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 $X$.


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0CTX. Beware of the difference between the letter 'O' and the digit '0'.