I'm trying to understand Böckle's proof of Theorem 2.1.1 in [his notes on deformation theory](http://www.iwr.uni-heidelberg.de/groups/arith-geom/boeckle/Deformations-Barca.pdf). 


Let's start with some motivation. Let $\Gamma$ be a profinite group (I'm thinking of an absolute Galois group), $k$ a finite field, and $\bar\rho\colon \Gamma\to \operatorname{GL}\_d(k)$ a continuous, absolutely irreducible representation. Let $\mathsf{Ar}\_k$ be the category of (commutative) finite local rings with residue field $k$. There is a *framed deformation functor* $\mathcal{X}\_{\bar\rho}^\square\colon \mathsf{Ar}\_k\to \mathsf{Set}$, that assigns to each $A\in \mathsf{Ar}\_k$ the set of continuous representations $\rho\colon \Gamma\to \operatorname{GL}\_d(A)$ such that $\rho\equiv \bar\rho\pmod{\mathfrak{m}\_A}$. Let $\widehat{\mathrm{PGL}}\_d(A) = \ker(\mathrm{PGL}\_d(A)\to \mathrm{PGL}\_d(k))$. Then $\widehat{\mathrm{PGL}}\_d$ acts freely on $\mathcal{X}^\square\_{\bar\rho}$. 


Put $\mathcal{X}\_{\bar\rho} = \mathcal{X}^\square\_{\bar\rho} / \widehat{\mathrm{PGL}}\_d$ (categorical quotient; you could also take quotient of flat sheaves). Böckle claims that [SGA 3, VIIb Théorème 1.4](http://www.math.cornell.edu/~dkmiller/bin/sga3-1.pdf) implies that $\mathcal{X}\_{\bar\rho}$ is actually the *presheaf* quotient of $\mathcal{X}\_{\bar\rho}^\square$ by $\widehat{\mathrm{PGL}}\_d$. For those who don't have a copy of SGA at hand, here's what the theorem says:



> 
> Let $k$ be a pseudocompact ring and $d\_0,d\_1\colon X\_1\rightrightarrows X$ an equivalence relation in $\mathsf{Vaf}\_k$ (cf. Exp V, §2.b) such that $d\_1$ is topologicaly flat. 
> 
> 
> (a) The canonical projection of $X$ onto $X/X\_1$ ($=\mathrm{Coker}(d\_0,d\_1)$) is surjective and topologically flat, and the morphism $X\_1\to X\times\_{X/X\_1} X$ with components $(d\_0,d\_1)$ is an isomorphism. 
> 
> 
> (b) If $X$ is topologically flat over $k$, then the same is true of $X/X\_1$. 
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> 
> 


Here $\mathsf{Vaf}\_k$ can be taken to be the category of pro-representable functors $\mathsf{Ar}\_k\to \mathsf{Set}$. The problem is, (a) seems to say merely that $X\to X/X\_1$ is surjective *on the underlying sets of points*, and that the "pointwise quotient" of $X$ by $X\_1$ embeds into $X/X\_1$. It's not at all clear to me how this implies what Böckle claims it does. 


Just to be clear, I'm not trying to understand why the "unframed" deformation functor $\mathcal X\_{\bar\rho}$ is pro-representable. I'm trying to understand how its representability follows from the general result in SGA. 



Representability of deformation functors via SGA
It turns out that the only thing Böckle is using is the smoothness of $\widehat{\mathrm{PGL}}\_d$. I claim that the following more general result is true. 



> 
> Suppose $R\rightrightarrows X$ is an equivalence relation in $\mathsf{Vaf}$, for which one of the arrows is flat and one is smooth. Then $X/R$ (categorical or flat quotient) is the presheaf quotient. 
> 
> 
> 


Essentially, all we need to do is prove that $X\to X/R$ is surjective on $A$-points for all $A$. Since we're working with (formal spectra of) pro-artinian rings, this follows from the formal smoothness of $X\to X/R$. But smoothness is a (flat-)local property, so it suffices to check that 
$R=X\times\_{X/R} X\to X$ is smooth, but we assumed this, QED.


