Suppose $D\subset\mathbb C$ is a unit disc and $\mathcal F$ is a torsion free analytic coherent sheaf over $D$. Define $S(\mathcal F)=\{x\in D|\mathcal F_{x}\, is \,not\, locally\, free\}$. Is $S(\mathcal F)$ always empty? I know there is a theorem in Friedman's book algebraic surfaces and holomorphic vector bundles shows that if the base is algebraic, $S(\mathcal F)$ is always codimension two. Is it still true if the base is analytic but not algebraic?
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1$\begingroup$ Yes. A finitely generated, torsion-free module over a discrete valuation ring is free. $\endgroup$– abxJan 8, 2016 at 6:55
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$\begingroup$ But here only the germ at a fixed point is a ring, how do we apply your results? $\endgroup$– user42804Jan 8, 2016 at 14:44
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$\begingroup$ A basis of $\mathcal{F}_x$ as an $\mathcal{O}_x$-module gives a basis of $\mathcal{F}$ in a neighborhood of $x$. $\endgroup$– abxJan 8, 2016 at 15:51
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