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.

Here is the link to the paper (sadly, not available openly–it’s in the 1980 Rocky Mountain Journal of Math). I’ll maybe talk about this tomorrow.