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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.

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  • $\begingroup$ For the first question, do you see why this is true in the affine case? $\endgroup$
    – loch
    Commented Jan 15, 2018 at 8:25
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    $\begingroup$ @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.
    Commented Jan 15, 2018 at 8:32
  • $\begingroup$ And I can proof the only if part of the first quesion,in general case. $\endgroup$
    – k.j.
    Commented Jan 15, 2018 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
    Commented Jan 15, 2018 at 11:14

1 Answer 1

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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 $\pi^*L$.

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  • $\begingroup$ 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.
    Commented Jan 16, 2018 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
    Commented Jan 16, 2018 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
    Commented Dec 31, 2020 at 15:26

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