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It is a standard fact that for any finite morphism of proper Noetherian $A$-schemes ($A$ being Noetherian), the pullback of an ample line bundle is ample. The usual proof of this fact is via Serre's cohomological criterion for ampleness. However, since the statement seems, on its face, to have nothing to do with cohomology, I thought the following question worth asking: Does anyone know a reasonable proof of this fact that does not go through cohomology? |
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Non-cohomological proof that the pullback of an ample bundle by a finite morphism is ampleIt is a standard fact that for any morphism of proper Noetherian $A$-schemes ($A$ being Noetherian), the pullback of an ample line bundle is ample. The usual proof of this fact is via Serre's cohomological criterion for ampleness. However, since the statement seems, on its face, to have nothing to do with cohomology, I thought the following question worth asking: Does anyone know a reasonable proof of this fact that does not go through cohomology?
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