Theorem on formal functions

In algebraic geometry, the theorem on formal functions states the following:

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 the projective limit of the pull-backs of ${\mathcal {F}}$ along $(X_{0},{\mathcal {O}}_{X}/{{\mathcal {J}}^{n+1}})\to X$, ${\mathcal {J}}$ defining $X_{0}$.
• ${\mathcal {F}}_{k}={\mathcal {F}}\otimes _{{\mathcal {O}}_{S}}({\mathcal {O}}_{S}/{\mathcal {I}}^{k+1}).$

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 $s\in S$,

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

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

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

Corollary: 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.

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

$R^{q}{\widehat {f}}_{*}{\widehat {\mathcal {F}}}\to \varprojlim R^{p}f_{*}{\mathcal {F}}_{n}$

where ${\mathcal {F}}_{n}$ are as before. The composition of the two maps is the one in the lede. (To see that the map we constructed is the same one as in EGA, see EGA III-1.)

References

• Luc Illusie, Topics in Algebraic Geometry
• Grothendieck, Alexandre; Dieudonné, Jean (1961). "Eléments de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents, Première partie". Publications Mathématiques de l'IHÉS. 11. doi:10.1007/bf02684274. MR 0217085.
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157