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ag.algebraic geometry – Projective modules restricted to flush curves Answer

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ag.algebraic geometry – Projective modules restricted to flush curves

I requested this query on Stack Exchange, however nobody answered this.

I need to show a coherent sheaf $M$ on $X$ is domestically free if and provided that that is undoubted for $M|_{X’}$ ,
for all flush curves $X’$ mapping to $X$. I cerebrate the provided that path is patent. For the if path, a coherent module is flat if and solely whether it is projective, for Dedekind domains if and provided that torsion free as effectively. So I’m considering of utilizing Tor$_1$. There is native gauge for flatness, however I’m not positive if this may ameliorate.

This is used on the birth of the proof of Proposition 5.13 of Gaitsgory’s lecture notes.

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