The Stacks project

Lemma 31.16.8. Let $X$ be a quasi-compact, regular scheme with affine diagonal. Then $X$ has an ample family of invertible modules (Morphisms, Definition 29.12.1.

Proof. Observe that $X$ is a finite disjoint union of integral schemes (Properties, Lemmas 28.9.4 and 28.7.6). Thus we may assume that $X$ is integral as well as Noetherian, regular, and having affine diagonal. Let $x \in X$. Choose an affine open neighbourhood $U \subset X$ of $x$. Since $X$ is integral, $U$ is dense in $X$. By More on Algebra, Lemma 15.118.2 the local rings of $X$ are UFDs. Hence by Lemma 31.16.7 we can find an effective Cartier divisor $D \subset X$ whose complement is $U$. Then the canonical section $s = 1_ D$ of $\mathcal{L} = \mathcal{O}_ X(D)$, see Definition 31.14.1, vanishes exactly along $D$ hence $U = X_ s$. Thus both conditions in Morphisms, Definition 29.12.1 hold and we are done. $\square$


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