The Stacks project

Lemma 31.15.4. Let $X$ be a Noetherian scheme. Let $D \subset X$ be an integral closed subscheme which is also an effective Cartier divisor. Then the local ring of $X$ at the generic point of $D$ is a discrete valuation ring.

Proof. By Lemma 31.13.2 we may assume $X = \mathop{\mathrm{Spec}}(A)$ and $D = \mathop{\mathrm{Spec}}(A/(f))$ where $A$ is a Noetherian ring and $f \in A$ is a nonzerodivisor. The assumption that $D$ is integral signifies that $(f)$ is prime. Hence the local ring of $X$ at the generic point is $A_{(f)}$ which is a Noetherian local ring whose maximal ideal is generated by a nonzerodivisor. Thus it is a discrete valuation ring by Algebra, Lemma 10.119.7. $\square$

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