Lemma 33.43.8. Let $k$ be a field. Let $X$ be a curve over $k$. Let $x \in X$ be a closed point. We think of $x$ as a (reduced) closed subscheme of $X$ with sheaf of ideals $\mathcal{I}$. The following are equivalent

1. $\mathcal{O}_{X, x}$ is regular,

2. $\mathcal{O}_{X, x}$ is normal,

3. $\mathcal{O}_{X, x}$ is a discrete valuation ring,

4. $\mathcal{I}$ is an invertible $\mathcal{O}_ X$-module,

5. $x$ is an effective Cartier divisor on $X$.

If $k$ is perfect or if $\kappa (x)$ is separable over $k$, these are also equivalent to

1. $X \to \mathop{\mathrm{Spec}}(k)$ is smooth at $x$.

Proof. Since $X$ is a curve, the local ring $\mathcal{O}_{X, x}$ is a Noetherian local domain of dimension $1$ (Lemma 33.20.3). Parts (4) and (5) are equivalent by definition and are equivalent to $\mathcal{I}_ x = \mathfrak m_ x \subset \mathcal{O}_{X, x}$ having one generator (Divisors, Lemma 31.15.2). The equivalence of (1), (2), (3), (4), and (5) therefore follows from Algebra, Lemma 10.119.7. The final statement follows from Lemma 33.25.8 in case $k$ is perfect. If $\kappa (x)/k$ is separable, then the equivalence follows from Algebra, Lemma 10.140.5. $\square$

Comments (2)

Comment #6645 by Laurent Moret-Bailly on

For condition (6) one could of course weaken the assumption "If $k$ is perfect" to "If $\kappa(x)/k$ is separable".

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• 4 comment(s) on Section 33.43: Curves

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