TITLE: Short exact sequences every mathematician should know
QUESTION [139 upvotes]: I'd like to have a big-list of "great" short exact sequences that capture some vital phenomena. I'm learning module theory, so I'd like to get a good stock of examples to think about. An elementary example I have in mind is the SES:
$$
0 \rightarrow I \cap J \rightarrow I \oplus J \rightarrow I + J \rightarrow 0
$$
from which one can recover the rank-nullity theorem for vector spaces and the Chinese remainder theorem.
I'm wondering what other 'bang-for-buck' short exact sequences exist which satisfy one of the criteria:

They portray some deep relationship between the objects in the sequence that is non-obvious, or
They describe an interesting relationship that is obvious, but is of important consequence.

REPLY [4 votes]: The defining short exact sequence for Milnor's $K_2(R)$ ($R$ any ring) is
$$0\rightarrow K_2(R)\rightarrow St(R)\rightarrow E(R)\rightarrow 0$$
where $St(R)$ and $E(R)$ are the Steinberg and elementary groups.  If we cheat a little on the definition of "short" this extends to
$$0\rightarrow K_2(R)\rightarrow St(R)\rightarrow GL(R)\rightarrow K_1(R)\rightarrow 0$$