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Let $\varphi:X \to Y$ be a smooth morphism of schemes.

There is a well-known local structure theorem: Zariski locally $\varphi$ is given by the composition of an etale morphism and an affine space (see https://stacks.math.columbia.edu/tag/039P).

What if $X$ and $Y$ are no longer schemes but algebraic stacks? Is there a similar structure theorem?

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    $\begingroup$ Just to clarify, are you asking for a description that is smooth local on $Y$ and etale local on $X$? The difficulty is that the diagonal morphism may not be affine (it could even be proper). If you assume that the diagonal is affine, it should locally be $[Z/\textbf{GL}_n]$ for $Z\to Y$ smooth and representable and an action of $\textbf{GL}_n$ on $Z$ over $Y$. $\endgroup$
    – Jason Starr
    Commented Mar 30, 2023 at 21:44

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