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Tag 00A5

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

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.

  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 $$ \mathop{\rm Mor}\nolimits_{\textit{Ab}(X)}(j_!\mathcal{F}, \mathcal{G}) = \mathop{\rm Mor}\nolimits_{\textit{Ab}(U)}(\mathcal{F}, j^{-1}\mathcal{G}) = \mathop{\rm Mor}\nolimits_{\textit{Ab}(U)}(\mathcal{F}, \mathcal{G}|_U) $$ bifunctorially in $\mathcal{F}$ and $\mathcal{G}$.
  3. Let $\mathcal{F}$ be an abelian sheaf on $U$. The stalks of the sheaf $j_!\mathcal{F}$ are described as follows $$ j_{!}\mathcal{F}_x = \left\{ \begin{matrix} 0 & \text{if} & x \not \in U \\ \mathcal{F}_x & \text{if} & x \in U \end{matrix} \right. $$
  4. On the category of abelian presheaves of $U$ we have $j_pj_{p!} = \text{id}$.
  5. On the category of abelian sheaves of $U$ we have $j^{-1}j_! = \text{id}$.

Proof. Omitted. $\square$

    The code snippet corresponding to this tag is a part of the file sheaves.tex and is located in lines 4612–4648 (see updates for more information).

    \begin{lemma}
    \label{lemma-j-shriek-abelian}
    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.
    \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
    $$
    \Mor_{\textit{Ab}(X)}(j_!\mathcal{F}, \mathcal{G})
    =
    \Mor_{\textit{Ab}(U)}(\mathcal{F}, j^{-1}\mathcal{G})
    =
    \Mor_{\textit{Ab}(U)}(\mathcal{F}, \mathcal{G}|_U)
    $$
    bifunctorially in $\mathcal{F}$ and $\mathcal{G}$.
    \item Let $\mathcal{F}$ be an abelian sheaf on $U$.
    The stalks of the sheaf $j_!\mathcal{F}$ are described
    as follows
    $$
    j_{!}\mathcal{F}_x =
    \left\{
    \begin{matrix}
    0 & \text{if} & x \not \in U \\
    \mathcal{F}_x & \text{if} & x \in U
    \end{matrix}
    \right.
    $$
    \item On the category of abelian presheaves of $U$
    we have $j_pj_{p!} = \text{id}$.
    \item On the category of abelian sheaves of $U$
    we have $j^{-1}j_! = \text{id}$.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Omitted.
    \end{proof}

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