Let $\mathcal{X}$ be an Artin stack over $S$.

In the paper by Jarod Alper, Good moduli spaces for Artin stacks, Ann. Inst. Fourier 63 (2013) 2349-2402, a tame moduli space is defined as a morphism $\phi:\mathcal{X}\rightarrow X$ such that

  1. $\phi$ is a good moduli space.
  2. For all geometric points $Spec(k)\rightarrow S$, the map $$[\mathcal{X}(k)]\rightarrow X(k)$$ is a bijection of sets.

Are there known examples of Artin stacks without finite inertia which admit tame moduli spaces?

  • 3
    $\begingroup$ I have not studied Alper's notion, but it seems to me like the stack $B \mathbb{G}_m$ satisfies all of the properties. $\endgroup$ – Jason Starr Feb 1 '18 at 13:16

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