Let $k$ be a field.
Consider the ring morphism $f: k[x,y,x^{-1}, y^{-1}] \to k[t,t^{-1}]$ where $x \to t$ and $y \to t$. How do we know if $f$ is flat or not?
Now let $R=k[x,y,x^{-1}, y^{-1}]$. Then $k[t,t^{-1}] = R/(x-y)$.
Now consider the exact sequence
$$(x-y) \to R \to R/(x-y).$$
Tensoring this exact sequence with $R/(x-y)$ we get
$$(x-y)/(x-y)^2 \to R/(x-y) \to R/(x-y).$$
But since $(x-y)/(x-y)^2$ is not $0$ the above sequence is not exact.
So, we conclude this morphism is not flat.
Is the above argument ok? Or am I missing something?