= Finite pullback of an ample line bundle
For a <finite morphism> $f:X\to Y$, the pullback of an <ample line bundle> is <ample>. If $f$ is also surjective, <ampleness> is equivalent in both directions. These results hold for proper schemes over a field, including nonreduced schemes. Cohomological <ampleness> and the exact finite pushforward prove the forward direction; coherent-sheaf devissage gives finite-surjective descent. In particular, a finite map to projective space makes the pulled-back hyperplane bundle <ample>.
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