Theorem on formal functions

In algebraic geometry, the theorem on formal functions states the following:^[1]

Let

f:X\to S

be a proper morphism of noetherian schemes with a coherent sheaf

{\mathcal {F}}

on X. Let

S_{0}

be a closed subscheme of S defined by

{\mathcal {I}}

and

{\widehat {X}},{\widehat {S}}

formal completions with respect to

X_{0}=f^{-1}(S_{0})

and

S_{0}

. Then for each

p\geq 0

the canonical (continuous) map:

(R^{p}f_{*}{\mathcal {F}})^{\wedge }\to \varprojlim _{k}R^{p}f_{*}{\mathcal {F}}_{k}

is an isomorphism of (topological)

{\mathcal {O}}_{\widehat {S}}

-modules, where

The left term is $\varprojlim R^{p}f_{*}{\mathcal {F}}\otimes _{{\mathcal {O}}_{S}}{\mathcal {O}}_{S}/{{\mathcal {I}}^{k+1}}$ .
${\mathcal {F}}_{k}={\mathcal {F}}\otimes _{{\mathcal {O}}_{S}}({\mathcal {O}}_{S}/{\mathcal {I}}^{k+1})$
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:^[2] For any $s\in S$ , topologically,

((R^{p}f_{*}{\mathcal {F}})_{s})^{\wedge }\simeq \varprojlim H^{p}(f^{-1}(s),{\mathcal {F}}\otimes _{{\mathcal {O}}_{S}}({\mathcal {O}}_{s}/{\mathfrak {m}}_{s}^{k}))

where the completion on the left is with respect to ${\mathfrak {m}}_{s}$ .

Corollary:^[3] Let r be such that $\operatorname {dim} f^{-1}(s)\leq r$ for all $s\in S$ . Then

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

Corollay:^[4] For each $s\in S$ , there exists an open neighborhood U of s such that

R^{i}f_{*}{\mathcal {F}}|_{U}=0,\quad i>\operatorname {dim} f^{-1}(s).

Corollary:^[5] If $f_{*}{\mathcal {O}}_{X}={\mathcal {O}}_{S}$ , then $f^{-1}(s)$ 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 $i':{\widehat {X}}\to X,i:{\widehat {S}}\to S$ be the canonical maps. Then we have the base change map of ${\mathcal {O}}_{\widehat {S}}$ -modules

i^{*}R^{q}f_{*}{\mathcal {F}}\to R^{p}{\widehat {f}}_{*}(i'^{*}{\mathcal {F}})

.

where ${\widehat {f}}:{\widehat {X}}\to {\widehat {S}}$ is induced by $f:X\to S$ . Since ${\mathcal {F}}$ is coherent, we can identify $i'^{*}{\mathcal {F}}$ with ${\widehat {\mathcal {F}}}$ . Since $R^{q}f_{*}{\mathcal {F}}$ is also coherent (as f is proper), doing the same identification, the above reads:

(R^{q}f_{*}{\mathcal {F}})^{\wedge }\to R^{p}{\widehat {f}}_{*}{\widehat {\mathcal {F}}}

.

Using $f:X_{n}\to S_{n}$ where $X_{n}=(X_{0},{\mathcal {O}}_{X}/{\mathcal {J}}^{n+1})$ and $S_{n}=(S_{0},{\mathcal {O}}_{S}/{\mathcal {I}}^{n+1})$ , one also obtains (after passing to limit):