theorem on formal functions について

:
*The canonical map is one obtained by passage to limit.
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 (with the notations as above) are:
Corollary: For any

s \in S

, topologically,
:

((R^p f_* \mathcal)_s)^\wedge \simeq \varprojlim H^p(f^(s), \mathcal\otimes_ (\mathcal_s/\mathfrak_s^k))

where the completion on the left is with respect to

\mathfrak_s

.
Corollary: Let ''r'' be such that

\operatorname f^(s) \le r

for all

s \in S

. Then
:

R^i f_* \mathcal = 0, \quad i > r.

Corollay:〔The same argument as in the preceding corollary〕 For each

s \in S

, there exists an open neighborhood ''U'' of ''s'' such that
:

R^i f_* \mathcal|_U = 0, \quad i > \operatorname f^(s).

Corollary: If

f_* \mathcal_X = \mathcal_S

, then

f^(s)

is connected for all

s \in S

.
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 (in particular, it yields algebralizability.)
Finally, it is possible to weaken the hypothesis in the theorem; cf. Illusie. According to Illusie (pg. 204), the proof given in EGA III is due to Serre. The original proof (due to Grothendieck) 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

i': \widehat \to X, i: \widehat \to S

be the canonical maps. Then we have the base change map of

\mathcal_ \to R^p \widehat_* (i'^* \mathcal)

.
where

\widehat: \widehat \to \widehat

is induced by

f: X \to S

. Since

\mathcal

is coherent, we can identify

i'^*\mathcal

with

\widehat

is also coherent (as ''f'' is proper), doing the same identification, the above reads:
:

(R^q f_* \mathcal)^\wedge \to R^p \widehat_* \widehat_X/\mathcal^)

and

S_n = (S_0, \mathcal_S/\mathcal^)

, one also obtains (after passing to limit):
:

R^q \widehat_* \widehat_n

where

\mathcal_n

are as before. One can verify that the composition of the two maps is the same map in the lede. (cf. EGA III-1, section 4)

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