Branching (algebra)
Branching is a mathematical term that connects the fields of algebra , algebraic geometry , algebraic number theory and complex analysis .
Naming example
Let it be a natural number and the function . If and is a (sufficiently small) neighborhood of , then the archetype of consists of connected components that emerge from one another through a rotation around , i.e. multiplication with a -th root of unity. If it moves , the archetypes also move towards 0, in order to then merge into a single archetype . So, to a certain extent, 0 is the branching point for the branches. (Note that the branches are not locally separated at 0, even if the 0 is removed.)
For the transition to an algebraic view, let us now assume a holomorphic function that is defined in a neighborhood of 0. If there is a -fold zero at 0 , then the function has been withdrawn
a -fold zero. This withdrawal of locally defined holomorphic functions corresponds to a ring homomorphism
(Here denotes the ring of the power series whose radius of convergence is positive.) The order of the zeros is a discrete evaluation on the rings involved, and it applies as I said
This property is characteristic of branch points.
Branching in the context of extensions of valued bodies
Let it be a body with a discrete (exponential) evaluation . Be further
- or.
the evaluation ring or the evaluation ideal of being a uniform, d. H. a producer of , and the remainder class field . Further, let be a finite extension of with discrete valuation that continues, i.e. H. . Finally, be analogous to above.
The branch index of is defined as
If it is equal to 1, the extension is called unbranched . Its counterpart is the degree of inertia .
properties
- If the extension is separable and runs through all possible extensions of , then the fundamental equation holds
- Is also fully, it is clearly determined as
- and it applies
- Let it be complete and Galois , and furthermore be separable . (These prerequisites are met for local bodies , for example .) Then is even Galois and there is a short exact sequence
- the core is called the inertia group . Its fixed field is the maximum unbranched partial extension of , and in the case of finite extensions we have
- In particular, if it is unbranched, then it is
- If the maximum unbranched extension (in a separable conclusion of ), then applies accordingly
- In the case of local fields, the latter group is canonically isomorphic to , so it has a particularly simple structure. Since the Galois group in Frobenius automorphism
- With
- has a canonical generator, there is also a canonical element, which is also called Frobenius automorphism .
Branching in the context of extensions of Dedekindring
Let it be a Dedekind ring with a quotient field , a finite separable extension of and the entire closure of in ; is again a Dedekindring.
One of the most important special cases is , , a number field and its wholeness ring .
Further be a maximum ideal of . Then be different in a unique way as a product of powers prime ideals of write:
The numbers are called branching indices , the degrees of the remainder field extensions are called degrees of inertia .
- If and the extension of the remainder class field is separable, then it is called unbranched . (In the case of number fields and function fields over finite fields , the remainder class field expansion is always separable.)
- Is , it is called purely branched .
- If all are unbranched, it is called unbranched . then breaks down into a product of various prime ideals.
- If all prime ideals (not equal to zero) of unbranched, the extension is called unbranched .
properties
- A prime ideal of above a prime ideal of is unbranched in the sense defined here if the extension with the evaluations defined by or is unbranched in the theoretical evaluation sense.
- The fundamental equation applies
- There are always only a finite number of branched prime ideals in . A prime ideal in is branched if and only if it divides the discriminant ; a prime ideal in is branched if and only if it divides the differences .
- The only unbranched extension of is self.
- Is a Galois extension global body and unbranched, so there is analogous to the local case for each prime ideal over a Frobenius automorphism , of the decomposition group of generated. It is the basis for the Artin symbol of the class field theory .
example
A relatively simple Dedekind ring is the ring of the Eisenstein numbers . If you look at it, as usual, as an extension of the whole numbers, then exactly the prime ideal generated by the prime number 3 (in the ring of whole numbers) is branched.
Unbranched schema morphisms
Let there be and schemes and a morphism of locally finite presentation . Then is called unbranched if one of the following equivalent conditions is met:
- For one (and therefore for everyone) morphism is
- surjective.
- The fibers of over points are disjoint unions of spectra of finite separable body extensions of .
- The diagonal is an open embedment.
- If there is an affine schema and a closed sub- schema , which is defined by a nilpotent ideal sheaf, then is the induced mapping
- injective.
Morphism is called unbranched in the point when there is an open environment of in so that is unbranched. Unbranchedness in a point can also be characterized differently (unless ):
- The diagonal is a local isomorphism at .
- is a field that is a finite separable extension of .
The unbranchedness of in the point depends only on the fiber .
properties
- Unbranched morphisms are locally quasi-indifferent .
- If connected and unbranched and separated , the sections of uniquely correspond to the connected components of , which are mapped by isomorphic to .
meaning
Algebraic Geometry
If a scheme is over a discretely evaluated body with an evaluation ring , then models of over are often considered, i.e. H. Schemes about with . If there is an unbranched extension and the evaluation ring of , then the morphism and thus also the morphism are étale and surjective, consequently many properties of are transferred to the model of .
literature
- Jürgen Neukirch: Algebraic number theory . Springer-Verlag, Berlin 1992. ISBN 3-540-54273-6 .
- A. Grothendieck , J. Dieudonné : Éléments de géométrie algébrique . Publications mathématiques de l'IHÉS 4, 8, 11, 17, 20, 24, 28, 32 (1960–1967)
swell
- ↑ Jürgen Neukirch: Algebraic Number Theory. Springer-Verlag, Berlin 1992. ISBN 3-540-54273-6 , sentence (II.8.5), p. 173
- ↑ Jürgen Neukirch: Algebraic Number Theory. Springer-Verlag, Berlin 1992. ISBN 3-540-54273-6 , Theorem (II.6.2), p. 150
- ↑ Jürgen Neukirch: Algebraic Number Theory. Springer-Verlag, Berlin 1992. ISBN 3-540-54273-6 , sentence (II.6.8), p. 157
- ↑ Jürgen Neukirch: Algebraic Number Theory. Springer-Verlag, Berlin 1992. ISBN 3-540-54273-6 , sentence (II.9.9), p. 181
- ↑ Jürgen Neukirch: Algebraic Number Theory. Springer-Verlag, Berlin 1992. ISBN 3-540-54273-6 , sentence (II.9.11), p. 182
- ↑ Jürgen Neukirch: Algebraic Number Theory. Springer-Verlag, Berlin 1992. ISBN 3-540-54273-6 , sentence (I.8.1), p. 47
- ↑ Jürgen Neukirch: Algebraic Number Theory. Springer-Verlag, Berlin 1992. ISBN 3-540-54273-6 , sentence (I.8.2), p. 48
- ↑ Jürgen Neukirch: Algebraic Number Theory. Springer-Verlag, Berlin 1992. ISBN 3-540-54273-6 , sentence (I.8.4), p. 52
- ↑ Jürgen Neukirch: Algebraic Number Theory. Springer-Verlag, Berlin 1992. ISBN 3-540-54273-6 , Korollar (III.2.12), p. 213
- ↑ Jürgen Neukirch: Algebraic Number Theory. Springer-Verlag, Berlin 1992. ISBN 3-540-54273-6 , Theorem (III.2.6), p. 210
- ↑ Jürgen Neukirch: Algebraic Number Theory. Springer-Verlag, Berlin 1992. ISBN 3-540-54273-6 , sentence (III.2.18), p. 218
- ↑ Jürgen Neukirch: Algebraic Number Theory. Springer-Verlag, Berlin 1992. ISBN 3-540-54273-6 , task I.9.2, p. 61, as well as section VI.7, p. 424ff.
- ↑ EGA IV, 17.4.2, 17.2.2, 17.1.1, 17.3.1
- ↑ EGA IV, 17.4.1
- ↑ EGA IV, April 17, 3
- ↑ EGA IV, April 17, 9