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- In algebraic geometry, the theorem on formal functions states the following: Let be a proper morphism of noetherian schemes with a coherent sheaf on X. Let be a closed subscheme of S defined by and formal completions with respect to and . Then for each the canonical (continuous) map:is an isomorphism of (topological) -modules, where
* The left term is .
*
* 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 , topologically, where the completion on the left is with respect to . Corollary: Let r be such that for all . Then Corollay: For each , there exists an open neighborhood U of s such that Corollary: If , then is connected for all . 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. (en)
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- 4458 (xsd:nonNegativeInteger)
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- In algebraic geometry, the theorem on formal functions states the following: Let be a proper morphism of noetherian schemes with a coherent sheaf on X. Let be a closed subscheme of S defined by and formal completions with respect to and . Then for each the canonical (continuous) map:is an isomorphism of (topological) -modules, where
* The left term is .
*
* The canonical map is one obtained by passage to limit. Corollary: For any , topologically, where the completion on the left is with respect to . Corollary: Let r be such that for all . Then (en)
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- Theorem on formal functions (en)
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