I have two questions about Liu Qing's Algebraic Geometry and Arithmetic Curves. On page38, Lemma 2.23, there is an equation about a composition of direct image and inverse image:
$j_*(j^{-1}(\mathcal{O}_X/\mathcal{J}))=\mathcal{O}_X/\mathcal{J}$
I don't understand why it is true.
The second one is about $V(\mathcal{J})=\left\{x\in X |\mathcal{J}_x\neq \mathcal{O}_{X,x}\right\}$.
What is $\mathcal{J}_x\neq \mathcal{O}_{X,x}$ really mean?
(1) One doesn't have $\mathscr{G} \simeq f_*f^{-1}\mathscr{G}$ in general (example?) but there is always at least a morphism $\mathscr{G} \to f_*f^{-1}\mathscr{G}$ by adjointness and in this particular case you can check that it's an isomorphism on stalks, using the fact that if $i\colon Z \to X$ is the inclusion of a closed subset then the stalk $(i_*\mathscr{F})_p$ is $\mathscr{F}_p$ if $p \in Z$ and zero otherwise.
More generally the same work shows that there is an equivalence between sheaves on $Z$ and sheaves on $X$ with support contained in $Z$.
(2) Whenever you have a subsheaf $\mathscr{F}'$ of $\mathscr{F}$ there is a natural inclusion of stalks $\mathscr{F}'_p \subset \mathscr{F}_p$. You can then ask whether equality holds at certain points.
Maybe it would be good to try this with an explicit example of a closed embedding, just to see what the maps in (1) and (2) look like.