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I want to ask a very general question because I am in the middle of my bachelor thesis and I need some help. If we have a morphism from a scheme to a stack, what properties of the stack can be deduced from the structure of the scheme? In the sense that, for instance, if it is true that if the scheme is smooth then the stack is too?

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    $\begingroup$ In general, one cannot deduce much from a map between a scheme and a stack. On the other hand, if the map is surjective and smooth, or surjective étale, one has that certain properties of the scheme "descend" to the stack. I believe this is covered in M. Olsson's book Algebraic Spaces and Stacks. $\endgroup$
    – Brian Shin
    Mar 11, 2021 at 17:50
  • $\begingroup$ To add to Brian's comment, there is nothing special about the stacky situation that causes you to need more hypotheses: it's just that as written, you allow constant morphisms which tend not to tell you a whole lot about much of anything. More generally, non-epic morphisms in any category will miss something about the target object. $\endgroup$ Mar 12, 2021 at 4:07

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