The Stacks project

Definition 6.31.2. Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset.

  1. Let $\mathcal{G}$ be a presheaf of sets, abelian groups or algebraic structures on $X$. The presheaf $j_ p\mathcal{G}$ described in Lemma 6.31.1 is called the restriction of $\mathcal{G}$ to $U$ and denoted $\mathcal{G}|_ U$.

  2. Let $\mathcal{G}$ be a sheaf of sets on $X$, abelian groups or algebraic structures on $X$. The sheaf $j^{-1}\mathcal{G}$ is called the restriction of $\mathcal{G}$ to $U$ and denoted $\mathcal{G}|_ U$.

  3. If $(X, \mathcal{O})$ is a ringed space, then the pair $(U, \mathcal{O}|_ U)$ is called the open subspace of $(X, \mathcal{O})$ associated to $U$.

  4. If $\mathcal{G}$ is a presheaf of $\mathcal{O}$-modules then $\mathcal{G}|_ U$ together with the multiplication map $\mathcal{O}|_ U \times \mathcal{G}|_ U \to \mathcal{G}|_ U$ (see Lemma 6.24.6) is called the restriction of $\mathcal{G}$ to $U$.

Comments (0)

There are also:

  • 2 comment(s) on Section 6.31: Open immersions and (pre)sheaves

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Your email address will not be published.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 00A1. Beware of the difference between the letter 'O' and the digit '0'.



View Definition 6.31.2 as pdf