Lemma 66.15.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Then

The support of $\mathcal{F}$ is closed.

For a geometric point $\overline{x}$ lying over $x \in |X|$ we have

\[ x \in \text{Supp}(\mathcal{F}) \Leftrightarrow \mathcal{F}_{\overline{x}} \not= 0 \Leftrightarrow \mathcal{F}_{\overline{x}} \otimes _{\mathcal{O}_{X, \overline{x}}} \kappa (\overline{x}) \not= 0. \]For any morphism of algebraic spaces $f : Y \to X$ the pullback $f^*\mathcal{F}$ is of finite type as well and we have $\text{Supp}(f^*\mathcal{F}) = f^{-1}(\text{Supp}(\mathcal{F}))$.

## Comments (0)