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6 votes

When quotient stacks (for nonsmooth group) are algebraic and related questions

About 1. : no, smoothness isn't essential. "Flat is enough" : De Jong's slogan to express this result due to M.Artin. https://www.math.columbia.edu/~dejong/wordpress/?p=1584 I quote : "Given a flat, f …
Niels's user avatar
  • 4,008
6 votes

Do equivariant morphisms induce representable maps of quotient stacks?

Because first, $Y\to [Y/G]$ is a chart second, the following diagram $$\require{AMScd} \begin{CD} X @>>> Y \\ @VVV @VVV \\ [X/G] @>>> [Y/G] \end{CD} $$ is cartesian (this is so because any morphis …
Niels's user avatar
  • 4,008
4 votes

Representability of morphism of stacks

This is http://stacks.math.columbia.edu/tag/04Y5 . I quote : " lemma Let $S$ be a scheme contained in $Sch_{fppf}$. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of algebraic stacks over …
Niels's user avatar
  • 4,008
4 votes
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Group action on a stack and fixed points

A standard reference is Romagny, Matthieu Group actions on stacks and applications. Michigan Math. J. 53 (2005), no. 1, 209–236. http://projecteuclid.org/euclid.mmj/1114021093 http://perso.univ-re …
Niels's user avatar
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3 votes

Explicit description of the stack associated to a groupoid

This is called "Stackification via torsors" in the book by Kai Behrend, Brian Conrad, Dan Edidin, William Fulton, Barbara Fantechi, Lothar Göttsche and Andrew Kresch. See Andrew Kresch's homepage: ht …
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