Let $X$ be a noetherian scheme over $\mathbb{C}$, and let $E$ be a locally free sheaf of finite rank over $X$. Then we have the projective bundle $f: \mathbb{P}(E)\rightarrow X$.


Now $f$ is a flat morphism and we have $f\_{\*}\mathcal{O}\_{\mathbb{P}(E)}=\mathcal{O}\_X$ and $R^if\_{\*}\mathcal{O}\_{\mathbb{P}(E)}=0$ for $i\geq 1$.


So according to this  [question](https://mathoverflow.net/questions/180121/sheaf-isomorphism)  and the following answers we should get for every coherent sheaf $H\in Coh(X)$ an isomorphism $f\_{\*}f^{\*}H\cong H$ and for $i\geq 1$ we should have $R^if\_{\*}f^{\*}H=0$. $(\*)$


Is there a direct way to see that we have the two facts in $(\*)$ for any noetherian $X$ without using the derived category? Or is this wrong in this generality?


I'm asking because in the answers to my question [here](https://mathoverflow.net/questions/220031/which-sheaves-on-a-projective-bundle-are-flat-over-the-base-scheme), it is sugested to use the projection formula for $f$ to see the vanishing of higher direct images of sheaves $G$ with the property $f^{\*}f\_{\*}G\cong G$. But the usual projection formula $R^if\_{\*}(G\otimes f^{\*}H)\cong (R^if\_{\*}G)\otimes H$, e.g. Hartshorne Exercise III.$8.3$ needs the sheaf $H$ on $X$ to be locally free. In this case we don't have an arbitrary morphism but a flat one and we have the fact $f\_{\*}\mathcal{O}\_{\mathbb{P}(E)}=\mathcal{O}\_X$ resp. $R^if\_{\*}\mathcal{O}\_{\mathbb{P}(E)}=0$ for $i\geq 1$ so maybe it works also for coherent sheaves on $X$.



Can one drop the locally free assumption in projection formula on a projective bundle?
Pick an affine covering of $X$ over which $E$ trivialises. For an affine $U = Spec(A)$ from the covering $\Gamma(U, R^if\_\*f^\*H) = H^i(f^{-1}(U), f^\*H)$. Now, use the standard covering of $f^{-1}(U)=Proj(A[x\_0,x\_1,\ldots,x\_n])$ to compute Cech cohomology.


