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.) ${\displaystyle n>1}$ ${\displaystyle f\colon \mathbb {C} \to \mathbb {C} }$ ${\displaystyle z\mapsto w=z^{n}}$ ${\displaystyle w\neq 0}$ ${\displaystyle U}$ ${\displaystyle w}$ ${\displaystyle U}$ ${\displaystyle n}$ ${\displaystyle 2\pi /n}$ ${\displaystyle n}$ ${\displaystyle w\to 0}$ ${\displaystyle w_{0}=0}$ ${\displaystyle \{0\}}$ ${\displaystyle n}$

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 If there is a -fold zero at 0 , then the function has been withdrawn ${\displaystyle g(w)}$ ${\displaystyle g}$ ${\displaystyle k}$

${\displaystyle f^{*}(g)=g\circ f,\quad z\mapsto g(z^{n})}$

a -fold zero. This withdrawal of locally defined holomorphic functions corresponds to a ring homomorphism ${\displaystyle nk}$

${\displaystyle f^{*}\colon \mathbb {C} \{w\}\to \mathbb {C} \{z\},\quad w\mapsto z^{n}.}$

(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 ${\displaystyle \mathbb {C} \{w\}}$

${\displaystyle \operatorname {ord} _{z=0}f^{*}(g)=n\cdot \operatorname {ord} _{w=0}g.}$

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 ${\displaystyle K}$ ${\displaystyle v\colon K^{\times }\to \mathbb {R} }$

${\displaystyle {\mathcal {O}}_{K}=\{x\in K\mid v(x)\geq 0\}}$ or. ${\displaystyle {\mathfrak {m}}_{K}=\{x\in K\mid v(x)>0\}}$

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. ${\displaystyle K}$ ${\displaystyle \pi _{K}}$ ${\displaystyle {\mathfrak {m}}_{K}}$ ${\displaystyle \kappa ={\mathcal {O}}_{K}/{\mathfrak {m}}_{K}}$ ${\displaystyle L}$ ${\displaystyle K}$ ${\displaystyle w\colon L^{\times }\to \mathbb {R} }$ ${\displaystyle v}$ ${\displaystyle w|_{K}=v}$ ${\displaystyle {\mathcal {O}}_{L},{\mathfrak {m}}_{L},\pi _{L},\lambda }$

The branch index of is defined as ${\displaystyle L/K}$

${\displaystyle e_{w/v}={\frac {v(\pi _{K})}{w(\pi _{L})}}=(w(L^{\times }):v(K^{\times }))}$

If it is equal to 1, the extension is called unbranched . Its counterpart is the degree of inertia . ${\displaystyle f_{w/v}=[\lambda :\kappa ]}$

properties

• If the extension is separable and runs through all possible extensions of , then the fundamental equation holds ${\displaystyle L/K}$ ${\displaystyle w}$ ${\displaystyle v}$

${\displaystyle \sum _{w/v}e_{w/v}f_{w/v}=[L:K].}$

• Is also fully, it is clearly determined as ${\displaystyle K}$ ${\displaystyle w}$

${\displaystyle w(x)={\frac {1}{[L:K]}}v(N_{L/K}(x)),}$

and it applies

${\displaystyle e_{w/v}f_{w/v}=[L:K].}$

• 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 ${\displaystyle K}$ ${\displaystyle L/K}$ ${\displaystyle \lambda /\kappa }$ ${\displaystyle \lambda /\kappa }$

${\displaystyle 1\to I\to \mathrm {Gal} (L/K)\to \mathrm {Gal} (\lambda /\kappa )\to 1;}$

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 ${\displaystyle I}$ ${\displaystyle T}$ ${\displaystyle L/K}$

${\displaystyle [L:T]=\#I=e,\quad [T:K]=f.}$

In particular, if it is unbranched, then it is ${\displaystyle L/K}$

${\displaystyle \mathrm {Gal} (L/K)\cong \mathrm {Gal} (\lambda /\kappa ).}$

If the maximum unbranched extension (in a separable conclusion of ), then applies accordingly ${\displaystyle K^{\mathrm {nr} }}$ ${\displaystyle K^{\mathrm {sep} }}$ ${\displaystyle K}$

${\displaystyle \mathrm {Gal} (K^{\mathrm {nr} }/K)\cong \mathrm {Gal} (\kappa ^{\mathrm {sep} }/\kappa ).}$

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 ${\displaystyle {\hat {\mathbb {Z} }}}$ ${\displaystyle \mathrm {Gal} (\kappa ^{\mathrm {sep} }/\kappa )}$

${\displaystyle x\mapsto x^{q}}$ With ${\displaystyle q=\#\kappa }$

has a canonical generator, there is also a canonical element, which is also called Frobenius automorphism . ${\displaystyle \mathrm {Gal} (K^{\mathrm {nr} }/K)}$

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. ${\displaystyle A}$ ${\displaystyle K}$ ${\displaystyle L}$ ${\displaystyle K}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle L}$ ${\displaystyle B}$

One of the most important special cases is , , a number field and its wholeness ring . ${\displaystyle A=\mathbb {Z} }$ ${\displaystyle K=\mathbb {Q} }$ ${\displaystyle L}$ ${\displaystyle B}$

Further be a maximum ideal of . Then be different in a unique way as a product of powers prime ideals of write: ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle A}$ ${\displaystyle {\mathfrak {p}}B}$ ${\displaystyle B}$

${\displaystyle {\mathfrak {p}}B={\mathfrak {P}}_{1}^{e_{1}}\cdots {\mathfrak {P}}_{k}^{e_{k}}.}$

The numbers are called branching indices , the degrees of the remainder field extensions are called degrees of inertia . ${\displaystyle e_{i}}$ ${\displaystyle f_{i}=[B/{\mathfrak {P}}_{i}:A/{\mathfrak {p}}]}$

• 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.) ${\displaystyle e_{i}=1}$ ${\displaystyle {\mathfrak {P}}_{i}}$
• Is , it is called purely branched . ${\displaystyle f_{i}=1}$ ${\displaystyle {\mathfrak {P}}_{i}}$
• If all are unbranched, it is called unbranched . then breaks down into a product of various prime ideals. ${\displaystyle {\mathfrak {P}}_{i}}$ ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle {\mathfrak {p}}}$
• If all prime ideals (not equal to zero) of unbranched, the extension is called unbranched . ${\displaystyle K}$ ${\displaystyle L/K}$

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. ${\displaystyle {\mathfrak {P}}}$ ${\displaystyle L}$ ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle K}$ ${\displaystyle L/K}$ ${\displaystyle {\mathfrak {P}}}$ ${\displaystyle {\mathfrak {p}}}$
• The fundamental equation applies

${\displaystyle [L:K]=\sum _{i=1}^{k}e_{i}f_{i}.}$

• 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 . ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle L}$
• The only unbranched extension of is self. ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Q} }$
• 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 . ${\displaystyle L/K}$ ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle {\mathfrak {P}}}$ ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle \varphi _{\mathfrak {P}}\in \mathrm {Gal} (L/K)}$ ${\displaystyle {\mathfrak {P}}}$

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: ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle f\colon X\to Y}$ ${\displaystyle f}$

• ${\displaystyle \Omega _{X/Y}^{1}=0}$
• For one (and therefore for everyone) morphism is ${\displaystyle g\colon Y\to Z}$

${\displaystyle f^{*}\Omega _{Y/Z}^{1}\to \Omega _{X/Z}^{1}}$

surjective.

• The fibers of over points are disjoint unions of spectra of finite separable body extensions of . ${\displaystyle f}$ ${\displaystyle y\in Y}$ ${\displaystyle \kappa (y)}$
• The diagonal is an open embedment. ${\displaystyle X\to X\times _{Y}X}$
• If there is an affine schema and a closed sub- schema , which is defined by a nilpotent ideal sheaf, then is the induced mapping ${\displaystyle T}$ ${\displaystyle T_{0}}$

${\displaystyle \mathrm {Hom} _{Y}(T,X)\to \mathrm {Hom} _{Y}(T_{0},X)}$

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 ): ${\displaystyle f}$ ${\displaystyle x\in X}$ ${\displaystyle U}$ ${\displaystyle x}$ ${\displaystyle X}$ ${\displaystyle f|_{U}}$ ${\displaystyle x}$ ${\displaystyle y=f(x)}$

• ${\displaystyle \Omega _{X/Y,x}^{1}=0}$
• The diagonal is a local isomorphism at . ${\displaystyle X\to X\times _{Y}X}$ ${\displaystyle x}$
• ${\displaystyle {\mathcal {O}}_{X,x}/{\mathfrak {m}}_{y}{\mathcal {O}}_{X,x}}$is a field that is a finite separable extension of . ${\displaystyle \kappa (y)}$

The unbranchedness of in the point depends only on the fiber . ${\displaystyle f}$ ${\displaystyle x}$ ${\displaystyle f^{-1}(y)}$

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 . ${\displaystyle Y}$ ${\displaystyle f\colon X\to Y}$ ${\displaystyle f}$ ${\displaystyle X}$ ${\displaystyle f}$ ${\displaystyle Y}$

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 . ${\displaystyle X}$ ${\displaystyle K}$ ${\displaystyle V}$ ${\displaystyle X}$ ${\displaystyle V}$ ${\displaystyle {\mathcal {X}}}$ ${\displaystyle V}$ ${\displaystyle X\cong {\mathcal {X}}\otimes _{V}K}$ ${\displaystyle L/K}$ ${\displaystyle W}$ ${\displaystyle L}$ ${\displaystyle \mathrm {Spec} \,W\to \mathrm {Spec} \,V}$ ${\displaystyle {\mathcal {X}}_{W}:={\mathcal {X}}\otimes _{V}W\to {\mathcal {X}}}$ ${\displaystyle {\mathcal {X}}}$ ${\displaystyle {\mathcal {X}}_{W}}$ ${\displaystyle X_{L}}$

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

1. ↑ Jürgen Neukirch: Algebraic Number Theory. Springer-Verlag, Berlin 1992. ISBN 3-540-54273-6 , sentence (II.8.5), p. 173
2. ↑ Jürgen Neukirch: Algebraic Number Theory. Springer-Verlag, Berlin 1992. ISBN 3-540-54273-6 , Theorem (II.6.2), p. 150
3. ↑ Jürgen Neukirch: Algebraic Number Theory. Springer-Verlag, Berlin 1992. ISBN 3-540-54273-6 , sentence (II.6.8), p. 157
4. ↑ Jürgen Neukirch: Algebraic Number Theory. Springer-Verlag, Berlin 1992. ISBN 3-540-54273-6 , sentence (II.9.9), p. 181
5. ↑ Jürgen Neukirch: Algebraic Number Theory. Springer-Verlag, Berlin 1992. ISBN 3-540-54273-6 , sentence (II.9.11), p. 182
6. ↑ Jürgen Neukirch: Algebraic Number Theory. Springer-Verlag, Berlin 1992. ISBN 3-540-54273-6 , sentence (I.8.1), p. 47
7. ↑ Jürgen Neukirch: Algebraic Number Theory. Springer-Verlag, Berlin 1992. ISBN 3-540-54273-6 , sentence (I.8.2), p. 48
8. ↑ Jürgen Neukirch: Algebraic Number Theory. Springer-Verlag, Berlin 1992. ISBN 3-540-54273-6 , sentence (I.8.4), p. 52
9. ↑ Jürgen Neukirch: Algebraic Number Theory. Springer-Verlag, Berlin 1992. ISBN 3-540-54273-6 , Korollar (III.2.12), p. 213
10. ↑ Jürgen Neukirch: Algebraic Number Theory. Springer-Verlag, Berlin 1992. ISBN 3-540-54273-6 , Theorem (III.2.6), p. 210
11. ↑ Jürgen Neukirch: Algebraic Number Theory. Springer-Verlag, Berlin 1992. ISBN 3-540-54273-6 , sentence (III.2.18), p. 218
12. ↑ 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.
13. ↑ EGA IV, 17.4.2, 17.2.2, 17.1.1, 17.3.1
14. ↑ EGA IV, 17.4.1
15. ↑ EGA IV, April 17, 3
16. ↑ EGA IV, April 17, 9