Difference between revisions of "Separable algebra"
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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|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|Associative rings and algebras]]) are separable field extensions of $k$ (cf. [[Separable extension|Separable extension]]). | 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|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|Associative rings and algebras]]) are separable field extensions of $k$ (cf. [[Separable extension|Separable extension]]). | ||
| − | ==== | + | ====Separable algebra over a ring==== |
An algebra $A$ over a commutative ring $R$ is separable if $A$ is projective as a left $A \otimes_R A^o = A^e$-module (cf. [[Projective module|Projective module]]). Here, $A^o$ is the opposite algebra of $A$. | An algebra $A$ over a commutative ring $R$ is separable if $A$ is projective as a left $A \otimes_R A^o = A^e$-module (cf. [[Projective module|Projective module]]). Here, $A^o$ is the opposite algebra of $A$. | ||
Revision as of 11:36, 26 April 2012
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 $A \otimes_R A^o = A^e$-module (cf. Projective module). Here, $A^o$ 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) |
| [CuRe] | C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962) |
| [AuGo] | M. Auslander, O. Goldman, "The Brauer group of a commutative ring" Trans. Amer. Math. Soc. , 97 (1960) pp. 367–409 |
| [MeIn] | F. de Meyer, E. Ingraham, "Separable algebras over commutative rings" , Lect. notes in math. , 181 , Springer (1971) |
| [KnuOj] | M.-A. Knus, M. Ojanguren, "Théorie de la descente et algèbres d'Azumaya" , Lect. notes in math. , 389 , Springer (1974) |
| [CaOy] | S. Caenepeel, F. van Oystaeyen, "Brauer groups and the cohomology of graded rings" , M. Dekker (1988) |
Separable algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Separable_algebra&oldid=25499