The Stacks project

Theorem 115.21.5. Let $S$ be a scheme. Let $f : X \to B$ be morphism of algebraic spaces over $S$. Assume that $f$ is of finite presentation and separated. Then $\textit{Coh}_{X/B}$ is an algebraic stack over $S$.

Proof. This theorem is a copy of Quot, Theorem 99.6.1. The reason we have this copy here is that with the material in this section we get a second proof (as discussed at the beginning of this section). Namely, we argue exactly as in the proof of Quot, Theorem 99.5.12 except that we substitute Lemma 115.21.4 for Quot, Lemma 99.5.11. $\square$

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