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Jan Weidner
Jan Weidner
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Maybe I'm totally confused, but a locally free sheaf over an affine variety should be nothing else than a freeprojective module over the ring of global functions. Push forward along the projection is then just restriction of scalars along the inclusion $$O_X\rightarrow O_X\otimes_k O_Y$$ Now $O_Y=\bigoplus k$ so $$O_X\otimes_k O_Y=O_X\otimes_k \bigoplus k=\bigoplus O_X$$ so free modules stay free and projective Modules=Summands of free modules stay projective.

Maybe I'm totally confused, but a locally free sheaf over an affine variety should be nothing else than a free module over the ring of global functions. Push forward along the projection is then just restriction of scalars along the inclusion $$O_X\rightarrow O_X\otimes_k O_Y$$ Now $O_Y=\bigoplus k$ so $$O_X\otimes_k O_Y=O_X\otimes_k \bigoplus k=\bigoplus O_X$$ so free modules stay free.

Maybe I'm totally confused, but a locally free sheaf over an affine variety should be nothing else than a projective module over the ring of global functions. Push forward along the projection is then just restriction of scalars along the inclusion $$O_X\rightarrow O_X\otimes_k O_Y$$ Now $O_Y=\bigoplus k$ so $$O_X\otimes_k O_Y=O_X\otimes_k \bigoplus k=\bigoplus O_X$$ so free modules stay free and projective Modules=Summands of free modules stay projective.

Source Link
Jan Weidner
Jan Weidner
  • 13.2k
  • 11
  • 61
  • 88

Maybe I'm totally confused, but a locally free sheaf over an affine variety should be nothing else than a free module over the ring of global functions. Push forward along the projection is then just restriction of scalars along the inclusion $$O_X\rightarrow O_X\otimes_k O_Y$$ Now $O_Y=\bigoplus k$ so $$O_X\otimes_k O_Y=O_X\otimes_k \bigoplus k=\bigoplus O_X$$ so free modules stay free.