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

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?
• If $G$ is not smooth, you should replace the étale topology by the fpqc topology in your definition of $[X/G]$. – Marc Hoyois Nov 3 '20 at 19:24
• @MarcHoyois Thank you. I edited my question. – 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.

https://www.math.columbia.edu/~dejong/wordpress/?p=1584

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."

• It is a beautiful theorem. Thank you. – Slup Nov 3 '20 at 18:33
• @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). – Ariyan Javanpeykar Nov 3 '20 at 19:36
• @AriyanJavanpeykar I am mainly interested in schemes over fields, but it is very interesting and worth knowing. Thank you. – Slup Nov 3 '20 at 19:43
• 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 $Y$. – David Roberts Nov 3 '20 at 22:20