How is a universal deformation ring in a 1-dimensional DM stack related to the complete etale local ring of coarse moduli scheme?

Question

Let $\mathcal{M}$ be a smooth 1-dimensional Deligne-Mumford stack with finite diagonal, and let $M$ be its coarse moduli scheme (all over some field $k$). Suppose $M$ is also smooth over $k$, and that all local deformation problems for $\mathcal{M}$ are pro-representable (I think this is already implied by the smooth + DM conditions?)

Let $x_0 : \text{Spec }\bar{k}\rightarrow\mathcal{M}$ a geometric point, and let $\bar{x_0}\in M$ be its image in $M$.

Let $\mathcal{R}$ be the completion of the etale local ring at $x_0$ (colimit of global sections of scheme etale neighborhoods of $x_0$), and let $R$ be the completion of the etale local ring of $M$ at $\bar{x_0}$.

This yields a natural map $R\rightarrow\mathcal{R}$ (both are isomorphic to a 1-variable power series ring over $\bar{k}$), whence a map $$p : \text{Spec }\mathcal{R}\rightarrow\text{Spec }R$$

I believe the infinitesimal lifting property of etale morphisms then implies that $\mathcal{R}$ is also the universal deformation ring of the object corresponding to $x_0$ (am I missing any conditions here?).

Let $X_0/\bar{k}$ be the object corresponding to $x_0$, and let $X^\text{univ}/\mathcal{R}$ be the universal deformation of $X_0$ over $\mathcal{R}$.

I would like to say something like: $p$ is finite flat totally ramified with ramification index equal the order of the group of automorphisms of $X_0$ modulo the subgroup of those automorphisms which extend to an automorphism of $X^\text{univ}/\mathcal{R}$.

Is this true? What is a good reference? If it's false, what is the relation between the universal deformation ring of $x_0$ and the complete etale local ring of $x_0$?

Actually, some calculations of mine seem to indicate that may be false, though it's possible I've made a mistake somewhere else...

Answer 1

I believe that this is true - that is, "$p$ is finite flat totally ramified with ramification index equal to the order of the group of automorphisms of $X_0$ modulo the subgroup of those which extend to an automorphism of $X^\text{univ}/\mathcal{R}$."

Specifically, let $G$ be the automorphism group of the object $x_0$. Let $\mathcal{R}$ be the etale local ring of $\mathcal{M}$ at $x_0$. Then there is a natural action of $G$ on $\mathcal{R}$. Let $R$ be the etale local ring of $\overline{x_0}\in M$. Let $\mathcal{M}_{(x_0)} := \mathcal{M}\times_M\text{Spec }R$. From the proof of Theorem 11.3.1 of Olsson's book "Algebraic Spaces and Stacks", we find that $$\mathcal{M}_{(x_0)} \cong [\text{Spec }\mathcal{R}/G]$$ and moreover the composition $$\text{Spec }\mathcal{R}\rightarrow[\text{Spec }\mathcal{R}/G]\rightarrow \text{Spec }R$$ is finite.

Since the map $\text{Spec }R\rightarrow M$ is flat, the projection $[\mathcal{R}/G]\cong\mathcal{X}_{(x_0)}\rightarrow\text{Spec }R$ identifies the target with the coarse moduli scheme of $[\mathcal{R}/G]$ (Theorem 11.1.2 of the same book). Thus, we have $R = \mathcal{R}/G = \mathcal{R}^G$. Since $M,\mathcal{M}$ are smooth and 1-dimensional, $R,\mathcal{R}$ are discrete valuation rings, and the map $R\rightarrow\mathcal{R}$ being a finite map between regular schemes must be flat (see Katz-Mazur's book, end of p507).

At this point standard ramification theory tells us that if $K$ is the kernel of the action of $G$ on $\mathcal{R}$, then since we're working with strictly henselian DVRs, the extension $\mathcal{R}/R$ is totally ramified with ramification index $|G/K|$.

Let $X_0$ be the object over $\overline{k}$ corresponding to $x_0$, and let $X/\mathcal{R}$ be the object corresponding to the canonical map $\text{Spec }\mathcal{R}\rightarrow\mathcal{M}$.

It remains to show that $K$ is precisely the subgroup of automorphisms of $X_0$ which extend to automorphisms of $X^\text{univ}/\mathcal{R}$. For this, we use the interpretation of $\mathcal{R}$ as the universal deformation ring which pro-represents the deformation functor $F_{X_0}$ which to every artinian local $\overline{k}$-algebra $A$ associates the set of isomorphism classes $$\{(X/A,\;\;\varphi : X_{\overline{k}}\cong X_0)\}/\cong$$ Then group $G$ acts naturally on these sets by acting on the isomorphism $\varphi$, ie $g\in G$ sends $(X,\varphi)$ to $(X,g\circ\varphi)$, and hence $G$ acts naturally on the functor $F_{X_0}$, and hence on the prorepresenting object $\mathcal{R}$. From this it is clear that an element $g\in G$ fixes $\mathcal{R}$ if and only if for every deformation $X/A$ over Artinian local $\overline{k}$-algebras $A$, there exists an extension of $g$ to an automorphism of $X/A$. Since $\mathcal{M}$ is assumed Deligne-Mumford, the diagonal is unramified (hence the automorphism group schemes of objects are unramified - in particular, formally unramified), so such extensions of $g$, if they exist, are unique. This implies that whenever $g$ extends to every $X/A$, the various extensions are compatible with each other, and hence gives an automorphism of $X^\text{univ}/\mathcal{R}$. Conversely, it's clear that automorphism of $X^\text{univ}/\mathcal{R}$ restricts to an automorphism $g\in G$ of $X_0$ which acts trivially on the deformation functor, and hence on $\mathcal{R}$.