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…)