The Stacks Project


Tag 00A0

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

Lemma 6.31.1. Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset $U$ into $X$.

  1. Let $\mathcal{G}$ be a presheaf of sets on $X$. The presheaf $j_p\mathcal{G}$ (see Section 6.21) is given by the rule $V \mapsto \mathcal{G}(V)$ for $V \subset U$ open.
  2. Let $\mathcal{G}$ be a sheaf of sets on $X$. The sheaf $j^{-1}\mathcal{G}$ is given by the rule $V \mapsto \mathcal{G}(V)$ for $V \subset U$ open.
  3. For any point $u \in U$ and any sheaf $\mathcal{G}$ on $X$ we have a canonical identification of stalks $$ j^{-1}\mathcal{G}_u = (\mathcal{G}|_U)_u = \mathcal{G}_u. $$
  4. On the category of presheaves of $U$ we have $j_pj_* = \text{id}$.
  5. On the category of sheaves of $U$ we have $j^{-1}j_* = \text{id}$.

The same description holds for (pre)sheaves of abelian groups, (pre)sheaves of algebraic structures, and (pre)sheaves of modules.

Proof. The colimit in the definition of $j_p\mathcal{G}(V)$ is over collection of all $W \subset X$ open such that $V \subset W$ ordered by reverse inclusion. Hence this has a largest element, namely $V$. This proves (1). And (2) follows because the assignment $V \mapsto \mathcal{G}(V)$ for $V \subset U$ open is clearly a sheaf if $\mathcal{G}$ is a sheaf. Assertion (3) follows from (2) since the collection of open neighbourhoods of $u$ which are contained in $U$ is cofinal in the collection of all open neighbourhoods of $u$ in $X$. Parts (4) and (5) follow by computing $j^{-1}j_*\mathcal{F}(V) = j_*\mathcal{F}(V) = \mathcal{F}(V)$.

The exact same arguments work for (pre)sheaves of abelian groups and (pre)sheaves of algebraic structures. $\square$

    The code snippet corresponding to this tag is a part of the file sheaves.tex and is located in lines 4380–4402 (see updates for more information).

    \begin{lemma}
    \label{lemma-j-pullback}
    Let $X$ be a topological space.
    Let $j : U \to X$ be the inclusion of an open subset $U$ into $X$.
    \begin{enumerate}
    \item Let $\mathcal{G}$ be a presheaf of sets on $X$.
    The presheaf $j_p\mathcal{G}$
    (see Section \ref{section-presheaves-functorial}) is given by the rule
    $V \mapsto \mathcal{G}(V)$ for $V \subset U$ open.
    \item Let $\mathcal{G}$ be a sheaf of sets on $X$.
    The sheaf $j^{-1}\mathcal{G}$ is given by the rule
    $V \mapsto \mathcal{G}(V)$ for $V \subset U$ open.
    \item For any point $u \in U$ and any sheaf $\mathcal{G}$ on $X$
    we have a canonical identification of stalks
    $$
    j^{-1}\mathcal{G}_u = (\mathcal{G}|_U)_u = \mathcal{G}_u.
    $$
    \item On the category of presheaves of $U$ we have $j_pj_* = \text{id}$.
    \item On the category of sheaves of $U$ we have $j^{-1}j_* = \text{id}$.
    \end{enumerate}
    The same description holds for (pre)sheaves of abelian groups,
    (pre)sheaves of algebraic structures, and (pre)sheaves of modules.
    \end{lemma}
    
    \begin{proof}
    The colimit in the definition of $j_p\mathcal{G}(V)$
    is over collection of all $W \subset X$ open such that $V \subset W$
    ordered by reverse inclusion.
    Hence this has a largest element, namely $V$. This proves (1).
    And (2) follows because the assignment $V \mapsto \mathcal{G}(V)$
    for $V \subset U$ open is clearly a sheaf if $\mathcal{G}$ is a
    sheaf. Assertion (3) follows from (2) since the collection
    of open neighbourhoods of $u$ which are contained in $U$ is cofinal
    in the collection of all open neighbourhoods of $u$ in $X$.
    Parts (4) and (5) follow by computing
    $j^{-1}j_*\mathcal{F}(V) = j_*\mathcal{F}(V) = \mathcal{F}(V)$.
    
    \medskip\noindent
    The exact same arguments work for (pre)sheaves of abelian groups
    and (pre)sheaves of algebraic structures.
    \end{proof}

    Comments (0)

    There are no comments yet for this tag.

    There is also 1 comment on Section 6.31: Sheaves on Spaces.

    Add a comment on tag 00A0

    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?