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Let $X$ be a smooth curve defined over a field and $F$ a coherent sheaf on $X$.

I would like to show that $F/F_{t}$ is locally free, for $F_{t}$ the torsion subsheaf of $F$.

Since $F$ is coherent it is enough to show the stalk $F_{p}$ is free as an $(\mathcal O_{X})_{p}$-module for all $p$. How do I see that a torsion-free module is free in general in this case?

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up vote 2 down vote accepted

This is a consequence of the structure theorem for modules over principal ideal domains: note in particular that the local rings at all points of a smooth curve are such.

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