Why consider ramification only over number fields?

Question

Is there a reason why one looks at ramification of prime ideals only over (rings of integers of) number fields? There surely are many more situations where one has rings with prime ideals.

Answer 1

I know at least one example other than number fields where ramification is studied. Ramification is studied for finite extensions of $\mathbb{Q}_p$, ($\mathbb{Q}_p$ is the field of $p$-adic numbers). We take a finite extension $K$ of $\mathbb{Q}_p$ and consider its integral closure over $\mathbb{Z}_p$. ($\mathbb{Z}_p$ is the analogue of $\mathbb{Z}$ in the case of $\mathbb{Q}_p$).

Constructing $\mathbb{Q}_p$ from $\mathbb{Q}$ requires some effort. We start with $\mathbb{Q}$ and define a rather peculiar norm on it. The completion of $\mathbb{Q}$ with respect to this norm gives us $\mathbb{Q}_p$. But when it comes to ramification, the picture is much simpler than $\mathbb{Q}$. This is because the integral closure of $K$ over $\mathbb{Z}_p$ happens to have only one prime ideal. This makes the study somewhat simpler than $\mathbb{Q}$.

Answer 2

If $\pi:A\to B$ is any map of commutative rings whatsoever, there is a good notion of ramification. The examples mentioned so far (rings of integers of number fields or local fields, Riemann surfaces) assume that $\pi$ is an extension of Dedekind domains. In this setting, we start with a prime ideal of $\mathfrak{p}\subset A$ and extend it to $B$, then factor it uniquely as $\mathfrak{p}B=\prod\mathfrak{p}_i^{e_i}$. We then say that $\mathfrak{p}$ ramifies in $B$ if we have some $e_i>1$.

But unique factorization of ideals into primes is something specific to Dedekind domains. In the more general setting, the thing to do is take the module (sheaf) of differentials $\Omega_{B/A}$ on $B$; its support on $B$ (set of primes $\mathfrak{q}\subset B$ for which $(\Omega_{B/A})_{\mathfrak{q}}\neq0$) is the ramification locus of the map $\pi$, and the prime ideals $\pi^{-1}(\mathfrak{q})\subset A$ form the branch locus of $\pi$; these are the ramified primes from before. Because the formation of $\Omega_{B/A}$ commutes with localization, a prime ideal $\mathfrak{p}\subset A$ ramifies in $B$ if and only if $\Omega_{B_\mathfrak{p}/A_\mathfrak{p}}\neq0$. In the case of rings of integers of number fields, we recover the definition from the previous paragraph.

Geometrically, the ramification and branch loci correspond to the (closed) loci on the source and target where the map of schemes (varieties) $Spec(B)\to Spec(A)$ fails to be smooth, given additional assumptions (namely that $A\to B$ is flat and locally of finite presentation). In the case where $A,B$ are Dedekind domains and finitely generated $\mathbb{C}$-algebras (i.e. $Spec(A)$ and $Spec(B)$ are smooth complex curves), we recover the geometric picture of branching for Riemann surfaces.