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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:

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

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

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

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

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

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

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 [[#References|[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

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

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====
<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>