Ramification index of a morphism of curves

Assume the Axiom of Choice (The Axiom of Choice), inherited through the finite curve-map and smooth-curve DVR interfaces and the unramifiedness comparison below. Let $k$ be a field and let $f:C\to D$ be a nonconstant morphism of smooth proper geometrically integral curves over $k$, of degree $\mathrm{deg}(f)$ (Degree of a nonconstant morphism of curves). Let $p\in C$ be a closed point and put $q=f(p)$. The local rings ${\mathcal{O}}_{C,p}$ and ${\mathcal{O}}_{D,q}$ are discrete valuation rings (Local rings at closed points of smooth curves are discrete valuation rings): they are Noetherian local domains of dimension one whose maximal ideals are principal, so they are discrete valuation rings in the sense of Discrete valuation rings. Let $t_q$ be a uniformizer of ${\mathcal{O}}_{D,q}$, that is, a generator of its maximal ideal.

Because $f$ is a morphism of $k$-schemes with $f(p)=q$, the comorphism $f_p^{\mathrm{♯}}:{\mathcal{O}}_{D,q}\to {\mathcal{O}}_{C,p}$ is a local homomorphism of local rings, so $f_p^{\mathrm{♯}}(t_q)$ lies in the maximal ideal of ${\mathcal{O}}_{C,p}$. The ramification index of $f$ at $p$ is $e_p:={\mathrm{ord}}_p(f_p^{\mathrm{♯}}(t_q))\in {\mathbb{Z}}_{>0},$ the order of vanishing at $p$ of the pullback of the local parameter (Order codimension one rational function), which is a positive integer because $f_p^{\mathrm{♯}}(t_q)$ is a nonzero element of the maximal ideal and the order of a uniformizer of a discrete valuation ring is one (Every nonzero fraction is a unit times a power of a uniformiser).

The definition is independent of the chosen uniformizer. If $t_q^{\mathrm{\prime }}$ is another uniformizer of ${\mathcal{O}}_{D,q}$, then $t_q^{\mathrm{\prime }}=u{t}_q$ for a unit $u\in {\mathcal{O}}_{D,q}^{\times }$; a local homomorphism carries units to units, so $f_p^{\mathrm{♯}}(u)$ is a unit of ${\mathcal{O}}_{C,p}$, and ${\mathrm{ord}}_p(f_p^{\mathrm{♯}}(t_q^{\mathrm{\prime }}))={\mathrm{ord}}_p(f_p^{\mathrm{♯}}(u))+{\mathrm{ord}}_p(f_p^{\mathrm{♯}}(t_q))=0+e_p=e_p$, by additivity of the order (Order codimension one rational function). Thus $e_p$ depends only on $f$ and $p$. Equivalently, in the notation of the structure of a local homomorphism of discrete valuation rings, $e_p$ is the unique positive integer with $t_q\mapsto u{t}_p^{e_p},u\in {\mathcal{O}}_{C,p}^{\times },$ where $t_p$ is a uniformizer of ${\mathcal{O}}_{C,p}$; this is the unique factorization of $f_p^{\mathrm{♯}}(t_q)$ supplied by Every nonzero fraction is a unit times a power of a uniformiser. The point $p$ is index-unramified over $q$ when $e_p=1$ and index-ramified when $e_p>1$. This terminology records the index only; it does not by itself assert that $f$ is unramified as a morphism.

For the scheme-theoretic notion, the exact criterion in this finite curve-map setting is $p\text{ is unramified for }f⟺e_p=1\text{ and }\kappa (p)\mathrm{/}\kappa (q)\text{ is separable}\mathrm{.}$ Here is the local route. Once the DVR structures are available, independence of the uniformizer uses no additional Choice; the comparison also inherits Choice through the cited residue and Nakayama lemmas. Write $A={\mathcal{O}}_{D,q}$ and $B={\mathcal{O}}_{C,p}$, with $t_q\mapsto u{t}_p^{e_p}$. If $f$ is unramified, then Unramified residue extensions are finite separable gives ${\mathfrak{m}}_{A}B={\mathfrak{m}}_B$ and a finite separable residue extension. Since ${\mathfrak{m}}_{A}B=(t_p^{e_p})$, its equality with $(t_p)$ gives $e_p=1$. Conversely, if $e_p=1$, then $B\mathrm{/}{t}_{q}B=\kappa (p)$. If $\kappa (p)\mathrm{/}\kappa (q)$ is separable, the finite separable field extension has zero Kähler differentials by Finite-type field extensions with zero Ω. Base change of differentials (Kähler differentials commute with scalar base change) gives ${\mathrm{\Omega }}_{B\mathrm{/}A}{\otimes }_B\kappa (p)=0$. Since $f$ is locally of finite type, ${\mathrm{\Omega }}_{B\mathrm{/}A}$ is a finite $B$-module; Nakayama's lemma gives ${\mathrm{\Omega }}_{B\mathrm{/}A}=0$, and the locally-finite-type criterion for unramifiedness is Unramified morphism. Thus the residue-field condition is essential whenever the index is used to describe ordinary unramifiedness.