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One of the fundamental properties that distinguishes schemes among all contravariant functors $\mathrm{Sch}^\circ \rightarrow \mathrm{Sets}$ is algebraization: a functor $F$ satisfies algebraization if, whenever $S$ is the spectrum of a complete noetherian local ring and $S_n$ are the infinitesimal neighborhoods of the central point in $S$,

$F(S) = \varprojlim_n F(S_n)$.

I only know of two basic algebraization results: (1) Grothendieck's existence theorem gives algebraization when $F$ is the stack of coherent sheaves on a proper scheme, and (2) SGA3.IX.7.1 gives algebraization for maps from tori into affine group schemes.

It is possible to deduce algebraization for many other functors from these. My question is: are there any other basic algebraization results (that don't eventually reduce to one of these) out there?

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This isn't an answer, just an opinion. I have never heard of any other algebraization theorems; I think they all reduce to the ones you already know. –  Jason Starr Jan 3 '12 at 4:32
    
That is also my impression. I guess that to get a feeling fo this, one should first look for algebraization results in the analytic world (ie over $\bf C$). Are there any general algebraization results for certain classes of open complex analytic manifolds ? (I don't know of any). –  Damian Rössler Jan 3 '12 at 7:51
    
@Roessler: There is an algebraization theorem due to Toen for analytic stacks. However, my impression is that this uses Grothendieck's existence theorem. –  Jason Starr Jan 3 '12 at 15:30
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Faltings proved a few algebraisation results in the late 70s and early 80s in the setting of local algebra that were considerably deeper than any previously known results; see MR0554381 for a lovely example involving algebraisation of formal cohomology groups. (He used these to prove new algebraisation (and topological) results in projective geometry.) –  anon Jan 1 '13 at 6:38
    
Jean-Benoît Bost reminded my that in SGA 2, Exposé IX, Grothendieck proves a comparison theorem between formal and algebraic cohomology which works beyond the proper case. In the following Exposés, he gives applications to fundamental groups and Picard groups. –  ACL Feb 21 '13 at 8:26
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2 Answers

There are algebraizations theorems in Diophantine Geometry of an apparently different nature. In fact, Jean-Benoît Bost has explained how to think of them as variants of Grothendieck's existence theorem over a compactification of $\mathop{\rm Spec} (\mathbf Z)$ and has developed this point of view in many papers.

Examples are:

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The recent effectivity result of Brown-Geraschenko (arXiv:1208.2882) for coherent sheaves on stacks with the resolution property and admitting good moduli spaces does not reduce to the usual ones. The strategy (resolving by algebraizable vector bundles) is somewhat similar as in the projective case (resolving by algebraizable ample line bundles) though.

Another non-standard result, I think, is the algebraization of proper formal schemes admitting an ample family of line bundles (Brenner–Schröer, Thm 6.1) + some extra conditions.

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