Assume $X$ is a noetherian scheme over $\mathbb{C}$ and $E$ a locally free sheaf of finite rank on $X$. Denote the the associated projective bundle by $f: \mathbb{P}(E)\rightarrow X$.

Are there any coherent sheaves on $\mathbb{P}(E)$ that are flat over $X$ except locally free ones?

I'm especially interested in such $G\in Coh(\mathbb{P}(E))$ with the property that the canonical morphism $f^{*}f_{*}G\rightarrow G$ is an isomorphism. For these $G$ we have $G_{|\mathbb{P}(E(x))}=\mathcal{O}_{\mathbb{P}(E(x))}^{r_x}$ for all closed points $x\in X$ and some $r_x\geq 1$. So we have $H^i(\mathbb{P}(E(x)),G_{|\mathbb{P}(E(x))})=\{0\}$ for all $i\geq 1$.

I'm trying to see that this implies $R^if_{*}G=0$ for all $i\geq 1$, for which flatness of $G$ over $X$ is needed. Maybe there is a version of the Cohomology and Base Change Theory that does not need a flatness assumption? Or is there another theorem which im implies the vanishing of the higher direct images in this case?

I asked this question in a similar way on Math.Stackexchange.

Assume $X$ is a noetherian scheme over $\mathbb{C}$ and $E$ a locally free sheaf of finite rank on $X$. Denote the the associated projective bundle by $f: \mathbb{P}(E)\rightarrow X$.

Are there any coherent sheaves on $\mathbb{P}(E)$ that are flat over $X$ except locally free ones?

I'm especially interested in such $G\in Coh(\mathbb{P}(E))$ with the property that the canonical morphism $f^{*}f_{*}G\rightarrow G$ is an isomorphism. For these $G$ we have $G_{|\mathbb{P}(E(x))}=\mathcal{O}_{\mathbb{P}(E(x))}^{r_x}$ for all closed points $x\in X$ and some $r_x\geq 1$. So we have $H^i(\mathbb{P}(E(x)),G_{|\mathbb{P}(E(x))})=\{0\}$ for all $i\geq 1$.

I'm trying to see that this implies $R^if_{*}G=0$ for all $i\geq 1$, for which flatness of $G$ over $X$ is needed. Maybe there is a version of the Cohomology and Base Change Theory that does not need a flatness assumption? Or is there another theorem which im implies the vanishing of the higher direct images in this case?

I asked this question in a similar way on Math.Stackexchange.