The Stacks Project


Tag 00A1

Chapter 6: Sheaves on Spaces > Section 6.31: Open immersions and (pre)sheaves

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 $U$.

    The code snippet corresponding to this tag is a part of the file sheaves.tex and is located in lines 4422–4443 (see updates for more information).

    \begin{definition}
    \label{definition-restriction}
    Let $X$ be a topological space.
    Let $j : U \to X$ be the inclusion of an open subset.
    \begin{enumerate}
    \item Let $\mathcal{G}$ be a presheaf of sets, abelian groups or
    algebraic structures on $X$. The presheaf $j_p\mathcal{G}$ described
    in Lemma \ref{lemma-j-pullback} is called
    the {\it restriction of $\mathcal{G}$ to $U$} and denoted $\mathcal{G}|_U$.
    \item 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 {\it restriction of $\mathcal{G}$ to $U$} and denoted $\mathcal{G}|_U$.
    \item If $(X, \mathcal{O})$ is a ringed space, then the pair
    $(U, \mathcal{O}|_U)$ is called the
    {\it open subspace of $(X, \mathcal{O})$ associated to $U$}.
    \item 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 \ref{lemma-pullback-module})
    is called the {\it restriction of $\mathcal{G}$ to $U$}.
    \end{enumerate}
    \end{definition}

    Comments (0)

    There are no comments yet for this tag.

    There are also 2 comments on Section 6.31: Sheaves on Spaces.

    Add a comment on tag 00A1

    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 lower-right corner).

    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 box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

    This captcha seems more appropriate than the usual illegible gibberish, right?