Lemma 51.22.3. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Then $\mathcal{F}$ is scheme theoretically supported on $Y_ c$ if and only if the canonical map $\mathcal{F} \to \mathcal{F}(c)$ is zero.

Proof. This is true because $\mathcal{O}_ X \to \mathcal{O}_ X(1)$ vanishes exactly along $Y$. $\square$

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