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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 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, 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 $X$.

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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.

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  • $\begingroup$ Thanks, i never thought of using Cech cohomology. Nice and easy. $\endgroup$ – Bernie Oct 9 '15 at 10:27

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