The Stacks project

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.