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Let $k$ be a field. Let $X$ be a scheme over $k.$ Let $G$ be an affine smooth group scheme over $k$ acting on $X.$ Suppose $X$ is of finite type over $k.$ Does this guarantee that the quotient stack $[X/G]$ is of finite type?

We know $[X/G]$ is locally of finite type over $k$. So the question is whether $[X/G]$ is quasi-compact.

Improve this question
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    $\begingroup$ Yes it is of finite type. $\endgroup$
    – Jason Starr
    Jul 11, 2022 at 23:13
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    $\begingroup$ How does one prove it? $\endgroup$
    – Yuhang Chen
    Jul 11, 2022 at 23:54
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    $\begingroup$ If X \to Y is a map of top spaces s.t. X qc, then so is f(X) (easy exercise). See also stacks.math.columbia.edu/tag/04YC $\endgroup$
    – crystalline
    Jul 12, 2022 at 5:02

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