Skolem–Noether theorem

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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[edit]

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 : AB

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[edit]

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[edit]

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

References[edit]