Lemma 33.43.8 (0B8Y)—The Stacks project

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

$\mathcal{O}_{X, x}$ is regular,
$\mathcal{O}_{X, x}$ is normal,
$\mathcal{O}_{X, x}$ is a discrete valuation ring,
$\mathcal{I}$ is an invertible $\mathcal{O}_ X$-module,
$x$ is an effective Cartier divisor on $X$.

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

$X \to \mathop{\mathrm{Spec}}(k)$ 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 October 12, 2021 at 02:14

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

Comment #6870 by Johan on January 14, 2022 at 12:52

Thanks and fixed here.

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