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

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

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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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