Ramification points, branch points and unramifiedness

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be any field and let $f:C\to D$ be a nonconstant morphism of smooth proper geometrically integral curves over $k$. Then $f$ is finite and surjective and has degree $\mathrm{deg}(f)$ (Degree of a nonconstant morphism of curves, Nonconstant morphisms of proper curves are finite and surjective). For a closed point $p\in C$ put $q=f(p)$ and let $e_p$ be the ramification index (Ramification index of a morphism of curves).

The index-ramification locus of $f$ is the set of closed points $R^{\mathrm{i}\mathrm{n}\mathrm{d}}(f)=\{p\in C:p\text{ is closed and }{e}_p>1\},$ and its image $f(R^{\mathrm{i}\mathrm{n}\mathrm{d}}(f))\subseteq D$ is the index-branch locus; when the index convention is used one speaks of the ramification locus and branch locus without further qualification. Independently, the differential-ramification locus is $R^{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}(f)=\mathrm{Supp}({\mathrm{\Omega }}_{C\mathrm{/}D}),$ the support of the sheaf of relative differentials (Sheaf of relative Kähler differentials), and its image in $D$ is the differential branch locus.

For every closed point $p$, the morphism $f$ is unramified at $p$ exactly when ${\mathrm{\Omega }}_{C\mathrm{/}D,p}=0$: a finite morphism is locally of finite type, and pointwise formal unramifiedness is equivalent to vanishing of this stalk (Unramified morphism, Étale equals flat and unramified in finite presentation). In this curve-map setting it is also étale at $p$. The finite presentation and flatness needed for this last equivalence follow as follows.

Choose an affine neighborhood $V=\mathrm{Spec}(A_0)\subseteq D$ of $q$ with $f^{-1}(V)=\mathrm{Spec}(S_0)$. Since $f$ is finite, $S_0$ is a finite $A_0$-module (Finite morphisms of schemes). The ring $A_0$ is Noetherian: $k$ is Noetherian and $A_0$ is a finite-type $k$-algebra (A field has only the zero ideal and itself, hence is Noetherian, Every algebra of finite type over a Noetherian ring is a Noetherian ring). A finite $A_0$-algebra is finite type as an algebra (Subalgebra generated by a subset, algebras of finite type, and module-finite algebras), so some presentation $A_0[x_1,\dots ,x_m]\twoheadrightarrow S_0$ has finitely generated kernel: the polynomial ring is Noetherian by If $R$ is Noetherian then $R[x_1,\dots ,x_n]$ is Noetherian for every $n\in \mathbb{N}$. Thus $S_0$ is finitely presented over $A_0$, and $f$ is locally of finite presentation at $p$.

For flatness, set $A=(A_0{)}_q$ and $S=S_0{\otimes }_{A_0}A$. The target local ring $A={\mathcal{O}}_{D,q}$ is a DVR (Local rings at closed points of smooth curves are discrete valuation rings). Since $f$ is dominant and $C,D$ are integral, $A\to S$ is injective and $S$ is a domain. The finite $A$-algebra $S$ is integral over $A$ (Integrality and finite-module characterizations for one element, Integral ring maps and integral extensions), so every maximal ideal of $S$ contracts to the maximal ideal of the local ring $A$ (Under an integral extension, a prime is maximal if and only if its contraction is maximal). Thus $S$ is semilocal: its maximal ideals correspond to those of the closed fibre $S\mathrm{/}{\mathfrak{m}}_{A}S$, which is finite-dimensional, hence Artinian, over $\kappa (q)$; it has finitely many maximal ideals (An Artinian ring is canonically the finite product of its localizations at its maximal ideals). Thus $S$ is a finite torsion-free $A$-module. A DVR is a PID and every finitely generated torsion-free module over a PID is free, so $S$ is free and flat over $A$ (Every DVR is a PID, Every finitely generated torsion-free module over a PID is free). Let $P\subset S$ be the prime corresponding to $p$. Then ${\mathcal{O}}_{C,p}=S_P$; localization of the flat $A$-algebra $S$ shows that ${\mathcal{O}}_{C,p}$ is flat over $A$. This argument uses the finite affine algebra $S$; the source local ring ${\mathcal{O}}_{C,p}$ itself need not be finite over $A$.

The published pointwise criterion Étale equals flat and unramified in finite presentation says that a locally finitely presented morphism is étale at $p$ exactly when it is flat and unramified at $p$, the latter equivalent to ${\mathrm{\Omega }}_{C\mathrm{/}D,p}=0$. The preceding chart and local-algebra arguments verify its finite-presentation and flatness hypotheses here. All these statements hold over arbitrary $k$; no perfectness, residue-separability, or characteristic restriction is imposed.

Assume now that the function-field extension $k(C)\mathrm{/}k(D)$ is separable. Then by Local support and index bound for the different of a curve map the sheaf ${\mathrm{\Omega }}_{C\mathrm{/}D}$ is coherent and torsion with finite support, and $R^{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}(f)=\{p\in C:p\text{ is closed and }(e_p>1\text{ or }\kappa (p)\mathrm{/}\kappa (f(p))\text{ is inseparable})\}\mathrm{.}$ In particular, at a closed point whose residue extension $\kappa (p)\mathrm{/}\kappa (f(p))$ is separable, the two loci agree: $p\in R^{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}(f)$ if and only if $e_p>1$, hence if and only if $p\in R^{\mathrm{i}\mathrm{n}\mathrm{d}}(f)$. If $k$ is perfect then every residue extension is separable (Local support and index bound for the different of a curve map), so $R^{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}(f)=R^{\mathrm{i}\mathrm{n}\mathrm{d}}(f)$ and the index and differential branch loci coincide. Over an imperfect field a closed point with $e_p=1$ and inseparable residue extension lies in the differential support but not in the index locus, so the two loci need not agree. If the function-field extension $k(C)\mathrm{/}k(D)$ is inseparable, neither the finite-support statement nor any comparison of the two loci is asserted.