 Ramification divisor

For a finite separable morphism $f:X\to Y$ of smooth projective curves over an algebraically closed field, the ramification divisor is the zero divisor of the nonzero differential map $f^{\ast }{\Omega }_Y^1\to {\Omega }_X^1$. WriteThese coefficients are the local different exponents. The canonical relation and degree formula areUnder tame ramification, $d_P=e_P-1$. In wild ramification it can be larger, so using $\sum (e_P-1)P$ without the tame hypothesis loses part of the divisor.^[ref]

For example, in characteristic $p$, the map ${\mathbb{P}}_t^1\to {\mathbb{P}}_x^1$ with $x=t^p-t$ is separable of degree $p$ and is unramified at finite points. Infinity has index $p$. Since $dx=-dt$, the differential canonical relation givesso $R_f=(2p-2)\infty$, exceeding $(p-1)\infty$. This verifies the genus-zero Riemann-Hurwitz identity and exhibits the wild correction explicitly. A purely inseparable map has identically zero differential, so this construction requires separability.