Theorem on formal functions

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

Let ${\displaystyle f:X\to S}$ be a proper morphism of noetherian schemes with a coherent sheaf ${\displaystyle {\mathcal {F}}}$ on X. Let ${\displaystyle S_{0}}$ be a closed subscheme of S defined by ${\displaystyle {\mathcal {I}}}$ and ${\displaystyle {\widehat {X}},{\widehat {S}}}$ formal completions with respect to ${\displaystyle X_{0}=f^{-1}(S_{0})}$ and ${\displaystyle S_{0}}$. Then for each ${\displaystyle p\geq 0}$ the canonical (continuous) map:

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

is an isomorphism of (topological) ${\displaystyle {\mathcal {O}}_{\widehat {S}}}$-modules, where

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

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

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

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

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

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

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

Corollary:^[5] If ${\displaystyle f_{*}{\mathcal {O}}_{X}={\mathcal {O}}_{S}}$, then ${\displaystyle f^{-1}(s)}$ is connected for all ${\displaystyle s\in S}$.

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.