The genus relation for unramified covers of curves

Given: AC; a field $k$; a finite etale morphism $f:C\to D$ of smooth proper geometrically integral curves over $k$ with $n=\mathrm{deg}(f)$; the different divisor $R_f$ of $f$.

A morphism is etale at a point when it is smooth of relative dimension zero there; in particular an etale morphism is flat and unramified at each point, and for a morphism of smooth curves unramifiedness at a closed point $p$ is equivalent to ${\mathrm{\Omega }}_{C\mathrm{/}D,p}=0$. (Étale morphism of schemes, Ramification points, branch points and unramifiedness)

Let $f:C\to D$ be a finite surjective morphism of smooth proper geometrically integral curves with separable function-field extension. For every closed point $p$ put $l_p={\mathrm{length}}_{{\mathcal{O}}_{C,p}}({\mathrm{\Omega }}_{C\mathrm{/}D,p})$; then $l_p$ is a nonnegative integer and the different divisor is the effective divisor $R_f={\sum }_p{l}_p[p]$, whose support is the differential ramification locus and whose coefficients satisfy $l_p\ge e_p-1$, with $l_p=0$ if and only if $e_p=1$ and the residue extension $\kappa (p)\mathrm{/}\kappa (f(p))$ is separable. (The different divisor of a generically separable morphism of curves, Local support and index bound for the different of a curve map)

Riemann-Hurwitz: for a finite surjective morphism $f:C\to D$ of smooth proper geometrically integral curves whose function-field extension is separable, with $n=\mathrm{deg}(f)$, one has $2g(C)-2=n(2g(D)-2)+{\mathrm{deg}}_k(R_f)$. (The Riemann-Hurwitz formula with the different)

The Axiom of Choice: every family of nonempty sets has a choice function. (The Axiom of Choice)

A finite morphism has affine inverse images of affine opens, and the coordinate algebra of each such inverse image is finite over the target ring. (Finite morphisms of schemes)

Assuming AC, a finite morphism is closed. In particular its image is a closed subset of the target. (Finite morphisms are integral and universally closed)

A curve over a field is nonempty and has chain dimension one; an integral scheme is irreducible and hence connected. Thus the given curves $C,D$ are nonempty connected integral curves of dimension one. (Curves over a field, Integral schemes)

Assuming AC, every proper closed subset of an integral finite-type curve of dimension one is a finite set of closed points. (Proper closed subsets of a curve are finite)

If $L\mathrm{/}K$ is a finitely generated field extension and ${\mathrm{\Omega }}_{L\mathrm{/}K}=0$, then $L\mathrm{/}K$ is finite and separable. (Finite-type field extensions with zero Ω)

For a smooth morphism of pure relative dimension $n$, the relative differential sheaf is locally free of rank $n$. Since the étale morphism in [F1] is smooth of relative dimension zero, its relative differentials vanish at every point. (Differentials of a smooth morphism)

On affine charts, the sheaf of relative differentials is the sheaf associated to the module of Kähler differentials; localizing that module at the generic point identifies the generic stalk with ${\mathrm{\Omega }}_{k(C)\mathrm{/}k(D)}$. (Affine charts recover the algebraic module of differentials, Kähler differentials commute with localization)