TITLE: Intuitive meaning of Exact Sequence
QUESTION [87 upvotes]: I'm currently learning about exact sequences in grad sch Algebra I course, but I really can't get the intuitive picture of the concept and why it is important at all.
Can anyone explain them for me? Thanks in advance.

REPLY [40 votes]: There are many good answers here.  I'd just like to add one example that made exact sequences 'click' for me, related to "Euler's Formula" relating the number of vertices ($V$), edges ($E$), and faces ($F$) of a simple non-self-intersecting polyhedron:
$$ |F| - |E| + |V| = 2$$
Now what does this have to do with exact sequences, you may well ask!  Well if you consider the free abelian groups generated by the set of faces, edges, and vertices separately, and create certain linear maps between them (see 'boundary maps' for simplicial homology), then you almost get an exact sequence:
$$
 \mathbb{Z}[F] \to \mathbb{Z}[E] \to \mathbb{Z}[V]
$$
In fact, this sequence is exact at the middle term.  If we append two rank $1$ groups on the left and right (one with a generator the the whole solid $S$, and one generated by the symbol $e =$ '$\emptyset$'), then you do get an exact sequence:
$$
  0 \to \mathbb{Z}[S] \to \mathbb{Z}[F] \to \mathbb{Z}[E] \to \mathbb{Z}[V]
  \to \mathbb{Z}[e] \to 0
$$
Then Euler's Formula is the statement just that the alternating sum of ranks is $0$ (because there is no torsion to keep track of).
$$
 -1 + |F| - |E| + |V| - 1 = 0,
$$
or
$$
  |F| - |E| + |V| = 2.
$$
Hope this helps!