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An operation defined on the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h0478401.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h0478402.png" />. One considers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h0478403.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h0478404.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h0478405.png" /> over a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h0478406.png" />. The derived functors (cf. [[Derived functor|Derived functor]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h0478407.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h0478408.png" /> over them may be combined with one another by means of four homomorphisms, known as homology products:
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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/h/h047/h047840/h0478409.png" /></td> </tr></table>
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{{TEX|auto}}
 +
{{TEX|done}}
  
<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/h/h047/h047840/h04784010.png" /></td> </tr></table>
+
An operation defined on the groups  $  \mathop{\rm Tor} $
 +
and  $  \mathop{\rm Ext} $.
 +
One considers  $  K $-
 +
algebras  $  R, S $
 +
and  $  T = R \otimes _ {K} S $
 +
over a commutative ring  $  K $.  
 +
The derived functors (cf. [[Derived functor|Derived functor]])  $  \mathop{\rm Tor} $
 +
and  $  \mathop{\rm Ext} $
 +
over them may be combined with one another by means of four homomorphisms, known as homology products:
  
<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/h/h047/h047840/h04784011.png" /></td> </tr></table>
+
$$
 +
\perp  :   \mathop{\rm Tor} _ {p}  ^ {R} ( A, A  ^  \prime  ) \otimes
 +
\mathop{\rm Tor} _ {q}  ^ {S} ( C, C  ^  \prime  )  \rightarrow \
 +
\mathop{\rm Tor} _ {p + q }  ^ {T} ( A \otimes C, A  ^  \prime  \otimes C  ^  \prime  ),
 +
$$
  
<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/h/h047/h047840/h04784012.png" /></td> </tr></table>
+
$$
 +
\perp  :   \mathop{\rm Ext} _ {T} ^ {p + q } ( A \otimes C,  \mathop{\rm Hom} ( A  ^  \prime  ,
 +
C  ^  \prime  )) \rightarrow
 +
$$
  
<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/h/h047/h047840/h04784013.png" /></td> </tr></table>
+
$$
 +
\rightarrow \
 +
\mathop{\rm Hom} (  \mathop{\rm Tor} _ {p}  ^ {R} ( A  ^  \prime  ,
 +
A),  \mathop{\rm Ext} _ {s}  ^ {q} ( C, C  ^  \prime  )),
 +
$$
  
<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/h/h047/h047840/h04784014.png" /></td> </tr></table>
+
$$
 +
\lor :   \mathop{\rm Ext} _ {R}  ^ {p} ( A, A  ^  \prime  ) \otimes
 +
\mathop{\rm Ext} _ {s}  ^ {q} ( C, C  ^  \prime  )  \rightarrow  \mathop{\rm Ext} _ {T} ^ {p + q } ( A \otimes C, A  ^  \prime  \otimes C  ^  \prime  ),
 +
$$
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h04784015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h04784016.png" /> are right or left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h04784017.png" />-modules, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h04784018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h04784019.png" /> are right or left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h04784020.png" />-modules, while the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h04784021.png" /> is omitted in all functors. The last two homomorphisms are defined only if the algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h04784022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h04784023.png" /> are projective over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h04784024.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h04784025.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h04784026.png" />. If certain supplementary restrictions are made, intrinsic products can be obtained which connect <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h04784027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h04784028.png" /> over the same ring.
+
$$
 +
\wedge :   \mathop{\rm Tor} _ {p + q }  ^ {T}
 +
(  \mathop{\rm Hom} ( A, C), A  ^  \prime  \otimes C  ^  \prime  ) \rightarrow
 +
$$
  
All four products can be obtained from formulas representing the functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h04784029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h04784030.png" /> by replacing the arguments by the corresponding resolutions [[#References|[1]]]. The multiplication <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h04784031.png" /> permits the following interpretation in terms of Yoneda products. Let
+
$$
 +
\rightarrow \
 +
\mathop{\rm Hom} (  \mathop{\rm Ext} _ {R}  ^ {p} ( A, A  ^  \prime  ),  \mathop{\rm Tor} _ {q}  ^ {S} ( C, C  ^  \prime  )).
 +
$$
  
<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/h/h047/h047840/h04784032.png" /></td> </tr></table>
+
Here,  $  A $
 +
and  $  A  ^  \prime  $
 +
are right or left  $  R $-
 +
modules,  $  C $
 +
and  $  C  ^  \prime  $
 +
are right or left  $  S $-
 +
modules, while the symbol  $  K $
 +
is omitted in all functors. The last two homomorphisms are defined only if the algebras  $  R $
 +
and  $  S $
 +
are projective over  $  K $
 +
and if  $  \mathop{\rm Tor} _ {n}  ^ {K} ( A, C) = 0 $
 +
for all  $  n > 0 $.  
 +
If certain supplementary restrictions are made, intrinsic products can be obtained which connect  $  \mathop{\rm Tor} $
 +
and  $  \mathop{\rm Ext} $
 +
over the same ring.
  
<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/h/h047/h047840/h04784033.png" /></td> </tr></table>
+
All four products can be obtained from formulas representing the functors  $  \otimes $
 +
and  $  \mathop{\rm Hom} $
 +
by replacing the arguments by the corresponding resolutions [[#References|[1]]]. The multiplication  $  \lor $
 +
permits the following interpretation in terms of Yoneda products. Let
  
be exact sequences of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h04784034.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h04784035.png" />-modules, respectively, that are representatives of the corresponding equivalence classes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h04784036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h04784037.png" />. Multiplying the former tensorially from the right by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h04784038.png" /> and the latter from the left by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h04784039.png" />, one obtains exact sequences
+
$$
 +
0 \rightarrow  A  ^  \prime  \rightarrow  X _ {1}  \rightarrow \dots \rightarrow  X _ {p}  \rightarrow  A  \rightarrow  0,
 +
$$
  
<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/h/h047/h047840/h04784040.png" /></td> </tr></table>
+
$$
 +
0 \rightarrow  C  ^  \prime  \rightarrow  Y _ {1}  \rightarrow \dots \rightarrow  Y _ {q}  \rightarrow  C  \rightarrow  0
 +
$$
  
<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/h/h047/h047840/h04784041.png" /></td> </tr></table>
+
be exact sequences of  $  R $-  
 +
and  $  S $-
 +
modules, respectively, that are representatives of the corresponding equivalence classes in  $  \mathop{\rm Ext} _ {R}  ^ {p} ( A, A  ^  \prime  ) $
 +
and  $  \mathop{\rm Ext} _ {S}  ^ {q} ( C, C  ^  \prime  ) $.
 +
Multiplying the former tensorially from the right by  $  C  ^  \prime  $
 +
and the latter from the left by  $  A $,
 +
one obtains exact sequences
 +
 
 +
$$
 +
0  \rightarrow  A  ^  \prime  \otimes C  ^  \prime  \rightarrow  X _ {1} \otimes C  ^  \prime  \rightarrow \dots \rightarrow \
 +
X _ {p} \otimes C  ^  \prime  \rightarrow  A \otimes C  ^  \prime  \rightarrow  0,
 +
$$
 +
 
 +
$$
 +
0  \rightarrow  A \otimes C  ^  \prime  \rightarrow  A \otimes Y _ {1}  \rightarrow \dots
 +
\rightarrow  A \otimes Y _ {q}  \rightarrow  A \otimes C  \rightarrow  0,
 +
$$
  
 
which can be combined into the exact sequence
 
which can be combined into the exact sequence
  
<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/h/h047/h047840/h04784042.png" /></td> </tr></table>
+
$$
 +
0  \rightarrow  A  ^  \prime  \otimes C  ^  \prime  \rightarrow \
 +
X _ {1} \otimes C  ^  \prime  \rightarrow \dots \rightarrow  A \otimes Y _ {q}  \rightarrow  A \otimes C  \rightarrow  0.
 +
$$
  
 
This sequence can be regarded as the representative of an equivalence class in the group
 
This sequence can be regarded as the representative of an equivalence class in the group
  
<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/h/h047/h047840/h04784043.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Exp} _ {R \otimes S }  ^ {p + q } ( A \otimes C, A  ^  \prime  \otimes C  ^  \prime  ).
 +
$$
  
The product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h04784044.png" /> in the cohomology space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h04784045.png" /> of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h04784046.png" /> with coefficients in the ring of integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h04784047.png" /> is known as the Alexander–Kolmogorov product or the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047840/h04784049.png" />-product.
+
The product $  \lor $
 +
in the cohomology space $  H ( X, \mathbf Z ) $
 +
of a topological space $  X $
 +
with coefficients in the ring of integers $  \mathbf Z $
 +
is known as the Alexander–Kolmogorov product or the $  \cup $-
 +
product.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Cartan,  S. Eilenberg,  "Homological algebra" , Princeton Univ. Press  (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. MacLane,  "Homology" , Springer  (1963)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Cartan,  S. Eilenberg,  "Homological algebra" , Princeton Univ. Press  (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. MacLane,  "Homology" , Springer  (1963)</TD></TR></table>

Revision as of 22:11, 5 June 2020


An operation defined on the groups $ \mathop{\rm Tor} $ and $ \mathop{\rm Ext} $. One considers $ K $- algebras $ R, S $ and $ T = R \otimes _ {K} S $ over a commutative ring $ K $. The derived functors (cf. Derived functor) $ \mathop{\rm Tor} $ and $ \mathop{\rm Ext} $ over them may be combined with one another by means of four homomorphisms, known as homology products:

$$ \perp : \mathop{\rm Tor} _ {p} ^ {R} ( A, A ^ \prime ) \otimes \mathop{\rm Tor} _ {q} ^ {S} ( C, C ^ \prime ) \rightarrow \ \mathop{\rm Tor} _ {p + q } ^ {T} ( A \otimes C, A ^ \prime \otimes C ^ \prime ), $$

$$ \perp : \mathop{\rm Ext} _ {T} ^ {p + q } ( A \otimes C, \mathop{\rm Hom} ( A ^ \prime , C ^ \prime )) \rightarrow $$

$$ \rightarrow \ \mathop{\rm Hom} ( \mathop{\rm Tor} _ {p} ^ {R} ( A ^ \prime , A), \mathop{\rm Ext} _ {s} ^ {q} ( C, C ^ \prime )), $$

$$ \lor : \mathop{\rm Ext} _ {R} ^ {p} ( A, A ^ \prime ) \otimes \mathop{\rm Ext} _ {s} ^ {q} ( C, C ^ \prime ) \rightarrow \mathop{\rm Ext} _ {T} ^ {p + q } ( A \otimes C, A ^ \prime \otimes C ^ \prime ), $$

$$ \wedge : \mathop{\rm Tor} _ {p + q } ^ {T} ( \mathop{\rm Hom} ( A, C), A ^ \prime \otimes C ^ \prime ) \rightarrow $$

$$ \rightarrow \ \mathop{\rm Hom} ( \mathop{\rm Ext} _ {R} ^ {p} ( A, A ^ \prime ), \mathop{\rm Tor} _ {q} ^ {S} ( C, C ^ \prime )). $$

Here, $ A $ and $ A ^ \prime $ are right or left $ R $- modules, $ C $ and $ C ^ \prime $ are right or left $ S $- modules, while the symbol $ K $ is omitted in all functors. The last two homomorphisms are defined only if the algebras $ R $ and $ S $ are projective over $ K $ and if $ \mathop{\rm Tor} _ {n} ^ {K} ( A, C) = 0 $ for all $ n > 0 $. If certain supplementary restrictions are made, intrinsic products can be obtained which connect $ \mathop{\rm Tor} $ and $ \mathop{\rm Ext} $ over the same ring.

All four products can be obtained from formulas representing the functors $ \otimes $ and $ \mathop{\rm Hom} $ by replacing the arguments by the corresponding resolutions [1]. The multiplication $ \lor $ permits the following interpretation in terms of Yoneda products. Let

$$ 0 \rightarrow A ^ \prime \rightarrow X _ {1} \rightarrow \dots \rightarrow X _ {p} \rightarrow A \rightarrow 0, $$

$$ 0 \rightarrow C ^ \prime \rightarrow Y _ {1} \rightarrow \dots \rightarrow Y _ {q} \rightarrow C \rightarrow 0 $$

be exact sequences of $ R $- and $ S $- modules, respectively, that are representatives of the corresponding equivalence classes in $ \mathop{\rm Ext} _ {R} ^ {p} ( A, A ^ \prime ) $ and $ \mathop{\rm Ext} _ {S} ^ {q} ( C, C ^ \prime ) $. Multiplying the former tensorially from the right by $ C ^ \prime $ and the latter from the left by $ A $, one obtains exact sequences

$$ 0 \rightarrow A ^ \prime \otimes C ^ \prime \rightarrow X _ {1} \otimes C ^ \prime \rightarrow \dots \rightarrow \ X _ {p} \otimes C ^ \prime \rightarrow A \otimes C ^ \prime \rightarrow 0, $$

$$ 0 \rightarrow A \otimes C ^ \prime \rightarrow A \otimes Y _ {1} \rightarrow \dots \rightarrow A \otimes Y _ {q} \rightarrow A \otimes C \rightarrow 0, $$

which can be combined into the exact sequence

$$ 0 \rightarrow A ^ \prime \otimes C ^ \prime \rightarrow \ X _ {1} \otimes C ^ \prime \rightarrow \dots \rightarrow A \otimes Y _ {q} \rightarrow A \otimes C \rightarrow 0. $$

This sequence can be regarded as the representative of an equivalence class in the group

$$ \mathop{\rm Exp} _ {R \otimes S } ^ {p + q } ( A \otimes C, A ^ \prime \otimes C ^ \prime ). $$

The product $ \lor $ in the cohomology space $ H ( X, \mathbf Z ) $ of a topological space $ X $ with coefficients in the ring of integers $ \mathbf Z $ is known as the Alexander–Kolmogorov product or the $ \cup $- product.

References

[1] H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956)
[2] S. MacLane, "Homology" , Springer (1963)
How to Cite This Entry:
Homology product. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Homology_product&oldid=18757
This article was adapted from an original article by V.E. Govorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article