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Theorem on formal functions
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<p>In <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, the theorem on formal functions states the following: </p> Let f : X → S {\displaystyle f:X\to S} be a <a href="/facts/Proper_morphism/EJHIgjnr">proper morphism</a> of <a href="/facts/Noetherian_scheme/amyCoMDf">noetherian schemes</a> with a coherent sheaf F {\displaystyle {\mathcal {F}}} on <i>X</i>. Let S 0 {\displaystyle S_{0}} be a closed subscheme of <i>S</i> defined by I {\displaystyle {\mathcal {I}}} and X ^ , S ^ {\displaystyle {\widehat {X}},{\widehat {S}}} <a href="/facts/Formal_completion/FCB3BMez">formal completions</a> 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 <ul><li>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}}} .</li> <li> 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})} </li> <li>The canonical map is one obtained by passage to limit.</li></ul> <p>The theorem is used to deduce some other important theorems: <a href="/facts/Stein_factorization/097PaJwk">Stein factorization</a> and a version of <a href="/facts/Zariski%2527s_main_theorem/p4QM6TN2">Zariski's main theorem</a> that says that a <a href="/facts/Proper_morphism/EJHIgjnr">proper</a> <a href="/facts/Birational_morphism/hsWqRhJ1">birational morphism</a> into a <a href="/facts/Normal_variety/TQ6cB9By">normal variety</a> is an isomorphism. Some other corollaries (with the notations as above) are: </p><p>Corollary: For any s ∈ S {\displaystyle s\in S} , topologically, </p> ( ( 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}))} <p>where the completion on the left is with respect to m s {\displaystyle {\mathfrak {m}}_{s}} . </p><p>Corollary: Let <i>r</i> be such that dim f − 1 ( s ) ≤ r {\displaystyle \operatorname {dim} f^{-1}(s)\leq r} for all s ∈ S {\displaystyle s\in S} . Then </p> R i f ∗ F = 0 , i > r . {\displaystyle R^{i}f_{*}{\mathcal {F}}=0,\quad i>r.} <p>Corollay: For each s ∈ S {\displaystyle s\in S} , there exists an open neighborhood <i>U</i> of <i>s</i> such that </p> R i f ∗ F | U = 0 , i > dim f − 1 ( s ) . {\displaystyle R^{i}f_{*}{\mathcal {F}}|_{U}=0,\quad i>\operatorname {dim} f^{-1}(s).} <p>Corollary: If f ∗ O X = O S {\displaystyle f_{*}{\mathcal {O}}_{X}={\mathcal {O}}_{S}} , then f − 1 ( s ) {\displaystyle f^{-1}(s)} is connected for all s ∈ S {\displaystyle s\in S} . </p><p>The theorem also leads to the <a href="/facts/Grothendieck_existence_theorem/Q3YC4G7J">Grothendieck existence theorem</a>, 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.) </p><p>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. </p>