I asked this question on [Stack Exchange](https://math.stackexchange.com/questions/4032940/projective-modules-restricted-to-smooth-curves), but no one answered this.


I want to prove a coherent sheaf $M$ on $X$ is locally free if and only if this is true for $M|\_{X'}$ ,
for all smooth curves $X'$ mapping to $X$. I think the only if direction is obvious. For the if direction, a coherent module is flat if and only if it is projective, for Dedekind domains if and only if torsion free as well. So I am thinking of using Tor$\_1$. There is [local criterion for flatness](https://en.wikipedia.org/wiki/Local_criterion_for_flatness), but I am not sure if this will help.


This is used at the beginning of the proof of Proposition 5.13 of [Gaitsgory's lecture notes](http://people.math.harvard.edu/%7Egaitsgde/267y/catO.pdf).



Projective modules restricted to smooth curves
Assume that $X$ is integral and smooth (as in Gaitsgory's notes). For any two points $x,y$ in $X$, there is a smooth connected curve $C$ passing through $x$ and $y$. Since $M\_{|C}$ is locally free, this implies $\dim M\_x/\mathfrak{m}\_xM\_x=\dim M\_y/\mathfrak{m}\_yM\_y$, where $\mathfrak{m}\_x$ is the maximal ideal of $\mathscr{O}\_{X,x}$. But this implies that $M$ is locally free.


