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 $x$.

## Comments (2)

Comment #6645 by Laurent Moret-Bailly on

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