These are rough transcripts of my meetings with my advisor. The notes have not been carefully prepared nor proof-read. So there might be many a wrong statement in it. I have restricted the notes to only the ones where I discuss work of other people. I have not included the transcript of meetings where I discuss my work.
The object of the following is to understand why cohomology of affine schemes vanishes. The main references were EGA - III and Godement's book on Sheaves and Algebraic topology.
The object of the following is the proof of Grothendieck's theorem of the existence of enough injectives in the category of sheaves of abelian groups. All discussion is from his Tôhoku paper. Sorry, I did not type up Abelian Categories I.
We have enough now to define sheaf cohomology and prove a nice theorem of Grothendieck that all higher cohomology groups of a noetherian topological space vanish.
Tate proved as a corollary of his famous isogeny theorem that the characteristic polynomial of Frobenius on Abelian varieties (over finite fields) determines its isogeny class. Inspired by certain comments in the paper, I was trying to see if there is a direct proof of this corollary for elliptic curves. Here is a sketch of the proof, I might fill in the details later.