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Tag 00AD

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
    Categories, Lemma \ref{categories-lemma-exact-adjoint}.
    
    \medskip\noindent
    On the other hand, we will see later (see
    Modules, Lemma \ref{modules-lemma-i-star-right-adjoint})
    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}

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