In algebraic geometry, the theorem on formal functions states the following:
Let f : X → S {\displaystyle f:X\to S} be a proper morphism of noetherian schemes with a coherent sheaf F {\displaystyle {\mathcal {F}}} on X. Let S 0 {\displaystyle S_{0}} be a closed subscheme of S defined by I {\displaystyle {\mathcal {I}}} and X ^ , S ^ {\displaystyle {\widehat {X}},{\widehat {S}}} formal completions with respect to X 0 = f − 1 ( S 0 ) {\displaystyle X_{0}=f^{-1}(S_{0})} and S 0 {\displaystyle S_{0}} . Then for each p ≥ 0 {\displaystyle p\geq 0} the canonical (continuous) map:
( R p f ∗ F ) ∧ → lim ← k R p f ∗ F k {\displaystyle (R^{p}f_{*}{\mathcal {F}})^{\wedge }\to \varprojlim _{k}R^{p}f_{*}{\mathcal {F}}_{k}}
is an isomorphism of (topological) O S ^ {\displaystyle {\mathcal {O}}_{\widehat {S}}} -modules, where The left term is lim ← R p f ∗ F ⊗ O S O S / I k + 1 {\displaystyle \varprojlim R^{p}f_{*}{\mathcal {F}}\otimes _{{\mathcal {O}}_{S}}{\mathcal {O}}_{S}/{{\mathcal {I}}^{k+1}}} .
F k = F ⊗ O S ( O S / 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: For any s ∈ S {\displaystyle s\in S} , topologically,
( ( R p f ∗ F ) s ) ∧ ≃ lim ← H p ( f − 1 ( s ) , F ⊗ O S ( O s / m s k ) ) {\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}))}
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