Now I try to solve the exercise 5.1.29 and 5.1.30 of Liu's book Algebraic Geometry and Arithmetic Curves, but I can't solve these at all. So please give me some hints or references.
First, let $ \pi : X' \to X$ be a faithfully flat morphism of schemes, and $\mathscr{F}$ a quasi-coherent sheaf on $X$. Then $\mathscr{F}$ is generated by its global sections iff $\pi^*\mathscr{F}$ is so.
Next, suppose that $X,X'$ are quasi-compact, and let $\mathscr{L}$ an invertible sheaf on $X$. If $\pi^*\mathscr{L}$ is ample then $\mathscr{L}$ is ample.
Finally, let $A,B$ be noetherian rings, $X$ a proper scheme over $A$, $ A \to B$ a faithfully flat homomorphism, $\pi : X_B \to X$ a base change of $X$ with respect to $ A \to B$, and $\mathscr{L}$ an invertible sheaf on $X$. Now, if $\pi^*\mathscr{L}$ is very ample with respect to $B$, then $\mathscr{L}$ is very ample with respect to $A$.
Please help.
Both the questions about schemes are false.
Let $X$ be an elliptic curve and and $\pi:X'\to X$ be an etale cover of degree 2. So, $\pi$ is faithfully flat. Then there is a two torsion line bundle $L$ of $X$ such that $\pi^*L=\mathcal{O}_{X'}$. $L$ has no sections, so it is not globally generated, but $\pi^*L$ is.
For the next, let $\pi^{-1}(P)=Q+R$ for some $P\in X$. Let $Y=X'-\{Q\}$. Then, $\pi:Y\to X$ is faithfully flat, $L$ is not ample on $X$, but since $Y$ is affine any line bundle is ample, in particular $\pi^*L$.