Exercise 5, § 1

Let $k$ be a field, $C=k[X,Y]$; consider the subrings $A=k[X^4,Y^4]$, $B=k[X^4,X^3Y,XY^3,Y^4]$ and $\widetilde B=k[X^4,X^3Y,X^2Y^2,XY^3,Y^4]$ of $C$. The inclusions $A\subset B\subset\widetilde B$ allow one to define on $B$ and $\widetilde B$ structures of $A$-module. Now prove that $\widetilde B$ is a flat $A$-module, whereas the $A$-module $B$ is not flat.