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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e0370301.png" /> over a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e0370302.png" />''
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A homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e0370303.png" /> of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e0370304.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e0370305.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e0370306.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e0370307.png" /> is an algebra with zero multiplication, then the extension is called singular. In this case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e0370308.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e0370309.png" />-module in a natural way. The set of all extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e03703010.png" /> with kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e03703011.png" /> admits an equivalence relation (the same as for groups, modules, etc.), and the set of equivalence classes of extensions is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e03703012.png" />. If the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e03703013.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e03703014.png" />-projective, then the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e03703015.png" /> splits into a direct sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e03703016.png" />-modules, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e03703017.png" />, and the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e03703018.png" /> can be written as pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e03703019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e03703020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e03703021.png" />, with multiplication given by
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e03703022.png" /></td> </tr></table>
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'' $  R $
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over a commutative ring  $  K $''
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e03703023.png" />. The associativity of multiplication imposes restrictions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e03703024.png" /> that make it into a cocycle. Mapping an extension to its cocycle defines an isomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e03703025.png" />-modules between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e03703026.png" /> and the second cohomology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e03703027.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e03703028.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e03703029.png" />.
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A homomorphism  $  \phi : S \rightarrow R $
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of a  $  K $-
 +
algebra  $  S $
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onto  $  R $.  
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If  $  \mathop{\rm Ker}  \phi = I $
 +
is an algebra with zero multiplication, then the extension is called singular. In this case,  $  I $
 +
is an  $  R $-
 +
module in a natural way. The set of all extensions of  $  R $
 +
with kernel  $  I $
 +
admits an equivalence relation (the same as for groups, modules, etc.), and the set of equivalence classes of extensions is denoted by  $  F ( R, I) $.  
 +
If the algebra  $  R $
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is  $  K $-
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projective, then the algebra  $  S $
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splits into a direct sum of $  K $-
 +
modules,  $  S = I + R $,
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and the elements of $  S $
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can be written as pairs  $  ( u , r) $,
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$  u \in I $,
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$  r \in R $,
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with multiplication given by
  
In a completely different context, any algebra containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e03703030.png" /> is also called an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e03703031.png" />. Such extensions are often connected with a specific construction (polynomials over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e03703032.png" />, localization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e03703033.png" />, ring of partial fractions of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e03703034.png" />, etc.). See also [[Extension of a field|Extension of a field]].
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$$
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( u _ {1} , r _ {1} ) ( u _ {2} , r _ {2} )  = \
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( u _ {1} r _ {2} + r _ {1} u _ {2} + a ( r _ {1} , r _ {2} ),\
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r _ {1} r _ {2} ),
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$$
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 +
where  $  a:  R \otimes R \rightarrow I $.
 +
The associativity of multiplication imposes restrictions on  $  a $
 +
that make it into a cocycle. Mapping an extension to its cocycle defines an isomorphism of  $  K $-
 +
modules between  $  F ( R, I) $
 +
and the second cohomology group  $  H  ^ {2} ( R, I) $
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of  $  R $
 +
with coefficients in  $  I $.
 +
 
 +
In a completely different context, any algebra containing $  R $
 +
is also called an extension of $  R $.  
 +
Such extensions are often connected with a specific construction (polynomials over $  R $,  
 +
localization of $  R $,  
 +
ring of partial fractions of the algebra $  R $,  
 +
etc.). See also [[Extension of a field|Extension of a field]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. MacLane,  "Homology" , Springer  (1963)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Hochschild,  "On the cohomology groups of an associative algebra"  ''Ann. of Math.'' , '''46'''  (1945)  pp. 58–67</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. MacLane,  "Homology" , Springer  (1963)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Hochschild,  "On the cohomology groups of an associative algebra"  ''Ann. of Math.'' , '''46'''  (1945)  pp. 58–67</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The cohomology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e03703035.png" /> is also known as the Hochschild cohomology (group) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e03703036.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037030/e03703037.png" />.
+
The cohomology group $  H  ^ {2} ( R , I ) $
 +
is also known as the Hochschild cohomology (group) of $  R $
 +
with values in $  I $.

Latest revision as of 19:38, 5 June 2020


$ R $ over a commutative ring $ K $

A homomorphism $ \phi : S \rightarrow R $ of a $ K $- algebra $ S $ onto $ R $. If $ \mathop{\rm Ker} \phi = I $ is an algebra with zero multiplication, then the extension is called singular. In this case, $ I $ is an $ R $- module in a natural way. The set of all extensions of $ R $ with kernel $ I $ admits an equivalence relation (the same as for groups, modules, etc.), and the set of equivalence classes of extensions is denoted by $ F ( R, I) $. If the algebra $ R $ is $ K $- projective, then the algebra $ S $ splits into a direct sum of $ K $- modules, $ S = I + R $, and the elements of $ S $ can be written as pairs $ ( u , r) $, $ u \in I $, $ r \in R $, with multiplication given by

$$ ( u _ {1} , r _ {1} ) ( u _ {2} , r _ {2} ) = \ ( u _ {1} r _ {2} + r _ {1} u _ {2} + a ( r _ {1} , r _ {2} ),\ r _ {1} r _ {2} ), $$

where $ a: R \otimes R \rightarrow I $. The associativity of multiplication imposes restrictions on $ a $ that make it into a cocycle. Mapping an extension to its cocycle defines an isomorphism of $ K $- modules between $ F ( R, I) $ and the second cohomology group $ H ^ {2} ( R, I) $ of $ R $ with coefficients in $ I $.

In a completely different context, any algebra containing $ R $ is also called an extension of $ R $. Such extensions are often connected with a specific construction (polynomials over $ R $, localization of $ R $, ring of partial fractions of the algebra $ R $, etc.). See also Extension of a field.

References

[1] S. MacLane, "Homology" , Springer (1963)
[2] G. Hochschild, "On the cohomology groups of an associative algebra" Ann. of Math. , 46 (1945) pp. 58–67

Comments

The cohomology group $ H ^ {2} ( R , I ) $ is also known as the Hochschild cohomology (group) of $ R $ with values in $ I $.

How to Cite This Entry:
Extension of an associative algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Extension_of_an_associative_algebra&oldid=46882
This article was adapted from an original article by t sequence','../s/s086840.htm')" style="background-color:yellow;">V.E. Govorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article