6.31 Open immersions and (pre)sheaves
Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset $U$ into $X$. In Section 6.21 we have defined functors $j_*$ and $j^{-1}$ such that $j_*$ is right adjoint to $j^{-1}$. It turns out that for an open immersion there is a left adjoint for $j^{-1}$, which we will denote $j_!$. First we point out that $j^{-1}$ has a particularly simple description in the case of an open immersion.
Lemma 6.31.1. Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset $U$ into $X$.
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.
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.
For any point $u \in U$ and any sheaf $\mathcal{G}$ on $X$ we have a canonical identification of stalks
\[ j^{-1}\mathcal{G}_ u = (\mathcal{G}|_ U)_ u = \mathcal{G}_ u. \]
On the category of presheaves of $U$ we have $j_ pj_* = \text{id}$.
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.
$\square$
Definition 6.31.2. Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset.
Let $\mathcal{G}$ be a presheaf of sets, abelian groups or algebraic structures on $X$. The presheaf $j_ p\mathcal{G}$ described in Lemma 6.31.1 is called the restriction of $\mathcal{G}$ to $U$ and denoted $\mathcal{G}|_ U$.
Let $\mathcal{G}$ be a sheaf of sets on $X$, abelian groups or algebraic structures on $X$. The sheaf $j^{-1}\mathcal{G}$ is called the restriction of $\mathcal{G}$ to $U$ and denoted $\mathcal{G}|_ U$.
If $(X, \mathcal{O})$ is a ringed space, then the pair $(U, \mathcal{O}|_ U)$ is called the open subspace of $(X, \mathcal{O})$ associated to $U$.
If $\mathcal{G}$ is a presheaf of $\mathcal{O}$-modules then $\mathcal{G}|_ U$ together with the multiplication map $\mathcal{O}|_ U \times \mathcal{G}|_ U \to \mathcal{G}|_ U$ (see Lemma 6.24.6) is called the restriction of $\mathcal{G}$ to $U$.
We leave a definition of the restriction of presheaves of modules to the reader. Ok, so in this section we will discuss a left adjoint to the restriction functor. Here is the definition in the case of (pre)sheaves of sets.
Definition 6.31.3. Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset.
Let $\mathcal{F}$ be a presheaf of sets on $U$. We define the extension of $\mathcal{F}$ by the empty set $j_{p!}\mathcal{F}$ to be the presheaf of sets on $X$ defined by the rule
\[ j_{p!}\mathcal{F}(V) = \left\{ \begin{matrix} \emptyset
& \text{if}
& V \not\subset U
\\ \mathcal{F}(V)
& \text{if}
& V \subset U
\end{matrix} \right. \]
with obvious restriction mappings.
Let $\mathcal{F}$ be a sheaf of sets on $U$. We define the extension of $\mathcal{F}$ by the empty set $j_!\mathcal{F}$ to be the sheafification of the presheaf $j_{p!}\mathcal{F}$.
Lemma 6.31.4. Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset.
The functor $j_{p!}$ is a left adjoint to the restriction functor $j_ p$ (see Lemma 6.31.1).
The functor $j_!$ is a left adjoint to restriction, in a formula
\[ \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (X)}(j_!\mathcal{F}, \mathcal{G}) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (U)}(\mathcal{F}, j^{-1}\mathcal{G}) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (U)}(\mathcal{F}, \mathcal{G}|_ U) \]
bifunctorially in $\mathcal{F}$ and $\mathcal{G}$.
Let $\mathcal{F}$ be a sheaf of sets on $U$. The stalks of the sheaf $j_!\mathcal{F}$ are described as follows
\[ j_{!}\mathcal{F}_ x = \left\{ \begin{matrix} \emptyset
& \text{if}
& x \not\in U
\\ \mathcal{F}_ x
& \text{if}
& x \in U
\end{matrix} \right. \]
On the category of presheaves of $U$ we have $j_ pj_{p!} = \text{id}$.
On the category of sheaves of $U$ we have $j^{-1}j_! = \text{id}$.
Proof.
To map $j_{p!}\mathcal{F}$ into $\mathcal{G}$ it is enough to map $\mathcal{F}(V) \to \mathcal{G}(V)$ whenever $V \subset U$ compatibly with restriction mappings. And by Lemma 6.31.1 the same description holds for maps $\mathcal{F} \to \mathcal{G}|_ U$. The adjointness of $j_!$ and restriction follows from this and the properties of sheafification. The identification of stalks is obvious from the definition of the extension by the empty set and the definition of a stalk. Statements (4) and (5) follow by computing the value of the sheaf on any open of $U$.
$\square$
Note that if $\mathcal{F}$ is a sheaf of abelian groups on $U$, then in general $j_!\mathcal{F}$ as defined above, is not a sheaf of abelian groups, for example because some of its stalks are empty (hence not abelian groups for sure). Thus we need to modify the definition of $j_!$ depending on the type of sheaves we consider. The reason for choosing the empty set in the definition of the extension by the empty set, is that it is the initial object in the category of sets. Thus in the case of abelian groups we use $0$ (and more generally for sheaves with values in any abelian category).
Definition 6.31.5. Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset.
Let $\mathcal{F}$ be an abelian presheaf on $U$. We define the extension $j_{p!}\mathcal{F}$ of $\mathcal{F}$ by $0$ to be the abelian presheaf on $X$ defined by the rule
\[ j_{p!}\mathcal{F}(V) = \left\{ \begin{matrix} 0
& \text{if}
& V \not\subset U
\\ \mathcal{F}(V)
& \text{if}
& V \subset U
\end{matrix} \right. \]
with obvious restriction mappings.
Let $\mathcal{F}$ be an abelian sheaf on $U$. We define the extension $j_!\mathcal{F}$ of $\mathcal{F}$ by $0$ to be the sheafification of the abelian presheaf $j_{p!}\mathcal{F}$.
Let $\mathcal{C}$ be a category having an initial object $e$. Let $\mathcal{F}$ be a presheaf on $U$ with values in $\mathcal{C}$. We define the extension $j_{p!}\mathcal{F}$ of $\mathcal{F}$ by $e$ to be the presheaf on $X$ with values in $\mathcal{C}$ defined by the rule
\[ j_{p!}\mathcal{F}(V) = \left\{ \begin{matrix} e
& \text{if}
& V \not\subset U
\\ \mathcal{F}(V)
& \text{if}
& V \subset U
\end{matrix} \right. \]
with obvious restriction mappings.
Let $(\mathcal{C}, F)$ be a type of algebraic structure such that $\mathcal{C}$ has an initial object $e$. Let $\mathcal{F}$ be a sheaf of algebraic structures on $U$ (of the give type). We define the extension $j_!\mathcal{F}$ of $\mathcal{F}$ by $e$ to be the sheafification of the presheaf $j_{p!}\mathcal{F}$ defined above.
Let $\mathcal{O}$ be a presheaf of rings on $X$. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}|_ U$-modules. In this case we define the extension by $0$ to be the presheaf of $\mathcal{O}$-modules which is equal to $j_{p!}\mathcal{F}$ as an abelian presheaf endowed with the multiplication map $\mathcal{O} \times j_{p!}\mathcal{F} \to j_{p!}\mathcal{F}$.
Let $\mathcal{O}$ be a sheaf of rings on $X$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}|_ U$-modules. In this case we define the extension by $0$ to be the $\mathcal{O}$-module which is equal to $j_!\mathcal{F}$ as an abelian sheaf endowed with the multiplication map $\mathcal{O} \times j_!\mathcal{F} \to j_!\mathcal{F}$.
It is true that one can define $j_!$ in the setting of sheaves of algebraic structures (see below). However, it depends on the type of algebraic structures involved what the resulting object is. For example, if $\mathcal{O}$ is a sheaf of rings on $U$, then $j_{!, rings}\mathcal{O} \not= j_{!, abelian}\mathcal{O}$ since the initial object in the category of rings is $\mathbf{Z}$ and the initial object in the category of abelian groups is $0$. In particular the functor $j_!$ does not commute with taking underlying sheaves of sets, in contrast to what we have seen so far! We separate out the case of (pre)sheaves of abelian groups, (pre)sheaves of algebraic structures and (pre)sheaves of modules as usual.
Lemma 6.31.6. Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset. Consider the functors of restriction and extension by $0$ for abelian (pre)sheaves.
The functor $j_{p!}$ is a left adjoint to the restriction functor $j_ p$ (see Lemma 6.31.1).
The functor $j_!$ is a left adjoint to restriction, in a formula
\[ \mathop{\mathrm{Mor}}\nolimits _{\textit{Ab}(X)}(j_!\mathcal{F}, \mathcal{G}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{Ab}(U)}(\mathcal{F}, j^{-1}\mathcal{G}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{Ab}(U)}(\mathcal{F}, \mathcal{G}|_ U) \]
bifunctorially in $\mathcal{F}$ and $\mathcal{G}$.
Let $\mathcal{F}$ be an abelian sheaf on $U$. The stalks of the sheaf $j_!\mathcal{F}$ are described as follows
\[ j_{!}\mathcal{F}_ x = \left\{ \begin{matrix} 0
& \text{if}
& x \not\in U
\\ \mathcal{F}_ x
& \text{if}
& x \in U
\end{matrix} \right. \]
On the category of abelian presheaves of $U$ we have $j_ pj_{p!} = \text{id}$.
On the category of abelian sheaves of $U$ we have $j^{-1}j_! = \text{id}$.
Proof.
Omitted.
$\square$
Lemma 6.31.7. Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset. Let $(\mathcal{C}, F)$ be a type of algebraic structure such that $\mathcal{C}$ has an initial object $e$. Consider the functors of restriction and extension by $e$ for (pre)sheaves of algebraic structure defined above.
The functor $j_{p!}$ is a left adjoint to the restriction functor $j_ p$ (see Lemma 6.31.1).
The functor $j_!$ is a left adjoint to restriction, in a formula
\[ \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (X, \mathcal{C})}(j_!\mathcal{F}, \mathcal{G}) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (U, \mathcal{C})}(\mathcal{F}, j^{-1}\mathcal{G}) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (U, \mathcal{C})}(\mathcal{F}, \mathcal{G}|_ U) \]
bifunctorially in $\mathcal{F}$ and $\mathcal{G}$.
Let $\mathcal{F}$ be a sheaf on $U$. The stalks of the sheaf $j_!\mathcal{F}$ are described as follows
\[ j_{!}\mathcal{F}_ x = \left\{ \begin{matrix} e
& \text{if}
& x \not\in U
\\ \mathcal{F}_ x
& \text{if}
& x \in U
\end{matrix} \right. \]
On the category of presheaves of algebraic structures on $U$ we have $j_ pj_{p!} = \text{id}$.
On the category of sheaves of algebraic structures on $U$ we have $j^{-1}j_! = \text{id}$.
Proof.
Omitted.
$\square$
Lemma 6.31.8. Let $(X, \mathcal{O})$ be a ringed space. Let $j : (U, \mathcal{O}|_ U) \to (X, \mathcal{O})$ be an open subspace. Consider the functors of restriction and extension by $0$ for (pre)sheaves of modules defined above.
The functor $j_{p!}$ is a left adjoint to restriction, in a formula
\[ \mathop{\mathrm{Mor}}\nolimits _{\textit{PMod}(\mathcal{O})}(j_{p!}\mathcal{F}, \mathcal{G}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{PMod}(\mathcal{O}|_ U)}(\mathcal{F}, \mathcal{G}|_ U) \]
bifunctorially in $\mathcal{F}$ and $\mathcal{G}$.
The functor $j_!$ is a left adjoint to restriction, in a formula
\[ \mathop{\mathrm{Mor}}\nolimits _{\textit{Mod}(\mathcal{O})}(j_!\mathcal{F}, \mathcal{G}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{Mod}(\mathcal{O}|_ U)}(\mathcal{F}, \mathcal{G}|_ U) \]
bifunctorially in $\mathcal{F}$ and $\mathcal{G}$.
Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules on $U$. The stalks of the sheaf $j_!\mathcal{F}$ are described as follows
\[ j_{!}\mathcal{F}_ x = \left\{ \begin{matrix} 0
& \text{if}
& x \not\in U
\\ \mathcal{F}_ x
& \text{if}
& x \in U
\end{matrix} \right. \]
On the category of sheaves of $\mathcal{O}|_ U$-modules on $U$ we have $j^{-1}j_! = \text{id}$.
Proof.
Omitted.
$\square$
Note that by the lemmas above, both the functors $j_*$ and $j_!$ are fully faithful embeddings of the category of sheaves on $U$ into the category of sheaves on $X$. It is only true for the functor $j_!$ that one can easily describe the essential image of this functor.
Lemma 6.31.9. Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset. The functor
\[ j_! : \mathop{\mathit{Sh}}\nolimits (U) \longrightarrow \mathop{\mathit{Sh}}\nolimits (X) \]
is fully faithful. Its essential image consists exactly of those sheaves $\mathcal{G}$ such that $\mathcal{G}_ x = \emptyset $ for all $x \in X \setminus U$.
Proof.
Fully faithfulness follows formally from $j^{-1} j_! = \text{id}$. 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
\[ j_! j^{-1} \mathcal{G} \to \mathcal{G} \]
is an isomorphism on all stalks and hence an isomorphism.
$\square$
Lemma 6.31.10. Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset. The functor
\[ j_! : \textit{Ab}(U) \longrightarrow \textit{Ab}(X) \]
is fully faithful. Its essential image consists exactly of those sheaves $\mathcal{G}$ such that $\mathcal{G}_ x = 0$ for all $x \in X \setminus U$.
Proof.
Omitted.
$\square$
Lemma 6.31.11. Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset. Let $(\mathcal{C}, F)$ be a type of algebraic structure such that $\mathcal{C}$ has an initial object $e$. The functor
\[ j_! : \mathop{\mathit{Sh}}\nolimits (U, \mathcal{C}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (X, \mathcal{C}) \]
is fully faithful. Its essential image consists exactly of those sheaves $\mathcal{G}$ such that $\mathcal{G}_ x = e$ for all $x \in X \setminus U$.
Proof.
Omitted.
$\square$
Lemma 6.31.12. Let $(X, \mathcal{O})$ be a ringed space. Let $j : (U, \mathcal{O}|_ U) \to (X, \mathcal{O})$ be an open subspace. The functor
\[ j_! : \textit{Mod}(\mathcal{O}|_ U) \longrightarrow \textit{Mod}(\mathcal{O}) \]
is fully faithful. Its essential image consists exactly of those sheaves $\mathcal{G}$ such that $\mathcal{G}_ x = 0$ for all $x \in X \setminus U$.
Proof.
Omitted.
$\square$
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