# Schanuel’s lemma fails for flats

Recall that Schanuel’s lemma asserts that if $0\to K_i \to P_i\to M\to 0$ are short exact sequences $i=1,2$ and $P_i$ is projective, then $P_1\oplus K_2\simeq P_2\oplus K_1$. Show this fails if the $P_i$ are only assumed to be flat.

1. *Hint* As in the previous post, the integers work. In fact, this fails even if only one of the $P_i$ is flat.