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Question

How does one define the complete local ring of an algebraic stack at a (geometric) point?

Including what the right definition might be, this is all I'm asking.

I can do this for schemes and Deligne--Mumford stacks using the pro-representing rings of deformation functors (i.e. functor of maps from artin rings). However, I don't know if these deformation functors are pro-representable if our algebraic stack admits no etale cover.

I present the solution for schemes and DM stacks for those interested.

Solution for schemes

Let $S$ be a locally noetherian scheme and pick a point $s = \operatorname{Spec} k \in S$. Fix an inclusion $k \to \bar k$ and take the corresponding geometric point $\bar s = \operatorname{Spec} \bar k \to S$. Let $\mathfrak{o}_k$ and $\mathfrak{o}_{\bar k}$ be the Cohen rings corresponding to $k$ and $\bar k$ (i.e. the coefficient rings of Cohen structure theorem). Note that the inclusion gives us a flat extension of local rings $\mathfrak{o}_k \to \mathfrak{o}_{\bar k}$.

Now we may define $\hat{\mathcal{O}}_{S,\bar s} := \hat{\mathcal{O}}_{S,s}\otimes_{\mathfrak{o}_k} \mathfrak{o}_{\bar k}$. This is a complete local ring with residue field $\bar k$. But this is cheating and holds no moduli content which we can exploit.

Instead the following observation suggests the general definition. Let $\operatorname{Art}_{\mathfrak{o}_{\bar k}}$ denote the category of local artinian $\mathfrak{o}_{\bar k}$-algebras of finite type over $\bar k$ (hence with residue field $\bar k$). Consider the functor $F: \operatorname{Art}_{\mathfrak{o}_{\bar k}} \to (\text{Sets})$ defined by $A \mapsto \hom_{\bar s}(\operatorname{Spec} A, S)$, where the subscript $\bar s$ indicates that the morphisms to $S$ restrict to $\bar s \to S$ over the residue field of $A$.

It is easy to see that $\hat{\mathcal{O}}_{S,\bar s}$ pro-represents $F$. This should be our definition in general.

Solution for DM stacks

Let $X \to S$ be a an algebraic stack defined over a base scheme $S$. Suppose $\bar s = \operatorname{Spec} \bar k \to S$ is a geometric point and $\bar x = \operatorname{Spec} \bar k \to X$ is a geometric point of $X$ lying over $\bar s$. Pick an etale cover $U \to X$ and a geometric point $\bar u \to U $ lying over $\bar x$.

This time consider the functor $F:\operatorname{Art}_{\hat{\mathcal{O}}_{S,\bar s}} \to (\text{Sets}) : A \mapsto \hom_{\bar x}(\operatorname{Spec} A,X)$. The geometric formal neighbourhood of a scheme is defined above, the notation $\hom_{\bar x}$ should really mean that we consider morphisms from $\operatorname{Spec} A$ which is 2-isomorphic to $\bar x \to X$ \emph{and} a 2-isomorphism between the two things.

Because $U \to X$ is etale, $F$ is isomorphic to the funtor $A \mapsto \hom_{\bar u}(\operatorname{Spec} A,U)$ which we studied above. So $F$ is pro-representable with representing ring $\hat{\mathcal{O}}_{U,\bar u}$.

In particular, the ring pro-representing $F$ can be defined without reference to an atlas. We can simply denote this ring by $\hat{\mathcal{O}}_{X,\bar x}$ and call it the complete local ring of the geometric point $\bar x$.

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  • $\begingroup$ Your intuition that it might not be pro-representable is correct. If you consider a simple example like $\mathbb A^1 / \mathbb G_m$ where $\mathbb G_m$ acts by translation, then formal extensions of the point $0$ to an Artin local ring $R$ are given by elements of the maximal ideal of $R$, with morphisms given by multiplication by units of $R$. Because some of the points have nontrivial automorphisms, this functor is clearly not representible. $\endgroup$ – Will Sawin Dec 20 '16 at 22:56
  • $\begingroup$ This seems to be related to the concept of versal deformations of singularities. In these cases, you are able to define a local ring of the moduli space of singularities, even though it is not a Deligne-Mumford stack. You could look at this literature - maybe it generalizes. $\endgroup$ – Will Sawin Dec 20 '16 at 22:58
  • $\begingroup$ @WillSawin Thanks for the suggestion. It turns out that the deformation functors of singularities can be shown to be pro-representable by checking Schlessinger's conditions. This is no longer an option for an arbitrary stack, unless of course by some hidden argument it is. $\endgroup$ – Emre Dec 21 '16 at 20:51
  • $\begingroup$ Even if an algebraic stack admits only a smooth cover by a scheme, the local functors will admit a miniversal family meaning a complete noetherian local ring with a smooth map to your local pro functor, which is an isomorphism on tangent spaces (note that in the scheme world smooth maps that are isomorphism on tangent spaces are étale, but not here). This ring will be unique (up to unique isomorphism). If you want, in this way deformation theory produces for you a "smallest" cover of your stack. $\endgroup$ – Rieux Jun 18 '17 at 18:41

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