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.

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    $\begingroup$ For example $A = \mathbb{Z}, B = \mathbb{Z} \oplus \mathbb{Z}, C = \mathbb{Z}$. The morphisms are pretty straightforward. $\endgroup$ – Cosmare Sep 10 '15 at 9:31
  • $\begingroup$ Fairly so but are there any less self-evident ones? Even though I missed that particular one. $\endgroup$ – Zelos Malum Sep 10 '15 at 9:33

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 $A\cap B$.

  • $\begingroup$ Do you know of a "simpler" one perhaps? $\endgroup$ – Zelos Malum Sep 10 '15 at 10:24

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.

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    $\begingroup$ I am working on a long sequence of homologies so thats where it is comming from $\endgroup$ – Zelos Malum Sep 10 '15 at 10:52

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