Projective modules restricted to smooth curves

Question

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, 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.

Answer 1

This question has been asked and answered here on MO. This is a community-wiki recording of the accepted answer there by user abx so that this question can also be marked as answered.

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.

Answer 2

In Borel's book "Algebraic D-modules" the following result is proved:

Propositon VI.1.7 Let $X$ be a regular scheme of finite type over an algebraically closed field $k$ of characteristic zero. If $M$ is a $D_X$-module that is coherent as $\mathcal{O}_X$-module, then $M$ is locally free.

It seems you do not need the connection to be flat for the result to hold. It also seems to hold over any field $k$ of characteristic zero.