Exact sequence

Suppose we have three groups connected by two homomorphisms $$f_1 : C_2 \to C_1,f_0 : C_1 \to C_0.$$ Then we say that the this sequence is exact if the image of $f_1$ is equal to the kernel of $f_0$.

Now, a sequence of homomorphisms of any length is said to be exact if every piece of it is exact: $$ ... → C_k → C_{k-1} → ... → C_1 → C_0 → 0.$$ In other words, $${\rm im} f_{k+1} = {\rm ker} f_k.$$

Observe that if we replace equality above with inclusion we arrive to the notion of chain complex.

As the main example, see homology exact sequence.

The notion can be generalized to any abelian category.