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Difference between revisions of "Separable algebra"

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(Texed, removed comment section and reorganized the article in two parts: over a field (which was the main part of the old art.) and over a ring (which was the old comment section))
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A finite-dimensional semi-simple associative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084450/s0844501.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084450/s0844502.png" /> that remains semi-simple under any extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084450/s0844503.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084450/s0844504.png" /> (that is, the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084450/s0844505.png" /> is semi-simple for any field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084450/s0844506.png" />, cf. [[Semi-simple algebra|Semi-simple algebra]]). An algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084450/s0844507.png" /> 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 extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084450/s0844508.png" /> (cf. [[Separable extension|Separable extension]]).
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====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.L. van der Waerden,  "Algebra" , '''1–2''' , Springer  (1967–1971)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.W. Curtis,  I. Reiner,  "Representation theory of finite groups and associative algebras" , Interscience  (1962)</TD></TR></table>
 
 
 
  
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====separable algebra over a field====
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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]]).
  
====Comments====
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====separable algebra over a ring====
An algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084450/s0844509.png" /> over a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084450/s08445010.png" /> is separable if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084450/s08445011.png" /> is projective as a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084450/s08445012.png" />-module (cf. [[Projective module|Projective module]]).
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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 that is separable over its centre is called an Azumaya algebra. These algebras are important in the theory of the [[Brauer group|Brauer group]] of a commutative ring or scheme.
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An algebra that is separable over its centre is called an [[Azumaya algebra|Azumaya algebra]]. These algebras are important in the theory of the [[Brauer group|Brauer group]] of a commutative ring or scheme.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Auslander,  O. Goldman,  "The Brauer group of a commutative ring"  ''Trans. Amer. Math. Soc.'' , '''97'''  (1960)  pp. 367–409</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"F. de Meyer,  E. Ingraham,  "Separable algebras over commutative rings" , ''Lect. notes in math.'' , '''181''' , Springer  (1971)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"M.-A. Knus,  M. Ojanguren,  "Théorie de la descente et algèbres d'Azumaya" , ''Lect. notes in math.'' , '''389''' , Springer  (1974)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"S. Caenepeel,  F. van Oystaeyen,  "Brauer groups and the cohomology of graded rings" , M. Dekker  (1988)</TD></TR></table>
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{|
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|-
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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|}}
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|-
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|valign="top"|{{Ref|CuRe}}||valign="top"| C.W. Curtis,  I. Reiner,  "Representation theory of finite groups and associative algebras" , Interscience  (1962) {{MR|}} {{ZBL|}}
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|-
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|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|}}
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|-
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|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|}}
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|-
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|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|}}
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|-
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|valign="top"|{{Ref|CaOy}}||valign="top"| S. Caenepeel,  F. van Oystaeyen,  "Brauer groups and the cohomology of graded rings" , M. Dekker  (1988) {{MR|}} {{ZBL|}}
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|-
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|}

Revision as of 11:07, 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)
How to Cite This Entry:
Separable algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Separable_algebra&oldid=25494
This article was adapted from an original article by L.A. Bokut' (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article