Definition 31.2.1. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$.

We say $x \in X$ is

*associated*to $\mathcal{F}$ if the maximal ideal $\mathfrak m_ x$ is associated to the $\mathcal{O}_{X, x}$-module $\mathcal{F}_ x$.We denote $\text{Ass}(\mathcal{F})$ or $\text{Ass}_ X(\mathcal{F})$ the set of associated points of $\mathcal{F}$.

The

*associated points of $X$*are the associated points of $\mathcal{O}_ X$.

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