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Let $(X,\mathcal{O}_X)$ and $(Y,\mathcal{O}_Y)$ be ringed spaces and $f: X\to Y$ be a morphism between them. We call $f$ flat at $x\in X$ if the natural morphism $\mathcal{O}_{Y,f(x)}\to \mathcal{O}_{X,x}$ is a flat map. We call $f$ is flat if $f$ is flat at all points in $X$.

My question is: if $f$ is flat, is it always true that $f_*\mathcal{O}_X$ is a flat $\mathcal{O}_Y$-module?

Edit: I found that the answer is negative in general, even for proper and flat morphism over schemes see this MO question and this MO question.

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  • $\begingroup$ Note that a module is flat if and only if all its localizations are flat. That should answer your question for open affine sets, now unfold definitions. $\endgroup$
    – assaferan
    Commented Dec 11, 2020 at 4:13
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    $\begingroup$ @assaferan There are two problems: first, I am interested in not just schemes but general ringed spaces; second, even if we consider schemes, the fibers of $f$ need not to be affine. $\endgroup$
    – Zhaoting Wei
    Commented Dec 11, 2020 at 4:23

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