Grothendieck Existence, which I imagine is the less well known result among the two, states the following:
Let $A$ be a noetherian ring that is complete w.r.t. a proper ideal $I$. Let $V$ be a proper $A$-scheme. Let $W$ be the inverse image of the locus of $I$ (as a subscheme of $V$). Let $\mathfrak{V}=(W,\mathcal{O}\_{\mathfrak{V}})$ be the formal completion of $V$ along $W$. Then the functor $\mathcal{F} \mapsto \hat{\mathcal{F}}$ from the category of coherent $\mathcal{O} \_ V$ -modules to the category of coherent $\mathcal{O}\_{\mathfrak{V}}$-modules is an equivalence of categories.


This results leaves a Schlessinger taste in my mouth. It seems like we're showing a certain deformation is effective. Is this a consequence of Schlessinger's criterion? Does it go the other way around? Are they just completely unrelated and I'm just seeing patterns where there are none?



How does Schlessinger's criterion sit with Grothendieck Existence (aka GFGA)?
Schlessinger's criterion is a criterion for the pro-representability of a functor. This is the same thing is getting something like an object on the formal scheme $\mathfrak V$. Grothendieck's result tells us that this object comes from an actual object on $V$. Usually this is the step after pro-representability to get actual representability (and is usually formulated as saying that the formal deformation is "effective", a property that goes beyond pro-representability). This is precisely the setup of Artin's criteria for representability by an algebraic space.


