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 $?



restrictions and flatness
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. 


