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) |
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