## 29.12 Families of ample invertible modules

A short section on the notion of a family of ample invertible modules.

reference
Definition 29.12.1. Let $X$ be a scheme. Let $\{ \mathcal{L}_ i\} _{i \in I}$ be a family of invertible $\mathcal{O}_ X$-modules. We say $\{ \mathcal{L}_ i\} _{i \in I}$ is an *ample family of invertible modules on $X$* if

$X$ is quasi-compact, and

for every $x \in X$ there exists an $i \in I$, an $n \geq 1$, and $s \in \Gamma (X, \mathcal{L}_ i^{\otimes n})$ such that $x \in X_ s$ and $X_ s$ is affine.

If $\{ \mathcal{L}_ i\} _{i \in I}$ is an ample family of invertible modules on a scheme $X$, then there exists a finite subset $I' \subset I$ such that $\{ \mathcal{L}_ i\} _{i \in I'}$ is an ample family of invertible modules on $X$ (follows immediately from quasi-compactness). A scheme having an ample family of invertible modules has an affine diagonal by the next lemma and hence is a fortiori quasi-separated.

Lemma 29.12.2. Let $X$ be a scheme such that for every point $x \in X$ there exists an invertible $\mathcal{O}_ X$-module $\mathcal{L}$ and a global section $s \in \Gamma (X, \mathcal{L})$ such that $x \in X_ s$ and $X_ s$ is affine. Then the diagonal of $X$ is an affine morphism.

**Proof.**
Given invertible $\mathcal{O}_ X$-modules $\mathcal{L}$, $\mathcal{M}$ and global sections $s \in \Gamma (X, \mathcal{L})$, $t \in \Gamma (X, \mathcal{M})$ such that $X_ s$ and $X_ t$ are affine we have to prove $X_ s \cap X_ t$ is affine. Namely, then Lemma 29.11.3 applied to $\Delta : X \to X \times X$ and the fact that $\Delta ^{-1}(X_ s \times X_ t) = X_ s \cap X_ t$ shows that $\Delta $ is affine. The fact that $X_ s \cap X_ t$ is affine follows from Properties, Lemma 28.26.4.
$\square$

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