The Stacks project

Exercise 111.32.5. Let $j : U \to X$ be an open immersion. Show that

  1. Pullback $j^{-1} : \mathop{\mathit{Sh}}\nolimits (X) \to \mathop{\mathit{Sh}}\nolimits (U)$ has a left adjoint $j_{!} : \mathop{\mathit{Sh}}\nolimits (U) \to \mathop{\mathit{Sh}}\nolimits (X)$ called extension by the empty set.

  2. Characterize the stalks of $j_{!}({\mathcal G})$ for $\mathcal{G} \in \mathop{\mathit{Sh}}\nolimits (U)$.

  3. Pullback $j^{-1} : \textit{Ab}(X) \to \textit{Ab}(U)$ has a left adjoint $j_{!} : \textit{Ab}(U) \to \textit{Ab}(X)$ called extension by zero.

  4. Characterize the stalks of $j_{!}({\mathcal G})$ for $\mathcal{G} \in \textit{Ab}(U)$.

Observe that extension by zero differs from extension by the empty set!


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