Lemma 29.5.3. Let $\mathcal{F}$ be a finite type quasi-coherent module on a scheme $X$. Then

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

For $x \in X$ we have

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

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