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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 S0{\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 X0=f−1(S0){\displaystyle X_{0}=f^{-1}(S_{0})} and S0{\displaystyle S_{0}}.

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