algebraization theorems 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?
 A: 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.
A: Bhargav Bhatt has recently proved a remarkable algebraization theorem:
http://arxiv.org/abs/1404.7483
It implies that formal maps into quasi-compact, quasi-separated schemes may be algebraized.  Another version of this result (with slightly different hypotheses) has been proved by Hall and Rydh:
http://arxiv.org/abs/1405.7680
A: 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: 


*

*Theorems of Chudnovsky, André, Bost, according to which some formal subgroups of Abelian varieties defined over number fields are algebraic. This works more generally for leaves of appropriate foliations in algebraic varieties.  See Bost's paper, Algebraic leaves of algebraic foliations over number fields. Publications Mathématiques de l'IHÉS, 93 (2001), p. 161-221. See also my Bourbaki talk on the subject, Théorèmes d'algébricité en géométrie diophantienne. Séminaire Bourbaki, 43 (2000-2001), Exposé No. 886.

*A Lefschetz type theorem, also due to Bost, proving that fundamental groups of some algebraic surfaces are trivial (come from the base), see Potential theory and Lefschetz theorems for arithmetic surfaces. Annales scientifiques de l'École Normale Supérieure, Sér. 4, 32 no. 2 (1999), p. 241-312 

*The theorem of Borel-Dwork-Polya-Bertrandias asserting that some power series with rational coefficients are rational is also of this kind, see my paper with Bost, Analytic curves in algebraic varieties over number fields. In: Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, 69–124,  Progr. Math., 269, Birkhäuser Boston, Inc., Boston, MA, 2009. 
(http://arxiv.org/abs/math/0702593)
