So last time, we introduced the first form of the formal function theorem. We said that if {X } was a proper scheme over {\mathrm{Spec} A} with structure morphism {f}, and {\mathcal{I} = f^*(I)} for some ideal {I \subset A}, then there were two constructions one could do on a coherent sheaf {\mathcal{F}} on {X} that were in fact the same. Namely, we could complete the cohomology {H^n(X, \mathcal{F})} with respect to {I}, and we could take the inverse limit { \varprojlim H^n(X, \mathcal{F}/\mathcal{I}^k \mathcal{F})}. The claim was that the natural map

\displaystyle \widehat{H^n(X, \mathcal{F})} \rightarrow \varprojlim H^n(X, \mathcal{F}/\mathcal{I}^k \mathcal{F})

was in fact an isomorphism. This is a very nontrivial statement, but in fact we saw yesterday that a reasonably straightforward proof could be given via diagram-chasing if one appeals to a strong form of the proper mapping theorem.

1. Formal functions, jazzed up

Now, however, we want to jazz this up a little. I won’t do this as much as possible because I don’t want to talk too much about formal schemes yet. On the other hand, I want to replace cohomology groups with higher direct images. (more…)