Question regarding morphism of ringed spaces

Question

I have recently started studying schemes, and I have encountered this passage from the book by Kenji Ueno:

My questions:

i) If $(X,O_X)$ is a local ringed space, why is $(X, i_*({O_X}_{|U}))$ also a local ringed space?

ii) why is $i^#$ a local homomorphism?

Thanks in advance.

Answer 1

I don't think the author means to claim that $(X,i_*(\mathscr{O}_X\vert_U))$ is a locally ringed space. He says that if $X$ is a locally ringed space then the map $i^\sharp:\mathscr{O}_X\to i_*(\mathscr{O}_X\vert_U)$ he's defined is a local homomorphism, but what he means is that the stalk maps $i_x^\sharp:\mathscr{O}_{X,x}\to (\mathscr{O}_X\vert_U)_x$ are local homomorphisms for each $x\in U$. And this is true: these maps are isomorphisms.

In fact for any ringed space, the stalk maps above are isomorphisms, which makes it clear that if $(X,\mathscr{O}_X)$ is a locally ringed space, $(U,\mathscr{O}_X\vert_U)$ is too.

Answer 2

It is just wrong. For example, let $U=\emptyset$. Then $\mathcal{O}_X|_U = 0$, hence $i_* \mathcal{O}_X|_U = 0$. The stalks are $0$, hence not local. More generally, if $U$ is open and closed, then for $x \in X \setminus U$ we have $(i_* \mathcal{O}_{X|_U})_x=0$.