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Whee~ I got a perfect score on two homeworks. But also, I just totally didn't do my biology project so in general things were okay, how are things with you?
Also moosehead_beer I'm not sure where to start. ^_^; "A → B" means a homomorphism from A to B (as abelian groups or modules or whatever.)
A sequence A → B → C is "exact at B" if the image of the map A → B is the same as the kernel of the map B → C.
If a sequence is "exact", it's exact everywhere (but not at the ends, because it doesn't make sense for a sequence to be exact at the ends).
Maps "A → 0" and "0 → A" are the unique maps to and from the abelian group with one element (or the module which just has the zero element). Exercise. What can you say about a map
which is exact at A?
A free resolution of A is a sequence
which is exact, and where every F_n is free. The F_n can be anything and the maps can be anything.
Also moosehead_beer I'm not sure where to start. ^_^; "A → B" means a homomorphism from A to B (as abelian groups or modules or whatever.)
A sequence A → B → C is "exact at B" if the image of the map A → B is the same as the kernel of the map B → C.
If a sequence is "exact", it's exact everywhere (but not at the ends, because it doesn't make sense for a sequence to be exact at the ends).
Maps "A → 0" and "0 → A" are the unique maps to and from the abelian group with one element (or the module which just has the zero element). Exercise. What can you say about a map
A free resolution of A is a sequence
