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In
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, the theorem on formal functions states the following: :Let f: X \to S be a proper morphism of noetherian schemes with a coherent sheaf \mathcal on ''X''. Let S_0 be a closed subscheme of ''S'' defined by \mathcal and \widehat, \widehat
formal completion In mathematics, specifically in algebraic geometry, a formal scheme is a type of space which includes data about its surroundings. Unlike an ordinary scheme, a formal scheme includes infinitesimal data that, in effect, points in a direction off of ...
s with respect to X_0 = f^(S_0) and S_0. Then for each p \ge 0 the canonical (continuous) map: ::(R^p f_* \mathcal)^\wedge \to \varprojlim_k R^p f_* \mathcal_k :is an isomorphism of (topological) \mathcal_-modules, where :*The left term is \varprojlim R^p f_* \mathcal \otimes_ \mathcal_S/. :*\mathcal_k = \mathcal \otimes_ (\mathcal_S/^) :*The canonical map is one obtained by passage to limit. The theorem is used to deduce some other important theorems:
Stein factorization In algebraic geometry, the Stein factorization, introduced by for the case of complex spaces, states that a proper morphism can be factorized as a composition of a finite mapping and a proper morphism with connected fibers. Roughly speaking, Stein ...
and a version of Zariski's main theorem that says that a proper
birational morphism In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational fu ...
into a normal variety is an isomorphism. Some other corollaries (with the notations as above) are: Corollary: For any s \in S, topologically, :((R^p f_* \mathcal)_s)^\wedge \simeq \varprojlim H^p(f^(s), \mathcal\otimes_ (\mathcal_s/\mathfrak_s^k)) where the completion on the left is with respect to \mathfrak_s. Corollary: Let ''r'' be such that \operatorname f^(s) \le r for all s \in S. Then :R^i f_* \mathcal = 0, \quad i > r. Corollay: For each s \in S, there exists an open neighborhood ''U'' of ''s'' such that :R^i f_* \mathcal, _U = 0, \quad i > \operatorname f^(s). Corollary: If f_* \mathcal_X = \mathcal_S, then f^(s) is connected for all s \in S. The theorem also leads to the
Grothendieck existence theorem In mathematics, the Grothendieck existence theorem, introduced by , gives conditions that enable one to lift infinitesimal deformations of a scheme to a deformation, and to lift schemes over infinitesimal neighborhoods over a subscheme of a scheme ' ...
, 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.


The construction of the canonical map

Let the setting be as in the lede. In the proof one uses the following alternative definition of the canonical map. Let i': \widehat \to X, i: \widehat \to S be the canonical maps. Then we have the base change map of \mathcal_-modules :i^* R^q f_* \mathcal \to R^p \widehat_* (i'^* \mathcal). where \widehat: \widehat \to \widehat is induced by f: X \to S. Since \mathcal is coherent, we can identify i'^*\mathcal with \widehat. Since R^q f_* \mathcal is also coherent (as ''f'' is proper), doing the same identification, the above reads: :(R^q f_* \mathcal)^\wedge \to R^p \widehat_* \widehat. Using f: X_n \to S_n where X_n = (X_0, \mathcal_X/\mathcal^) and S_n = (S_0, \mathcal_S/\mathcal^), one also obtains (after passing to limit): :R^q \widehat_* \widehat \to \varprojlim R^p f_* \mathcal_n where \mathcal_n 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)


Notes


References

* *


Further reading

*{{cite web , author-link=Luc Illusie , first=Luc , last=Illusie , url=http://staff.ustc.edu.cn/~yiouyang/Illusie.pdf , title=Topics in Algebraic Geometry Theorems in algebraic geometry