Notes now through Feb 23

They're still here. We're getting to Frobenius splitting soon!

Feb 3,5,8

Obviously I'm not keeping up the blog too well; the notes are more reliably up-to-date.
Anyway:

Derived functors. Injective resolutions. Acyclic objects for a functor.
Theorem: a resolution by acyclic objects is good enough for computing derived functors.

The category of sheaves has enough injectives. Flasque sheaves. Injective sheaves are flasque, flasque are acyclic.

Feb 1

Finished horseshoe lemma.
Used classification of indecomposable reps of the A_n quiver to motivate the definition of homotopy operator.
Lemma: if phi_1, phi_2 : (P_i) -> (M_i) are chain maps inducing the same map on cohomology, and the (P_i) are projective, then there exists a homotopy operator.

Cor: if (P_i),(Q_i) are two projective resolutions, then they're homotopic. Hence when we apply a right exact functor to them, those complexes are still homotopic, so have the same cohomology.

Def. Injective modules.
Stated (but didn't prove) the analogue of the "TFAE" theorem we gave for projective modules.
Gave some examples injective Z-modules.