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Let $f:X \to Y$ be a flat proper morphism of noetherian projective schemes and $\mathcal{F}$ is a coherent sheaf on $X$. Suppose for all $y \in Y$, $\mathcal{F} \otimes_{\mathcal{O}_Y} \mathcal{O}_y$ is a locally free sheaf on $X_y$. Is $\mathcal{F}$ a locally free sheaf on $X$?

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    $\begingroup$ No. Take $f$ to be identity and $F$ to be the structure sheaf of a point. $\endgroup$ – Sasha Apr 16 '15 at 19:00
  • $\begingroup$ @Sasha: I am a bit a confused. By $y\in Y$, I do not mean just closed points $y \in Y$. $\endgroup$ – Chen Apr 16 '15 at 19:21
  • $\begingroup$ even with this --- take $X = Y = {\mathbb A}^1$. Then the only nonclosed point is the generic point and for it the tensor product is zero. $\endgroup$ – Sasha Apr 16 '15 at 19:25

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