I was following Ravi Vakil notes FOAG (http://math.stanford.edu/~vakil/216blog/FOAGjun1113public.pdf) and I cannot understand Th.18.2.4, at least not all of it.
What I can follow is the first part: you extent the Cech complex, add $\textrm{Spec}(R) = U_0$ to your open cover, make a clever short exact sequence of complexes, and the long exact sequence in cohomology does the rest.
After this particular case, he want to show that the extended Cech complex is still exact even if $\textrm{Spec}(R)$ is not one of the open sets of the covering. And this general case I cannot understand.
I see the main idea is to show that we can reduce to the particular case, but I will be grateful if someone can break this down into pieces.
Thank you very much.