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Suppose $ X $ and $ T $ are complex algebraic varieties and let $ Y $ be a subvariety of $ X $ . If we have a coherent sheaf $\mathcal {F} $ on the product $ X\times T $ flat over $ X $, is it true that the restriction of $\mathcal {F} $ to $ Y\times T $ is also flat over $ Y $?

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  • $\begingroup$ Thank you! Do I have some hope in the case $ Y $ is a determinatal variety? $\endgroup$
    – Claude
    Commented Jun 4, 2014 at 11:31

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Actually the answer is yes, and this is true in a much more general setting: any morphism $Z\rightarrow X$, $\mathcal{F}$ a quasi-coherent sheaf on $Z$ flat over $X$, any base change $Y\rightarrow X$; then the pull back of $\mathcal{F}$ to $Y\times _XZ$ is flat over $Y$. This is EGA IV.2, Prop. 2.1.4.

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  • $\begingroup$ Dear abx, Thank you Very much, this is Just what I was looking for! $\endgroup$
    – Claude
    Commented Jun 4, 2014 at 12:43

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