I am now going to discuss Kempf’s proof of the theorem of Serre. Note that this is lifted verbatim of some notes I have been taking, so apologies if the style seems out of place as a result. I use the (highly nonstandard) notation for global sections of a sheaf for entirely logistical (and typo-errorgraphical) reasons. Since this is really better suited to a PDF, I’ll also post that.

(Note: You really should read the PDF, since some diagrams are missing from this post.)

Theorem 1 (Serre)Let be an affine scheme, a quasi-coherent sheaf. Then for .

We shall prove this result following Kempf. The idea is that has a very nice basis: namely, the family of all sets . These are themselves affine, and moreover the intersection of any two elements in this basis is still in this basis. For .

** 0.1. A lemma of Kempf (more…)**