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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+1 = 37 n = 0 |
| + | $#C+1 = 37 : ~/encyclopedia/old_files/data/E037/E.0307030 Extension of an associative algebra |
| + | Automatically converted into TeX, above some diagnostics. |
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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>
| + | '' $ R $ |
| + | 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" />.
| + | 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 |
| | | |
− | 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]]. | + | $$ |
| + | ( u _ {1} , r _ {1} ) ( u _ {2} , r _ {2} ) = \ |
| + | ( u _ {1} r _ {2} + r _ {1} u _ {2} + a ( r _ {1} , r _ {2} ),\ |
| + | r _ {1} r _ {2} ), |
| + | $$ |
| + | |
| + | 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|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> |
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| ====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 |
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=17005
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