Ramification (mathematics)

In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign.

The term is also used from the opposite perspective (branches coming together) as when a covering map degenerates at a point of a space, with some collapsing of the fibers of the mapping.

In complex analysis, the basic model can be taken as the z → zn mapping in the complex plane, near z = 0.

This is the standard local picture in Riemann surface theory, of ramification of order n. It occurs for example in the Riemann–Hurwitz formula for the effect of mappings on the genus.

In a covering map the Euler–Poincaré characteristic should multiply by the number of sheets; ramification can therefore be detected by some dropping from that.

The z → zn mapping shows this as a local pattern: if we exclude 0, looking at 0 < |z| < 1 say, we have (from the homotopy point of view) the circle mapped to itself by the n-th power map (Euler–Poincaré characteristic 0), but with the whole disk the Euler–Poincaré characteristic is 1, n – 1 being the 'lost' points as the n sheets come together at z = 0.

In geometric terms, ramification is something that happens in codimension two (like knot theory, and monodromy); since real codimension two is complex codimension one, the local complex example sets the pattern for higher-dimensional complex manifolds.

In complex analysis, sheets can't simply fold over along a line (one variable), or codimension one subspace in the general case.

The ramification set (branch locus on the base, double point set above) will be two real dimensions lower than the ambient manifold, and so will not separate it into two 'sides', locally―there will be paths that trace round the branch locus, just as in the example.

Ramification in algebraic number theory means a prime ideal factoring in an extension so as to give some repeated prime ideal factors.

be the ring of integers of an algebraic number field

This ideal may or may not be prime, but for finite

, it has a factorization into prime ideals: where the

are distinct prime ideals of

has a non-zero nilpotent element: it is not a product of finite fields.

The analogy with the Riemann surface case was already pointed out by Richard Dedekind and Heinrich M. Weber in the nineteenth century.

and is divisible by the prime ideal

are all relatively prime to the residue characteristic p of

This condition is important in Galois module theory.

A finite generically étale extension

of Dedekind domains is tame if and only if the trace

The more detailed analysis of ramification in number fields can be carried out using extensions of the p-adic numbers, because it is a local question.

In that case a quantitative measure of ramification is defined for Galois extensions, basically by asking how far the Galois group moves field elements with respect to the metric.

A sequence of ramification groups is defined, reifying (amongst other things) wild (non-tame) ramification.

In valuation theory, the ramification theory of valuations studies the set of extensions of a valuation of a field K to an extension field of K. This generalizes the notions in algebraic number theory, local fields, and Dedekind domains.

There is also corresponding notion of unramified morphism in algebraic geometry.

It serves to define étale morphisms.

The support of the quasicoherent sheaf

, is called the branch locus of

is also of locally finite presentation we say that