# Skolem-Noether theorem

2010 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].

How to Cite This Entry:
Skolem-Noether theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Skolem-Noether_theorem&oldid=25509
This article was adapted from an original article by S. Caenepeel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article