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This is one more question on flat morphisms that I have been thinking about.

Suppose $f:X\rightarrow S$ is a flat projective morphism of noetherian schemes. Both $X$ and $S$ are smooth and irreducible. Let $T$ be a smooth closed subscheme of $S$.

Suppose $F$ is a coherent $O_X$ module such that $F_T$ is flat over $T$.

Is there some additional condition which will ensure that $F$ is $S$-flat?

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  • $\begingroup$ If $T$ is a point, then $F_T$ is flat over $T$ for any $F$. So, what kind of conditions are you expecting? $\endgroup$
    – Mohan
    Commented Apr 3, 2017 at 1:26
  • $\begingroup$ @Mohan, is there some sheaf theoretic condition? I heard that there is a condition in terms of ideal of the closed subscheme. But I don't know what it is. $\endgroup$
    – user349424
    Commented Apr 3, 2017 at 1:42

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No. Flatness is local condition. We can always ensure $F$ is misbehaved at some place outside of $T$.

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