Algebraic Künneth formula over the integers

Example 6.7.4. For \(R = \Z \), the Künneth formula specializes to the classical result for abelian groups. Given chain complexes \(A\) and \(B\) of abelian groups, the homology of the derived tensor product fits into the short exact sequence \[ 0 \to \bigoplus _{i+j=n} H_i(A) \otimes _{\Z } H_j(B) \to H_n(A \otimes ^{\bL }_{\Z } B) \to \bigoplus _{i+j=n-1} \Tor _1^{\Z }(H_i(A), H_j(B)) \to 0. \] Since \(\Tor _1^{\Z }(M,N)\) consists of the common torsion of \(M\) and \(N\) (see Exercise 6.5.8), this shows that the homology of the derived tensor product is determined by the homology groups of \(A\) and \(B\), up to an extension involving their torsion.