Skolem–Noether theorem

In ring theory, a branch of mathematics, the Skolem–Noether 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 paper Zur Theorie der assoziativen Zahlensysteme (German: On the theory of associative number systems) and later rediscovered by Emmy Noether.

Statement

In a general formulation, let A and B be simple rings, and let k be the centre of B. Notice that k is a field since given x nonzero in k, the simplicity of B implies that the nonzero two-sided ideal Bx is the whole of B, and hence that x is a unit. Suppose further that the dimension of B over k is finite, i.e. that B is a central simple algebra. Then given k-algebra homomorphisms

f, g : A → B

there exists a unit b in B such that for all a in A[1][2]

g(a) = b · f(a) · b−1.

In particular, every automorphism of a central simple k-algebra is an inner automorphism.[3][4]

Proof

First suppose \( B = \operatorname{M}_n(k) = \operatorname{End}_k(k^n) \). Then f and g define the actions of A on \( k^n; let\( V_f, V_g \) denote the A-modules thus obtained. Any two simple A-modules are isomorphic and \( V_f, V_g \) are finite direct sums of simple A-modules. Since they have the same dimension, it follows that there is an isomorphism of A-modules b: \( V_g \to V_f \). But such b must be an element of \( \operatorname{M}_n(k) = B \). For the general case, note that \( B \otimes B^{\text{op}} \) is a matrix algebra and thus by the first part this algebra has an element b such that

\( (f \otimes 1)(a \otimes z) = b (g \otimes 1)(a \otimes z) b^{-1} \)

for all \( a \in A and\( z \in B^{\text{op}} \). Taking a = 1, we find

\( 1 \otimes z = b (1\otimes z) b^{-1} \)

for all z. That is to say, b is in \( Z_{B \otimes B^{\text{op}}}(k \otimes B^{\text{op}}) = B \otimes k \) and so we can write \( b = b' \otimes 1 \). Taking z = 1 this time we find

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

which is what was sought.

Notes

Lorenz (2008) p.173
Farb, Benson; Dennis, R. Keith (1993). Noncommutative Algebra. Springer. ISBN 9780387940571.
Gille & Szamuely (2006) p.40

Lorenz (2008) p.174

References

Skolem, Thoralf (1927). "Zur Theorie der assoziativen Zahlensysteme". Skrifter Oslo (in German) (12): 50. JFM 54.0154.02.
A discussion in Chapter IV of Milne, class field theory [1]
Gille, Philippe; Szamuely, Tamás (2006). Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics 101. Cambridge: Cambridge University Press. ISBN 0-521-86103-9. Zbl 1137.12001.
Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. ISBN 978-0-387-72487-4. Zbl 1130.12001.

Quadratic forms with the same core form are said to be similar or Witt equivalent.

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