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.
In complex analysis, the basic model can be taken as the "z" → "z""n" 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. See also branch point.
In algebraic topology.
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" → "z""n" 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. In algebraic geometry over any field, by analogy, it also happens in algebraic codimension one.
In algebraic number theory.
In algebraic extensions of formula_1.
Ramification in algebraic number theory means a prime ideal factoring in an extension so as to give some repeated prime ideal factors. Namely, let formula_2 be the ring of integers of an algebraic number field formula_3, and formula_4 a prime ideal of formula_2. For a field extension formula_6 we can consider the
ring of integers formula_7 (which is the integral closure of formula_2 in formula_9), and the ideal formula_10 of formula_7. This ideal may or may not be prime, but for finite formula_12, it has a factorization into prime ideals:
where the formula_14 are distinct prime ideals of formula_7. Then formula_4 is said to ramify in formula_9 if formula_18 for some formula_19; otherwise it is . In other words, formula_4 ramifies in formula_9 if the ramification index formula_22 is greater than one for some formula_14. An equivalent condition is that formula_24 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.
The ramification is encoded in formula_3 by the relative discriminant and in formula_9 by the relative different. The former is an ideal of formula_2 and is divisible by formula_4 if and only if some ideal formula_14 of formula_7 dividing formula_4 is ramified. The latter is an ideal of formula_7 and is divisible by the prime ideal formula_14 of formula_7 precisely when formula_14 is ramified.
The ramification is tame when the ramification indices formula_22 are all relatively prime to the residue characteristic "p" of formula_4, otherwise wild. This condition is important in Galois module theory. A finite generically étale extension formula_38 of Dedekind domains is tame if and only if the trace formula_39 is surjective.
In local fields.
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. This goes beyond the geometric analogue.
In algebra.
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.
In algebraic geometry.
There is also corresponding notion of unramified morphism in algebraic geometry. It serves to define étale morphisms.
Let formula_40 be a morphism of schemes. The support of the quasicoherent sheaf formula_41 is called the ramification locus of formula_42 and the image of the ramification locus, formula_43, is called the branch locus of formula_42. If formula_45 we say that formula_42 is formally unramified and if formula_42 is also of locally finite presentation we say that formula_42 is unramified (see ).