# The Stacks Project

## Tag 00AE

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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