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.

  • $\begingroup$ For the first question, do you see why this is true in the affine case? $\endgroup$
    – loch
    Jan 15 '18 at 8:25
  • 1
    $\begingroup$ @user37864 Yes, but I'm not confidence. If these are affine, then $\mathscr{F} = \tilde{M}$ for some $A$-module, and $\pi^* \mathscr{F} = M \otimes_A B$. $\endgroup$
    – k.j.
    Jan 15 '18 at 8:32
  • $\begingroup$ And I can proof the only if part of the first quesion,in general case. $\endgroup$
    – k.j.
    Jan 15 '18 at 8:39
  • $\begingroup$ Maybe you can translate the fact that $\mathcal{F}$ is generated by global sections into a sheaf theoretic statement? $\endgroup$
    – Ahr
    Jan 15 '18 at 11:14

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 $\pi^*L$.

  • $\begingroup$ Really? I don't know etale coverings, so I can't understand it. But if so, is Liu's book wrong? He uses these propositions in his book... $\endgroup$
    – k.j.
    Jan 16 '18 at 0:47
  • $\begingroup$ @k.j. I do not know Liu's book, but I doubt such a mistake would be made. Please read all the assumptions in the text. $\endgroup$
    – Mohan
    Jan 16 '18 at 2:06
  • $\begingroup$ So just to be sure: In Liu's book "Algebraic Geometry and Arithmetic curves", Chapter 5, Exercises 1.29 and 1.30, are those true or not in the way they are stated? The comment from k.j. about the affine case in 1.19 is correct, I think this proves the affine case. But what about Exercise 1.30? Without the correctness of (at least) parts of those exercises (maybe under more restrictive assumptions), major parts of Chapter 7 would turn out to be not proven. $\endgroup$
    – J224
    Dec 31 '20 at 15:26

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