Ampleness on reduced components
= Ampleness on reduced components
On a <projective scheme>, a <line bundle> is <ample> exactly when its restriction to every reduced irreducible component is <ample>. The disjoint union of those components maps finitely and surjectively to the scheme, so finite-surjective descent of the <finite pullback of an ample line bundle> proves this. The same argument applies to restrictions to any closed subscheme. Thus nilpotent structure does not affect <ampleness>.