Cohomolgy-Some Module theory

 Preliminaries:
Given two modules $B,C$ we seek a module $A$ such that $B$ contains an isomorphic copy of $A$ such that resulting quotient module $B/A$ is isomorphic to $C$.
Clearly, $B$ contains an isomorphic copy of $A$ is same as saying that there is an injective homomorphism $\psi:A \rightarrow B$. This can be expressed as
$$A\equiv \psi (A)\subset B$$
To say $C$ is isomorphic to quotient means that there is a surjective homomorphism $\phi: B\rightarrow C$ with $ker\;\phi=\psi(A)$.
This gives us a pair of homomorphisms
$$A\stackrel{\psi }{\to }B\stackrel{\varphi }{\to }C$$
such that $im\;\psi=ker\;\phi$.

These homorphisms such that above holds are known as "exact".

Examples:

Using direct sum of modules $A,C$ with $B=A \oplus C$, the following exact sequence can be constructed.
$$0\to A\stackrel{i}{\to }A\oplus C\stackrel{\pi }{\to }C\to 0$$
where $i(a)=(a,0)$ and $\pi(a,c) = c$. Notice that $pi \circ i=\pi(a,c)\circ(a,0)=\pi(a,0)=0$. Thus $\partial^2=0$ map is satisfied.

When $A=Z$ a $Z$ module with $C=Z/nZ$, above sequence becomes
$$\begin{array}{r}0\to Z\stackrel{i}{\to }Z\oplus Z/nZ\stackrel{\pi }{\to }Z/nZ\to 0\\ 0\to Z\stackrel{n}{\to }Z\stackrel{\pi }{\to }Z/nZ\to 0\end{array}$$
If we consider $A=Z$ and $C=Z/nZ$, we can consider these as extension of $C$ by $A$.

For homomorphism $\phi$, we may form the following.
$$0\to K\stackrel{i}{\to }F(S)\stackrel{\varphi }{\to }M\to 0$$
Here $\phi$ is a unique $R$-module homomorphism which is identity in $S$ - set of generators for $M$ an $R$-module.k