Why is it natural or useful to organize objects (of some appropriate category) into exact sequences? Exact sequences are ubiquitous - and I've encountered them enough to know that they can provide a very useful and efficient framework to work within. However, I have no idea what this framework truly is, or why it is effective.

So, my questions are:

  1. What makes exact sequences natural objects to deal with?

  2. What do they encode, generally speaking? Or if you are unable to think of a satisfactory answer in general, what are some specific examples of exact sequences encoding some desirable property?

Please set me straight! It seems like all of the references that I've come across only encyclopedically develop the idea of an exact sequence, sparing the reader of any qualification or exposition.

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    $\begingroup$ A more general question one might ask is "what are chain complexes, metaphysically speaking" (as exact sequences are just the chain complexes with trivial homology) and then one answer is that chain complexes are a "linearization" of simplicial complexes in a fairly precise sense (see ncatlab.org/nlab/show/Dold-Kan+correspondence ). $\endgroup$ – Qiaochu Yuan May 5 '12 at 2:55
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    $\begingroup$ Exact sequences are an easy way of specifying many requirements at once. $\endgroup$ – Zhen Lin May 5 '12 at 6:27
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    $\begingroup$ Can I just add that this is something that seems to be left out of every textbook like some sort of secret code? I've never once seen a development that attempted to justify studying exact sequences before (or after) introducing the formal definition. $\endgroup$ – Daniel McLaury May 10 '12 at 5:02
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    $\begingroup$ @Joseph: It's fine to correct old typos, and improve the post. But please do not remove thank you messages. This goes against the spirit of this community. $\endgroup$ – Asaf Karagila Nov 21 '13 at 23:20

This was too long to put as a comment, I apologize if it doesn't help.

I don't know how totally accurate this is, but I like to think of (short) exact sequences as being algebraified versions of fiber bundles. Thus, putting $X$ in a short exact sequence $0\to Y\to X\to Z\to0$ indicates to me that $X$ is put together in some way from $Y$ and $Z$, and in such a way that, in a perfect world where everything is nice, is just the product of $Y$ and $Z$. Therefore, $X$ is some kind of "twisted product" of $Y$ and $Z$.

Thus, any time we are able to put $X$ into an exact sequene we should (in spirit) be able to tell properties of $X$ from properties of $Y$ and $Z$.

For example, knowing that $B$ is an abelian groups such that

$$0\to A\to B\to C\to 0$$

is a SES for $B,C$ also abelian groups tells me that $\text{rank}(B)=\text{rank}(A)+\text{rank}(C)$ (or more generally this works nicely for modules over PIDs).

The reason that SESs are such a convenient framework to deal with the notion of "put-togetheredness" is that we live in a fundamentally arrow obsessed world. Things phrased entirely in terms of arrows make us happy, because they are often easy to deal with.

  • $\begingroup$ This does help. Thanks. -- If you're able or willing to play the devil's advocate, can you think of an instance of an exact sequence for which it might be unreasonable to regard it as an expression of 'put-togetheredness'? $\endgroup$ – Joshua Seaton May 5 '12 at 15:15
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    $\begingroup$ @JSeaton: short exact sequences (in nice enough categories) are definitely "B is put together from A and C". Resolutions are longer sequences that either go off to the left or to the right, and are more loosely "C is B with something like A removed, except the thing removed is only like A with something else removed...". Exact chain complexes that go on forever in both directions are even more loosely described as "We put things in, and we take things out, and we haven't left anything out, but it's pretty hard to say where anything actually went." $\endgroup$ – Jack Schmidt Jun 25 '12 at 4:04
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    $\begingroup$ Possibly helpful example of a twisting: $0 \to \mathbb{Z}_2 \to \mathbb{Z}_2 \oplus \mathbb{Z}_2 \to \mathbb{Z}_2 \to 0$ and $0 \to \mathbb{Z}_2 \to \mathbb{Z}_4 \to \mathbb{Z}_2 \to 0$ have different middles but the same components. One is the direct product, and the other is a semidirect product. $\endgroup$ – leewz Sep 25 '14 at 4:33
  • $\begingroup$ Lovely answer, Alex! $\endgroup$ – Brenin Jan 17 '15 at 10:56
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    $\begingroup$ @leewangzhong 's exact sequences are good examples of trivial vs. nontrivial SES. So are direct vs. semidirect products. But the second example (with $\mathbb Z_4$ in the middle) is not a semidirect product because it does not have a section. Just wanted to clarify. $\endgroup$ – SMoore Jan 29 '16 at 23:39

One algebraic answer is that exact sequences are a natural abstraction of the notion of generators and relations. That is, let $R$ be a ring and $M$ a left $R$-module with generating set $S$. Then there is a canonical surjection $$R^S \xrightarrow{f} M \to 0.$$

The kernel of this surjection describes all the possible relations in $S$ and gives rise to a short exact sequence $$0 \to \text{ker}(f) \to R^S \xrightarrow{f} M \to 0.$$

If $R$ is a PID, then $\text{ker}(f)$ is free, so picking a basis for $\text{ker}(f)$ gives an irredundant set of relations among the generators. However, if $\text{ker}(f)$ is not free, then picking a defining set of relations $T$ (that is, a generating set in $\text{ker}(f)$) instead gives rise to an exact sequence $$0 \to \text{ker}(g) \to R^T \xrightarrow{g} R^S \xrightarrow{f} M \to 0.$$

If $\text{ker}(g)$ is not free, then... and so on. From this perspective we are thinking of exact sequences as resolutions.


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