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The Stacks project

Lemma 31.13.4. Let S be a scheme. Let D \subset S be an effective Cartier divisor. Let U = S \setminus D. Then U \to S is an affine morphism and U is scheme theoretically dense in S.

Proof. Affineness is Lemma 31.13.3. The density question is local on S, see Morphisms, Lemma 29.7.5. Thus we may assume S = \mathop{\mathrm{Spec}}(A) and D corresponding to the nonzerodivisor f \in A, see Lemma 31.13.2. Thus A \subset A_ f which implies that U \subset S is scheme theoretically dense, see Morphisms, Example 29.7.4. \square


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