Lemma 108.4.5 (0DNA)—The Stacks project

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Table of contents
Part 8: Topics in Moduli Theory
Chapter 108: Moduli Stacks
Section 108.4: Properties of the stack of coherent sheaves
Lemma 108.4.5 (cite)

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Lemma 108.4.5. Let $B$ be an algebraic space. Let $\pi : X \to Y$ be an open immersion of algebraic spaces which are separated and of finite presentation over $B$ . Then the morphism $\mathcal{C}\! \mathit{oh}_{X/B} \to \mathcal{C}\! \mathit{oh}_{Y/B}$ of Lemma 108.4.4 is an open immersion.

Proof. Omitted. Hint: If $\mathcal{F}$ is an object of $\mathcal{C}\! \mathit{oh}_{Y/B}$ over $T$ and for $t \in T$ we have $\text{Supp}(\mathcal{F}_ t) \subset |X_ t|$ , then the same is true for $t' \in T$ in a neighbourhood of $t$ . $\square$

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