# 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 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

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

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 ${\displaystyle i':{\widehat {X}}\to X,i:{\widehat {S}}\to S}$ be the canonical maps. Then we have the base change map of ${\displaystyle {\mathcal {O}}_{\widehat {S}}}$-modules

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

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

${\displaystyle (R^{q}f_{*}{\mathcal {F}})^{\wedge }\to R^{p}{\widehat {f}}_{*}{\widehat {\mathcal {F}}}}$.
${\displaystyle R^{q}{\widehat {f}}_{*}{\widehat {\mathcal {F}}}\to \varprojlim R^{p}f_{*}{\mathcal {F}}_{n}}$

where ${\displaystyle {\mathcal {F}}_{n}}$ 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)

## Notes

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4. The same argument as in the preceding corollary
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