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If $X \xrightarrow{f} Y$ is a morphism of schemes then the scheme theoretic image of $f$ is the smallest closed subscheme $Z \subset Y$ through which $f$ factors through.

Is this notion defined for algebraic stacks (at least under reasonable assumptions)? If so, can someone provide a reference?

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    $\begingroup$ This definition in the scheme case is problematic because it is not local on the base (consider $\coprod {\rm{Spec}}(k[x]/(x^n)) \rightarrow \mathbf{A}^1_k$). So it seems better to just focus on qcqs $f$, where (as in EGA) you can define it to correspond to $\ker(O_Y \rightarrow f_{\ast}(O_X))$ (quasi-coherent since $f_{\ast}(O_X)$ is, as $f$ is qcqs), and then show by local calculation that this has the desired topological space and mapping property. Similar arguments work for Artin stacks. So the upshot is: consider requiring $f$ to be qcqs. $\endgroup$
    – user36938
    Aug 7, 2013 at 12:54
  • $\begingroup$ In fact qc is enough, in either the scheme or Artin stack context. $\endgroup$
    – tracing
    Mar 27, 2015 at 1:56

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