In Liu's book, Chapter 7, Proposition 4.4, the question is about closed embeddings (or: closed immersions) coming from very ample divisors. The theorems are quite well-known when the ground field $k$ is algebraically closed (e.g., Hartshorne's Proposition 7.3 and Sec. 7, Chap. 2 there, in general). But Liu does not assume $k$ to be algebraically closed because of the Exercises 5.1.29 and 5.1.30. and not assuming $k$ to be algebraically closed is also a major difference to various other texbooks!
Here it is said that the statements in the exercises could be incorrect. Can one prove them (at least) in special cases?
Ex. 5.1.29: Let $\pi: X'\rightarrow X$ be a faithfully flat morphism of schemes, $F$ quasi-coherent sheaf on $X$.
(a) There is a morphism $\phi: \mathcal{O}_{X}^{(I)} \rightarrow F$ such that $\phi(X)$ is surjective. Further, $F$ is generated by global sections iff $\phi$ is surjective.
(b) $F$ generated by global sections iff $\pi^{\ast}F$ generated by global sections.
(c) Now $X,X'$ quasi-compact, $L$ invertible sheaf on $X$. If $\pi^{\ast}L$ is ample, then $L$ is ample.
Ex. 5.1.30 (light version): Let $X$ be aproper scheme over $\mathbb{R}$, $L$ invertible sheaf on $X$, $\pi: X_{\mathbb{C}} \rightarrow X$ projection (to base change $\mathbb{R}$ to $\mathbb{C}$) and let $\pi^{\ast}L$ be very ample.
(a) Show that $L$ is ample and generated by some global sections $s_0, \ldots , s_n$. Let $f: X \rightarrow Y:=\mathbb{P}^{n}_\mathbb{R}$ be the morphism associated to $L$ and the $s_i$. Show that $f_\mathbb{C}: X_\mathbb{C} \rightarrow Y_\mathbb{C} $ is a closed immersion.
(b) Let $y \in Y$ and let $y' \in Y_\mathbb{C}$ be a point lying over $y$. Show that the fiber of $X_\mathbb{C}$ over $y'$ is isomorphic to $X_y \times_{Spec(k(y))} Spec(k(y')) $. Deduce from this that $f^{-1}(y)$ consists of at most one point, and that $f$ is a topological closed immersion.
(c) Show that $\mathcal{O}_Y \rightarrow f_{\ast}\mathcal{O}_X$ is surjective and that $f$ is a closed immersion.
Best Answer
Exercise 5.1.29(b) and (c) are just plain wrong. Mohan's counterexamples are completely correct, and some further exposition of the general case of a cyclic cover of a curve can be found here. The key issue is that there is no good way to go from a surjective map $\mathcal{O}_{X'}^I\to \pi^*\mathcal{F}$ to a surjective map $\mathcal{O}_{X}^I\to \mathcal{F}$.
On the other hand, when we are in the situation of exercise 5.1.30, everything works out. The big idea that lets us bypass the issue from exercise 5.1.30 is that $H^0(X,\mathcal{L})\otimes_A B \cong H^0(X_B,\mathcal{L}_B)$ by flat base change (ref Stacks 02KH, for instance). So pick a generating set $\{s_i\}_{i\in I}$ for $H^0(X,\mathcal{L})$, construct the obvious morphism $g:\mathcal{O}_X^I\to \mathcal{L}$, and then note that $\{s_i\otimes 1\}_{i\in I}$ is a generating set for $H^0(X,\mathcal{L})\otimes_A B = H^0(X_B,\mathcal{L}_B)$. Thus $\pi^*g:\pi^*\mathcal{O}_X^I\to\pi^*\mathcal{L}$ is a map $\mathcal{O}_{X_B}^I\to\mathcal{L}_B$ which is surjective on global sections. So $\mathcal{L}_B$ is globally generated iff $\pi^*g$ is surjective, which happens if and only if $g:\mathcal{O}_X^I\to\mathcal{L}$ is surjective by the argument in Niven's post addressing 5.1.29(b). This fixes all of the issues in Niven's outline of a solution to exercise 5.1.30.