Tame stacks in positive characteristic

Good moduli spaces for Artin stacks

“…was proved in the case of tame Artin stacks in [34] by using the local structure theorem of [1] to reduce to the case of a quotient stack of an affine scheme by a finite linearly reductive group scheme. As in [34, Theorem 1.7 and 1.8], part (iii) directly implies that that the cohomology and base change theorem and semicontinuity theorem hold for noetherian Artin stacks admitting a proper good moduli space.…”

“…-If X is an Artin stack over S with finite inertia stack I X → X then by the Keel-Mori Theorem ( [19]) and its generalizations ( [5], [39]), there exists a coarse moduli space φ : X → Y . Abramovich, Olsson and Vistoli in [1] define X to be a tame stack if φ is cohomologically affine in which case φ is a good moduli space. Of those Artin stacks with finite inertia, only tame stacks admit good moduli spaces.…”

“…We define the notion of a good moduli space (see Definition 4.1) which was inspired by and generalizes the existing notions of a good GIT quotient and tame stack (see [1]). The definition is strikingly simple:…”

“…Keel and Mori gave a precise construction of the coarse moduli space in [19]. Recently, Abramovich, Olsson and Vistoli in [1] have distinguished a subclass of stacks with finite inertia, called tame stacks, whose coarse moduli space has additional desired properties such as its formation commutes with arbitrary base change. Artin stacks without finite inertia rarely admit coarse moduli spaces.…”

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“…To prove G/H has no pseudoreflections at the origin, it suffices by Lemma 2.3 to replace k by its algebraic closure. By [Abramovich et al 2008, Lemma 2.11], we see then that G is the semidirect product of its identity where q (u) = (u, qu). We see that Z q is a closed subscheme of U and that Z q (T ) is the set of u ∈ U (T ) which are fixed by q.…”