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Tag 00AE

Chapter 6: Sheaves on Spaces > Section 6.32: Closed immersions and (pre)sheaves

Lemma 6.32.1. Let $X$ be a topological space. Let $i : Z \to X$ be the inclusion of a closed subset $Z$ into $X$. Let $\mathcal{F}$ be a sheaf of sets on $Z$. The stalks of $i_*\mathcal{F}$ are described as follows $$ i_*\mathcal{F}_x = \left\{ \begin{matrix} \{*\} & \text{if} & x \not \in Z \\ \mathcal{F}_x & \text{if} & x \in Z \end{matrix} \right. $$ where $\{*\}$ denotes a singleton set. Moreover, $i^{-1}i_* = \text{id}$ on the category of sheaves of sets on $Z$. Moreover, the same holds for abelian sheaves on $Z$, resp. sheaves of algebraic structures on $Z$ where $\{*\}$ has to be replaced by $0$, resp. a final object of the category of algebraic structures.

Proof. If $x \not \in Z$, then there exist arbitrarily small open neighbourhoods $U$ of $x$ which do not meet $Z$. Because $\mathcal{F}$ is a sheaf we have $\mathcal{F}(i^{-1}(U)) = \{*\}$ for any such $U$, see Remark 6.7.2. This proves the first case. The second case comes from the fact that for $z \in Z$ any open neighbourhood of $z$ is of the form $Z \cap U$ for some open $U$ of $X$. For the statement that $i^{-1}i_* = \text{id}$ consider the canonical map $i^{-1}i_*\mathcal{F} \to \mathcal{F}$. This is an isomorphism on stalks (see above) and hence an isomorphism.

For sheaves of abelian groups, and sheaves of algebraic structures you argue in the same manner. $\square$

    The code snippet corresponding to this tag is a part of the file sheaves.tex and is located in lines 4863–4884 (see updates for more information).

    \begin{lemma}
    \label{lemma-stalks-closed-pushforward}
    Let $X$ be a topological space.
    Let $i : Z \to X$ be the inclusion of a closed subset $Z$ into $X$.
    Let $\mathcal{F}$ be a sheaf of sets on $Z$.
    The stalks of $i_*\mathcal{F}$ are described as follows
    $$
    i_*\mathcal{F}_x =
    \left\{
    \begin{matrix}
    \{*\} & \text{if} & x \not \in Z \\
    \mathcal{F}_x & \text{if} & x \in Z
    \end{matrix}
    \right.
    $$
    where $\{*\}$ denotes a singleton set. Moreover,
    $i^{-1}i_* = \text{id}$ on the category of sheaves
    of sets on $Z$. Moreover, the same holds for abelian
    sheaves on $Z$, resp.\ sheaves of algebraic structures on $Z$
    where $\{*\}$ has to be replaced by $0$, resp.\ a
    final object of the category of algebraic structures.
    \end{lemma}
    
    \begin{proof}
    If $x \not \in Z$, then there exist arbitrarily small open
    neighbourhoods $U$ of $x$ which do not meet $Z$.
    Because $\mathcal{F}$ is a sheaf
    we have $\mathcal{F}(i^{-1}(U)) = \{*\}$ for any such $U$,
    see Remark \ref{remark-confusion}. This proves the first case.
    The second case comes from the fact that for $z \in Z$
    any open neighbourhood of $z$ is of the form $Z \cap U$ for
    some open $U$ of $X$. For the statement that
    $i^{-1}i_* = \text{id}$ consider the canonical map
    $i^{-1}i_*\mathcal{F} \to \mathcal{F}$. This is an isomorphism
    on stalks (see above) and hence an isomorphism.
    
    \medskip\noindent
    For sheaves of abelian groups, and sheaves of algebraic structures
    you argue in the same manner.
    \end{proof}

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