Let $X$ and $Y \ $ be smooth varieties over a field or - depending on the answers - more general nice schemes (I don't know what one needs exactly as conditions). Let $p: X\times Y \rightarrow X$ be the projection morphism. It has constant fiber dimension $n:=dim(Y)$ over$X$, so is smooth of relative dimension n.
Is it then right that one has a vanishing of the higher direct images for any quasicoherent sheaf $\mathcal F$ on $X\times Y$:
$\mathcal R^i \mathcal F =0$ for all $i > n$ ?
Note that by Grothendieck's Vanishing Theorem one surely has it for $i>n+dim(X)$.
Also I really want Quasicoherence, not Coherence, so Semicontinuity Arguments don't seem to be at hand.
Furthermore, how far can one generalize this?