Riemann-Hurwitz theorem

First we define the different divisor of an extension of function fields. Let $K$ be a function field over a field $F$ and let $L$ be a finite separable extension of $K$ . Let $\mathcal{O}_{P}$ be a prime of $K$ , i.e. a discrete valuation ring with $F\subset\mathcal{O}_{P}$ , maximal ideal $P$ and quotient field equal to $K$ . Let $R_{P}$ be the integral closure of $\mathcal{O}_{P}$ in $L$ . Notice that if $\mathfrak{p}$ is a prime ideal of $R_{P}$ , then the localization $\mathcal{O}_{\mathfrak{p}}=(R_{P})_{\mathfrak{p}}$ is a prime of $L$ (which is said to be lying over $\mathcal{O}_{P}$ ). The maximal ideal of $\mathcal{O}_{\mathfrak{p}}$ is $\mathfrak{p}(R_{P})_{\mathfrak{p}}$ .

Let $\mathcal{O}_{\mathfrak{P}}$ be any prime of $L$ , then it lays over some prime ideal $P$ of $K$ and in fact, if $\mathfrak{p}=R_{P}\cap\mathfrak{P}$ then $\mathcal{O}_{\mathfrak{p}}\cong\mathcal{O}_{\mathfrak{P}}$ . Let $\delta(\mathfrak{P})$ be the exact power of $\mathfrak{p}$ dividing the different of $R_{P}$ over $\mathcal{O}_{P}$ (the different of an extension of Dedekind domains is a fractional ideal). We define the different divisor of $L/K$ as follows:

D_{L/K}=\sum_{\mathfrak{P}}\delta({\mathfrak{P}})\mathfrak{P}

as an element of the free abelian group generated by the prime ideals of $L$ .

Theorem (Riemann-Hurwitz).

Let $L/K$ be a finite, separable, geometric extension of function fields and suppose the genus of $K$ is $g_{K}$ . Then the genus of $L$ is given by the formula:

2g_{L}-2=[L:K](2g_{K}-2)+\deg_{L}D_{L/K}.

Title	Riemann-Hurwitz theorem
Canonical name	RiemannHurwitzTheorem
Date of creation	2013-03-22 15:34:40
Last modified on	2013-03-22 15:34:40
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	4
Author	alozano (2414)
Entry type	Theorem
Classification	msc 11R58
Defines	different divisor of an extension of function fields