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While reading Qing Liu's book on Algebraic geometry, I came across the following claim in page 38: Let $(X,\mathscr{O}_X)$ be a locally ringed space, and let $\mathscr{I}$ be a sheaf of ideals of $\mathscr{O}_X$. Then the set $V(\mathscr{I}):=\{x\in X |\mathscr{I}_x\neq \mathscr{O}_{X,x}\} $ is a closed subset (easy to check). Let $j:V(\mathscr{I})\rightarrow X$ denote the inclusion. Then

$\mathscr{O}_X/\mathscr{I} = j_*(j^{-1}(\mathscr{O}_X/\mathscr{I} )).$

I can't prove this. All I can see that there is a map $j_*(j^{-1}(\mathscr{O}_X/\mathscr{I} ))\rightarrow \mathscr{O}_X/\mathscr{I}$ following definitions, but I can't see how that will help. Thanks in advance.

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    Can you prove that the map is an isomorphism on the stalks? – 2017-01-03
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    @LeonSot: That worked! thanks! basically both the stalks are same for $x\in V(\mathscr{I})$, and zero for $x$ outside $V(\mathscr{I}).$ – 2017-01-03

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