Let $k$ be a field. Consider a group $k$-scheme $G$ and let $X$ be a $k$-scheme equipped with an action of $G$. Then one can define the quotient stack $[X/G]$. Objects of $[X/G]$ over $k$-scheme $T$ are pairs $(\pi, \alpha)$ such that $\pi:P\rightarrow T$ is a locally (with respect to fpqc topology) trivial $G$-bundle and $\alpha:P\rightarrow X$ is a $G$-equivariant morphism.

Now in Olson's book Algebraic spaces and stacks in Example 8.1.12 the author assumes that $G$ is smooth in order to derive that the canonical map $X\rightarrow [X/G]$ is smooth and in result to infer that $[X/G]$ is an algebraic stack. It seems that the other part of his argument, which shows that the diagonal $\Delta_{[X/G]}:[X/G]\rightarrow [X/G]\times_k[X/G]$ is representable holds for any group $k$-scheme.

Here are some immediate questions.

  1. Under what conditions on $G$ the stack $[X/G]$ is algebraic? Is smoothness essential?
  2. If $G$ is affine over $k$, then is $\Delta_{[X/G]}$ representable by quasi-affine morphism of algebraic spaces? If not, then what one should impose on $X$ to know that this is the case?
  • $\begingroup$ If $G$ is not smooth, you should replace the étale topology by the fpqc topology in your definition of $[X/G]$. $\endgroup$ – Marc Hoyois Nov 3 '20 at 19:24
  • $\begingroup$ @MarcHoyois Thank you. I edited my question. $\endgroup$ – Slup Nov 3 '20 at 19:25

About 1. : no, smoothness isn't essential.

"Flat is enough" : De Jong's slogan to express this result due to M.Artin.


I quote :

"Given a flat, finitely presented group scheme $G$ over $S$ acting on a scheme $X$ over $S$, then the quotient stack $[X/G]$ is algebraic."

  • $\begingroup$ It is a beautiful theorem. Thank you. $\endgroup$ – Slup Nov 3 '20 at 18:33
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    $\begingroup$ @Slup You might find the following interesting: If I recall correctly, if $G$ is a locally finitely presented group scheme over a scheme $S$, then $BG :=[S/G]$ is algebraic if and only if $G$ is flat. A proof of this can be found on the Stacks Project (but might also have appeared on MO before). $\endgroup$ – Ariyan Javanpeykar Nov 3 '20 at 19:36
  • $\begingroup$ @AriyanJavanpeykar I am mainly interested in schemes over fields, but it is very interesting and worth knowing. Thank you. $\endgroup$ – Slup Nov 3 '20 at 19:43
  • $\begingroup$ I think this is subsumed by stacks.math.columbia.edu/tag/0DLS, which talks about the case of algebraic spaces and also a further layer of 'relativeness': $X\to Y \to S$ and a group algebraic space $G\to Y$ acting on $X$ over $Y$. $\endgroup$ – David Roberts Nov 3 '20 at 22:20

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