Theorem on formal functions


In algebraic geometry, the theorem on formal functions states the following:
The theorem is used to deduce some other important theorems: Stein factorization and a version of Zariski's main theorem that says that a proper birational morphism into a normal variety is an isomorphism. Some other corollaries are:
Corollary: For any, topologically,
where the completion on the left is with respect to.
Corollary: Let r be such that for all. Then
Corollay: For each, there exists an open neighborhood U of s such that
Corollary: If, then is connected for all.
The theorem also leads to the Grothendieck existence theorem, which gives an equivalence between the category of coherent sheaves on a scheme and the category of coherent sheaves on its formal completion
Finally, it is possible to weaken the hypothesis in the theorem; cf. Illusie. According to Illusie, the proof given in EGA III is due to Serre. The original proof was never published.

The construction of the canonical map

Let the setting be as in the lede. In the proof one uses the following alternative definition of the canonical map.
Let be the canonical maps. Then we have the base change map of -modules
where is induced by. Since is coherent, we can identify with. Since is also coherent, doing the same identification, the above reads:
Using where and, one also obtains :
where are as before. One can verify that the composition of the two maps is the same map in the lede.