Lemma 31.16.7. Let X be a Noetherian scheme with affine diagonal. Let U \subset X be a dense affine open. If \mathcal{O}_{X, x} is a UFD for all x \in X \setminus U, then there exists an effective Cartier divisor D \subset X with U = X \setminus D.
Proof. Since X is Noetherian, the complement X \setminus U has finitely many irreducible components D_1, \ldots , D_ r (Properties, Lemma 28.5.7 applied to the reduced induced subscheme structure on X \setminus U). We view D_ i as a reduced closed subscheme of X. Let X = \bigcup _{j \in J} X_ j be an affine open covering of X. For all j in J, set U_ j = U \cap X_ j. Since X has affine diagonal, the scheme
U_ j = X \times _{(X \times X)} (U \times X_ j)
is affine. Therefore, as X_ j is separated, it follows from Lemma 31.16.6 and its proof that for all j \in J and 1 \leq i \leq r the intersection D_ i \cap X_ j is either empty or an effective Cartier divisor in X_ j. Thus D_ i \subset X is an effective Cartier divisor (as this is a local property). Hence we can take D = D_1 + \ldots + D_ r. \square
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