Let $T$ be a scheme over $\mathbb{C}$ and $\mathcal{F}$ be any $\mathcal{O}_T$ module. Suppose for all morphisms $Spec~R\rightarrow T$, (where R is a DVR) the restriction $\mathcal{F}_{Spec~R}$ is flat over $Spec~R$, then is it possible to conclude that $\mathcal{F}$ is flat over $T$?
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3$\begingroup$ No, that is not true. Consider the case where $T$ is $\text{Spec}\ \mathbb{C}[\epsilon]/\langle \epsilon^2 \rangle$ and $\mathcal{F}$ is the ideal sheaf generated by $\epsilon$. If you assume that $T$ is reduced, then the Valuative Criterion of Flatness gives a positive answer. $\endgroup$– Jason StarrCommented Nov 22, 2017 at 16:27

$\begingroup$ The answer is basically the same as the answer to the following MathOverflow question: mathoverflow.net/questions/282655/… $\endgroup$– Jason StarrCommented Nov 22, 2017 at 16:28
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