# Do canonical stacks exist over Spec(Z)?

Suppose a scheme $X$ has tame quotient singularities. Does there exist a smooth DM stack $\mathcal X$ with coarse space $X$ so that the coarse space morphism $\mathcal X\to X$ is an isomorphism away from codimension 2?

If $X$ is finite type over a field, the answer is yes, by the following argument. Theorem 4.6 of Fantechi-Mann-Nironi's Smooth toric DM stacks shows that such stacks have a universal property, so it suffices to show that they exist étale-locally (the universal property ensures the locally constructed canonical stacks will glue). Étale locally, we may assume $X=U/G$, where $U$ is smooth and $G$ is a finite group acting tamely on $U$—that's the definition of "tame quotient singularities". Let $H\subseteq G$ be the (normal) subgroup acting by pseudoreflections (through a given closed point $x\in U$; shrinking $U$ if necessary). Then by the Chevalley-Shephard-Todd theorem, $T_x/H$ is smooth, where $T_x$ is the tangent space at $x\in U$. If $X$ is defined over the residue field $k(x)$, then we can construct an $H$-equivariant morphism $U\to T_x$ sending $x$ to the origin, which is étale at $x$. Since $T_x/H$ is smooth, so is $U/H$. (If $X$ isn't defined over $k(x)$, base change to $k(x)$ and get smoothness of $U/H$ by descent.) The action of $G/H$ on $U/H$ is free away from codimension 2, so $\mathcal X = [(U/H)/(G/H)]$ does the trick.

As far as I can tell, the only place we had to use that $X$ is defined over a field was in showing smoothness of $U/H$. Without it, we can't construct the étale morphism $U/H\to T_x/H$ we need to deduce smoothness of $U/H$ from smoothness of $T_x/H$.

Can this "over a field" condition be relaxed to get the existence of canonical stacks in an absolute setting (i.e. over $\mathrm{Spec}(\mathbb Z)$)?

Context: my more general goal is to understand if canonical stacks exist over an arbitrary base $S$. That is, suppose $X$ is a scheme (probably locally of finite type) over a base $S$ which has an étale cover by a disjoint union of schemes of the form $U/G$, where $U$ is smooth over $S$, and $G$ is an abstract group acting on $U$ (over $S$), with order relatively prime to all residue fields of $U$. Then does there exist a stack $\mathcal X$ which is smooth over $S$ and has coarse space $X$, such that the coarse space morphism is an isomorphism away from codimension 2 (on both $X$ and $\mathcal X$)?

Cover your $X$ in the étale topology with schemes of the form $U/G$, where $G$ is prime to all the residue characteristics of $U$, and $U$ is smooth over $S$. Choose a geometric point $p$ of $U$; we can restrict to the stabilizer of $G$, and assume that $G$ leaves $p$ fixed. Call $H$ the subgroup generated by the elements of $G$ that are pseudoreflections, or the identity, when restricted to the fiber along $p$; this is normal in $G$. The scheme $U/H$ is flat over $S$, because of the tameness hypothesis. Furthermore taking quotients by $H$ commutes with base change on $S$, again because of tameness; hence the geometric fiber of $U/H \to S$ along the image of $p$ is the quotient of the geometric fiber, which is smooth, because of Cartan’s and Chevalley’s theorems. In this way we can assume that $G$ stabilizes $p$, and the restriction of $G$ to the fiber through $p$ contains no pseudoreflexions.
Now look at the locus in which $U \to X$ is étale. Notice that if the restriction of $U \to X$ to the fiber is étale at $p$, then $U \to X$ is étale at $p$, by the local criterion of flatness. The locus on which $U \to X$ is étale is open in X; its complement must have codimension larger than $1$, because otherwise it would intersect the fiber in codimension $1$.
Now take two of these charts $U \to X$ and $V \to X$; the normalization of the part of the fibered product $U \times_X V$ that dominates $X$ is étale over $U$ and $V$, by purity of the branch locus. These data give a Q-variety, in the sense of Mumford; from them you get an étale groupoid that defines the stack that you are looking for.
• Thanks. Could you explain how the "regular" and "excellent" hypotheses on $S$ are being used? I think excellence of $S$ is needed to get the étale locus of $U\to X$ to be open and of codimension 1. Is regularity of $S$ needed to get $U/H$ to be smooth over $S$ at $p$? Feb 8, 2015 at 20:22
• No, I don’t think that $S$ being regular is necessary to conclude that $U/H$ is smooth; flatness follows from the fact that $H$ is tame, and then smoothness can be checked on the fibers. The fact that $S$ is regular is used in the last step, when you normalize the fiber product $U \times_S V$, and use purity of the branch locus to conclude that it is étale over$U$ and $V$. Feb 8, 2015 at 21:13