Let $k$ be an algebraically closed field of characteristic zero. Let $X, Y, Z$ be $k$-schemes of finite type. Let $F$ be a coherent sheaf on $X \times Y \times Z$, flat over $Y \times Z$. Is $\mathrm{pr}_{23_*}F$ flat over $Z$, where $\mathrm{pr}_{23}:X \times Y \times Z \to Y \times Z$ is the natural projection map?
Is the pushforward of a flat modules flat
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algebraic-geometry
commutative-algebra
coherent-sheaves
flatness
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3I think you must at least recognize that you could write $Y\times Z=T$ and $Y,Z$ do not play any individual roles. This is false in general and not difficult to construct examples. I suggest you look at the case of $X$ a smooth curve, $T=\mathrm{Pic}^{2g-2} X$, where $g\geq 2$, the genus of $X$ with the Poincare line bundle. – 2017-01-19
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0@Mohan What is $Z$ in this case? Just to be clear I am not asking if $\mathrm{pr}_{23_*}F$ is flat as a $Y \times Z$-module. – 2017-01-19
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0As I said, $Y,Z$ do not play any separate roles here. So, to go from $T$ to $Y\times Z$, you can simply take $T=Y\times Z$, since $T$ was arbitrary. – 2017-01-19