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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