Difference between revisions of "Separable algebra"
From Encyclopedia of Mathematics
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| − | ==== | + | ====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|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. [[ | + | 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 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 $A \otimes_R A^\textrm{o} = A^e$-module (cf. [[Projective module|Projective module]]). Here, $A^\textrm{o}$ is the [[opposite algebra]] of $A$. |
| − | An algebra that is separable over its centre is called an [[ | + | 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==== | ====References==== | ||
{| | {| | ||
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| − | |valign="top"|{{Ref|Wae}}||valign="top"| B.L. van der Waerden, "Algebra" , '''1–2''' , Springer (1967–1971) (Translated from German) {{MR|}} {{ZBL|}} | + | |valign="top"|{{Ref|Wae}}||valign="top"| B.L. van der Waerden, "Algebra" , '''1–2''' , Springer (1967–1971) (Translated from German) {{MR|1541390}} {{ZBL|0192.33002}} |
|- | |- | ||
| − | |valign="top"|{{Ref|CuRe}}||valign="top"| C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962) {{MR|}} {{ZBL|}} | + | |valign="top"|{{Ref|CuRe}}||valign="top"| C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962) {{MR|0144979}} {{ZBL|0131.25601}} |
|- | |- | ||
| − | |valign="top"|{{Ref|AuGo}}||valign="top"| M. Auslander, O. Goldman, "The Brauer group of a commutative ring" ''Trans. Amer. Math. Soc.'' , '''97''' (1960) pp. 367–409 {{MR|}} {{ZBL|}} | + | |valign="top"|{{Ref|AuGo}}||valign="top"| M. Auslander, O. Goldman, "The Brauer group of a commutative ring" ''Trans. Amer. Math. Soc.'' , '''97''' (1960) pp. 367–409 {{MR|0121392}} {{ZBL|0100.26304}} |
|- | |- | ||
| − | |valign="top"|{{Ref|MeIn}}||valign="top"| F. de Meyer, E. Ingraham, "Separable algebras over commutative rings" , ''Lect. notes in math.'' , '''181''' , Springer (1971) {{MR|}} {{ZBL|}} | + | |valign="top"|{{Ref|MeIn}}||valign="top"| F. de Meyer, E. Ingraham, "Separable algebras over commutative rings" , ''Lect. notes in math.'' , '''181''' , Springer (1971) {{MR|0280479}} {{ZBL|0215.36602}} |
|- | |- | ||
| − | |valign="top"|{{Ref|KnuOj}}||valign="top"| M.-A. Knus, M. Ojanguren, "Théorie de la descente et algèbres d'Azumaya" , ''Lect. notes in math.'' , '''389''' , Springer (1974) {{MR|}} {{ZBL|}} | + | |valign="top"|{{Ref|KnuOj}}||valign="top"| M.-A. Knus, M. Ojanguren, "Théorie de la descente et algèbres d'Azumaya" , ''Lect. notes in math.'' , '''389''' , Springer (1974) {{MR|0417149}} {{ZBL|0284.13002}} |
|- | |- | ||
| − | |valign="top"|{{Ref|CaOy}}||valign="top"| S. Caenepeel, F. van Oystaeyen, "Brauer groups and the cohomology of graded rings" , M. Dekker (1988) {{MR|}} {{ZBL|}} | + | |valign="top"|{{Ref|CaOy}}||valign="top"| S. Caenepeel, F. van Oystaeyen, "Brauer groups and the cohomology of graded rings" , M. Dekker (1988) {{MR|0972258}} {{ZBL|0702.13001}} |
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|} | |} | ||
Latest revision as of 21:57, 29 November 2014
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 |
| [AuGo] | M. Auslander, O. Goldman, "The Brauer group of a commutative ring" Trans. Amer. Math. Soc. , 97 (1960) pp. 367–409 MR0121392 Zbl 0100.26304 |
| [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 |
How to Cite This Entry:
Separable algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Separable_algebra&oldid=25495
Separable algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Separable_algebra&oldid=25495
This article was adapted from an original article by L.A. Bokut' (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article