11.6 Skolem-Noether

Theorem 11.6.1. Let $A$ be a finite central simple $k$-algebra. Let $B$ be a simple $k$-algebra. Let $f, g : B \to A$ be two $k$-algebra homomorphisms. Then there exists an invertible element $x \in A$ such that $f(b) = xg(b)x^{-1}$ for all $b \in B$.

Proof. Choose a simple $A$-module $M$. Set $L = \text{End}_ A(M)$. Then $L$ is a skew field with center $k$ which acts on the left on $M$, see Lemmas 11.3.2 and 11.4.6. Then $M$ has two $B \otimes _ k L^{op}$-module structures defined by $m \cdot _1 (b \otimes l) = lmf(b)$ and $m \cdot _2 (b \otimes l) = lmg(b)$. The $k$-algebra $B \otimes _ k L^{op}$ is simple by Lemma 11.4.7. Since $B$ is simple, the existence of a $k$-algebra homomorphism $B \to A$ implies that $B$ is finite. Thus $B \otimes _ k L^{op}$ is finite simple and we conclude the two $B \otimes _ k L^{op}$-module structures on $M$ are isomorphic by Lemma 11.4.6. Hence we find $\varphi : M \to M$ intertwining these operations. In particular $\varphi$ is in the commutant of $L$ which implies that $\varphi$ is multiplication by some $x \in A$, see Lemma 11.4.6. Working out the definitions we see that $x$ is a solution to our problem. $\square$

Lemma 11.6.2. Let $A$ be a finite central simple $k$-algebra. Any automorphism of $A$ is inner. In particular, any automorphism of $\text{Mat}(n \times n, k)$ is inner.

Proof. Note that $A$ is a finite central simple algebra over the center of $A$ which is a finite field extension of $k$, see Lemma 11.4.2. Hence the Skolem-Noether theorem (Theorem 11.6.1) applies. $\square$

Comment #4168 by Patrick Chu on

lemma 11.6.2 (074R) seems incorrect

Field extensions with non-trivial galois groups have automorphisms which aren't inner, but are still simple $k$-algebras.

Any automorphisms which are linear over the center field are certainly inner by the theorem (so matrices of fields are still fine), but an automorphism could be linear over the original field $k$ and not linear over the center.

Comment #4370 by on

Yep, I agree the word "central" was missing. Thanks and fixed here.

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