Skolem-Noether theorem

In ring theory , Skolem-Noether's theorem characterizes the automorphisms of simple rings . It is a fundamental result in the theory of central simple algebras .

The theorem was first published by Thoralf Skolem in 1927 in his work on the theory of associative number systems and was later rediscovered by Emmy Noether .

claim

Be and simple rings and be the center of . Note that there is a body . It is further assumed that the dimension of over is finite. ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle k}$ ${\displaystyle B}$ ${\displaystyle k}$ ${\displaystyle B}$ ${\displaystyle k}$

So be a central simple finite-dimensional algebra, also called Azumaya algebra . There are also -algebra homomorphisms . ${\displaystyle B}$ ${\displaystyle k}$ ${\displaystyle f,g\colon A\rightarrow B}$

Then there exists such that: ${\displaystyle b\in B\setminus \{0\}}$

${\displaystyle \forall a\in A\colon g(a)=bf(a)b^{-1}}$

In particular, every automorphism of a central simple algebra is an inner automorphism. ${\displaystyle k}$

proof

Be . Then define and act from on . denote the - modules obtained from this . Any two simple -modules are isomorphic and are direct sums of simple -modules. Since these have the same dimension, it follows that there is an isomorphism of modules. But one must be in . In the general case, it is a matrix algebra and therefore with the first part this algebra contains an element such that: ${\displaystyle B=M_{n}(k)=End_{k}(k^{n})}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle A}$ ${\displaystyle k^{n}}$ ${\displaystyle V_{f},V_{g}}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle V_{f},V_{g}}$ ${\displaystyle A}$ ${\displaystyle b\colon V_{g}\rightarrow V_{f}}$ ${\displaystyle A}$ ${\displaystyle b}$ ${\displaystyle M_{n}(k)=B}$ ${\displaystyle B\otimes B^{op}}$ ${\displaystyle b}$

${\displaystyle \forall a\in A,~\forall z\in B^{op}\colon (f\otimes 1)(a\otimes z)=b(g\otimes 1)(a\otimes z)b^{-1}}$

With we get ${\displaystyle a=1}$

${\displaystyle \forall z\in B^{op}\colon 1\otimes z=b(a\otimes z)b^{-1}}$.

It applies , so we can write. With results ${\displaystyle b\in T_{B\otimes B^{op}}(k\otimes B^{op})=B\otimes k}$ ${\displaystyle b=b'\otimes 1}$ ${\displaystyle z=1}$

${\displaystyle f(a)=b'g(a)b'^{-1}}$,

what was to be shown.

literature

• Thoralf Skolem : On the theory of associative number systems . In: Skrifter utgitt av Det Norske Videnskaps-Akademi i Oslo, No. 12 . 1927, p. 50.
• Discussion in Chapter IV by James Milne : Class field theory. On-line.
• Philippe Gille, Tamás Szamuely: Central simple algebras and Galois cohomology  (= Cambridge Studies in Advanced Mathematics), Volume 101. Cambridge University Press , Cambridge 2006, ISBN 0-521-86103-9 .
• Falko Lorenz: Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics . Springer, 2008, ISBN 978-0-387-72487-4 .
• Ina Kersten : Brewer groups. Universitätsdrucke Göttingen, Göttingen 2007, p. 38, PDF (accessed on July 18, 2016).

Individual evidence

1. ↑ Lorenz (2008) p. 173
2. ^ Benson Farb, R. Keith Dennis: Noncommutative Algebra . Springer, 1993, ISBN 9780387940571 .
3. ↑ Gille & Szamuely (2006), p. 40.
4. ↑ Lorenz (2008), p. 174.