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The Riemann-Hurwitz formula with the different

Statement

Assume the Axiom of Choice as inherited from the ramification and duality suppliers. Let $f : C \to D$ be a finite surjective morphism of smooth proper geometrically integral curves over a field $k$ whose function-field extension $k (C) / k (D)$ is separable (equivalently, a nonconstant morphism whose generic fibre is separable), with $n = \deg (f)$ and different divisor $R_{f}$ . Then $2 g (C) - 2 = n (2 g (D) - 2) + \deg_{k} (R_{f}),$ where $g (C)$ and $g (D)$ are the genera. Equivalently, for canonical divisors, $K_{C}$ is linearly equivalent to $f^{*} K_{D} + R_{f}$ , and the displayed identity is the degree identity obtained from it.

Facts & Assumptions

Given: A field $k$ ; a finite surjective morphism $f : C \to D$ of smooth proper geometrically integral curves whose function-field extension $k (C) / k (D)$ is separable; $n = \deg (f)$ ; the different divisor $R_{f}$ on $C$ .

[F1]

Under the separability hypothesis, the natural map $f^{*} ω_{D} \to ω_{C}$ has cokernel $Ω_{C / D}$ and there is a canonical isomorphism $ω_{C} ≅ f^{*} ω_{D} \otimes O_{C} (R_{f})$ ; equivalently $K_{C}$ is linearly equivalent to $f^{*} K_{D} + R_{f}$ for canonical divisors, where $R_{f}$ is the different divisor. The different is defined by $R_{f} = \sum_{p} l_{p} [p]$ with $l_{p} = {length}_{O_{C, p}} (Ω_{C / D, p})$ a nonnegative integer vanishing exactly off the support of $Ω_{C / D}$ , so that $R_{f}$ is an effective divisor supported on the differential ramification locus with $l_{p} \geq e_{p} - 1$ . (Canonical bundle formula with the different, The different divisor of a generically separable morphism of curves)

[F2]

A nonconstant morphism of smooth proper geometrically integral curves is finite and surjective and has a positive degree $n = \deg (f) = [k (C) : k (D)]$ ; for such a morphism, if $E$ is a divisor on $D$ then $f^{*} E$ is defined and $\deg_{k} (f^{*} E) = n \deg_{k} (E)$ , and for every invertible $O_{D}$ -module $M$ one has $\deg (f^{*} M) = n \deg (M)$ . (Nonconstant morphisms of proper curves are finite and surjective, Degree of a nonconstant morphism of curves, Fibres, pullbacks and degrees of divisors under a finite morphism of curves)

[F3]

For a smooth proper geometrically integral curve $C$ of genus $g (C)$ and any canonical divisor $K_{C}$ one has $\deg_{k} (K_{C}) = 2 g (C) - 2$ ; likewise $\deg_{k} (K_{D}) = 2 g (D) - 2$ on $D$ . (The canonical divisor has degree 2g - 2, Canonical bundle and canonical divisors)

[F4]

On a smooth proper geometrically integral curve, divisors are finite sums of closed points with additive degree $\deg_{k} (D) = \sum_{x} n_{x} [κ (x) : k]$ , the Weil and Cartier descriptions agree, and every invertible sheaf is $O_{C} (D)$ for a divisor $D$ well defined modulo linear equivalence, so degrees of invertible sheaves are computed by $\deg_{k}$ of any associated divisor. (Divisors on a smooth proper curve, Degree divisor proper curve, Cartier and Weil divisors agree on a smooth curve)

[F5]

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

Proof

Proof technique: direct; take the degree of the canonical ramification formula and evaluate with $\deg ω = 2 g - 2$ on both curves.

1.1F1F2F4given

(Set-up.) By [F2] the morphism $f$ is finite and surjective with $n = \deg (f) = [k (C) : k (D)] \geq 1$ , and pullback of divisors along $f$ is defined with $\deg_{k} (f^{*} E) = n \deg_{k} (E)$ for divisors $E$ on $D$ ; by [F1] the different $R_{f} = \sum_{p} l_{p} [p]$ is an effective divisor on $C$ determined by the lengths of the torsion module $Ω_{C / D}$ , and by [F4] divisors on $C$ and $D$ have additive degrees and any invertible sheaf has a well-defined degree given by any associated divisor.

2.1F1F4step 1.1

(Canonical formula.) Since $k (C) / k (D)$ is separable, [F1] provides the isomorphism $ω_{C} ≅ f^{*} ω_{D} \otimes O_{C} (R_{f})$ ; with $K_{C}$ and $K_{D}$ canonical divisors, $ω_{C} ≅ O_{C} (K_{C})$ , $ω_{D} ≅ O_{D} (K_{D})$ and $O_{C} (R_{f})$ the sheaf of the effective divisor $R_{f}$ by [F1] and [F4], this says exactly that $K_{C}$ is linearly equivalent to $f^{*} K_{D} + R_{f}$ .

3.1F2F4step 2.1

(Degree identity.) Taking degrees of the two isomorphic invertible sheaves of step 2.1 using the degree conventions of [F4] gives $\deg_{k} (K_{C}) = \deg_{k} (f^{*} K_{D} + R_{f}) = \deg_{k} (f^{*} K_{D}) + \deg_{k} (R_{f}) = n \deg_{k} (K_{D}) + \deg_{k} (R_{f})$ , where the middle step is additivity of $\deg_{k}$ [F4] and the last step is the pullback formula of [F2].

4.1F3step 2.1step 3.1

(Genera.) By [F3] one has $\deg_{k} (K_{C}) = 2 g (C) - 2$ and $\deg_{k} (K_{D}) = 2 g (D) - 2$ , so substituting into step 3.1 gives $2 g (C) - 2 = n (2 g (D) - 2) + \deg_{k} (R_{f})$ , which is the displayed Riemann-Hurwitz identity, obtained exactly as the degree identity of the linear equivalence $K_{C} \sim f^{*} K_{D} + R_{f}$ of step 2.1.

5.1F5step 2.1step 4.1∎

The Axiom of Choice [F5] is used exactly through the ramification and duality suppliers cited above, each of which assumes it; together steps 2.1 and 4.1 prove both the linear equivalence and the numerical identity.

Depends on

Used by

Dependency tree · two levels

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Sources

William Fulton, Algebraic Curves (Internet Archive copy), Ch. 8 (standard reference, not scraped)
Ravi Vakil, The Rising Sea (version of October 21, 2025) (standard reference, not scraped)
The Stacks Project, Algebraic Curves (tag 0BRV) (standard reference, not scraped)