[IV Corollary 9.6.4, EGA]

Lemma 37.49.3. Let $f : X \to Y$ be a proper morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $y \in Y$ be a point such that $\mathcal{L}_ y$ is ample on $X_ y$. Then there is an open neighbourhood $V \subset Y$ of $y$ such that $\mathcal{L}|_{f^{-1}(V)}$ is ample on $f^{-1}(V)/V$.

Proof. We may assume $Y$ is affine. Then we find a directed set $I$ and an inverse system of morphisms $X_ i \to Y_ i$ of schemes with $Y_ i$ of finite type over $\mathbf{Z}$, with affine transition morphisms $X_ i \to X_{i'}$ and $Y_ i \to Y_{i'}$, with $X_ i \to Y_ i$ proper, such that $X \to Y = \mathop{\mathrm{lim}}\nolimits (X_ i \to Y_ i)$. See Limits, Lemma 32.13.3. After shrinking $I$ we can assume we have a compatible system of invertible $\mathcal{O}_{X_ i}$-modules $\mathcal{L}_ i$ pulling back to $\mathcal{L}$, see Limits, Lemma 32.10.3. Let $y_ i \in Y_ i$ be the image of $y$. Then $\kappa (y) = \mathop{\mathrm{colim}}\nolimits \kappa (y_ i)$. Hence for some $i$ we have $\mathcal{L}_{i, y_ i}$ is ample on $X_{i, y_ i}$ by Limits, Lemma 32.4.15. By Cohomology of Schemes, Lemma 30.21.4 we find an open neigbourhood $V_ i \subset Y_ i$ of $y_ i$ such that $\mathcal{L}_ i$ restricted to $f_ i^{-1}(V_ i)$ is ample relative to $V_ i$. Letting $V \subset Y$ be the inverse image of $V_ i$ finishes the proof (hints: use Morphisms, Lemma 29.37.9 and the fact that $X \to Y \times _{Y_ i} X_ i$ is affine and the fact that the pullback of an ample invertible sheaf by an affine morphism is ample by Morphisms, Lemma 29.37.7). $\square$

Comment #2688 by on

A reference is EGA IV_3, Corollary 9.6.4

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