# if the base chage of all fibres of a morphism is faithfully flat, do we have flatness of the morphism?

Let $$f:X\rightarrow Y$$ be a morphism of schemes over $$S$$ (with possibly Noetherian conditions all over the place).

For every point $$s\in S$$, the morphism base changed to the fiber $$f_s:X_s\rightarrow Y_s$$ is faithfully flat.

Do we have that $$f$$ is flat? or even faithfully flat?

Edit: what if moreover both $$X$$ and $$Y$$ are flat over $$S$$?

• If $Y=S$ and $f:X\to S$ is onto, then for every $s\in S, f_s:X_s\to s$ is faithfully flat since $s$ is the spectrum of the field. But $f$ is obviously not necessarily flat. – Roland Jan 26 at 12:41
• @Roland what if both of $X$ and $Y$ are flat over $S$? Then your example doesn't apply. – Lance Wu Jan 26 at 13:11