Skolem-Noether theorem

2020 Mathematics Subject Classification: Primary: 16-XX [MSN][ZBL]

In its classical form, the Skolem–Noether theorem can be stated as follows. Let $A$ and $B$ be finite-dimensional algebras over a field $k$, and assume that $A$ is simple and $B$ is central simple (cf. also Simple algebra; Central algebra; Field). If $f,g:A\to B$ are $k$-algebra homomorphisms, then there exists an invertible $u\in B$ such that

$$f(a) = u^{-1}g(a) u$$ for all $a\in A$. A proof can be found, for example, in [Ke], p. 21, or [He], Chap, 4. In particular, every $k$-algebra automorphism of a central simple algebra is inner (cf. also Inner automorphism). This can be generalized to an Azumaya algebra $A$ over a commutative ring $R$ (cf. also Separable algebra): There is an exact sequence, usually called the Rosenberg–Zelinsky exact sequence:

$$\def\Inn{\textrm{Inn}}\def\Aut{\textrm{Aut}}\def\Pic{\textrm{Pic}} 0\to\Inn(A)\to \Aut(A)\to \Pic(R),$$ where $\Pic(R)$ is the Picard group of $R$, $\Aut(A)$ is the group of $k$-algebra automorphisms of $A$ and $\Inn(A)$ is the subgroup consisting of inner automorphisms. The proof is an immediate application of the categorical characterization of Azumaya algebras: An $R$-algebra $A$ is Azumaya if and only if the categories of $R$-modules and $A$-bimodules are equivalent via the functors sending an $R$-module $N$ to $A\otimes N$, and sending an $A$-bimodule $M$ to

$$M^A = \{m\in M\;|\; am = ma \textrm{ for all } a\in A \}$$ (see, e.g., [KnOj], IV.1, for details).

The Skolem–Noether theorem plays a crucial role in the theory of the Brauer group; for example, it is used in the proof of the Hilbert 90 theorem (cf. also Hilbert theorem) and the cross product theorem. There exist versions of the Skolem–Noether theorem (and the Rosenberg–Zelinsky exact sequence) for other generalized types of Azumaya algebras; in particular, for Azumaya algebras over schemes [Gr], Azumaya algebras relative to a torsion theory [VaOyVe], III.3.26, and Long's $H$-dimodule Azumaya algebras [Be], [Ca].

References

[Be]	M. Beattie, "Automorphisms of $G$-Azumaya algebras" Canad. J. Math., 37 (1985) pp. 1047–1058 MR0828833 Zbl 0571.16003
[Ca]	S. Caenepeel, "Brauer groups, Hopf algebras and Galois theory", K-Monographs Math., 4, Kluwer Acad. Publ. (1998) MR1610222 Zbl 0898.16001
[Gr]	A. Grothendieck, "Le groupe de Brauer I", Dix Exposés sur la cohomologie des schémas, North-Holland (1968) MR0244269 MR0244270 MR0244271 Zbl 0193.21503
[He]	I.N. Herstein, "Noncommutative rings", Carus Math. Monographs, 15, Math. Assoc. Amer. (1968) MR1535024 MR0227205 Zbl 0177.05801
[Ke]	I. Kersten, "Brauergruppen von Körpern", Aspekte der Math., D6, Vieweg (1990) MR1137014 Zbl 0709.13001
[KnOj]	M.A. Knus, M. Ojanguren, "Théorie de la descente et algèbres d'Azumaya", Lecture Notes in Mathematics, 389, Springer (1974) MR0417149 Zbl 0284.13002
[VaOyVe]	F. Van Oystaeyen, A. Verschoren, "Relative invariants of rings I", Monographs and Textbooks in Pure and Appl. Math., 79, M. Dekker (1983) MR0722296 Zbl 0555.16001