The Stacks project

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

    \[ \mathop{\mathrm{Mor}}\nolimits _{\textit{Ab}(X)}(j_!\mathcal{F}, \mathcal{G}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{Ab}(U)}(\mathcal{F}, j^{-1}\mathcal{G}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{Ab}(U)}(\mathcal{F}, \mathcal{G}|_ U) \]

    bifunctorially in $\mathcal{F}$ and $\mathcal{G}$.

  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. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 6.31: Open immersions and (pre)sheaves

Post a comment

Your email address will not be published. Required fields are marked.

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 toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 00A5. Beware of the difference between the letter 'O' and the digit '0'.