# Homology product

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=47261
This article was adapted from an original article by V.E. Govorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article