A question about tensoring and retaining exactness

Question

This is a question from this document on the Universal Coefficient Theorem.

We have the following chain complex:

We then tensor each module with $G$, and get the following complex:

How come exactness is still preserved, if we haven't assumed that $G$ is flat?

Answer 1

In the attached paper, the $B_n$ (and $C_n$) are free Abelian groups. Therefore each horizontal short exact sequence splits, and so remains a split short exact sequence after tensoring with $G$.