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 15.121.2 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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