• Let $M$ be an $R$ module.
• A resolution of $M$ is an exact chain complex ... -> M2 -> M1 -> M0 -> M -> 0
• A projective resolution of P* 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 P* to Q*. We are told that this set is in bijection with maps [H0(P*), H0(Q*)]. That is, the map takes f* to H0[f*] is a bijection.

§ Corollary: two projective resolutions are chain homotopy equivalent

• Let 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:

     ker e        |        |   e        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 P1!

    ker e    ^   |    |   v  eP1---   P0 -> M -> 0

• Composing these two maps gives us P1 -> P0 -> M. Iterate until your heart desires.

§ Chain homotopy classes of chain maps