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).
Read more about this topic: Exact Sequence
Famous quotes containing the words special and/or cases:
“A successful woman preacher was once asked “what special obstacles have you met as a woman in the ministry?” “Not one,” she answered, “except the lack of a minister’s wife.””
—Anna Garlin Spencer (1851–1931)
“I do not believe in lawyers, in that mode of attacking or defending a man, because you descend to meet the judge on his own ground, and, in cases of the highest importance, it is of no consequence whether a man breaks a human law or not. Let lawyers decide trivial cases.”
—Henry David Thoreau (1817–1862)