The Stacks project

Lemma 29.5.1. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $\mathop{\mathrm{Spec}}(A) = U \subset X$ be an affine open, and set $M = \Gamma (U, \mathcal{F})$. Let $x \in U$, and let $\mathfrak p \subset A$ be the corresponding prime. The following are equivalent

  1. $\mathfrak p$ is in the support of $M$, and

  2. $x$ is in the support of $\mathcal{F}$.

Proof. This follows from the equality $\mathcal{F}_ x = M_{\mathfrak p}$, see Schemes, Lemma 26.5.4 and the definitions. $\square$

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