\tau \left( \tilde{f}_\# (\varphi ') \right)=\tau \left(\varphi ' \circ \tilde{f}_{n+1}\right)=\left(\varphi ' \circ \tilde{f}_{n+1} \circ q, \beta \circ \varphi ' \circ \tilde{f}_{n+1} \right) $ and $\left((f_\#,\tilde{f}_\#) \circ \tau ' \right) \left(\varphi '\right) = \left(f_\#,\tilde{f}_\#\right) \left( \tau ' \left(\varphi '\right) \right)=\left(f_\#,\tilde{f}_\#\right) \left(\varphi ' \circ q',\beta \circ \varphi'\right)=(\varphi ' \circ  q' \circ f_{n+1},\beta \circ  \varphi ' \circ \tilde{f}_{n+1}).$  Since $\tilde{f}_{n+1} \circ q =q' \circ f_{n+1},$ we have $\varphi ' \circ \tilde{f}_{n+1} \circ q = \varphi ' \circ q' \circ f_{n+1}.$ Hence, $\tau \circ \tilde{f}_\# = (f_\#, \tilde{f}_\#) \circ \tau '$.  
	
	{\bf b. $\mu \circ (f_\#, \tilde{f}_\#)=\tilde{f} \circ \mu '$.} By the definition, we have $\left(\mu \circ (f_\#, \tilde{f}_\#)\right)(\varphi ' , \varphi '' )=\mu \left((f_\#, \tilde{f}_\#)(\varphi ', \varphi '')\right)= \mu \left(\varphi ' \circ f_{n+1}, \varphi '' \circ \tilde{f}_{n+1}\right)=(\varphi ' \circ f_{n+1} \circ \delta, \beta \circ \varphi ' \circ f_{n+1}- \varphi'' \circ \tilde{f}_{n+1} \circ q )$ and $\left( \tilde{f} \circ \mu ' \right)(\varphi ', \varphi '')= \tilde{f} \left( \mu ' (\varphi ', \varphi '')\right)=\tilde{f} \left(\varphi ' \circ \delta ', \beta \circ \varphi' - \varphi '' \circ q' \right)=(\varphi ' \circ \delta ' \circ f_n, \beta \circ \varphi ' \circ f_{n+1}-\varphi '' \circ  q' \circ f_{n+1}).$ Since $\tilde{f}_{n+1} \circ q= q' \circ f_{n+1}$ and $\delta ' \circ f_n = f_{n+1} \circ \delta,$ there are qualities  $ \varphi ' \circ f_{n+1} \circ \delta = \varphi ' \circ \delta ' \circ f_n$ and  $\varphi '' \circ \tilde{f}_{n+1} \circ q= \varphi '' \circ q' \circ f_{n+1}.$ Hence, $\mu \circ (f_\#, \tilde{f}_\#)=\tilde{f} \circ \mu '$.
\end{proof}
Let $\{\operatorname{Hom}(C_\gamma^{n+1};G') \oplus \operatorname{Hom}(C_\gamma^{n+1}/B_\gamma^{n+1}; G'') \}$ 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
\begin{equation}\label{eq43}
\begin{tikzpicture}\footnotesize

\node (A) {$0$};
\node (B) [node distance=2cm, right of=A] {$\operatorname{Hom}(C^{n+1}_\gamma;G')$};
\node (C) [node distance=5cm, right of=B] {$\operatorname{Hom}(C^{n+1}_\gamma;G') \oplus \operatorname{Hom}(C^{n+1}_\gamma/B^{n+1}_\gamma;G'')$};
\node (D) [node distance=5.4cm, right of=C] {$\operatorname{Hom}(C^{n+1}_\gamma/B^{n+1}_\gamma;G'')$};
\node (E) [node distance=2.5cm, right of=D] {$0$.};

\draw[->] (A) to node [above]{}(B);
\draw[->] (B) to node [above]{$\tau$}(C);
\draw[->] (C) to node [above]{$\mu$}(D);
\draw[->] (D) to node [above]{}(E);

\end{tikzpicture}
\end{equation}
Hence, by the main property of the derived functors $\varprojlim ^{(i)}$ there is a long exact sequence:
\begin{equation}\label{eq44}
\begin{tikzpicture}\footnotesize

\node (A) {$\dots$};
\node (B) [node distance=2.5cm, right of=A] {$\varprojlim ^{(i)} \operatorname{Hom}(C^{n+1}_\gamma;G')$};
\node (C) [node distance=6cm, right of=B] {$\varprojlim ^{(i)} \left( \operatorname{Hom}(C^{n+1}_\gamma;G') \oplus \operatorname{Hom}(C^{n+1}_\gamma/B^{n+1}_\gamma;G'')\right)$};
\node (D) [node distance=3.5cm, right of=C] {};
\node (E) [node distance=1.5cm, right of=D] {};
\node (B1) [node distance=1.5cm, below of=B] {};
\node (C1) [node distance=1.5cm, below of=C] {$\varprojlim ^{(i)} \operatorname{Hom}(C^{n+1}_\gamma/B^{n+1}_\gamma;G'')$};
\node (D1) [node distance=1.5cm, below of=D] {$\dots$.};

\draw[->] (A) to node [above]{}(B);
\draw[->] (B) to node [above]{$\tilde{\tau}$}(C);
\draw[->] (C) to node [above]{$\tilde{\mu}$}(E);
\draw[->] (B1)+(3,0) to node [above]{$\tilde{\mu}$}(C1);
\draw[->] (C1) to node [above]{}(D1);
\end{tikzpicture}
\end{equation}
On the other hand, since for each injective abelian groups  $G$,  $\varprojlim ^{(i)} \{\operatorname{Hom}(C^{n+1};G)\}=0,$ $i\ge 1$ (see Lemma 1.3 \cite{6}), we obtain the following result. 
\begin{corollary}
	For each injective abelian groups $G'$ and $G''$, there is the following equality
	\begin{equation}\label{eq45} \footnotesize
	{\varprojlim} ^ {(i)} \left( \operatorname{Hom}(C^{n+1}_\gamma;G') \oplus \operatorname{Hom}(C^{n+1}_\gamma/B^{n+1}_\gamma;G'') \right)=0, ~~ i\ge 1.
	\end{equation}
\end{corollary}

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

\begin{lemma} 
	For each integer $i \ge1$, there is an equality
	\begin{equation}\label{eq46} \footnotesize
	{\varprojlim} ^ {(i)} Ker \xi_\gamma = 0.
	\end{equation}
\end{lemma}
\begin{proof}
	By Lemma 1, for each $\xi_\gamma$, there is  a short exact sequence
	\begin{equation}\label{eq47}
	\begin{tikzpicture}\footnotesize
	
	\node (A) {$0$};
	\node (B) [node distance=2.3cm, right of=A] {$\operatorname{Hom}(C^{n+1}_\gamma/B^{n+1}_\gamma;G')$};
	\node (C) [node distance=5.4cm, right of=B] {$\operatorname{Hom}(C^{n+1}_\gamma;G') \oplus \operatorname{Hom}(C^{n+1}_\gamma/B^{n+1}_\gamma;G'')$};
	\node (D) [node distance=4.2cm, right of=C] {$Ker \xi_\gamma$};
	\node (E) [node distance=1.2cm, right of=D] {$0$.};
	
	
	\draw[->] (A) to node [above]{}(B);
	\draw[->] (B) to node [above]{$\sigma_\gamma$}(C);
	\draw[->] (C) to node [above]{$\omega_\gamma$}(D);
	\draw[->] (D) to node [above]{}(E);
	
	\end{tikzpicture}
	\end{equation}
	By the main property of a derived functor $\varprojlim ^{(i)}$, there is a long exact sequence	
	
	\begin{equation}\label{eq48}
	\begin{tikzpicture}\footnotesize
	
	\node (A) {$\dots$};
	\node (B) [node distance=2.9cm, right of=A] {$ \varprojlim ^{(i)} \operatorname{Hom}(C^{n+1}_\gamma/B^{n+1}_\gamma;G')$};
	\node (C) [node distance=6.4cm, right of=B] {$\varprojlim ^{(i)} \left( \operatorname{Hom}(C^{n+1}_\gamma;G') \oplus \operatorname{Hom}(C^{n+1}_\gamma/B^{n+1}_\gamma;G'') \right)$};
	\node (D) [node distance=5.2cm, right of=C] {$ \varprojlim ^{(i)}  \operatorname{Ker} \xi_\gamma $};
	\node (E) [node distance=1.7cm, right of=D] {$\dots$.};
	
	\draw[->] (A) to node [above]{}(B);
	\draw[->] (B) to node [above]{}(C);
	\draw[->] (C) to node [above]{}(D);
	\draw[->] (D) to node [above]{}(E);
	
	\end{tikzpicture}
	\end{equation}
	By Lemma 1.3 \cite{6} for each $i \ge 0,$ there is an equality ${\varprojlim} ^ {(i)} \operatorname{Hom}(C_\gamma^{n+1}/B_\gamma^{n+1};G') = 0$ and by Corollary 2,  for each $i \ge 1$ we obtain
	\begin{equation}\label{eq49} \footnotesize
	{\varprojlim} ^ {(i)} \left(\operatorname{Hom} (C_\gamma^{n+1};G') \oplus \operatorname{Hom}(C_\gamma^{n+1}/B_\gamma^{n+1};G'') \right)= 0.
	\end{equation}
	Hence, by the long exact sequence \eqref{eq48} we obtain that ${\varprojlim} ^ {(i)}  Ker \xi_\gamma =0$, $i\ge 1$.
\end{proof}
\begin{corollary}
	For each integer $i\ge1,$ there is an isomorphism
	\begin{equation}\label{eq50} \footnotesize
	{\varprojlim} ^ {(i)} \bar{Z}_n^\gamma \simeq  {\varprojlim} ^ {(i)} \operatorname{Hom}(H^n(C^*_\gamma) ;G).
	\end{equation}
\end{corollary}
\begin{proof}
	By {\bf 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
	\begin{equation}\label{eq51}
	\begin{tikzpicture}
	\footnotesize
	\node (A) {$0$};
	\node (B) [node distance=2cm, right of=A] {$Ker \xi_\gamma$};
	\node (C) [node distance=1.5cm, right of=B] {$\bar{Z}_n^\gamma$};
	\node (D) [node distance=2.5cm, right of=C] {$\operatorname{Hom}(H^n(C^*_\gamma);G)$};
	\node (E) [node distance=2.5cm, right of=D] {$0$.};
	
	\draw[->] (A) to node [above]{}(B);
	\draw[->] (B) to node [above]{}(C);
	\draw[->] (C) to node [above]{$\xi_\gamma$}(D);
	\draw[->] (D) to node [above]{}(E);
	
	\end{tikzpicture}
	\end{equation}
	Consequently, it induces the following long exact sequence
	\begin{equation}\label{eq52}
	\begin{tikzpicture}
	\footnotesize
	\node (A) {$\dots$};
	\node (B) [node distance=2cm, right of=A] {$ {\varprojlim} ^ {(i)}  Ker \xi_\gamma $};
	\node (C) [node distance=2.4cm, right of=B] {${\varprojlim} ^ {(i)} \bar{Z}_n^\gamma $};
	\node (D) [node distance=3.4cm, right of=C] {$ {\varprojlim} ^ {(i)} \operatorname{Hom}(H^n(C^*_\gamma);G) $};
	\node (E) [node distance=3.9cm, right of=D] {${\varprojlim} ^ {(i+1)}  Ker \xi_\gamma $};
	\node (F) [node distance=2.2cm, right of=E] {$\dots$.};
	
	\draw[->] (A) to node [above]{}(B);
	\draw[->] (B) to node [above]{}(C);
	\draw[->] (C) to node [above]{}(D);
	\draw[->] (D) to node [above]{}(E);
	\draw[->] (E) to node [above]{}(F);
	
	\end{tikzpicture}
	\end{equation}
	On the other hand, by Lemma 4,  ${\varprojlim} ^ {(i)} \{Ker \xi_\gamma \}= 0, ~~ i\ge 1.$ Therefore, for $i\ge 1,$ we have  an isomorphism ${\varprojlim} ^ {(i)} \bar{Z}_n^\gamma \simeq  {\varprojlim} ^ {(i)} \operatorname{Hom}(H^n(C^*_\gamma) ;G).$
\end{proof}

Note that for each $\gamma$, there is a natural commutative triangle
\begin{equation}\label{eq53}
\begin{tikzpicture}
\footnotesize
\node (A) {$\bar{Z}_n^\gamma$};
\node (B) [node distance=2cm, right of=A] {};
\node (C) [node distance=2cm, right of=B] {$\operatorname{Hom}(H^n(C^*_\gamma);G)$};
\node (B') [node distance=1.5cm, below of=B] {$\bar{H}_n(C^*_\gamma ; G)$.};


\draw[->] (A) to node [above]{$\xi_\gamma$}(C);
\draw[->] (A) to node [above]{$\bar{p}_\gamma$}(B');
\draw[->] (B') to node [above]{$\bar{\xi}_\gamma$}(C);
\end{tikzpicture}
\end{equation}
Therefore, if we take ${\varprojlim} ^ {(i)}$ of this diagram, then by Corollary 2, we obtain the following result. 

\begin{corollary}
	For each integer $i\ge 1$, ${\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.
\end{corollary}

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

\begin{theorem}
	For each integer $i\ge 0$, there is a short exact sequence:
	\begin{equation}\label{eq54}
	\begin{tikzpicture}\footnotesize
	
	\node (A) {$0$};
	\node (B) [node distance=2.5cm, right of=A] {${\varprojlim} ^ {(i)} \operatorname{Ext}(H^{n+1}(C^*_\gamma);G) $};
	\node (C) [node distance=4cm, right of=B] {${\varprojlim} ^ {(i)} \bar{H}_n(C^*_\gamma;G)  $};
	\node (D) [node distance=4cm, right of=C] {${\varprojlim} ^ {(i)} \operatorname{Hom}(H^n(C^*_\gamma);G) $};
	\node (E) [node distance=2.5cm, right of=D] {$0,$};
	
	\draw[->] (A) to node [above]{}(B);
	\draw[->] (B) to node [above]{}(C);
	\draw[->] (C) to node [above]{$\xi_\gamma$}(D);
	\draw[->] (D) to node [above]{}(E);
	
	\end{tikzpicture}
	\end{equation}
	and this sequence splits naturally for $i\ge 1$.	
\end{theorem}
\begin{proof} Using the commutative diagram \eqref{eq53}, for each $\gamma$ we have a commutative diagram with exact rows:
	\begin{equation}\label{eq55}	
	\begin{tikzpicture}\footnotesize
	
	\node (A) {$0$};
	\node (B) [node distance=2.2cm, right of=A] {$Ker \xi _\gamma$};
	\node (C) [node distance=2.7cm, right of=B] {$\bar{Z}_n^\gamma$};
	\node (D) [node distance=2.7cm, right of=C] {$\operatorname{Hom}(H^n(C^*_\gamma) ;G)$};
	\node (E) [node distance=2.2cm, right of=D] {$0$};
	
	\draw[->] (A) to node [left]{}(B);
	\draw[->] (B) to node [above]{}(C);
	\draw[->] (C) to node [above]{$\xi_\gamma$}(D);
	\draw[->] (D) to node [left]{}(E);
	
	\node (A1) [node distance=1.5cm, below of=A] {$0$};
	\node (B1) [node distance=1.5cm, below of=B] {$\operatorname{Ext}(H^{n+1}(C^*_\gamma) ;G)$};
	\node (C1) [node distance=1.5cm, below of=C] {$\bar{H}_n(C^*_\gamma;G)$};
	\node (D1) [node distance=1.5cm, below of=D] {$\operatorname{Hom}(H^n(C^*_\gamma) ;G)$};
	\node (E1) [node distance=1.5cm, below of=E] {$0$.};
	
	\draw[->] (A1) to node [left]{}(B1);
	\draw[->] (B1) to node [above]{}(C1);
	\draw[->] (C1) to node [above]{$\bar{\xi}_\gamma$}(D1);
	\draw[->] (D1) to node [left]{}(E1);
	
	\draw[->] (B) to node [left]{}(B1);
	\draw[->] (C) to node [left]{$\bar{p}_\gamma $}(C1);
	\draw[->] (D) to node [left]{$1$}(D1);
	
	\end{tikzpicture}
	\end{equation}
	This induces the following commutative diagram with exact rows:
	\begin{equation}\label{eq56}	
	\begin{tikzpicture}\footnotesize
	
	\node (A) {$\dots$};
	\node (B) [node distance=2.6cm, right of=A] {${\varprojlim} ^ {(i)} Ker \xi _\gamma$};
	\node (C) [node distance=3.7cm, right of=B] {${\varprojlim} ^ {(i)} \bar{Z}_n^\gamma$};
	\node (D) [node distance=4cm, right of=C] {${\varprojlim} ^ {(i)} \operatorname{Hom}(H^n(C^*_\gamma) ;G)$};
	\node (E) [node distance=4.4cm, right of=D] {${\varprojlim} ^ {(i+1)} Ker \xi _\gamma$};
	\node (F) [node distance=3cm, right of=E] {$\dots$};
	
	\draw[->] (A) to node [left]{}(B);
	\draw[->] (B) to node [above]{}(C);
	\draw[->] (C) to node [above]{${\varprojlim} ^ {(i)} \xi_\gamma$}(D);
	\draw[->] (D) to node [left]{}(E);
	\draw[->] (E) to node [left]{}(F);
	
	\node (A1) [node distance=1.5cm, below of=A] {$\dots$};
	\node (B1) [node distance=1.5cm, below of=B] {$ {\varprojlim} ^ {(i)} \operatorname{Ext}(H^{n+1}(C^*_\gamma) ;G)$};
	\node (C1) [node distance=1.5cm, below of=C] {${\varprojlim} ^ {(i)} \bar{H}_n(C^*_\gamma;G)$};
	\node (D1) [node distance=1.5cm, below of=D] {${\varprojlim} ^ {(i)} \operatorname{Hom}(H^n(C^*_\gamma) ;G)$};
	\node (E1) [node distance=1.5cm, below of=E] {${\varprojlim} ^ {(i+1)} \operatorname{Ext}(H^{n+1}(C^*_\gamma) ;G)$};
	\node (F1) [node distance=1.5cm, below of=F] {$\dots$~.};
	
	\draw[->] (A1) to node [left]{}(B1);
	\draw[->] (B1) to node [above]{}(C1);
	\draw[->] (C1) to node [above]{${\varprojlim} ^ {(i)} \bar{\xi}_\gamma$}(D1);
	\draw[->] (D1) to node [left]{}(E1);
	\draw[->] (E1) to node [left]{}(F1);
	
	\draw[->] (B) to node [left]{}(B1);
	\draw[->] (C) to node [left]{${\varprojlim} ^ {(i)} \bar{p}_\gamma $}(C1);
	\draw[->] (D) to node [left]{$1$}(D1);
	\draw[->] (E) to node [left]{}(E1);
	
	\end{tikzpicture}
	\end{equation}
	By Lemma 4, ${\varprojlim} ^ {(i)} Ker \xi_\gamma = 0,$ for $i\ge 1,$  and so the beginning of the  diagram \eqref{eq56} of the following form: 
	
	\begin{equation}\label{eq57}	
	\begin{tikzpicture}\footnotesize
	
	\node (A) {$0$};
	\node (B) [node distance=2.5cm, right of=A] {${\varprojlim} Ker \xi _\gamma$};
	\node (C) [node distance=3.5cm, right of=B] {${\varprojlim} \bar{Z}_n^\gamma$};
	\node (D) [node distance=3.5cm, right of=C] {${\varprojlim}  \operatorname{Hom}(H^n(C^*_\gamma) ;G)$};
	\node (E) [node distance=2.5cm, right of=D] {$0$};
	
	\draw[->] (A) to node [left]{}(B);
	\draw[->] (B) to node [above]{}(C);
	\draw[->] (C) to node [above]{${\varprojlim}  \xi_\gamma$}(D);
	\draw[->] (D) to node [left]{}(E);
	
	\node (A1) [node distance=1.5cm, below of=A] {$0$};
	\node (B1) [node distance=1.5cm, below of=B] {$ {\varprojlim}  \operatorname{Ext}(H^{n+1}(C^*_\gamma) ;G)$};
	\node (C1) [node distance=1.5cm, below of=C] {${\varprojlim}  \bar{H}_n(C^*_\gamma;G)$};
	\node (D1) [node distance=1.5cm, below of=D] {${\varprojlim}  \operatorname{Hom}(H^n(C^*_\gamma) ;G)$};
	\node (E1) [node distance=1.5cm, below of=E] {$\dots$ ~.};
	
	\draw[->] (A1) to node [left]{}(B1);
	\draw[->] (B1) to node [above]{}(C1);
	\draw[->] (C1) to node [above]{${\varprojlim}  \bar{\xi}_\gamma$}(D1);
	\draw[->] (D1) to node [left]{}(E1);
	
	\draw[->] (B) to node [left]{}(B1);
	\draw[->] (C) to node [left]{${\varprojlim} ^ {(i)} \bar{p}_\gamma $}(C1);
	\draw[->] (D) to node [left]{$1$}(D1);
	
	\end{tikzpicture}
	\end{equation}
	Therefore, the following sequence is exact: 
	\begin{equation}\label{eq58}
	\begin{tikzpicture}\footnotesize
	
	\node (A) {$0$};
	\node (B) [node distance=2.5cm, right of=A] {${\varprojlim} \operatorname{Ext}(H^{n+1}(C^*_\gamma);G) $};
	\node (C) [node distance=3.8cm, right of=B] {${\varprojlim}  \bar{H}_n(C^*_\gamma;G)  $};
	\node (D) [node distance=3.8cm, right of=C] {${\varprojlim}  \operatorname{Hom}(H^n(C^*_\gamma);G) $};
	\node (E) [node distance=2.5cm, right of=D] {$0$};
	
	\draw[->] (A) to node [above]{}(B);
	\draw[->] (B) to node [above]{}(C);
	\draw[->] (C) to node [above]{$\xi_\gamma$}(D);
	\draw[->] (D) to node [above]{}(E);
	
	\end{tikzpicture}
	\end{equation}
	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 $i \ge1,$ the result follows from the commutativity of the diagram \eqref{eq56} and Corollary 2 and 3. 
\end{proof}

\section{The main theorem} 

Here we have formulated the main result of the paper.

\begin{theorem}
	Let ${\mathbf C }^*=\{C^*_\gamma\}$ be a direct system of cochain complexes. Then, there is a natural exact sequence:
	\begin{equation}\label{eq59}
	\begin{tikzpicture}\footnotesize
	
	\node (A) {$\dots$};
	\node (B) [node distance=1.8cm, right of=A] {$ {\varprojlim} ^ {(3)}   \bar{H}^\gamma_{n+2} $};
	\node (C) [node distance=2.5cm, right of=B] {${\varprojlim} ^ {(1)}  \bar{H}^\gamma_{n+1} $};
	\node (D) [node distance=2.5cm, right of=C] {$ \bar{H}_n \left( {\varinjlim}   C^*_\gamma ;G \right) $};
	\node (E) [node distance=2.5cm, right of=D] {${\varprojlim}  \bar{H}^\gamma_{n} $};
	\node (F) [node distance=2cm, right of=E] {${\varprojlim} ^ {(2)}  \bar{H}^\gamma_{n+1} $};
	\node (H) [node distance=1.8cm, right of=F] {$\dots$};
	
	\draw[->] (A) to node [above]{}(B);
	\draw[->] (B) to node [above]{}(C);
	\draw[->] (C) to node [above]{}(D);
	\draw[->] (D) to node [above]{}(E);
	\draw[->] (E) to node [above]{}(F);
	\draw[->] (F) to node [above]{}(H);
	
	\end{tikzpicture}
	\end{equation}
	where $\bar{H}_*^\gamma=\bar{H}_* (C^*_\gamma ;G).$
\end{theorem}
\begin{proof} By Proposition 1.2 of \cite{6}, for the inverse system $\{H_\gamma^{n+1}\}$ we have an exact sequence:
	\begin{equation}\label{eq62}
	\begin{tikzpicture}\footnotesize
	
	\node (A) {$0$};
	\node (B) [node distance=2.3cm, right of=A] {$ {\varprojlim} ^ {(1)}   \operatorname{Hom}(H_\gamma^{n+1} ; G) $};
	\node (C) [node distance=3.5cm, right of=B] {$\operatorname{Ext}(\varinjlim H_\gamma^{n+1}; G ) $};
	\node (E) [node distance=6.8cm, right of=B] {$\varprojlim \operatorname{Ext}(H_\gamma^{n+1} ;G)$};
	\node (F) [node distance=3.5cm, right of=E] {${\varprojlim} ^ {(2)}  \operatorname{Hom}(H_\gamma^{n+1} ; G) $};
	\node (H) [node distance=2.3cm, right of=F] {$0$};
	
	\draw[->] (A) to node [above]{}(B);
	\draw[->] (B) to node [above]{}(C);
	\draw[->] (C) to node [above]{}(E);
	\draw[->] (E) to node [above]{}(F);
	\draw[->] (F) to node [above]{}(H);
	
	\end{tikzpicture}
	\end{equation}
	and 
	\begin{equation}\label{eq072} \footnotesize
	{\varprojlim} ^ {(i)} \operatorname{Ext}(H^{n+1}_\gamma ;G) \simeq {\varprojlim} ^ {(i+2)}  \operatorname{Hom}(H^{n+1}_\gamma ; G), \text{ for } i\ge 1.
	\end{equation}
	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:
	\begin{equation}\label{eq63}
	\begin{tikzpicture}\footnotesize
	
	\node (A) {$0$};
	\node (B) [node distance=2.4cm, right of=A] {$ {\varprojlim} ^ {(1)}   \operatorname{Hom}(H_\gamma^{n+1} ; G) $};
	\node (C) [node distance=3.6cm, right of=B] {$\operatorname{Ext}( H^{n+1}(C^*); G ) $};
	\node (E) [node distance=6.9cm, right of=B] {$\varprojlim \operatorname{Ext}(H_\gamma^{n+1} ;G)$};
	\node (F) [node distance=3.5cm, right of=E] {${\varprojlim} ^ {(2)}  \operatorname{Hom}(H_\gamma^{n+1} ; G) $};
	\node (H) [node distance=2.2cm, right of=F] {$0$.};
	
	\draw[->] (A) to node [above]{}(B);
	\draw[->] (B) to node [above]{}(C);
	\draw[->] (C) to node [above]{}(E);
	\draw[->] (E) to node [above]{}(F);
	\draw[->] (F) to node [above]{}(H);
	
	\end{tikzpicture}
	\end{equation}
	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:
	\begin{equation}\label{eq063}	
	\begin{tikzpicture}\footnotesize
	
	\node (A) {$0$};
	\node (B) [node distance=2.4cm, right of=A] {$\operatorname{Ext}(H^{n+1}(C^*);G)$};
	\node (C) [node distance=3.5cm, right of=B] {$\bar{H}_n(C^*;G)$};
	\node (D) [node distance=3.5cm, right of=C] {$\operatorname{Hom}(H^n(C^*) ;G)$};
	\node (E) [node distance=2.2cm, right of=D] {$0$};
	
	\draw[->] (A) to node [left]{}(B);
	\draw[->] (B) to node [above]{}(C);
	\draw[->] (C) to node [above]{}(D);
	\draw[->] (D) to node [left]{}(E);
	
	\node (A1) [node distance=1.5cm, below of=A] {$0$};
	\node (B1) [node distance=1.5cm, below of=B] {$\operatorname{Ext}(H^{n+1}(C^*_\gamma);G)$};
	\node (C1) [node distance=1.5cm, below of=C] {$  \bar{H}_n(C^*_\gamma;G)$};
	\node (D1) [node distance=1.5cm, below of=D] {$  \operatorname{Hom}(H^n(C^*_\gamma) ;G)$};
	\node (E1) [node distance=1.5cm, below of=E] {$0$.};
	
	\draw[->] (A1) to node [left]{}(B1);
	\draw[->] (B1) to node [above]{}(C1);
	\draw[->] (C1) to node [above]{}(D1);
	\draw[->] (D1) to node [left]{}(E1);
	
	\draw[->] (B) to node [left]{$\tilde{\pi}_\gamma$}(B1);
	\draw[->] (C) to node [left]{$\pi_\gamma$}(C1);
	\draw[->] (D) to node [left]{$\bar{\pi}_\gamma$}(D1);
	
	\end{tikzpicture}
	\end{equation}
	The diagram \eqref{eq063} generates the following diagram: 
	
	\begin{equation}\label{eq163}	
	\begin{tikzpicture}\footnotesize
	
	\node (A) {$0$};
	\node (B) [node distance=2.4cm, right of=A] {$\operatorname{Ext}(H^{n+1}(C^*);G)$};
	\node (C) [node distance=3.5cm, right of=B] {$\bar{H}_n(C^*;G)$};
	\node (D) [node distance=3.5cm, right of=C] {$\operatorname{Hom}(H^n(C^*) ;G)$};
	\node (E) [node distance=2.2cm, right of=D] {$0$};
	
	\draw[->] (A) to node [left]{}(B);
	\draw[->] (B) to node [above]{}(C);
	\draw[->] (C) to node [above]{}(D);
	\draw[->] (D) to node [left]{}(E);
	
	\node (A1) [node distance=1.5cm, below of=A] {$0$};
	\node (B1) [node distance=1.5cm, below of=B] {$\varprojlim \operatorname{Ext}(H^{n+1}_\gamma;G)$};
	\node (C1) [node distance=1.5cm, below of=C] {$\varprojlim  \bar{H}_n^\gamma$};
	\node (D1) [node distance=1.5cm, below of=D] {$\varprojlim  \operatorname{Hom}(H^n_\gamma ;G)$};
	\node (E1) [node distance=1.5cm, below of=E] {$0$.};
	
	\draw[->] (A1) to node [left]{}(B1);
	\draw[->] (B1) to node [above]{}(C1);
	\draw[->] (C1) to node [above]{}(D1);
	\draw[->] (D1) to node [left]{}(E1);
	
	\draw[->] (B) to node [left]{$\tilde{\pi}$}(B1);
	\draw[->] (C) to node [left]{$\pi$}(C1);
	\draw[->] (D) to node [left]{$\simeq$}(D1);
	
	\end{tikzpicture}
	\end{equation}
	Therefore, we have $Ker \tilde{\pi} \simeq Ker \pi$ and $Coker \tilde{\pi} \simeq Coker \pi$ and so, the following diagram is commutative: 
	\begin{equation}\label{eq64}	
	\begin{tikzpicture}\footnotesize
	
	\node (A) {$0$};
	\node (B) [node distance=2.4cm, right of=A] {$\operatorname{Ext}(H^{n+1}(C);G)$};
	\node (C) [node distance=3.5cm, right of=B] {$\bar{H}_n(C;G)$};
	\node (D) [node distance=3.5cm, right of=C] {$\operatorname{Hom}(H^n(C);G)$};
	\node (E) [node distance=2cm, right of=D] {$0$};
	
	\draw[->] (A) to node [left]{}(B);
	\draw[->] (B) to node [above]{}(C);
	\draw[->] (C) to node [above]{$$}(D);
	\draw[->] (D) to node [left]{}(E);
	
	
	\node (B2) [node distance=1.5cm, above of=B] {$Ker \tilde{\pi}$};
	\node (C2) [node distance=1.5cm, above of=C] {$Ker \pi$};
	\node (B3) [node distance=1.5cm, above of=B2] {$0$};
	\node (C3) [node distance=1.5cm, above of=C2] {$0$};
	
	\draw[->] (B3) to node [left]{}(B2);
	\draw[->] (C3) to node [left]{}(C2);
	\draw[->] (B2) to node [left]{}(B);
	\draw[->] (C2) to node [left]{}(C);
	\draw[->] (B2) to node [above]{$\simeq$}(C2);
	
	\node (A1) [node distance=2cm, below of=A] {$0$};
	\node (B1) [node distance=2cm, below of=B] {$ \varprojlim \operatorname{Ext}(H^{n+1}_\gamma ;G)$};
	\node (C1) [node distance=2cm, below of=C] {$\varprojlim \bar{H}_n^\gamma$};
	\node (D1) [node distance=2cm, below of=D] {$\varprojlim \operatorname{Hom}(H^n_\gamma ;G)$};
	\node (E1) [node distance=2cm, below of=E] {$0$~.};
	
	\draw[->] (A1) to node [left]{}(B1);
	\draw[->] (B1) to node [above]{}(C1);
	\draw[->] (C1) to node [above]{$$}(D1);
	\draw[->] (D1) to node [left]{}(E1);
	
	\draw[->] (B) to node [left]{$\tilde{\pi}$}(B1);
	\draw[->] (C) to node [left]{$\pi $}(C1);
	\draw[->] (D) to node [left]{$\simeq $}(D1);
	
	\node (B4) [node distance=1.5cm, below of=B1] {$Coker \tilde{\pi}$};
	\node (C4) [node distance=1.5cm, below of=C1] {$Coker \pi$};
	\node (B5) [node distance=1.5cm, below of=B4] {$0$};
	\node (C5) [node distance=1.5cm, below of=C4] {$0$};
	
	\draw[->] (B1) to node [left]{}(B4);
	\draw[->] (C1) to node [left]{}(C4);
	\draw[->] (B4) to node [left]{}(B5);
	\draw[->] (C4) to node [left]{}(C5);
	\draw[->] (B4) to node [above]{$\simeq$}(C4);
	\end{tikzpicture}
	\end{equation}
	Using the exact sequence \eqref{eq63} and diagram \eqref{eq64}, we obtain a four-term exact sequence:	
	\begin{equation}\label{eq65}
	\begin{tikzpicture}\footnotesize
	
	\node (A) {$0$};
	\node (B) [node distance=2.5cm, right of=A] {$ {\varprojlim} ^ {(1)}   \operatorname{Hom}(H_\gamma^{n+1} ; G) $};
	\node (C) [node distance=3cm, right of=B] {$\bar{H}_n(C;G)$};
	\node (E) [node distance=5cm, right of=B] {$\varprojlim \bar{H}_n^\gamma$};
	\node (F) [node distance=3cm, right of=E] {${\varprojlim} ^ {(2)}  \operatorname{Hom}(H_\gamma^{n+1} ; G) $};
	\node (H) [node distance=2.5cm, right of=F] {$0$.};
	
	\draw[->] (A) to node [above]{}(B);
	\draw[->] (B) to node [above]{}(C);
	\draw[->] (C) to node [above]{}(E);
	\draw[->] (E) to node [above]{}(F);
	\draw[->] (F) to node [above]{}(H);
	\end{tikzpicture}
	\end{equation}
	Using the exact sequence \eqref{eq65}, Theorem 3 and isomorphism \eqref{eq072}, we obtain the following diagram, which contains the long exact sequence of the theorem:
	\begin{equation}\label{eq66}
	\begin{tikzpicture}\footnotesize
	
	\node (A) {$0$};
	\node (B) [node distance=2.5cm, right of=A] {$ {\varprojlim} ^ {(1)}   \operatorname{Hom}(H_\gamma^{n+1} ; G) $};
	\node (C) [node distance=3cm, right of=B] {$\bar{H}_n(C;G)$};
	\node (E) [node distance=5cm, right of=B] {$\varprojlim \bar{H}_n^\gamma$};
	\node (F) [node distance=3cm, right of=E] {${\varprojlim} ^ {(2)}  \operatorname{Hom}(H_\gamma^{n+1} ; G) $};
	\node (H) [node distance=2.5cm, right of=F] {$0$.};
	
	\draw[->] (A) to node [above]{}(B);
	\draw[->] (B) to node [above]{}(C);
	\draw[->] (C) to node [above]{}(E);
	\draw[->] (E) to node [above]{}(F);
	\draw[->] (F) to node [above]{}(H);
	
	
	\node (A1) [node distance=1.5cm, below of=A] {$0$};
	\node (B1) [node distance=1.5cm, below of=B] {${\varprojlim} ^ {(2)} \operatorname{Ext}(H_\gamma^{n+2} ; G)$};
	\node (E1) [node distance=1.5cm, below of=E] {${\varprojlim} ^ {(2)} \bar{H}_{n+1}^\gamma$};
	\node (F1) [node distance=1.5cm, below of=F] {${\varprojlim} ^ {(2)} \operatorname{Hom}(H_\gamma^{n+1} ; G)$};
	\node (H1) [node distance=1.5cm, below of=H] {$0$};
	
	\draw[->] (H1) to node [above]{}(F1);
	\draw[->] (F1) to node [above]{}(E1);
	\draw[->] (E1) to node [above]{}(B1);
	\draw[->] (B1) to node [above]{}(A1);
	
	\node (A2) [node distance=1.5cm, below of=A1] {$0$};
	\node (B2) [node distance=1.5cm, below of=B1] {${\varprojlim} ^ {(4)} \operatorname{Hom}(H_\gamma^{n+2} ; G)$};
	\node (E2) [node distance=1.5cm, below of=E1] {${\varprojlim} ^ {(4)} \bar{H}_{n+2}^\gamma$};
	\node (F2) [node distance=1.5cm, below of=F1] {${\varprojlim} ^ {(4)} \operatorname{Ext}(H_\gamma^{n+3} ; G)$};
	\node (H2) [node distance=1.5cm, below of=H1] {$0$};
	
	\node (EE2) [node distance=1.5cm, below of=E2] {$\vdots$};
	
	
	\draw[->] (A2) to node [above]{}(B2);
	\draw[->] (B2) to node [above]{}(E2);
	\draw[->] (E2) to node [above]{}(F2);
	\draw[->] (F2) to node [above]{}(H2);
	
	\node (A3) [node distance=1.5cm, above of=A] {$0$};
	\node (B3) [node distance=1.5cm, above of=B] {${\varprojlim} ^ {(1)} \operatorname{Hom}(H_\gamma^{n+1} ; G)$};
	\node (E3) [node distance=1.5cm, above of=E] {${\varprojlim} ^ {(1)} \bar{H}_{n+1}^\gamma$};
	\node (F3) [node distance=1.5cm, above of=F] {${\varprojlim} ^ {(1)} \operatorname{Ext}(H_\gamma^{n+2} ; G)$};
	\node (H3) [node distance=1.5cm, above of=H] {$0$};
	
	\draw[->] (H3) to node [above]{}(F3);
	\draw[->] (F3) to node [above]{}(E3);
	\draw[->] (E3) to node [above]{}(B3);
	\draw[->] (B3) to node [above]{}(A3);
	
	
	\node (A4) [node distance=1.5cm, above of=A3] {$0$};
	\node (B4) [node distance=1.5cm, above of=B3] {${\varprojlim} ^ {(3)} \operatorname{Ext}(H_\gamma^{n+3} ; G)$};
	\node (E4) [node distance=1.5cm, above of=E3] {${\varprojlim} ^ {(3)} \bar{H}_{n+2}^\gamma$};
	\node (F4) [node distance=1.5cm, above of=F3] {${\varprojlim} ^ {(3)} \operatorname{Hom}(H_\gamma^{n+2} ; G)$};
	\node (H4) [node distance=1.5cm, above of=H3] {$0$};
	
	\node (EE4) [node distance=1.5cm, above of=E4] {$\vdots$};
	
	
	\draw[->] (A4) to node [above]{}(B4);
	\draw[->] (B4) to node [above]{}(E4);
	\draw[->] (E4) to node [above]{}(F4);
	\draw[->] (F4) to node [above]{}(H4);
	
	\draw[dashed, thick, ->] (EE4) to node [above]{}(E4);
	\draw[dashed, thick, ->] (E4) to node [above]{}(E3);
	\draw[dashed, thick, ->] (E3) to node [above]{}(C);
	\draw[dashed, thick, ->] (E) to node [above]{}(E1);
	\draw[dashed, thick, ->] (E1) to node [above]{}(E2);
	\draw[dashed, thick, ->] (E2) to node [above]{}(EE2);
	
	\draw[->] (F4) to node [left]{$\simeq $}(F3);
	\draw[->] (B3) to node [left]{$\simeq $}(B);
	\draw[->] (F) to node [left]{$\simeq $}(F1);
	\draw[->] (B1) to node [left]{$\simeq $}(B2);
	\end{tikzpicture}
	\end{equation}
\end{proof}

\begin{corollary}
	Let ${\mathbf C }^*=\{C^*_\gamma\}$ be a direct system of cochain complexes. Then, for each injective abelian group $G,$ there is an isomorphism
	\begin{equation}\label{eq59} \footnotesize
	\bar{H}_n \left( {\varinjlim}   C^*_\gamma ;G \right)  \simeq {\varprojlim} \bar{H}_* (C^*_\gamma ;G).
	\end{equation}
\end{corollary}

\section{Applications} 
{\bf 1. }Let  $C_c^*(X,G)$ be the cochain complex of Massey \cite{Mas}. 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 \cite{Mas}). 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 \cite{Mas} and theorem of universal coefficient 4.1 \cite{Mac}):
\begin{equation}\label{eq93} \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.
\end{equation}
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:
\begin{equation}\label{eq94} \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.
\end{equation}
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:

\begin{corollary}
	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:
	\begin{equation}\label{eq59}
	\begin{tikzpicture}\footnotesize
	
	\node (A) {$\dots$};
	\node (B) [node distance=2.8cm, right of=A] {$ {\varprojlim} ^ {(2k+1)}   \bar{H}^M_{n++k+1}(N _\alpha) $};
	\node (C) [node distance=2.8cm, right of=B] {$\dots$};
	\node (D) [node distance=2.3cm, right of=C] {$ {\varprojlim} ^ {(3)}   \bar{H}^M_{n+2}(N _\alpha) $};
	\node (E) [node distance=3cm, right of=D] {$ {\varprojlim} ^ {(1)}   \bar{H}^M_{n+1}(N _\alpha) $};
	\node (F) [node distance=2cm, right of=E] {};
	
	\node (A1) [node distance=1cm, below of=A] {};
	\node (B1) [node distance=1.5cm, right of=A1] {$\bar{H}^M_n(A;G)$};
	\node (C1) [node distance=2.5cm, right of=B1] {$ {\varprojlim}   \bar{H}^M_{n}(N _\alpha) $};
	\node (D1) [node distance=3cm, right of=C1] {$ {\varprojlim}^{(2)}   \bar{H}^M_{n+1}(N _\alpha) $};
	\node (E1) [node distance=2.3cm, right of=D1] {$ \dots $};
	\node (F1) [node distance=2.5cm, right of=E1] {$ {\varprojlim}^{(2k)}   \bar{H}^M_{n+k}(N _\alpha) $};
	\node (G1) [node distance=2.5cm, right of=F1] {$ \dots~~. $};

	
	\draw[->] (A) to node [above]{}(B);
	\draw[->] (B) to node [above]{}(C);
	\draw[->] (C) to node [above]{}(D);
	\draw[->] (D) to node [above]{}(E);
	\draw[->] (E) to node [above]{}(F);

	\draw[->] (A1) to node [above]{}(B1);
	\draw[->] (B1) to node [above]{$i_*$}(C1);
	\draw[->] (C1) to node [above]{}(D1);
	\draw[->] (D1) to node [above]{}(E1);
	\draw[->] (E1) to node [above]{}(F1);
	\draw[->] (F1) to node [above]{}(G1);
	\end{tikzpicture}
	\end{equation}
	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:
		\begin{equation}\label{eq59}
	\begin{tikzpicture}\footnotesize
	
	\node (A) {$\dots$};
	\node (B) [node distance=2.8cm, right of=A] {$ {\varprojlim} ^ {(2k+1)}   \bar{H}^M_{n++k+1}(U _\alpha) $};
	\node (C) [node distance=2.8cm, right of=B] {$\dots$};
	\node (D) [node distance=2.3cm, right of=C] {$ {\varprojlim} ^ {(3)}   \bar{H}^M_{n+2}(U _\alpha) $};
	\node (E) [node distance=3cm, right of=D] {$ {\varprojlim} ^ {(1)}   \bar{H}^M_{n+1}(U _\alpha) $};
	\node (F) [node distance=2cm, right of=E] {};
	
	\node (A1) [node distance=1cm, below of=A] {};
	\node (B1) [node distance=1.5cm, right of=A1] {$\bar{H}^M_n(A;G)$};
	\node (C1) [node distance=2.5cm, right of=B1] {$ {\varprojlim}   \bar{H}^M_{n}(U _\alpha) $};
	\node (D1) [node distance=3cm, right of=C1] {$ {\varprojlim}^{(2)}   \bar{H}^M_{n+1}(U _\alpha) $};
	\node (E1) [node distance=2.3cm, right of=D1] {$ \dots $};
	\node (F1) [node distance=2.5cm, right of=E1] {$ {\varprojlim}^{(2k)}   \bar{H}^M_{n+k}(U _\alpha) $};
	\node (G1) [node distance=2.5cm, right of=F1] {$ \dots~~. $};
	
	
	\draw[->] (A) to node [above]{}(B);
	\draw[->] (B) to node [above]{}(C);
	\draw[->] (C) to node [above]{}(D);
	\draw[->] (D) to node [above]{}(E);
	\draw[->] (E) to node [above]{}(F);
	
	\draw[->] (A1) to node [above]{}(B1);
	\draw[->] (B1) to node [above]{$i_*$}(C1);
	\draw[->] (C1) to node [above]{}(D1);
	\draw[->] (D1) to node [above]{}(E1);
	\draw[->] (E1) to node [above]{}(F1);
	\draw[->] (F1) to node [above]{}(G1);
	\end{tikzpicture}
	\end{equation}
\end{corollary} 

Note that the formula \eqref{eq89.1} is a generalization of Theorem 1.4.2 of \cite{Mas}.

{\bf 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 \cite{15} 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
\begin{equation}\label{eq90} \footnotesize
i: \varinjlim \bar{H}^n (U_\alpha;G) \longrightarrow \bar{H}^n(A;G).
\end{equation}
By Theorem 6.6.2 \cite{15}, if $A$ is a closed subspace of a paracompact Hausdorff space $X$, then \eqref{eq90} 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 
\begin{equation}\label{eq91} \footnotesize
i:  {H}_n(A;G) \longrightarrow \varprojlim {H}_n (U_\alpha;G).
\end{equation}
The question whether the homomorphism \eqref{eq91} 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 \eqref{eq91}.

\begin{corollary}
	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:
		\begin{equation}\label{eq59}
	\begin{tikzpicture}\footnotesize
	
	\node (A) {$\dots$};
	\node (B) [node distance=2.8cm, right of=A] {$ {\varprojlim} ^ {(2k+1)}   \bar{H}_{n++k+1}(U _\alpha) $};
	\node (C) [node distance=2.8cm, right of=B] {$\dots$};
	\node (D) [node distance=2.3cm, right of=C] {$ {\varprojlim} ^ {(3)}   \bar{H}_{n+2}(U _\alpha) $};
	\node (E) [node distance=3cm, right of=D] {$ {\varprojlim} ^ {(1)}   \bar{H}_{n+1}(U _\alpha) $};
	\node (F) [node distance=2cm, right of=E] {};
	
	\node (A1) [node distance=1cm, below of=A] {};
	\node (B1) [node distance=1.5cm, right of=A1] {$\bar{H}_n(A;G)$};
	\node (C1) [node distance=2.5cm, right of=B1] {$ {\varprojlim}   \bar{H}_{n}(U _\alpha) $};
	\node (D1) [node distance=3cm, right of=C1] {$ {\varprojlim}^{(2)}   \bar{H}_{n+1}(U _\alpha) $};
	\node (E1) [node distance=2.3cm, right of=D1] {$ \dots $};
	\node (F1) [node distance=2.5cm, right of=E1] {$ {\varprojlim}^{(2k)}   \bar{H}_{n+k}(U _\alpha) $};
	\node (G1) [node distance=2.5cm, right of=F1] {$ \dots~~. $};
	
	
	\draw[->] (A) to node [above]{}(B);
	\draw[->] (B) to node [above]{}(C);
	\draw[->] (C) to node [above]{}(D);
	\draw[->] (D) to node [above]{}(E);
	\draw[->] (E) to node [above]{}(F);
	
	\draw[->] (A1) to node [above]{}(B1);
	\draw[->] (B1) to node [above]{$i_*$}(C1);
	\draw[->] (C1) to node [above]{}(D1);
	\draw[->] (D1) to node [above]{}(E1);
	\draw[->] (E1) to node [above]{}(F1);
	\draw[->] (F1) to node [above]{}(G1);
	\end{tikzpicture}
	\end{equation}
\end{corollary}

{\bf 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:
\begin{equation}\label{eq31.0}
\footnotesize
i^*:\bar{H}_*(X;G) \overset{\simeq}{\longrightarrow} \bar{H}^M_*(X;G).
\end{equation}
On the other hand, since on the category of compact metric spaces the Steenrod homology $H^{St}_*$ \cite{16}, \cite{EH1}, \cite{EH2}  and the Massey homology are isomorphic, using the isomorphism \eqref{eq31.0}, we will obtain that
\begin{equation}\label{eq31.1}
\footnotesize
\bar{H}_*(X;G) \simeq {H}^{St}_*(X;G).
\end{equation}
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
\begin{equation}\label{eq31.2}
\footnotesize
\bar{H}_*(X;G) \simeq {H}^{Mil}_*(X;G).
\end{equation}
If $H_*^{BM}(-;G)$ is the Borel-Moore homology with coefficient in $G$, then by Theorem 3 \cite{Kuz}, we have the isomororphism
\begin{equation}\label{}
\footnotesize
\bar{H}_*(X;G) \simeq {H}^{BM}_*(X;G).
\end{equation}

{\bf 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
\begin{equation}\label{eq92}
\footnotesize
\pi_{*} :H_*(X,A) \to \varprojlim H_*(X_\alpha , A_ \alpha).
\end{equation}
\begin{definition} 
	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:
		\begin{equation}
	\begin{tikzpicture}\footnotesize
	
	\node (A) {$\dots$};
	\node (B) [node distance=3cm, right of=A] {$ {\varprojlim} ^ {(2k+1)}   \bar{H}^M_{n++k+1}(X_\alpha , A_\alpha) $};
	\node (C) [node distance=3cm, right of=B] {$\dots$};
	\node (D) [node distance=2.5cm, right of=C] {$ {\varprojlim} ^ {(3)}   \bar{H}^M_{n+2}(X_\alpha , A_\alpha) $};
	\node (E) [node distance=3.7cm, right of=D] {$ {\varprojlim} ^ {(1)}   \bar{H}^M_{n+1}(X_\alpha , A_\alpha) $};
	\node (F) [node distance=2.2cm, right of=E] {};
	
	\node (A1) [node distance=1.2cm, below of=A] {};
	\node (B1) [node distance=1.8cm, right of=A1] {$\bar{H}^M_n(X,A;G)$};
	\node (C1) [node distance=2.8cm, right of=B1] {$ {\varprojlim}   \bar{H}^M_{n}(X_\alpha , A_\alpha) $};
	\node (D1) [node distance=3.5cm, right of=C1] {$ {\varprojlim}^{(2)}   \bar{H}^M_{n+1}(X_\alpha , A_\alpha) $};
	\node (E1) [node distance=2.5cm, right of=D1] {$ \dots $};
	\node (F1) [node distance=2.5cm, right of=E1] {$ {\varprojlim}^{(2k)}   \bar{H}^M_{n+k}(X_\alpha , A_\alpha) $};
	\node (G1) [node distance=2.5cm, right of=F1] {$ \dots~~. $};
	
	
	\draw[->] (A) to node [above]{}(B);
	\draw[->] (B) to node [above]{}(C);
	\draw[->] (C) to node [above]{}(D);
	\draw[->] (D) to node [above]{}(E);
	\draw[->] (E) to node [above]{}(F);
	
	\draw[->] (A1) to node [above]{}(B1);
	\draw[->] (B1) to node [above]{$i_*$}(C1);
	\draw[->] (C1) to node [above]{}(D1);
	\draw[->] (D1) to node [above]{}(E1);
	\draw[->] (E1) to node [above]{}(F1);
	\draw[->] (F1) to node [above]{}(G1);
	\end{tikzpicture}
	\end{equation}
\end{definition}

\begin{definition}
	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
	\begin{equation}\label{eq93} \footnotesize
	\varinjlim H^*(C^*_\alpha) \to H^*(C^*)
	\end{equation}
	is an isomorphism.
\end{definition}

\begin{lemma}
	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:
			\begin{equation}\label{lem1}
	\begin{tikzpicture}\footnotesize
	
	\node (A) {$\dots$};
	\node (B) [node distance=3cm, right of=A] {$ {\varprojlim} ^ {(2k+1)}   \bar{H}^M_{n++k+1}(C^*_\alpha;G)  $};
	\node (C) [node distance=3cm, right of=B] {$\dots$};
	\node (D) [node distance=2.5cm, right of=C] {$ {\varprojlim} ^ {(3)}   \bar{H}^M_{n+2}(C^*_\alpha;G)  $};
	\node (E) [node distance=3.7cm, right of=D] {$ {\varprojlim} ^ {(1)}   \bar{H}^M_{n+1}(C^*_\alpha;G)  $};
	\node (F) [node distance=2.2cm, right of=E] {};
	
	\node (A1) [node distance=1.2cm, below of=A] {};
	\node (B1) [node distance=1.8cm, right of=A1] {$\bar{H}^M_n(C^*;G)$};
	\node (C1) [node distance=2.8cm, right of=B1] {$ {\varprojlim}   \bar{H}^M_{n}(C^*_\alpha;G) $};
	\node (D1) [node distance=3.5cm, right of=C1] {$ {\varprojlim}^{(2)}   \bar{H}^M_{n+1}(C^*_\alpha;G)  $};
	\node (E1) [node distance=2.5cm, right of=D1] {$ \dots $};
	\node (F1) [node distance=2.5cm, right of=E1] {$ {\varprojlim}^{(2k)}   \bar{H}^M_{n+k}(C^*_\alpha;G)  $};
	\node (G1) [node distance=2.5cm, right of=F1] {$ \dots~~. $};
	
	
	\draw[->] (A) to node [above]{}(B);
	\draw[->] (B) to node [above]{}(C);
	\draw[->] (C) to node [above]{}(D);
	\draw[->] (D) to node [above]{}(E);
	\draw[->] (E) to node [above]{}(F);
	
	\draw[->] (A1) to node [above]{}(B1);
	\draw[->] (B1) to node [above]{$i_*$}(C1);
	\draw[->] (C1) to node [above]{}(D1);
	\draw[->] (D1) to node [above]{}(E1);
	\draw[->] (E1) to node [above]{}(F1);
	\draw[->] (F1) to node [above]{}(G1);
	\end{tikzpicture}
	\end{equation}
	where $\bar{H}_{*}(C^*)=H_*\left((\operatorname{Hom}(C^*;\beta_\#)\right)$ and $\bar{H}_{*}(C^*_\alpha)=H_*\left(\operatorname{Hom}(C^*_\alpha;\beta_\#)\right).$
\end{lemma}

\begin{proof}
	By theorem 4, There is a natural exact sequence:
	\begin{equation}\label{lem2} \footnotesize
	\begin{tikzpicture}\footnotesize
	
	\node (A) {$\dots$};
	\node (B) [node distance=2cm, right of=A] {$ {\varprojlim} ^ {(3)}   \bar{H}^\alpha_{n+2} $};
	\node (C) [node distance=2.5cm, right of=B] {${\varprojlim} ^ {(1)}  \bar{H}^\alpha_{n+1} $};
	\node (D) [node distance=2.8cm, right of=C] {$ \bar{H}_n \left( {\varinjlim}   C^*_\alpha ;G \right) $};
	\node (E) [node distance=2.5cm, right of=D] {${\varprojlim}  \bar{H}^\alpha_{n} $};
	\node (F) [node distance=2cm, right of=E] {${\varprojlim} ^ {(2)}  \bar{H}^\alpha_{n+1} $};
	\node (H) [node distance=1.9cm, right of=F] {$\dots$~,};
	
	\draw[->] (A) to node [above]{}(B);
	\draw[->] (B) to node [above]{}(C);
	\draw[->] (C) to node [above]{}(D);
	\draw[->] (D) to node [above]{}(E);
	\draw[->] (E) to node [above]{}(F);
	\draw[->] (F) to node [above]{}(H);
	
	\end{tikzpicture}
	\end{equation}
	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 
	\begin{equation}\label{lem3} \footnotesize
	\bar{H}_*(\varinjlim C^*_\alpha;G) \simeq  \varinjlim \bar{H}^*(C^*_\alpha;G) \overset{\simeq}{\longrightarrow} \bar{H}_*(C^*;G).
	\end{equation}
	On the other hand, by Universal Coefficient Formula, we have the following commutative diagram with exact rows:
	\begin{equation}\label{lem4}	
	\begin{tikzpicture}\footnotesize
	
	\node (A) {$0$};
	\node (B) [node distance=2.4cm, right of=A] {$\operatorname{Ext}(H^{n+1}(C^*);G)$};
	\node (C) [node distance=3.5cm, right of=B] {$\bar{H}_n(C^*;G)$};
	\node (D) [node distance=3.5cm, right of=C] {$\operatorname{Hom}(H^n(C^*) ;G)$};
	\node (E) [node distance=2.2cm, right of=D] {$0$};
	
	\draw[->] (A) to node [left]{}(B);
	\draw[->] (B) to node [above]{}(C);
	\draw[->] (C) to node [above]{}(D);
	\draw[->] (D) to node [left]{}(E);
	
	\node (A1) [node distance=2cm, below of=A] {$0$};
	\node (B1) [node distance=2cm, below of=B] {$\operatorname{Ext}(H^{n+1}(\varinjlim C^*_\alpha);G)$};
	\node (C1) [node distance=2cm, below of=C] {$  \bar{H}_n(\varinjlim C^*_\alpha;G)$};
	\node (D1) [node distance=2cm, below of=D] {$  \operatorname{Hom}(H^n(\varinjlim C^*_\alpha) ;G)$};
	\node (E1) [node distance=2cm, below of=E] {$0$.};
	
	\draw[->] (A1) to node [left]{}(B1);
	\draw[->] (B1) to node [above]{}(C1);
	\draw[->] (C1) to node [above]{}(D1);
	\draw[->] (D1) to node [left]{}(E1);
	
	\draw[->] (B) to node [left]{$\simeq$}(B1);
	\draw[->] (C) to node [left]{$\bar{\pi}_n$}(C1);
	\draw[->] (D) to node [left]{$\simeq$}(D1);
	
	\end{tikzpicture}
	\end{equation}
	Hence, the homomorphism $\bar{\pi}_n$ is an isomorphism for all $n \in \mathbb{Z}.$ Using the exact sequence \eqref{lem2} and the isomorphism $\bar{\pi}_n,$ we obtain an infinite exact sequence \eqref{lem1}.	 
\end{proof}

\begin{corollary}
	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:
		\begin{equation}
	\begin{tikzpicture}\footnotesize
	
	\node (A) {$\dots$};
	\node (B) [node distance=3cm, right of=A] {$ {\varprojlim} ^ {(2k+1)}   \bar{H}_{n+k+1}(X_\alpha , A_\alpha) $};
	\node (C) [node distance=3cm, right of=B] {$\dots$};
	\node (D) [node distance=2.5cm, right of=C] {$ {\varprojlim} ^ {(3)}   \bar{H}_{n+2}(X_\alpha , A_\alpha) $};
	\node (E) [node distance=3.7cm, right of=D] {$ {\varprojlim} ^ {(1)}   \bar{H}_{n+1}(X_\alpha , A_\alpha) $};
	\node (F) [node distance=2.2cm, right of=E] {};
	
	\node (A1) [node distance=1.2cm, below of=A] {};
	\node (B1) [node distance=1.8cm, right of=A1] {$\bar{H}_n(X,A;G)$};
	\node (C1) [node distance=2.8cm, right of=B1] {$ {\varprojlim}   \bar{H}_{n}(X_\alpha , A_\alpha) $};
	\node (D1) [node distance=3.5cm, right of=C1] {$ {\varprojlim}^{(2)}   \bar{H}_{n+1}(X_\alpha , A_\alpha) $};
	\node (E1) [node distance=2.5cm, right of=D1] {$ \dots $};
	\node (F1) [node distance=2.5cm, right of=E1] {$ {\varprojlim}^{(2k)}   \bar{H}_{n+k}(X_\alpha , A_\alpha) $};
	\node (G1) [node distance=2.5cm, right of=F1] {$ \dots~~. $};
	
	
	\draw[->] (A) to node [above]{}(B);
	\draw[->] (B) to node [above]{}(C);
	\draw[->] (C) to node [above]{}(D);
	\draw[->] (D) to node [above]{}(E);
	\draw[->] (E) to node [above]{}(F);
	
	\draw[->] (A1) to node [above]{}(B1);
	\draw[->] (B1) to node [above]{$\pi_*$}(C1);
	\draw[->] (C1) to node [above]{}(D1);
	\draw[->] (D1) to node [above]{}(E1);
	\draw[->] (E1) to node [above]{}(F1);
	\draw[->] (F1) to node [above]{}(G1);
	\end{tikzpicture}
	\end{equation}
\end{corollary}

\begin{corollary}
	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:
	\begin{equation} \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 .
	\end{equation}
\end{corollary}

\begin{corollary}
	If $\bar{H}_*$ is the homology theory generated by the Alexander-Spanier cochains, then there is an exact sequence:
	\begin{equation} \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 ,
	\end{equation}
	where $ (X,A)=\varprojlim (K_\alpha,L_\alpha)$ and $(K_\alpha,L_\alpha)$ are finite polyhedral pairs.
\end{corollary}


{\bf 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:
\begin{equation}\label{eq31} \footnotesize
j_*:\bar{H}^s_*(X;G) \overset{\simeq}{\longrightarrow} \bar{H}_*(X;G).
\end{equation}





















