# The Stacks Project

## Tag 00A0

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}

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