Local rings at closed points of smooth curves are discrete valuation rings

Choose an affine open $\mathrm{Spec}A\subseteq C$ containing $x$; then $A$ is a finite-type $k$-domain [F1], the point $x$ corresponds to a maximal ideal $\mathfrak{m}\subset A$, and ${\mathcal{O}}_{C,x}=A_{\mathfrak{m}}$ [F4]. The function field of $C$ satisfies $k(C)=\mathrm{Frac}(A)=\mathrm{Frac}(A_{\mathfrak{m}})$ [F8].

The local ring ${\mathcal{O}}_{C,x}$ has dimension one. Indeed the local dimension formula [F3] applied to $\mathfrak{q}=\mathfrak{m}$ gives ${\dim }_{\mathfrak{m}}\mathrm{Spec}A=\dim A_{\mathfrak{m}}+{\mathrm{trdeg}}_k\kappa (\mathfrak{m})$, and ${\mathrm{trdeg}}_k\kappa (\mathfrak{m})=0$ because $\kappa (\mathfrak{m})$ is a finite extension of $k$ [F3]. Every open neighbourhood of the closed point $x$ contains the generic point $\eta$ [F1], so each such neighbourhood has chain dimension one, and hence the infimum ${\dim }_{\mathfrak{m}}\mathrm{Spec}A$ of the dimensions of these neighbourhoods equals $1$; therefore $\dim A_{\mathfrak{m}}=1$.

The local ring is regular. Under the equivalence of [F2], smoothness of $C$ over $k$ applies to the field extension $k\mathrm{/}k$ itself, so every local ring of $C_k=C$ is regular; in particular ${\mathcal{O}}_{C,x}$ is a regular local ring in the sense of [F5].

The ring ${\mathcal{O}}_{C,x}=A_{\mathfrak{m}}$ is a Noetherian local ring: $A$ is Noetherian by [F4] and localisations of Noetherian rings are Noetherian [F4]. It is nonzero because $A$ is a domain and $\mathfrak{m}$ is a prime.

By steps 1.2, 1.3 and 2.1 the ring ${\mathcal{O}}_{C,x}$ is a nonzero Noetherian regular local ring of dimension one; under Choice [F9] the criterion [F6] shows that ${\mathcal{O}}_{C,x}$ is a discrete valuation ring.

By [F7] there is a discrete valuation $v:k(C{)}^{\times }\to \mathbb{Z}$ with ${\mathcal{O}}_{C,x}=V_v$, and there is $t_x\in k(C)$ with $v(t_x)=1$. Define ${\mathrm{ord}}_x(f):=v(f)$ for $f\in k(C{)}^{\times }$; this is a well-defined element of $\mathbb{Z}$ because $v$ is a function on $k(C{)}^{\times }$. For $0\mathrm{\ne }f\in {\mathcal{O}}_{C,x}$ put $n={\mathrm{ord}}_x(f)$, so that $n\ge 0$ because $f\in V_v$, and set $u=f\cdot t_x^{-n}$. Then $v(u)=v(f)-nv(t_x)=0$, so $u$ is a unit of ${\mathcal{O}}_{C,x}$ by [F7], and $f=u{t}_x^n$. In particular ${\mathfrak{m}}_x=(t_x)$, since $f\in {\mathfrak{m}}_x$ if and only if $v(f)>0$, which by the display means $f\in (t_x)$.

Steps 3.1 and 4.1 give every clause of the statement: ${\mathcal{O}}_{C,x}$ is Noetherian, regular and one-dimensional (steps 1.2, 1.3 and 2.1), hence a discrete valuation ring with maximal ideal generated by the uniformizer $t_x$ (steps 3.1, 4.1), and orders and the normal form $f=u{t}_x^n$ are well defined (step 4.1). Choice is inherited through the local-dimension formula [F3] in step 1.2, the smoothness characterization [F2] in step 1.3, and the DVR criterion [F6] in step 3.1.