Consider the obviously non-flat morphism of schemes $$ \text{Spec}(\mathbb{C}[x,y]/(xy)) \to \text{Spec}(\mathbb{C}[x]) $$ How can I show that it is not flat using local algebra? I've looked at a few local algebra morphisms $$ \begin{align} \mathbb{C}[x]_{(x)} \to (\mathbb{C}[x,y]/(xy))_{(x)} \\ \mathbb{C}[x]_{(x)} \to (\mathbb{C}[x,y]/(xy))_{(x,y)} \\ \mathbb{C}[x]_{(x)} \to (\mathbb{C}[x,y]/(xy))_{(x,y-1)} \end{align} $$ and tried to come up with an exact sequence where they do not preserve injectivity, but haven't been able to find one. Do you have any hints?
It seems like parts 2 and 4 of proposition 5 in https://math.berkeley.edu/~ogus/Math%20_256B--09/Supplements/flat.pdf should help, but I'm not convinced that the tensor of the residue field is zero for any of the previous modules.
As per Mohan's suggestion, we can take the sequence $$ 0 \to \mathbb{C}[x]_{(x)} \xrightarrow{\cdot x} \mathbb{C}[x]_{(x)} $$ and tensor it with our local algebra. Since $xy = 0$ in the module we have that $$ (\mathbb{C}[x,y]/(xy))_{(x)} \xrightarrow{\cdot x}(\mathbb{C}[x,y]/(xy))_{(x)} $$ is not injective.