Preamble which you can skip
I am reading Proposition 2.24 in Liu's "algebraic geometry and arithmetic curves" book and there is a confusing reference to exercise 2.13 in the proof.
In said exercise he claims that:
the natural map $f^{-1}f_*\mathcal{F} \to \mathcal{F}$ is an isomorphism whenever $f$ is an open immersion and that the natural map $\mathcal{G} \to f_*f^{-1}\mathcal{G}$ is surjective whenever $f$ is a closed immersion.
But in proposition 2.24 $f$ is a closed immersion and he is using the isomorphism $f^{-1}f_*\mathcal{F} \cong \mathcal{F}$ referring to the exercise which instead requires $f$ to be open.
Question
I believe the correct statement for closed immersions should be the following:
Let $f:Y \to X$ be a topological closed immersion. Then:
- for every $\mathcal{F} \in \operatorname{Sh}(Y)$, the natural map $f^{-1}f_*\mathcal{F} \to \mathcal{F}$ is an isomorphism
and
- for every $\mathcal{G}\in \operatorname{Sh}(X)$ such that $\mathcal{G}_x =0$ iff $x \notin f(Y)$ the natural map $\mathcal{G} \to f_*f^{-1}\mathcal{G}$ is an isomorphism.
What I said above follows, I believe, from the following fact about topological closed immersion:
Let $f: Y \to X$ be a topological closed immersion and let $\mathcal{F}\in \operatorname{Sh}(Y),$ then $f_*\mathcal{F}_x = 0$ if $x \notin f(Y)$ and $f_*\mathcal{F}_x = \mathcal{F}_y$ if $x=f(y)\in f(Y).$
Q: Is my statement above correct? Is there a typo in Q.Liu's exercise?