Are there any homological checks I can use to check if a projective morphism is flat? For example, I would expect the following projective morphism to be flat
$$
\textbf{Proj}\left( \frac{\mathbb{C}[s][x,y,z]}{x^3 + y^3 - xyz + sz^3} \right) \to \textbf{Spec}(\mathbb{C}[s])
$$
while
$$
\textbf{Proj}\left( \frac{\mathbb{C}[s][x,y,z]}{s(x^4 + y^4 + z^4)} \right) \to \textbf{Spec}(\mathbb{C}[s])
$$
would not be flat.



How can I show flatness for projective morphisms?
For a scheme to be flat over $A^1\_s = Spec(C[s])$ is equivalent to not having irreducible (or embedded) components supported over (closed) points of $A^1\_s$. Since your schemes are hypersurfaces in $A^1\_s \times A^3\_{x,y,z}$, having such a component is equivalent to containing $\{a\} \times A^3\_{x,y,z}$ as a subvariety for some point $a \in A^1\_s$. The later is equivalent to the equation of the hypersurface being nonzero after substitution $s = a$. So, it easily follows that the first hypersurface is flat, while the second is not.


