The Stacks project

Definition 17.23.1. Let $(X, \mathcal{O}_ X)$ be a ringed space and let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. The annihilator of $\mathcal{F}$, denoted $\text{Ann}_{\mathcal{O}_ X}(\mathcal{F})$ is the kernel of the map $\mathcal{O}_ X \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{F})$ discussed above.