Lemma 37.3.2. Let (f, f') : (X \subset X') \to (S \subset S') be a morphism of thickenings. Let \mathcal{L}' be an invertible sheaf on X' and denote \mathcal{L} the restriction to X. Then \mathcal{L}' is f'-ample if and only if \mathcal{L} is f-ample.
Proof. Recall that being relatively ample is a condition for each affine open in the base, see Morphisms, Definition 29.37.1. By Lemma 37.2.3 there is a 1-to-1 correspondence between affine opens of S and S'. Thus we may assume S and S' are affine and we reduce to proving that \mathcal{L}' is ample if and only if \mathcal{L} is ample. This is Limits, Lemma 32.11.4. \square
Comments (0)