So last time, we introduced the first form of the formal function theorem. We said that if was a proper scheme over with structure morphism , and for some ideal , then there were two constructions one could do on a coherent sheaf on that were in fact the same. Namely, we could complete the cohomology with respect to , and we could take the inverse limit . The claim was that the natural map

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…)