Generalization of Künneth’s relation. (Russian. English summary) Zbl 0534.18008
For a chain complex \(C_*\) in an Abelian category with enough injectives, and a left exact functor T, one has the classical Künneth map \(\phi_ n: H_ nTC_*\to TH_ nC_*\). By imposing the strong restriction that \(T^{(i)}(Z_ nC\to H_ nC)\) is a split monomorphism for every i, and assuming as usual that \(C_ n\) is T-acyclic for all n, the author writes down a long exact sequence which incorporates \(\phi_ n\), and whose other terms are of the form \(T^{(j)}H_{n+j}\). The proof is elementary and straightforward. Applications are indicated to singular and Massey cohomology theories of topological spaces.
Reviewer: L.L.Avramov
MSC:
18G15 | Ext and Tor, generalizations, Künneth formula (category-theoretic aspects) |
55N10 | Singular homology and cohomology theory |
18G35 | Chain complexes (category-theoretic aspects), dg categories |
18E10 | Abelian categories, Grothendieck categories |
55U25 | Homology of a product, Künneth formula |
18E25 | Derived functors and satellites (MSC2010) |