Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 101 / 1 / ii / d / Solution

The statement is false. Give $\mathbb{C}[X,Y]$ its $\mathbb{C}[T]$-algebra structure byand use the quotient mapsThe induced maps send $T$ to $X$ and $Y$, respectively, so both polynomial rings are free of rank one, hence flat modules, over $\mathbb{C}[T]$.

Their tensor product over the middle ring isHere $T$ acts by zero, so this is the torsion module $\mathbb{C}[T]/(T)$. It is not flat: tensoring the injection $\mathbb{C}[T]\stackrel{T}{\to }\mathbb{C}[T]$ with it produces the zero map on a nonzero module.