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Schematic depiction of ramification: the fibers of almost all points in Y below consist of three points, except for two points in Y marked with dots, where the fibers consist of one and two points (marked in black), respectively. The map f is said to be ramified in these points of Y.

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 together of the fibers of the mapping.

In complex analysis

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. 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 → 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. In algebraic geometry over any field, by analogy, it also happens in algebraic codimension one.

In algebraic number theory

In algebraic extensions of Qdisplaystyle mathbb Q

Ramification in algebraic number theory means a prime ideal factoring in an extension so as to give some repeated prime ideal factors. Namely, let OKdisplaystyle mathcal O_K be the ring of integers of an algebraic number field Kdisplaystyle K, and pdisplaystyle mathfrak p a prime ideal of OKdisplaystyle mathcal O_K. For a field extension L/Kdisplaystyle L/K we can consider the
ring of integers OLdisplaystyle mathcal O_L (which is the integral closure of OKdisplaystyle mathcal O_K in Ldisplaystyle L), and the ideal pOLdisplaystyle mathfrak pmathcal O_L of OLdisplaystyle mathcal O_L. This ideal may or may not be prime, but for finite [L:K]displaystyle [L:K], it has a factorization into prime ideals:

p⋅OL=p1e1⋯pkekdisplaystyle mathfrak pcdot mathcal O_L=mathfrak p_1^e_1cdots mathfrak p_k^e_k

where the pidisplaystyle mathfrak p_i are distinct prime ideals of OLdisplaystyle mathcal O_L. Then pdisplaystyle mathfrak p is said to ramify in Ldisplaystyle L if ei>1displaystyle e_i>1 for some idisplaystyle i; otherwise it is unramified. In other words, pdisplaystyle mathfrak p ramifies in Ldisplaystyle L if the ramification index eidisplaystyle e_i is greater than one for some pidisplaystyle mathfrak p_i. An equivalent condition is that OL/pOLdisplaystyle mathcal O_L/mathfrak pmathcal O_L 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 Kdisplaystyle K by the relative discriminant and in Ldisplaystyle L by the relative different. The former is an ideal of OKdisplaystyle mathcal O_K and is divisible by pdisplaystyle mathfrak p if and only if some ideal pidisplaystyle mathfrak p_i of OLdisplaystyle mathcal O_L dividing pdisplaystyle mathfrak p is ramified. The latter is an ideal of OLdisplaystyle mathcal O_L and is divisible by the prime ideal pidisplaystyle mathfrak p_i of OLdisplaystyle mathcal O_L precisely when pidisplaystyle mathfrak p_i is ramified.

The ramification is tame when the ramification indices eidisplaystyle e_i are all relatively prime to the residue characteristic p of pdisplaystyle mathfrak p, otherwise wild. This condition is important in Galois module theory. A finite generically étale extension B/Adisplaystyle B/A of Dedekind domains is tame if and only if the trace Tr:B→Adisplaystyle operatorname Tr :Bto A 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 f:X→Ydisplaystyle f:Xto Y be a morphism of schemes. The support of the quasicoherent sheaf ΩX/Ydisplaystyle Omega _X/Y is called the ramification locus of fdisplaystyle f and the image of the ramification locus, f(Supp⁡ΩX/Y)displaystyle fleft(operatorname Supp Omega _X/Yright), is called the branch locus of fdisplaystyle f. If ΩX/Y=0displaystyle Omega _X/Y=0 we say that fdisplaystyle f is formally unramified and if fdisplaystyle f is also of locally finite presentation we say that fdisplaystyle f is unramified [see Vakil's notes].

See also

• Eisenstein polynomial


• Newton polygon


• Puiseux expansion


• Branched covering

Look up ramification (mathematics) in Wiktionary, the free dictionary.

References

• 
  Neukirch, Jürgen (1999). Algebraic Number Theory. Grundlehren der mathematischen Wissenschaften. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021..mw-parser-output cite.citationfont-style:inherit.mw-parser-output qquotes:"""""""'""'".mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em
  


• Vakil, Ravi, "Foundations of Algebraic Geometry", Lecture Notes, http://math.stanford.edu/~vakil/216blog/
  

External links

• 
  "Ramification in number fields". PlanetMath.