How does Schlessinger's criterion sit with Grothendieck Existence (aka GFGA)?

Question

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?

Answer 1

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.