Exact Sequence - Special Cases

Special Cases

To make sense of the definition, it is helpful to consider what it means in relatively simple cases where the sequence is finite and begins or ends with 0.

• The sequence 0 → A → B is exact at A if and only if the map from A to B has kernel {0}, i.e. if and only if that map is a monomorphism (one-to-one).
• Dually, the sequence B → C → 0 is exact at C if and only if the image of the map from B to C is all of C, i.e. if and only if that map is an epimorphism (onto).
• A consequence of these last two facts is that the sequence 0 → X → Y → 0 is exact if and only if the map from X to Y is an isomorphism.

Important are short exact sequences, which are exact sequences of the form

By the above, we know that for any such short exact sequence, f is a monomorphism and g is an epimorphism. Furthermore, the image of f is equal to the kernel of g. It is helpful to think of A as a subobject of B with f being the embedding of A into B, and of C as the corresponding factor object B/A, with the map g being the natural projection from B to B/A (whose kernel is exactly A).