The Stacks project

Lemma 108.9.2. The diagonal of $\mathrm{Pic}_{X/B}$ over $B$ is a quasi-compact immersion.

Proof. The diagonal is an immersion by Quot, Lemma 99.11.9. To finish we show that the diagonal is quasi-compact. The diagonal of $\mathcal{P}\! \mathit{ic}_{X/B}$ is quasi-compact by Lemma 108.8.1 and $\mathcal{P}\! \mathit{ic}_{X/B}$ is a gerbe over $\mathrm{Pic}_{X/B}$ by Lemma 108.9.1. We conclude by Morphisms of Stacks, Lemma 101.28.14. $\square$

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