Let $X$, $S$ be integral quasi-projective schemes (over $\Bbb C$). Let $\mathcal F$ be a coherent sheaf on $X\times S$, flat on $S$. Suppose that $x\in X$, $s\in S$ are closed points, and ${\mathcal F}(s)={\mathcal F}_{\rvert X\times\{s\}}$. Suppose that ${\mathcal F}(s)_x$ is a free ${\mathcal O}_{X,x}$-module. Does this imply that ${\mathcal F}_{x,s}$ is a free ${\mathcal O}_{X\times S,(x,s)}$-module ?
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