The Stacks Project

6.32. Closed immersions and (pre)sheaves

Let $X$ be a topological space. Let $i : Z \to X$ be the inclusion of a closed subset $Z$ into $X$. In Section 6.21 we have defined functors $i_*$ and $i^{-1}$ such that $i_*$ is right adjoint to $i^{-1}$.

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. $\square$

Lemma 6.32.2. Let $X$ be a topological space. Let $i : Z \to X$ be the inclusion of a closed subset. The functor $$i_* : \mathop{\textit{Sh}}\nolimits(Z) \longrightarrow \mathop{\textit{Sh}}\nolimits(X)$$ is fully faithful. Its essential image consists exactly of those sheaves $\mathcal{G}$ such that $\mathcal{G}_x = \{*\}$ for all $x \in X \setminus Z$.

Proof. Fully faithfulness follows formally from $i^{-1} i_* = \text{id}$. We have seen that any sheaf in the image of the functor has the property on the stalks mentioned in the lemma. Conversely, suppose that $\mathcal{G}$ has the indicated property. Then it is easy to check that $$\mathcal{G} \to i_* i^{-1} \mathcal{G}$$ is an isomorphism on all stalks and hence an isomorphism. $\square$

Lemma 6.32.3. Let $X$ be a topological space. Let $i : Z \to X$ be the inclusion of a closed subset. The functor $$i_* : \textit{Ab}(Z) \longrightarrow \textit{Ab}(X)$$ is fully faithful. Its essential image consists exactly of those sheaves $\mathcal{G}$ such that $\mathcal{G}_x = 0$ for all $x \in X \setminus Z$.

Proof. Omitted. $\square$

Lemma 6.32.4. Let $X$ be a topological space. Let $i : Z \to X$ be the inclusion of a closed subset. Let $(\mathcal{C}, F)$ be a type of algebraic structure with final object $0$. The functor $$i_* : \mathop{\textit{Sh}}\nolimits(Z, \mathcal{C}) \longrightarrow \mathop{\textit{Sh}}\nolimits(X, \mathcal{C})$$ is fully faithful. Its essential image consists exactly of those sheaves $\mathcal{G}$ such that $\mathcal{G}_x = 0$ for all $x \in X \setminus Z$.

Proof. Omitted. $\square$

Remark 6.32.5. Let $i : Z \to X$ be a closed immersion of topological spaces as above. Let $x \in X$, $x \not \in Z$. Let $\mathcal{F}$ be a sheaf of sets on $Z$. Then $(i_*\mathcal{F})_x = \{ * \}$ by Lemma 6.32.1. Hence if $\mathcal{F} = * \amalg *$, where $*$ is the singleton sheaf, then $i_*\mathcal{F}_x = \{*\} \not = i_*(*)_x \amalg i_*(*)_x$ because the latter is a two point set. According to our conventions in Categories, Section 4.23 this means that the functor $i_*$ is not right exact as a functor between the categories of sheaves of sets. In particular, it cannot have a right adjoint, see Categories, Lemma 4.24.5.

On the other hand, we will see later (see Modules, Lemma 17.6.3) that $i_*$ on abelian sheaves is exact, and does have a right adjoint, namely the functor that associates to an abelian sheaf on $X$ the sheaf of sections supported in $Z$.

Remark 6.32.6. We have not discussed the relationship between closed immersions and ringed spaces. This is because the notion of a closed immersion of ringed spaces is best discussed in the setting of quasi-coherent sheaves, see Modules, Section 17.13.

The code snippet corresponding to this tag is a part of the file sheaves.tex and is located in lines 4853–4997 (see updates for more information).

\section{Closed immersions and (pre)sheaves}
\label{section-closed-immersions}

\noindent
Let $X$ be a topological space.
Let $i : Z \to X$ be the inclusion of a closed subset $Z$ into $X$.
In Section \ref{section-presheaves-functorial} we have defined
functors $i_*$ and $i^{-1}$ such that $i_*$ is right adjoint to
$i^{-1}$.

\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}

\begin{lemma}
\label{lemma-equivalence-categories-closed}
Let $X$ be a topological space.
Let $i : Z \to X$ be the inclusion of a closed subset.
The functor
$$i_* : \Sh(Z) \longrightarrow \Sh(X)$$
is fully faithful. Its essential image consists exactly
of those sheaves $\mathcal{G}$ such that
$\mathcal{G}_x = \{*\}$ for all $x \in X \setminus Z$.
\end{lemma}

\begin{proof}
Fully faithfulness follows formally from $i^{-1} i_* = \text{id}$.
We have seen that any sheaf in the image of the functor has
the property on the stalks mentioned in the lemma. Conversely, suppose
that $\mathcal{G}$ has the indicated property.
Then it is easy to check that
$$\mathcal{G} \to i_* i^{-1} \mathcal{G}$$
is an isomorphism on all stalks and hence an isomorphism.
\end{proof}

\begin{lemma}
\label{lemma-equivalence-categories-closed-abelian}
Let $X$ be a topological space.
Let $i : Z \to X$ be the inclusion of a closed subset.
The functor
$$i_* : \textit{Ab}(Z) \longrightarrow \textit{Ab}(X)$$
is fully faithful. Its essential image consists exactly
of those sheaves $\mathcal{G}$ such that
$\mathcal{G}_x = 0$ for all $x \in X \setminus Z$.
\end{lemma}

\begin{proof}
Omitted.
\end{proof}

\begin{lemma}
\label{lemma-equivalence-categories-closed-structures}
Let $X$ be a topological space.
Let $i : Z \to X$ be the inclusion of a closed subset.
Let $(\mathcal{C}, F)$ be a type of algebraic structure
with final object $0$. The functor
$$i_* : \Sh(Z, \mathcal{C}) \longrightarrow \Sh(X, \mathcal{C})$$
is fully faithful. Its essential image consists exactly
of those sheaves $\mathcal{G}$ such that
$\mathcal{G}_x = 0$ for all $x \in X \setminus Z$.
\end{lemma}

\begin{proof}
Omitted.
\end{proof}

\begin{remark}
\label{remark-i-star-not-exact}
Let $i : Z \to X$ be a closed immersion of topological spaces as above.
Let $x \in X$, $x \not \in Z$. Let $\mathcal{F}$ be a sheaf of sets
on $Z$. Then $(i_*\mathcal{F})_x = \{ * \}$
by Lemma \ref{lemma-stalks-closed-pushforward}.
Hence if $\mathcal{F} = * \amalg *$, where
$*$ is the singleton sheaf, then
$i_*\mathcal{F}_x = \{*\} \not = i_*(*)_x \amalg i_*(*)_x$
because the latter is a two point set.
According to our conventions in
Categories, Section \ref{categories-section-exact-functor}
this means that the functor $i_*$ is not right exact
as a functor between the categories of sheaves of sets.
In particular, it cannot have a right adjoint, see

\medskip\noindent
On the other hand, we will see later (see
that $i_*$ on abelian sheaves is exact, and does have a right
adjoint, namely the functor that associates to an abelian sheaf on $X$
the sheaf of sections supported in $Z$.
\end{remark}

\begin{remark}
\label{remark-closed-immersion-spaces}
We have not discussed the relationship between closed immersions
and ringed spaces. This is because the notion of a closed immersion
of ringed spaces is best discussed in the setting of quasi-coherent
sheaves, see Modules, Section \ref{modules-section-closed-immersion}.
\end{remark}

There are no comments yet for this tag.

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).