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The lower shriek functor is defined by

$$f_{!}F(U)=\{s\in\Gamma(f^{-1}(U),F)\;:\; f|_{\mathrm{supp}(s)}:\mathrm{supp}(s)\rightarrow U\text{ is proper}\}$$

On the other hand, if $j:V\subset X$ is the inclusion of an open set, the extension by zero functor is defined by $$ j_{!}F(U)=\begin{cases} F(jU)=F(U) & U\subset V\\ 0 & \text{otherwise} \end{cases}$$

How can I prove these definitions coincide for inclusions of open sets? I don't know anything about base-change theorems, so I'd like to avoid them.

  • What is $f$ and $F$? – Babai Dec 19 '15 at 19:20
  • @Babai $f$ is a continuous map $X\rightarrow Y$ of topological spaces. $F$ is some sheaf on $X$. –  Dec 19 '15 at 22:00
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    I think there is a typo; it should state that $j_!F(U) = F(U\cap V)$ – 54321user Jul 25 '16 at 01:33
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    Your second description of $j_!$ is not correct. In general it only describes a presheaf, not a sheaf. To get a sheaf, you need to sheafify. Equivalently, you can define $(j_! F)(U)$ as the set of those sections $s \in F(U \cap V)$ such that locally $s = 0$ or $U \subseteq V$ (by which I mean that there exists an open covering $U = \bigcup_i U_i$ such that for each $i$, $s|_{U_i \cap V} = 0$ or $U_i \subseteq V$). – Ingo Blechschmidt Nov 08 '16 at 12:17
  • Very well said, @Ingo. – Georges Elencwajg Nov 11 '16 at 09:45

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