Do line bundles descend to coarse moduli spaces of Artin stacks with finite inertia?

According to the Keel-Mori theorem if $\mathcal{X}$ is a separated Artin stack of finite type over a scheme S with finite inertia, then it admits a coarse moduli space $X$ (which is an algebraic space). See http://math.stanford.edu/~conrad/papers/coarsespace.pdf for the precise statement.

My question is the following: if there is a line bundle $L$ on $\mathcal{X}$, does some positive power of $L$ descend to $X$? I'm primarily interested in the case when $S=\mathrm{Spec}\ k$ for an algebraically closed field $k$ of positive characteristic, and $S= \mathrm{Spec}\ A$ for some finitely generated $\mathbb{Z}$-algebra $A$ would be the next important case.

• The zeroth power descends :) May 4, 2015 at 23:52
• You changed the question. May 5, 2015 at 0:14
• Yes, I did. I excluded the zeroth power, and added the second case of the base, which I realized is also important for my application. Btw, thanks for pointing it out. May 5, 2015 at 0:24
• You can probably answer positively (for S quasi compact) by using Alper's criterion : namely a vector bundle descends to the moduli space iff the stabilizers act trivially on the stalks. See his paper "Good moduli spaces for Artin stacks" Theorem 10.3 . May 5, 2015 at 7:52
• I think Alper's criterion is on the fibers, not on the stalks. In either case, unfortunately Alper's criterion does not help. This was pointed out to me by Alper himself: there are non-trivial representations of infinitesmial group schemes over the dual numbers that are trivial when restricted over the residue field. In particular, the coarse moduli spaces in the above question are not good in general in Alper's sense. May 5, 2015 at 13:10

As noted in the comments, the question has a positive answer for tame Artin stacks (see e.g. [Ols12, Prop 6.1]) and also for non-tame Deligne–Mumford stacks (see [KV04, Lem. 2]). It is however also true in general. Throughout, let $\mathscr{X}$ be an algebraic stack with finite inertia and let $\pi\colon \mathscr{X}\to X$ denotes its coarse moduli space.

Proposition 1. The map $\pi^*\colon \mathrm{Pic}(X)\to \mathrm{Pic}(\mathscr{X})$ is injective and if $\mathscr{X}$ is quasi-compact, then $\mathrm{coker}(\pi^*)$ has finite exponent, i.e., there exists a positive integer $n$ such that $\mathcal{L}^{n}\in \mathrm{Pic}(X)$ for every $\mathcal{L}\in \mathrm{Pic}(\mathscr{X})$.

For simplicity, assume that $\pi$ is of finite type (as is the case if $\mathscr{X}$ is of finite type over a noetherian base scheme) although this is not necessary for any of the results (the only property that is used is that $\mathscr{X}\to X$ is a universal homeomorphism and that invertible sheaves are trivial over semi-local rings etc).

Lemma 1. The functor $\pi^*\colon \mathbf{Pic}(X)\to \mathbf{Pic}(\mathscr{X})$ is fully faithful. In particular:

1. If $\mathcal{L}\in \mathrm{Pic}(X)$, then the adjunction map $\mathcal{L}\to\pi_*\pi^*\mathcal{L}$ is an isomorphism and the natural map $H^0(X,\mathcal{L})\to H^0(\mathscr{X},\pi^*\mathcal{L})$ is an isomorphism.

2. The natural map $\pi^*\colon \mathrm{Pic}(X)\to \mathrm{Pic}(\mathscr{X})$ is injective.

Moreover, a line bundle on $\mathscr{X}$ that is locally trivial on $X$ comes from $X$, that is:

1. Let $g\colon X'\to X$ be faithfully flat and locally of finite presentation and let $f\colon \mathscr{X}':=\mathscr{X}\times_X X'\to \mathscr{X}$ denote the pull-back. If $\mathcal{L}\in \mathrm{Pic}(\mathscr{X})$ is such that $f^*\mathcal{L}$ is in the image of $\pi'^*\colon\mathrm{Pic}(X')\to \mathrm{Pic}(\mathscr{X}')$, then $\mathcal{L}\in \mathrm{Pic}(X)$.

Proof. Statement 1 follows immediately from the isomorphism $\mathcal{O}_X\to \pi_*\mathcal{O}_{\mathscr{X}}$.

For the other statements, let $\mathcal{L}\in \mathrm{Pic}(\mathscr{X})$ and identify $\mathcal{L}$ with a class $c_\mathcal{L}\in H^1(\mathscr{X},\mathcal{O}_{\mathscr{X}}^*)$. If $\mathcal{L}$ is in the image of $\pi^*$ or locally in its image, then there exists an fppf covering $g\colon X'\to X$ such that $f^*\mathcal{L}$ is trivial. This means that we can represent $c_\mathcal{L}$ by a Čech $1$-cocycle for $f\colon \mathscr{X}'\to \mathscr{X}$. But since $H^0(\mathscr{X}\times_X U,\mathcal{O}_{\mathscr{X}}^*)=H^0(U,\mathcal{O}_X^*)$ for any flat $U\to X$, this means that $c_\mathcal{L}$ is given by a Čech $1$-cocycle for the covering $X'\to X$ giving a unique class in $H^1(X,\mathcal{O}_X^*)$. QED

As mentioned in the comments, when $\mathscr{X}$ is tame, then 3. can be replaced with: $\mathcal{L}\in \mathrm{Pic}(\mathscr{X})$ is in the image of $\pi^*$ if and only if the restriction to the residual gerbe $\mathcal{L}|_{\mathscr{G}_x}$ is trivial for every $x\in |\mathscr{X}|$ (see [Alp13, Thm 10.3] or [Ols12, Prop 6.1]).

Lemma 2. If there exists an algebraic space $Z$ and a finite morphism $p\colon Z\to \mathscr{X}$ such that $p_*\mathcal{O}_Z$ is locally free of rank $n$, then the cokernel of $\pi^*\colon \mathrm{Pic}(X)\to \mathrm{Pic}(\mathscr{X})$ is $n$-torsion.

Proof. If $\mathcal{L}\in\mathrm{Pic}(\mathscr{X})$, then $\mathcal{L}^{n}=N_p(p^*\mathcal{L})$ (the norm is defined and behaves as expected since $p$ is flat). Since $Z\to X$ is finite, we can trivialize $p^* \mathcal{L}$ étale-locally on $X$. This implies that the norm is trivial étale-locally on $X$, i.e., $\mathcal{L}^{n}$ is trivial étale-locally on $X$. The result follows from 3. in the previous lemma. QED

Proof of Proposition 1. There exist an étale covering $\{X'_i\to X\}_{i=1}^r$ such that $\mathscr{X}'_i:=\mathscr{X}\times_X X'_i$ admits a finite flat covering $Z_i\to \mathscr{X}'_i$ of some constant rank $n_i$ for every $i$. By the two lemmas above, the integer $n=\mathrm{lcm}(n_i)$ then kills every element in the cokernel of $\pi^*\colon \mathrm{Pic}(X)\to \mathrm{Pic}(\mathscr{X})$. QED

It is also simple to prove things like:

Proposition 2. Let $f\colon \mathscr{X}'\to \mathscr{X}$ be a representable morphism and let $\pi'\colon \mathscr{X}'\to X'$ denote the coarse moduli space and $g\colon X'\to X$ the induced morphism between coarse moduli spaces. If $\mathscr{X}$ is quasi-compact and $\mathcal{L}\in \mathrm{Pic}(\mathscr{X}')$ is $f$-ample, then there exists an $n$ such that $\mathcal{L}^{n}=\pi'^*\mathcal{M}$ and $\mathcal{M}$ is $g$-ample.

Proof. The question is local on $X$ so we may assume that $X$ is affine and $\mathscr{X}$ admits a finite flat morphism $p\colon Z\to \mathscr{X}$ of constant rank $n$ with $Z$ affine. Let $p'\colon Z'\to \mathscr{X}'$ be the pull-back. Then $p'^*\mathcal{L}$ is ample. We have seen that $\mathcal{L}^{n}=\pi'^*\mathcal{M}$ for $\mathcal{M}\in \mathrm{Pic}(X')$. It is enough to show that sections of $\mathcal{M}^m=\mathcal{L}^{mn}$ for various $m$ defines a basis for the topology of $X'$. Thus let $U'\subseteq X'$ be an open subset and pick any $x'\in U'$. Since $\mathcal{L}|_{Z'}$ is ample, we may find $s\in \Gamma(Z',\mathcal{L}^m)$ such that $D(s)=\{s\neq 0\}$ is an open neighborhood of the preimage of $x'$ (which is finite) contained in the preimage of $U'$. Let $t=N_p(s)$. Then $t\in H^0(\mathscr{X}',\mathcal{L}^{mn})=H^0(X',\mathcal{M}^m)$. But $\mathscr{X}\setminus D(t)=p(Z\setminus D(s))$ so $D(t)=X\setminus \pi(p(Z\setminus D(s)))$ is an open neighborhood of $x'$ contained in $U'$. QED

Acknowledgments I am grateful for comments from Jarod Alper and Daniel Bergh.

References

[Alp13] Alper, J. Good moduli spaces for Artin stacks, Ann. Inst. Fourier (Grenoble) 63 (2013), no. 6, 2349–2402.

[KV04] Kresch, A. and Vistoli, A. On coverings of Deligne–Mumford stacks and surjectivity of the Brauer map, Bull. London Math. Soc. 36 (2004), no. 2, 188–192.

[Ols12] Olsson, M. Integral models for moduli spaces of G-torsors, Ann. Inst. Fourier (Grenoble) 62 (2012), no. 4, 1483–1549.

• this is a great answer ! May 11, 2015 at 11:22
• Thanks for the answer ! In proof of Lemma 2 : "Since $Z\to X$ is finite, we can trivialize $\mathcal{L}$ étale-locally on $X$". I don't get it - could you elaborate ? Sorry if I am missing something obvious. May 12, 2015 at 9:37
• Niels: If $X=\mathrm{Spec}(A)$ where $A$ is a local ring, then $Z=\mathrm{Spec}(B)$ where $B$ is semi-local. Any locally free sheaf is free over the spectrum of a semi-local ring. The general case is obtained from this case by a limit argument: for every point $x\in X$, the line bundle $\mathcal{L}$ becomes trivial after passing to the henselian local ring at $x$ (or rather to the pull-back $Z\times_X \mathrm{Spec}(\mathcal{O}_{X,x})$), hence becomes trivial after passing to an étale neighborhood of $x$. May 12, 2015 at 11:06
• @David Rydh: thanks for the explanation. May 25, 2015 at 13:17