If $\pi:X \rightarrow Y$ is a flat morphism, and $Z$ is a closed subscheme of $Y$, then we could construct the blow up $\text{Bl}_Z Y$. Since $\pi$ is flat, the pullback of $\text{Bl}_Z Y$, i.e. $\text{Bl}_Z Y \times_Y X$ is isomrphic to $\text{Bl}_{Z \times_Y X} X$, and this is just the part (a) of ex 24.2 P, which I know how to prove. \begin{array}{c} Z \times_Y X & \rightarrow & Z \\ \downarrow & & \downarrow \\ X & \rightarrow & Y \\ \end{array}
\begin{array}{ccc} \text{Bl}_Z Y \times_Y X \cong \text{Bl}_{Z \times_Y X} X & \rightarrow & \text{Bl}_Z Y \\ \downarrow & & \downarrow \\ X & \rightarrow & Y \\ \end{array} Part (b) is to find an example to show that this is not true generally if $\pi$ is not flat, any one knows an example?
