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:
“When we walk the streets at night in safety, it does not strike us that this might be otherwise. This habit of feeling safe has become second nature, and we do not reflect on just how this is due solely to the working of special institutions. Commonplace thinking often has the impression that force holds the state together, but in fact its only bond is the fundamental sense of order which everybody possesses.”
—Georg Wilhelm Friedrich Hegel (1770–1831)
“In most cases a favorite writer is more with us in his book than he ever could have been in the flesh; since, being a writer, he is one who has studied and perfected this particular mode of personal incarnation, very likely to the detriment of any other. I should like as a matter of curiosity to see and hear for a moment the men whose works I admire; but I should hardly expect to find further intercourse particularly profitable.”
—Charles Horton Cooley (1864–1929)