Canonical bundle formula with the different

Given: A finite surjective morphism $f\text{\hspace{0.17em}⁣}:C\to D$ of smooth proper geometrically integral curves over a field $k$ with separable function-field extension $k(C)\mathrm{/}k(D)$; the Axiom of Choice is assumed for the coherence and differential suppliers.

A curve over $k$ is geometrically integral, separated, of finite type and of chain dimension one; it is integral and Noetherian. Under Choice, a smooth curve has discrete valuation rings at its closed points, and $f$ finite surjective forces the function-field extension $k(C)\mathrm{/}k(D)$ to be finite of degree $\mathrm{deg}(f)=[k(C):k(D)]\ge 1$. (Curves over a field, Local rings at closed points of smooth curves are discrete valuation rings, Finite morphisms of schemes, The different divisor of a generically separable morphism of curves)

The canonical bundles ${\omega }_C={\mathrm{\Omega }}_{C\mathrm{/}k}$ and ${\omega }_D={\mathrm{\Omega }}_{D\mathrm{/}k}$ are invertible $\mathcal{O}$-modules, being locally free of rank one for smooth curves of relative dimension one; a nonzero rational differential $\omega$ on $C$ defines the canonical divisor $K_C=\mathrm{div}(\omega )={\sum }_x{\mathrm{ord}}_x(\omega )[x]$, any two canonical divisors differ by a principal divisor, and the rational-section dictionary identifies ${\mathcal{O}}_C(K_C)\cong {\omega }_C$; the pullback $f^{\ast }{\omega }_D$ of an invertible sheaf along $f$ is invertible. (Canonical bundle and canonical divisors, Differentials of a smooth morphism, Invertible sheaves, Divisors of rational differentials form one linear equivalence class)

For the composition $C\stackrel{f}{\to }D\to \mathrm{Spec}k$ the sequence of ${\mathcal{O}}_C$-modules $f^{\ast }{\mathrm{\Omega }}_{D\mathrm{/}k}⟶{\mathrm{\Omega }}_{C\mathrm{/}k}⟶{\mathrm{\Omega }}_{C\mathrm{/}D}⟶0$ is exact, where the first map is the base change of the universal derivation of $D\mathrm{/}k$ along $f$ and the second is induced by the universal derivation of $C$ over $D$; on affine charts it is the transitivity sequence of Kähler differentials, and affineness of $f$ exhibits the charts compatibly with the sheaves of differentials. (Transitivity sequence for differentials, Affine charts recover the algebraic module of differentials, Sheaf of relative Kähler differentials, Relative differentials commute with scheme base change)

If the function-field extension $k(C)\mathrm{/}k(D)$ is separable, the sheaf ${\mathrm{\Omega }}_{C\mathrm{/}D}$ of relative differentials is coherent and torsion: it vanishes at the generic point of $C$, and at every closed point $p$ its stalk is a module of finite length $l_p={\mathrm{length}}_{{\mathcal{O}}_{C,p}}({\mathrm{\Omega }}_{C\mathrm{/}D,p})$ over the discrete valuation ring ${\mathcal{O}}_{C,p}$, zero for all but finitely many $p$; moreover $l_p=0$ if and only if $e_p=1$ and the residue extension $\kappa (p)\mathrm{/}\kappa (f(p))$ is separable, and the support of ${\mathrm{\Omega }}_{C\mathrm{/}D}$ is the differential-ramification locus of $f$. (Local support and index bound for the different of a curve map, Coherent module sheaves, Quasi-coherent module on a scheme)

The different divisor of $f$ is the effective divisor $R_f={\sum }_p{l}_p[p]$ on $C$ determined by the lengths $l_p$ of the relative differentials. (The different divisor of a generically separable morphism of curves, Divisors on a smooth proper curve)

Let $0\to \mathcal{L}\to \mathcal{M}\to \mathcal{Q}\to 0$ be an exact sequence of ${\mathcal{O}}_C$-modules with $\mathcal{L},\mathcal{M}$ invertible and $\mathcal{Q}$ a torsion sheaf of finite length $l_p$ at the closed points $p$ and zero generic stalk; then $\mathcal{M}\cong \mathcal{L}{\otimes }_{{\mathcal{O}}_C}{\mathcal{O}}_C({\sum }_p{l}_p[p])$. (An invertible quotient of an invertible subsheaf by a torsion sheaf is a twist by an effective divisor)

The current Cartier interfaces give ${\omega }_C\cong {\mathcal{O}}_C(K_C)$ for canonical divisors, identify tensor products with divisor addition, define the pullback Cartier divisor and identify its associated sheaf with the pulled-back line bundle, and identify the kernel of the divisor-to-Picard map with principal Cartier divisors. (Invertible sheaf of cartier divisor, Linear equivalence cartier divisors, Rational sections of line bundles are Cartier divisors, Addition of Cartier divisors is tensor product of their sheaves, Pullback of a Cartier divisor, Pullback of a Cartier divisor computes the pullback of its line bundle, On an integral scheme, Cartier divisors modulo principal divisors compute the Picard group)

The Axiom of Choice is assumed, here inherited from the coherence, differential and divisor suppliers; no further selection is made. (The Axiom of Choice)

Under Choice, for a finite dominant morphism between smooth integral curves, over a closed point $q$ the finite local algebra of the source is a torsion-free module over the target DVR ${\mathcal{O}}_{D,q}$, hence free. The local calculation is given in Degree of a nonconstant morphism of curves. At the generic point the local map is a field extension, so the morphism is flat. (Finite morphisms of schemes, Integral schemes, Local rings at closed points of smooth curves are discrete valuation rings, Every DVR is a PID, Every finitely generated torsion-free module over a PID is free)