Separable algebra

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

Separable algebra over a field

A finite-dimensional semi-simple associative algebra $A$ over a field $k$ that remains semi-simple under any extension $K$ of $k$ (that is, the algebra $K \otimes_k A$ is semi-simple for any field $K \supseteq k$, cf. Semi-simple algebra). An algebra $A$ is separable if and only if the centres of the simple components of this algebra (see Associative rings and algebras) are separable field extensions of $k$ (cf. Separable extension).

Separable algebra over a ring

An algebra $A$ over a commutative ring $R$ is separable if $A$ is projective as a left $A \otimes_R A^\textrm{o} = A^e$-module (cf. Projective module). Here, $A^\textrm{o}$ is the opposite algebra of $A$.

An algebra that is separable over its centre is called an Azumaya algebra. These algebras are important in the theory of the Brauer group of a commutative ring or scheme.

References

[Wae]	B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German) MR1541390 Zbl 0192.33002
[CuRe]	C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962) MR0144979 Zbl 0131.25601
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[MeIn]	F. de Meyer, E. Ingraham, "Separable algebras over commutative rings" , Lect. notes in math. , 181 , Springer (1971) MR0280479 Zbl 0215.36602
[KnuOj]	M.-A. Knus, M. Ojanguren, "Théorie de la descente et algèbres d'Azumaya" , Lect. notes in math. , 389 , Springer (1974) MR0417149 Zbl 0284.13002
[CaOy]	S. Caenepeel, F. van Oystaeyen, "Brauer groups and the cohomology of graded rings" , M. Dekker (1988) MR0972258 Zbl 0702.13001