Lemma 108.4.8. In Situation 108.4.7 the stack $\mathcal{C}\! \mathit{oh}^ P_{X/B}$ is algebraic and
is a flat closed immersion. If $I$ is finite or $B$ is locally Noetherian, then $\mathcal{C}\! \mathit{oh}^ P_{X/B}$ is an open and closed substack of $\mathcal{C}\! \mathit{oh}_{X/B}$.
Proof. This is immediately clear if $I$ is finite, because the functions $t \mapsto \chi _ i(t)$ are locally constant. If $I$ is infinite, then we write
and we denote $P' = P|_{I'}$. Then we have
Therefore, $\mathcal{C}\! \mathit{oh}^ P_{X/B}$ is always an algebraic stack and the morphism $\mathcal{C}\! \mathit{oh}^ P_{X/B} \subset \mathcal{C}\! \mathit{oh}_{X/B}$ is always a flat closed immersion, but it may no longer be an open substack. (We leave it to the reader to make examples). However, if $B$ is locally Noetherian, then so is $\mathcal{C}\! \mathit{oh}_{X/B}$ by Lemma 108.4.2 and Morphisms of Stacks, Lemma 101.17.5. Hence if $U \to \mathcal{C}\! \mathit{oh}_{X/B}$ is a smooth surjective morphism where $U$ is a locally Noetherian scheme, then the inverse images of the open and closed substacks $\mathcal{C}\! \mathit{oh}^{P'}_{X/B}$ have an open intersection in $U$ (because connected components of locally Noetherian topological spaces are open). Thus the result in this case. $\square$
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