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A binary operation on the set of classes of [[Extension of a module|extensions of modules]], proposed by R. Baer [[#References|[1]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015020/b0150201.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015020/b0150202.png" /> be arbitrary modules. An extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015020/b0150203.png" /> with kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015020/b0150204.png" /> is an [[exact sequence]]:
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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/b/b015/b015020/b0150205.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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A binary operation on the set of classes of [[Extension of a module|extensions of modules]], proposed by R. Baer [[#References|[1]]]. Let  $  A $
 +
and  $  B $
 +
be arbitrary modules. An extension of  $  A $
 +
with kernel  $  B $
 +
is an [[exact sequence]]:
 +
 
 +
$$ \tag{1 }
 +
0  \rightarrow  B  \rightarrow  X  \rightarrow  A  \rightarrow  0.
 +
$$
  
 
The extension (1) is called ''equivalent'' to the extension
 
The extension (1) is called ''equivalent'' to the extension
  
<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/b/b015/b015020/b0150206.png" /></td> </tr></table>
+
$$
 +
0  \rightarrow  B  \rightarrow  X _ {1}  \rightarrow  A  \rightarrow  0
 +
$$
 +
 
 +
if there exists a homomorphism  $  \alpha : X \rightarrow X _ {1} $
 +
forming part of the commutative diagram
 +
 
 +
$$
  
if there exists a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015020/b0150207.png" /> forming part of the commutative diagram
+
\begin{array}{ccccc}
 +
{}  &{}  & X  &{}  &{}  \\
 +
{}  &\nearrow  &{}  &\searrow  &{}  \\
 +
B  &{}  &\downarrow  &{}  & A  \\
 +
{}  &\searrow  &{}  &\nearrow  &{}  \\
 +
{}  &{}  &X _ {1}  &{}  &{}  \\
 +
\end{array}
  
<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/b/b015/b015020/b0150208.png" /></td> </tr></table>
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$$
  
The set of equivalence classes of extensions is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015020/b0150209.png" />. The Baer multiplication on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015020/b01502010.png" /> is induced by the operation of products of extensions defined as follows. Let
+
The set of equivalence classes of extensions is denoted by $  \mathop{\rm Ext} (A, B) $.  
 +
The Baer multiplication on $  \mathop{\rm Ext} (A, B) $
 +
is induced by the operation of products of extensions defined as follows. Let
  
<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/b/b015/b015020/b01502011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$ \tag{2 }
 +
0 \rightarrow  B  \mathop \rightarrow \limits ^  \beta    X  \mathop \rightarrow \limits ^  \alpha    A  \rightarrow 0,
 +
$$
  
<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/b/b015/b015020/b01502012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
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$$ \tag{3 }
 +
0 \rightarrow  B  \rightarrow ^ { {\beta _ 1} }  Y  \rightarrow ^ { {\alpha _ 1} }  A  \rightarrow  0
 +
$$
  
be two extensions. In the direct sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015020/b01502013.png" /> the submodules
+
be two extensions. In the direct sum $  X \oplus Y $
 +
the submodules
  
<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/b/b015/b015020/b01502014.png" /></td> </tr></table>
+
$$
 +
= \{ {(x, y) } : {
 +
\alpha (x) = \alpha _ {1} (y) } \}
 +
$$
  
 
and
 
and
  
<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/b/b015/b015020/b01502015.png" /></td> </tr></table>
+
$$
 +
= \{ {(-x, y) } : {
 +
x = \beta (b),\
 +
y = \beta _ {1} (b) } \}
 +
$$
  
are selected. Clearly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015020/b01502016.png" />, so that one can define the quotient module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015020/b01502017.png" />. The Baer product of the extensions (2) and (3) is the extension
+
are selected. Clearly, $  D \subset  C $,  
 +
so that one can define the quotient module $  Z = C/D $.  
 +
The Baer product of the extensions (2) and (3) is the extension
  
<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/b/b015/b015020/b01502018.png" /></td> </tr></table>
+
$$
 +
0 \rightarrow  B  \rightarrow ^ { {\beta _ 2} }  Z  \rightarrow ^ { {\alpha _ 2} }  A  \rightarrow  0,
 +
$$
  
 
where
 
where
  
<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/b/b015/b015020/b01502019.png" /></td> </tr></table>
+
$$
 +
\beta _ {2} (b)  = \
 +
[ \beta (b), 0]  = \
 +
[0, \beta  ^  \prime  (b)],
 +
$$
  
 
and
 
and
  
<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/b/b015/b015020/b01502020.png" /></td> </tr></table>
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$$
 +
\alpha _ {2} [x, y]  = \
 +
\alpha (x)  = \
 +
\alpha _ {1} (y).
 +
$$
  
 
====References====
 
====References====

Latest revision as of 10:26, 27 April 2020


A binary operation on the set of classes of extensions of modules, proposed by R. Baer [1]. Let $ A $ and $ B $ be arbitrary modules. An extension of $ A $ with kernel $ B $ is an exact sequence:

$$ \tag{1 } 0 \rightarrow B \rightarrow X \rightarrow A \rightarrow 0. $$

The extension (1) is called equivalent to the extension

$$ 0 \rightarrow B \rightarrow X _ {1} \rightarrow A \rightarrow 0 $$

if there exists a homomorphism $ \alpha : X \rightarrow X _ {1} $ forming part of the commutative diagram

$$ \begin{array}{ccccc} {} &{} & X &{} &{} \\ {} &\nearrow &{} &\searrow &{} \\ B &{} &\downarrow &{} & A \\ {} &\searrow &{} &\nearrow &{} \\ {} &{} &X _ {1} &{} &{} \\ \end{array} $$

The set of equivalence classes of extensions is denoted by $ \mathop{\rm Ext} (A, B) $. The Baer multiplication on $ \mathop{\rm Ext} (A, B) $ is induced by the operation of products of extensions defined as follows. Let

$$ \tag{2 } 0 \rightarrow B \mathop \rightarrow \limits ^ \beta X \mathop \rightarrow \limits ^ \alpha A \rightarrow 0, $$

$$ \tag{3 } 0 \rightarrow B \rightarrow ^ { {\beta _ 1} } Y \rightarrow ^ { {\alpha _ 1} } A \rightarrow 0 $$

be two extensions. In the direct sum $ X \oplus Y $ the submodules

$$ C = \{ {(x, y) } : { \alpha (x) = \alpha _ {1} (y) } \} $$

and

$$ D = \{ {(-x, y) } : { x = \beta (b),\ y = \beta _ {1} (b) } \} $$

are selected. Clearly, $ D \subset C $, so that one can define the quotient module $ Z = C/D $. The Baer product of the extensions (2) and (3) is the extension

$$ 0 \rightarrow B \rightarrow ^ { {\beta _ 2} } Z \rightarrow ^ { {\alpha _ 2} } A \rightarrow 0, $$

where

$$ \beta _ {2} (b) = \ [ \beta (b), 0] = \ [0, \beta ^ \prime (b)], $$

and

$$ \alpha _ {2} [x, y] = \ \alpha (x) = \ \alpha _ {1} (y). $$

References

[1] R. Baer, "Erweiterung von Gruppen und ihren Isomorphismen" Math. Z. , 38 (1934) pp. 374–416
[2] H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956)
How to Cite This Entry:
Baer multiplication. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Baer_multiplication&oldid=45581
This article was adapted from an original article by V.E. Govorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article