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Let $X$ be a smooth variety (over an algebraically closed field, if it makes a difference), and $\mathscr{F}$ a coherent sheaf on $X$. I have heard it claimed that $\mathscr{F}$ is locally free if and only if for all morphisms $f : C \to X$ with $C$ a smooth curve, the pullback $f^*\mathscr{F}$ is locally free. How does one prove such a thing?

It is enough to show something like the following: if $M$ is a finitely generated module over a regular local ring $A$, and for every $A$-algebra $B$ which is a discrete valuation ring the tensor product $B \otimes_A M$ is free, then $M$ is free. But that's as far as I can see.

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(I read incorrectly the statement at a first time).

  1. If $F$ is locally free on $X$, then for any morphism $f : C\to X$ for any scheme $C$, the pull-back $f^*F$ is locally free. Indeed, this is a local question, so you can suppose $F$ is free. Then $F$ is a direct sum of copies of $O_X$, hence $f^*F$ is a direct sum of copies of $f^*O_X=O_C$.

  2. As $X$ is reduced, it is enough to show that the map $x\mapsto d(x)=\dim F\otimes k(x)$ is locally constant on $X$. We can suppose $X$ is connected. Then it is known that through any pair of points $x_1, x_2$ of $X$ it passes a smooth curve $C$. As $F|_C$ is flat, $\dim F\otimes k(t)=d(t)$ is locally constant (hence constant) on $C$. Therefore $d(x_1)=d(x_2)$ and $d$ is constant.

EDIT 2'. I didn't find a reference for the fact stated above (existence of a smooth curve $C$ passing through $x_1, x_2$). However, in Mumford, Abelian Varieties, Lemma p. 56, it is proved that it passes an irreducible curve $D$ through $x_1, x_2$. Now consider the normalization map $\pi : C\to D$ of $D$. Then $\pi^*F$ is locally free by hypothesis. So $d(t)=\dim \pi^*F\otimes k(t)=\dim F\otimes k(\pi(t))$ is constant on $C$. This implies that $d(x_1)=d(x_2)$.

Well, the existence of a smooth curve $C$ inside $X$ is a consequence of Bertini. This is in a paper of Kleiman and Altman "Bertini theorems for hypersurface sections containing a subschemes" (over infinite fields), an in a paper of Poonen "Smooth hypersurface sections containing a given subscheme over a finite field" over finite fields. But as we saw above, we don't need such strong results. The one in Mumford is pretty easy.

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