(Ramification index)
Let \(f : X \to Y\) be a finite morphism of curves, \(p \in X\), \(q = f(p)\), and \(t\) a uniformizer of the discrete valuation ring \(\OO_{Y,q}\). The ramification index is \[ e_p \definedas v_p(f^\sharp t) , \] the valuation at \(p\) of the pulled-back uniformizer. \(f\) is ramified at \(p\) if \(e_p > 1\) and unramified there if \(e_p = 1\); the image \(q\) of a ramification point is a branch point.
(Pullback of divisors)
For \(f : X \to Y\) a finite morphism of nonsingular curves, \(f^* : \operatorname{Div} Y \to \operatorname{Div} X\) is the homomorphism with \[ f^* q = \sum_{p \in f^{-1}(q)} e_p \, p . \] It preserves linear equivalence and satisfies \(f^* \mcl(D) \cong \mcl(f^* D)\), so it induces \(f^* : \Pic Y \to \Pic X\), and \(\deg f^* D = \deg f \cdot \deg D\). [@Har10a, §IV.2, Proposition II.6.9, Exercise II.6.8]
(Ramification divisor and branch locus)
For \(f \colon X \to Y\) a finite separable morphism of smooth projective curves, the ramification divisor is \[ R = \sum_{p \in X} \length (\Omega_{X/Y})_p \cdot p , \] which is \(\sum_p (e_p - 1)\, p\) when all ramification is tame. The branch locus is the finite set \(f(\supp R)\) of branch points, and the branch divisor is the pushforward \(f_* R = \sum_p \length(\Omega_{X/Y})_p \cdot f(p)\).
(Tame and wild)
Let \(\characteristic k = p_0\). The ramification at \(p\) is tame if \(p_0 \nmid e_p\), and wild if \(p_0 \mid e_p\). In characteristic \(0\) all ramification is tame.
(The length formula)
For \(f\) finite separable, \(\Omega_{X/Y}\) is a torsion sheaf supported exactly on the ramification locus, its stalks are principal \(\OO_p\)-modules, and \[ \length (\Omega_{X/Y})_p \begin{cases} = e_p - 1 & \text{at a tamely ramified } p, \\ > e_p - 1 & \text{at a wildly ramified } p . \end{cases} \]
(Where the dichotomy comes from)
The stalk length is computed by one derivative. With \(u\) a uniformizer at \(p\) and \(t\) one at \(q\), write \(f^\sharp t = c u^{e_p}(1 + \cdots)\); then \[ f^* dt = \left( e_p \, c \, u^{e_p - 1} + \cdots \right) du , \] and \(\length(\Omega_{X/Y})_p\) is the valuation of that coefficient. If \(p_0 \nmid e_p\) the leading term survives and the valuation is \(e_p - 1\). If \(p_0 \mid e_p\) the leading term is killed, the valuation is determined by whatever comes next, and it is strictly larger. So the ramification at \(p\) is tame if and only if \(e_p\) is invertible in \(k\).
The map \(u \mapsto t=u^n\) with \(p_0 \nmid n\) is tamely ramified at \(u=0\), where \(R\) has coefficient \(n-1\). For a wild example take \(\characteristic k = p_0\) and \(f : \PP^1 \to \PP^1\) with \(t = u^{p_0} - u\), which is separable since \(dt = -du\). It is unramified on the affine line and totally ramified at infinity with \(e = p_0\), which is wild, and the local computation there gives length \(2p_0 - 2 > p_0 - 1\). Riemann–Hurwitz confirms it: \(-2 = p_0 \cdot (-2) + \deg R\) forces \(\deg R = 2p_0 - 2\), carried by that one point.
(Ramification and Riemann--Hurwitz)
Riemann–Hurwitz (T-LKT0U) holds with \(\deg R = \sum_p \length (\Omega_{X/Y})_p\), which equals \(\sum_p (e_p - 1)\) when all ramification is tame. At a wildly ramified point the local length exceeds \(e_p-1\), so \(\deg R>\sum_p(e_p-1)\). For the Artin–Schreier map \(u\mapsto u^{p_0}-u\) of degree \(p_0\), the tame formula would give \(\deg R=p_0-1\) at its single ramification point, while Riemann–Hurwitz requires \(\deg R=2p_0-2\).
The condition \(e_p = 1\) at every \(p\) is unramifiedness in the sense of D-MORUNR; a nonconstant morphism of smooth curves is flat, so it is étale if and only if \(e_p=1\) at every \(p\).
Dependencies
D-MORUNRUnramified morphismsuses
Backlinks
PR-IV2DEGREVENThe ramification divisor has even degreeuses