Sep 19

The notes have been updated.

First post!

The notes for 7310 have been updated.

Notes now through Apr 12

We made it through most of the "Higher direct images" section, which is remarkably painless.

Notes now through Apr 5

Including tiny fix to the long, tricky proof today: at one point I claimed we were computing the cohomology of one complex, but thankfully it was of a different, easier complex.

Notes now through Mar 10

Frobenius splitting done, moving on to Serre duality.

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.

Jan 27 and 29

In the notes I'm marking things as 1/27/10, 1/29/10.

Basically, we studied projective modules, talked about long exact sequences on cohomology, defined Exts, and got started on the horseshoe lemma. Also we defined K^0(ring) using projective modules.

Math 7670 begins

January 25: we covered section 1 of the notes.

HW: Exercises 1.1-1.5 from the notes.