The Stacks project

Definition 17.5.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules.

  1. The support of $\mathcal{F}$ is the set of points $x \in X$ such that $\mathcal{F}_ x \not= 0$.

  2. We denote $\text{Supp}(\mathcal{F})$ the support of $\mathcal{F}$.

  3. Let $s \in \Gamma (X, \mathcal{F})$ be a global section. The support of $s$ is the set of points $x \in X$ such that the image $s_ x \in \mathcal{F}_ x$ of $s$ is not zero.

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View Definition 17.5.1 as pdf