A short section on the notion of a family of ample invertible modules.
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. $\square$
Remark 29.12.3. In Properties, Lemma 28.26.7 we see that a scheme which has an ample invertible module is separated. This is wrong for schemes having an ample family of invertible modules. Namely, let $X$ be as in Schemes, Example 26.14.3 with $n = 1$, i.e., the affine line with zero doubled. We use the notation of that example except that we write $x$ for $x_1$ and $y$ for $y_1$. There is, for every integer $n$, an invertible sheaf $\mathcal{L}_ n$ on $X$ which is trivial on $X_1$ and $X_2$ and whose transition function $U_{12} \to U_{21}$ is $f(x) \mapsto y^ n f(y)$. The global sections of $\mathcal{L}_ n$ are pairs $(f(x), g(y)) \in k[x] \oplus k[y]$ such that $y^ n f(y) = g(y)$. The sections $s = (1, y)$ of $\mathcal{L}_1$ and $t = (x, 1)$ of $\mathcal{L}_{-1}$ determine an open affine cover because $X_ s = X_1$ and $X_ t = X_2$. Therefore $X$ has an ample family of invertible modules but it is not separated.
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