The different divisor of a generically separable morphism of curves

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be any field and let $f:C\to D$ be a finite surjective morphism of smooth proper geometrically integral curves over $k$ (Curves over a field, Finite morphisms of schemes) whose function-field extension $k(C)\mathrm{/}k(D)$ is separable. Let ${\mathrm{\Omega }}_{C\mathrm{/}D}$ be the sheaf of relative differentials (Sheaf of relative Kähler differentials). By Local support and index bound for the different of a curve map the sheaf ${\mathrm{\Omega }}_{C\mathrm{/}D}$ is a coherent ${\mathcal{O}}_C$-module of torsion with finite support: it vanishes at the generic point, and at every closed point $p$ of $C$ the stalk ${\mathrm{\Omega }}_{C\mathrm{/}D,p}$ is a finite-length module over the discrete valuation ring ${\mathcal{O}}_{C,p}$ (Local rings at closed points of smooth curves are discrete valuation rings, Composition series and length of a module). Put $l_p:={\mathrm{length}}_{{\mathcal{O}}_{C,p}}\left({\mathrm{\Omega }}_{C\mathrm{/}D,p}\right)\in {\mathbb{Z}}_{\ge 0}\mathrm{.}$

The different divisor of $f$ is the divisor $R_f:={\sum }_{p\in C\text{ closed}}{l}_p[p]$ on $C$ (Divisors on a smooth proper curve). It is well defined: each $l_p$ is a nonnegative integer, and $l_p=0$ for all but finitely many closed points $p$, so the sum is finite and $R_f\ge 0$ is an effective divisor on $C$. The definition depends only on $f$, since the relative differentials and the lengths $l_p$ are attached to $f$.

By Local support and index bound for the different of a curve map the coefficient $l_p$ satisfies $l_p\ge e_p-1$ for the ramification index $e_p$ of $f$ at $p$ (Ramification index of a morphism of curves), and $l_p=0$ exactly when $e_p=1$ and the residue extension $\kappa (p)\mathrm{/}\kappa (f(p))$ is separable. Consequently $\mathrm{Supp}(R_f)=\mathrm{Supp}({\mathrm{\Omega }}_{C\mathrm{/}D})=\{p\in C:p\text{ is closed and }(e_p>1\text{ or }\kappa (p)\mathrm{/}\kappa (f(p))\text{ is inseparable})\},$ the differential-ramification locus of $f$ (Ramification points, branch points and unramifiedness); it contains the index-ramification locus $\{p\in C:p\text{ is closed and }{e}_p>1\}$ and agrees with it when $k$ is perfect, while over an imperfect field a point with $e_p=1$ and inseparable residue extension is in $\mathrm{Supp}(R_f)$ but not in the index locus. The different $R_f$ is the divisor-theoretic correction term in the canonical-bundle comparison between ${\omega }_C$ and the pullback of ${\omega }_D$ (Canonical bundle and canonical divisors); the comparison is stated in the companion canonical-bundle theorem.