Let \(X\) be a scheme of finite type over a Noetherian ring \(A\) and \(\mcl\) invertible. The following agree:
\(\mcl\) is ample;
for every coherent \(\mcf\) there is \(n_0\) with \(\mcf \tensor \mcl\tensorpower{}{n}\) globally generated for all \(n \geq n_0\).
If \(X\) is proper over \(A\), both are equivalent to: for every coherent \(\mcf\) there is \(n_0\) with \(H^i(X, \mcf \tensor \mcl\tensorpower{}{n}) = 0\) for all \(i > 0\) and all \(n \geq n_0\).
This criterion expresses ampleness as eventual global generation and, in the proper case, eventual vanishing of higher cohomology. Every argument that starts “twist by \(\OO(n)\) for \(n \gg 0\)” is an appeal to it.
On a projective scheme \(\OO(1)\) is ample, and the criterion is then Serre vanishing. The proper case is where the equivalence has content — it is how ampleness gets checked without exhibiting an embedding, and it is why ampleness is detectable from numerical data such as intersection numbers.
Dependencies
D-DIVAMPLEAmple and very ample divisorsuses