Lemma 11.6.2 (074R)—The Stacks project

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Lemma 11.6.2. Let $A$ be a finite 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. $\square$

Comments (3)

Comment #8883 by Awllower on March 28, 2024 at 05:45

I do not understand why the proof tries to show that $A$ is a finite central simple algebra over the center of $A$ , which should be $k$ by the assumptions.

Comment #9206 by Stacks project on May 15, 2024 at 13:13

Ha! Thanks for pointing this out. Fixed it here by removing the word "central" from the statement.

Comment #9571 by Kevin Buzzard on August 20, 2024 at 12:20

Re comment 9206: you have now removed the word "central" from the statement of 074R and now the lemma is incorrect, as was pointed out in comment 4168 in 2019.

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3 comment(s) on Section 11.6: Skolem-Noether

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