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Tag 00A3

Chapter 6: Sheaves on Spaces > Section 6.31: Open immersions and (pre)sheaves

Lemma 6.31.4. Let $X$ be a topological space. Let $j : U \to X$ be the inclusion of an open subset.

  1. The functor $j_{p!}$ is a left adjoint to the restriction functor $j_p$ (see Lemma 6.31.1).
  2. The functor $j_!$ is a left adjoint to restriction, in a formula $$ \mathop{\rm Mor}\nolimits_{\mathop{\textit{Sh}}\nolimits(X)}(j_!\mathcal{F}, \mathcal{G}) = \mathop{\rm Mor}\nolimits_{\mathop{\textit{Sh}}\nolimits(U)}(\mathcal{F}, j^{-1}\mathcal{G}) = \mathop{\rm Mor}\nolimits_{\mathop{\textit{Sh}}\nolimits(U)}(\mathcal{F}, \mathcal{G}|_U) $$ bifunctorially in $\mathcal{F}$ and $\mathcal{G}$.
  3. Let $\mathcal{F}$ be a sheaf of sets on $U$. The stalks of the sheaf $j_!\mathcal{F}$ are described as follows $$ j_{!}\mathcal{F}_x = \left\{ \begin{matrix} \emptyset & \text{if} & x \not \in U \\ \mathcal{F}_x & \text{if} & x \in U \end{matrix} \right. $$
  4. On the category of presheaves of $U$ we have $j_pj_{p!} = \text{id}$.
  5. 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 $U$. $\square$

    The code snippet corresponding to this tag is a part of the file sheaves.tex and is located in lines 4476–4508 (see updates for more information).

    \begin{lemma}
    \label{lemma-j-shriek}
    Let $X$ be a topological space.
    Let $j : U \to X$ be the inclusion of an open subset.
    \begin{enumerate}
    \item The functor $j_{p!}$ is a left adjoint to the
    restriction functor $j_p$ (see Lemma \ref{lemma-j-pullback}).
    \item The functor $j_!$ is a left adjoint to restriction,
    in a formula
    $$
    \Mor_{\Sh(X)}(j_!\mathcal{F}, \mathcal{G})
    =
    \Mor_{\Sh(U)}(\mathcal{F}, j^{-1}\mathcal{G})
    =
    \Mor_{\Sh(U)}(\mathcal{F}, \mathcal{G}|_U)
    $$
    bifunctorially in $\mathcal{F}$ and $\mathcal{G}$.
    \item Let $\mathcal{F}$ be a sheaf of sets on $U$.
    The stalks of the sheaf $j_!\mathcal{F}$ are described
    as follows
    $$
    j_{!}\mathcal{F}_x =
    \left\{
    \begin{matrix}
    \emptyset & \text{if} & x \not \in U \\
    \mathcal{F}_x & \text{if} & x \in U
    \end{matrix}
    \right.
    $$
    \item On the category of presheaves of $U$ we have $j_pj_{p!} = \text{id}$.
    \item On the category of sheaves of $U$ we have $j^{-1}j_! = \text{id}$.
    \end{enumerate}
    \end{lemma}
    
    \begin{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 \ref{lemma-j-pullback}
    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 $U$.
    \end{proof}

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