The Stacks project

Definition 71.2.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $x \in |X|$.

  1. We say $x$ is weakly associated to $\mathcal{F}$ if the equivalent conditions (1), (2), and (3) of Lemma 71.2.1 are satisfied.

  2. We denote $\text{WeakAss}(\mathcal{F})$ the set of weakly associated points of $\mathcal{F}$.

  3. The weakly associated points of $X$ are the weakly associated points of $\mathcal{O}_ X$.

If $X$ is locally Noetherian we will say $x$ is associated to $\mathcal{F}$ if and only if $x$ is weakly associated to $\mathcal{F}$ and we set $\text{Ass}(\mathcal{F}) = \text{WeakAss}(\mathcal{F})$. Finally (still assuming $X$ is locally Noetherian), we will say $x$ is an associated point of $X$ if and only if $x$ is a weakly associated point of $X$.

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