Theorem on formal functions

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}}} .

Source: Wikipedia — Theorem on formal functions (CC BY-SA 4.0)

Theorem on formal functions

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}}} .

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Source: Wikipedia "Theorem on formal functions" · CC BY-SA 4.0

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