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I am wondering about the following: Suppose $X$ is a reasonable scheme or stack with the resolution property. (So, all quasi-coherent sheaves admit a surjection from a flat sheaf.) Then I believe ...

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]...