In the algebraic Künneth theorem, see https://ncatlab.org/nlab/show/Künneth+theorem theorem 2.2 We assume we have two complexes, one of which is free over some PID.
Now this gives some short exact sequence whose left map is
$\oplus{H_k(C) \otimes H_{n-k}(C')} \to \oplus {H_n(C \otimes C')} $ And this injection is split noncanonically.
Now my question is about the proof of this splitting.
From all the sources I looked at, the first step is to notice the splitting of
$0 \to Z \to C \to B \to 0$
Now this induces a splitting after tensoring by $C'$ or $Z'$ however I cannot really finish the proof.
I would appreciate someone spelling out the details of the last bit, I looked at different sources but none are really detailed. Hatcher seems to assume C' are also free? Please do not assume any properties of C', the nLab statement doesn't impose any conditions on C'.
Edit: just realized nLab doesn't even talk about splitting... Rest assured that both Weibel and Rotman writes split for the conditions I gave