Decker’s sharper Künneth formula. (English) Zbl 0655.18010

Mac Lane, Saunders

Categorical algebra and its applications, Proc. 1st Conf., Louvain-la- Neuve/Belg. 1987, Lect. Notes Math. 1348, 242-256 (1988).

[For the entire collection see Zbl 0644.00009.]
The Künneth formula gives the homology of \(K_ 1\otimes K_ 2\), where \(K_ 1\) and \(K_ 2\) are chain complexes of abelian groups with \(K_ 1\) having no torsion, by a short exact sequence: \[ 0\to \sum_{p+q=n}H_ p(K_ 1)\otimes H_ q(K_ 2)\to H_ n(K_ 1\otimes K_ 2)\to \sum_{p+q=n-1}Tor\quad (H_ p(K_ 1),H_ q(K_ 2))\to 0. \] This sequence splits but with a homomorphism which is not natural.
The present paper describes additional data, adapted of a technique introduced in the unpublished thesis of G. J. Decker (Univ. of Chicago, 1974), which give a natural splitting and thus present the homology of the tensor product complex as a direct sum of graded abelian groups.

Reviewer: G.Hoff

MSC:

18G35	Chain complexes (category-theoretic aspects), dg categories
18G15	Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)

Keywords:

Künneth formula; homology; chain complexes; splitting; tensor product

Citations:

Zbl 0644.00009