Ramification

Branch is a mathematical term, which connects the areas algebra, algebraic geometry and complex analysis together.

• 3.1 Properties
• 3.2 Example

• 4.1 properties

• 5.1 Algebraic Geometry

Named giving example

It is a natural number and the function. Is now and a ( sufficiently small ) neighborhood, there is the archetype of connected components from which emerge by a rotation around so multiplication by a -th root of unity apart. Moves, then move the archetypes to 0, and then to merge for a single archetype. 0 So in a way the branch point for the branches. ( Note that the branches are not locally separated at 0, even if the 0 is removed. )

Now is a holomorphic function defined in a neighborhood of 0 for the transition to an algebraic point of view. Has at 0 - a multiple zero, so has the retracted function

One times zero. This retraction locally defined holomorphic functions corresponds to a ring homomorphism

( Here denotes the ring of power series whose radius of convergence is positive. ) The zeros of order is a discrete valuation on the participating rings, and it is as I said

This property is characteristic of branch points.

Branching in the context of extensions weighted body

It is a body with a discrete ( exponential ) valuation. Next were

The valuation ring and the evaluation of ideal, a uniformizing, that is a producer of, and the residue class field. Next is a finite extension of discrete valuation, which continues, that is. Finally, are analogous to above.

The branching index is defined as

If it is equal to 1, ie the extension unbranched. Its counterpart is the degree of inertia.

Properties

• If the extension is separable, and goes through all the possible continuations of, the fundamental equation

• If, in addition completely, so is uniquely determined as

• There are now complete and a Galois field, and also be separable. ( These conditions are met, for example, for local body. ) Then even a Galois field, and there is a short exact sequence

Branching in the context of extensions of Dedekind rings

It should be a Dedekind ring with quotient field, a finite separable extension of and the whole concluding in; is a Dedekind ring.

One of the most important special cases, is a number field and his wholeness ring.

Next is a maximal ideal of. Then can be written uniquely as a product of powers of different prime ideals of:

The numbers are called branching indices, the degree of residue class field extensions inertia degrees.

• Is and separable extension of the residue field, so called unbranched. ( In the case of number fields and function fields over finite fields, the Restklassenkörperweiterung is always separable. )
• Is so called pure branched.
• Are all unbranched, so called unbranched. then decomposes into a product of different primes.
• Are all prime ideals ( non-zero ) of unbranched, it means the extension unbranched.

Properties

• A prime ideal of more than one prime ideal of if and only unbranched in the sense defined here, when the expansion of the area defined by reviews or unbranched in the valuation theoretical sense.
• It is the fundamental equation

• There are always only finitely many prime ideals in branched. A prime ideal if and only branched if it divides the discriminant; a prime ideal if and only branched if it shares the Differente.
• There is no unbranched extensions.
• If a Galois global field and unbranched, so there is analogous to the local case, for each prime ideal over a Frobenius automorphism which generates the decomposition group of. It is the basis for the Artinsymbol of class field theory.

Example

A relatively simple Dedekind ring is the ring of Eisenstein numbers. Looking at them, as usual, as an extension of the integers, then here is exactly of the prime number 3 generated ( in the ring of integers ) prime ideal branched.

Unbranched Schemamorphismen

Let and schemes and a morphism locally of finite presentation. Then called unbranched if one of the following equivalent conditions is satisfied:

• Is for one ( and hence for every ) morphism

• The fibers of about points are disjoint unions of spectra of finite separable field extensions of.
• The diagonal is an open embedding.
• Is an affine scheme and a closed subscheme which is defined by a nilpotent sheaf of ideals, then the induced map

The morphism is called unbranched in point is if there is an open neighborhood of in such that unbranched. Unbranched awareness in one point can be characterized by different ( it ):

• The diagonal is a local isomorphism at.
• Is a body that is separable extension of a finite.

Unbranched in the plurality of point depends only on the fiber.

Properties

• Unbranched morphisms are locally quasi- finite.
• Is coherent and unbranched and separated, so the corresponding sections of the uniquely connected components of which are represented by isomorphic to.

Importance

Algebraic Geometry

Is a scheme over a discretely valued field with valuation ring, so models are often viewed from above, that is, schemes over with. Is now an unramified extension and the valuation ring of, the morphism and hence the morphism is étale and surjective, hence to transfer many properties of the model.