§ Fundamental theorem of homological algebra [TODO ]
- Let be an module.
- A resolution of is an exact chain complex
... -> M2 -> M1 -> M0 -> M -> 0
- A projective resolution of
M is a resolution such that all the
P* are projective.
§ Fundamental theorem
- 1. Every
R module has projective resolution.
- 2. Let
P* be a chain complex of proj. R modules. Let
Q* be a chain complex with vanishing homology in degree greater than zero. Let
[P*, Q*] be the group of chain homotopoloy classes of chain maps from
Q*. We are told that this set is in bijection with maps
[H0(P*), H0(Q*)]. That is, the map takes
H0[f*] is a bijection.
§ Corollary: two projective resolutions are chain homotopy equivalent
P1 -> P0 -> M and
... -> Q1 -> Q0 -> M be two projective resolutions.
H0(P*) has an epi mono factorization
P0 ->> H0(P*) and
H0(P*) ~= M.
§ Proof of existence of projective resolution
- Starting with
M there always exists a free module
P0 that is epi onto
M, given by taking the free module of all elements of
M. So we get
P0 -> M -> 0.
- Next, we take the kernel, which gives us:
vP0 -> M -> 0
- The next
P1 must be projective, and it must project onto
ker e for homology to vanish. So we choose the free module generated by elements of
ker e to be
| v e
P1--- P0 -> M -> 0
- Composing these two maps gives us
P1 -> P0 -> M. Iterate until your heart desires.
§ Chain homotopy classes of chain maps