I'm trying to understand Böckle's proof of Theorem 2.1.1 in his notes on deformation theory.

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 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 $X/X_1$.

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.

  • 1
    $\begingroup$ Any formal torsor (for the flat topology, say) for a formally smooth formal affine group over a pseudo-compact local ring is trivial when there's a section over the residue field. In Boeckle's setting the relevant local ring has finite residue field, hence trivial Brauer group, so any ${\rm{PGL}}_d$-torsor over such a field is trivial. $\endgroup$
    – user74230
    Apr 1 '15 at 14:03

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.


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