flat chain complexes

Question

Let $X$ be a flat chain complex of R modules where R is a PID. Let $Z_n=\ker(d_n)$ and $B_n=\operatorname{Im}(d_{n+1})$ where $d_n :\to X_n \to X_{n-1}$. Then how can I conclude that the short exact sequence $0 \to Z_n \to X_n \to B_{n-1} \to 0$ is split? How can we write $X_n=Z_n \oplus B_{n-1}$?

Answer 1

This isn't true. For instance, let $Q$ be any $R$-module that is flat but not projective, and choose a free resolution $0\to K\to F\to Q\to 0$ of $Q$. Considering this as a chain complex $X$ with $X_0=Q$, we have $Z_1=K$, $X_1=F$, and $B_{n-1}=Q$. But the sequence $0\to K\to F\to Q\to 0$ cannot split since $Q$ is not projective.