Homology product

An operation defined on the groups and . One considers -algebras and over a commutative ring . The derived functors (cf. Derived functor) and over them may be combined with one another by means of four homomorphisms, known as homology products:

Here, and are right or left -modules, and are right or left -modules, while the symbol is omitted in all functors. The last two homomorphisms are defined only if the algebras and are projective over and if for all . If certain supplementary restrictions are made, intrinsic products can be obtained which connect and over the same ring.

All four products can be obtained from formulas representing the functors and by replacing the arguments by the corresponding resolutions [1]. The multiplication permits the following interpretation in terms of Yoneda products. Let

be exact sequences of - and -modules, respectively, that are representatives of the corresponding equivalence classes in and . Multiplying the former tensorially from the right by and the latter from the left by , one obtains exact sequences

which can be combined into the exact sequence

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

The product in the cohomology space of a topological space with coefficients in the ring of integers is known as the Alexander–Kolmogorov product or the -product.

References

[1]	H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956)
[2]	S. MacLane, "Homology" , Springer (1963)