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 $\mathcal{G}m$ 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 ${X = \mathrm{Spec} A}$ be an affine scheme, ${\mathcal{F}}$ a quasi-coherent sheaf. Then ${H^i(X, \mathcal{F}) = 0}$ for ${i \geq 1}$.

We shall prove this result following Kempf. The idea is that ${X}$ has a very nice basis: namely, the family of all sets ${D(f), f \in A}$. These are themselves affine, and moreover the intersection of any two elements in this basis is still in this basis. For ${D(fg) = D(f) \cap D(g)}$.

0.1. A lemma of Kempf (more…)

There is a theorem of Serre that the higher cohomology groups of quasi-coherent sheaves on an affine scheme all vanish.  This is proved in Hartshorne for noetherian rings by showing that the sheaves associated to injective modules are flasque, so can be used to compute cohomology; this proof makes the annoying noetherian hypothesis though. There is a paper of Kempf where he explains how to avoid this, and in fact use pretty much nothing more about sheaf cohomology than its trivalty on flasque sheaves. I’ve been reading it as of late, and I recommend it. Perhaps it will become a blog post shortly.