We prove that an algebraic stack, locally of finite presentation and quasi-separated over a quasi-separated algebraic space with affine stabilizers, is étale locally a quotient stack around any point with a linearly reductive stabilizer. This result generalizes the main result of [AHR19] to the relative setting and the main result of [AOV11] to the case of non-finite inertia. We also provide various coherent completeness and effectivity results for algebraic stacks as well as structure theorems for linearly reductive groups schemes. Finally, we provide several applications of these results including generalizations of Sumihiro's theorem on torus actions and Luna'setale slice theorem to the relative setting.
14D23 (Primary), 14L15, 14L24, 14L30 (Secondary)
|Original language||English (US)|
|State||Published - Dec 12 2019|
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