I have a problem with Proposition 4.3.9 of Qing Liu's algebraic geometry book. It says if R is a Dedekind ring and X is a reduced scheme and we have a dominant morphism $f:X\to \operatorname{spec}R$, then f is flat.
I don't understand the proof and I think $f:\mathbb Z_2\to \frac{\mathbb Z_2[x]}{\left<2x\right>}$ is a counter example. Clearly $\frac{\mathbb Z_2[x]}{\left<2x\right>}$ is reduced, and the morphism isn't flat.