I am working on some homological algebra and I struggle to find an example of a short exact sequence of chain complexes.
That is if $$0\to A.\to B. \to C.\to 0$$ then what can $A.$,$B.$, $C.$ be along with the morphisms inbetween? Are there any good examples I can look at to get a feel for them? I'll add abelian groups as an example is the most appriciated.
For clearification, I'll also add that I would want to have non-trivial morphisms between the objects in each chain complex.
A very important example of SES of chain complexes arising in a general setting is the Mayer-Vietoris sequence in homology theory, which could also be taken as a motivation of why it is important to understand SESs.
It goes as follows: let $X$ be a topological space with $A,B\subset X$ two open subsets. Then we obtain a natural SES of chain complexes $$0\longrightarrow S_\bullet(A\cap B)\stackrel{i}{\longrightarrow}S_\bullet(A)\oplus S_\bullet(B)\stackrel{j}{\longrightarrow}S_\bullet(X)\longrightarrow0$$ with $i$ given by the direct sum of the inclusions, and $j(\alpha\oplus\beta) = \alpha - \beta$ (looking at $\alpha,\beta$ as singular chains on $X$). Passing to the homology gives us the Mayer-Vietoris long exact sequence, which often allows us to compute the homology of $X$ starting from the homology of $A$, $B$ and $A\cap B$.
An example is the "trivial" quotient: Consider a complex $B_*$ and $A_*$ is a subcomplex of $B_*$. Then, at each dimension we have a short exact sequence $$0\to A_n \to B_n \to B_n/A_n\to 0$$ which together give a sort exact sequence of chain complexes $$0\to A_*\to B_*\to B_*/A_*\to 0.$$
Now that may sound silly (as it is simply the definition), but a more detailed example can be $B_*=S_*(X)$ the singular chain complex of a space $X$, and $A_*=S_*(Y)$ the singular chain complex of a subspace $Y$ of $X$. In this case $B_*/A_*=S_*(X,Y)$ is the relative chain complex of the pair $(X,Y)$. And we have the SES $$0\to S_*(X)\to S_*(Y)\to S_*(X,Y)\to 0.$$
Another example is to look at $B_n$ as $$\cdots \to \mathbb Z \stackrel0\to \mathbb Z\stackrel{\times 2}\to \mathbb Z\stackrel{0}\to 0.$$ $A_*=2B_*\cong \bigoplus \mathbb Z$. Now this settings give the short exact sequence $$0\to\bigoplus\mathbb Z\stackrel{\times 2}\to \bigoplus \mathbb Z\to \bigoplus \mathbb Z/2\to 0.$$
Finally, it's worth mentioning that the SES of chain complexes are significant when we consider the Long exact sequences of homologies.
For example, the first example here gives the relative homology $H_*(X,Y)$ and also cellular homology theory. The second example gives the Bockstein operations.
A fairly natural "toy" example involves $0\to A\to B\to C\to 0$ with $A$ consisting of linear polynomials with (e.g.) real coefficients, $B$ consisting of quadratic polynomials with real coefficients, and the natural inclusion, with $C$ the quotient. The operator $T=d/dx$ stabilizes each of these, and the (very short) "vertical" complexes $T:A\to A$, $T:B\to B$, and $T:C\to C$ do indeed a short exact sequence of (very short) complexes.
The long exact sequence obtained via the Snake Lemma is $$ 0\to \ker T|_A\to \ker T|_B \to \ker T|_C\to {A\over TA} \to {B\over TB}\to {C\over TC}\to 0$$
This is $$ 0\to \{constants\}\to \{constants\} \to \{cx^2\}\to {\{linear\}\over \{constants\}} \to {B\over TB}\to {C\over TC}\to 0$$
Since the first two joints are exact, the image of constants in $\{cx^2\}$ is $\{0\}$, so the (connecting homomorphism) map of $\{x^2\}$ to $\{linear\}/\{constants\}$ must be non-zero.
It's not toooo hard to work out the details. :)
