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| In algebraic geometry, the '''theorem on formal functions''' states the following:<ref>{{harvnb|EGA III-1|loc=4.1.5}}</ref>
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| :Let <math>f: X \to S</math> be a [[proper morphism]] of noetherian schemes with a coherent sheaf <math>\mathcal{F}</math> on ''X''. Let <math>S_0</math> be a closed subscheme of ''S'' defined by <math>\mathcal{I}</math> and <math>\widehat{X}, \widehat{S}</math> [[formal completion]]s with respect to <math>X_0 = f^{-1}(S_0)</math> and <math>S_0</math>. Then for each <math>p \ge 0</math> the canonical (continuous) map:
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| ::<math>(R^p f_* \mathcal{F})^\wedge \to \varprojlim_k R^p f_* \mathcal{F}_k</math>
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| :is an isomorphism of (topological) <math>\mathcal{O}_{\widehat{S}}</math>-modules, where
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| :*The left term is <math>\varprojlim R^p f_* \mathcal{F} \otimes_{\mathcal{O}_S} \mathcal{O}_S/{\mathcal{I}^{k+1}}</math>.
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| :*<math>\mathcal{F}_k = \mathcal{F} \otimes_{\mathcal{O}_S} (\mathcal{O}_S/{\mathcal{I}}^{k+1})</math>
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| :*The canonical map is one obtained by passage to limit.
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| 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 morphism|proper]] [[birational morphism]] into a [[normal variety]] is an isomorphism. Some other corollaries (with the notations as above) are:
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| '''Corollary''':<ref>{{harvnb|EGA III-1|loc=4.2.1}}</ref> For any <math>s \in S</math>, topologically,
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| :<math>((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))</math> | |
| where the completion on the left is with respect to <math>\mathfrak{m}_s</math>.
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| '''Corollary''':<ref>{{harvnb|Hartshorne|loc=Ch. III. Corollary 11.2}}</ref> Let ''r'' be such that <math>\operatorname{dim} f^{-1}(s) \le r</math> for all <math>s \in S</math>. Then
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| :<math>R^i f_* \mathcal{F} = 0, \quad i > r.</math>
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| '''Corollay''':<ref>The same argument as in the preceding corollary</ref> For each <math>s \in S</math>, there exists an open neighborhood ''U'' of ''s'' such that
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| :<math>R^i f_* \mathcal{F}|_U = 0, \quad i > \operatorname{dim} f^{-1}(s).</math>
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| '''Corollary''':<ref>{{harvnb|Hartshorne|loc=Ch. III. Corollary 11.3}}</ref> If <math>f_* \mathcal{O}_X = \mathcal{O}_S</math>, then <math>f^{-1}(s)</math> is connected for all <math>s \in S</math>.
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| 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.)
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| 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.
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| == The construction of the canonical map ==
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| Let the setting be as in the lede. In the proof one uses the following alternative definition of the canonical map.
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| Let <math>i': \widehat{X} \to X, i: \widehat{S} \to S</math> be the canonical maps. Then we have the [[base change map]] of <math>\mathcal{O}_{\widehat{S}}</math>-modules
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| :<math>i^* R^q f_* \mathcal{F} \to R^p \widehat{f}_* (i'^* \mathcal{F})</math>.
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| where <math>\widehat{f}: \widehat{X} \to \widehat{S}</math> is induced by <math>f: X \to S</math>. Since <math>\mathcal{F}</math> is coherent, we can identify <math>i'^*\mathcal{F}</math> with <math>\widehat{\mathcal{F}}</math>. Since <math>R^q f_* \mathcal{F}</math> is also coherent (as ''f'' is proper), doing the same identification, the above reads:
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| :<math>(R^q f_* \mathcal{F})^\wedge \to R^p \widehat{f}_* \widehat{\mathcal{F}}</math>.
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| Using <math>f: X_n \to S_n</math> where <math>X_n = (X_0, \mathcal{O}_X/\mathcal{J}^{n+1})</math> and <math>S_n = (S_0, \mathcal{O}_S/\mathcal{I}^{n+1})</math>, one also obtains (after passing to limit):
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| :<math>R^q \widehat{f}_* \widehat{\mathcal{F}} \to \varprojlim R^p f_* \mathcal{F}_n</math>
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| where <math>\mathcal{F}_n</math> are as before. One can verify that the composition of the two maps is the same map in the lede. (cf. EGA III-1, section 4)
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| == Notes == | |
| {{reflist}}
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| == References ==
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| *[[Luc Illusie]], [http://staff.ustc.edu.cn/~yiouyang/Illusie.pdf Topics in Algebraic Geometry]
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| *{{EGA|book=III-1}}
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| *{{Hartshorne AG}}
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| [[Category:Algebraic geometry]]
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