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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.

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1 Answer 1

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**Question: "I'm especially interested in such G∈Coh(P(E)) with the property that the canonical morphism $f^∗f_∗(G)\rightarrow G$ is an isomorphism."

Answer: If $S:=Spec(k)$ with $k$ the complex number field and $V:=k\{e_0,..,e_n\}$ it follows $\mathbb{P}(V^*) \cong \mathbb{P}^n_k$ is complex projective $n$-space. Let

$$\pi: \mathbb{P}^n_k \rightarrow S$$

be the projection morphism and let $G$ be a coherent sheaf on $\mathbb{P}^n_k$. It follows

$$\pi_*(G) \cong H^0(\mathbb{P}^n_k,G)$$

is the $k$-vector space of global sections of $G$ - let this vector space have basis $B:=\{u_i\}_{i \in I}$. It follows $\pi_*(G)$ is a free $\mathcal{O}_S$-module on the basis $u_i$. Since the pull back of a free module is a free module, it follows there is an isomorphism

$$\pi^*\pi_*(G) \cong \oplus_{i\in I}\mathcal{O}_{\mathbb{P}^n_k}u_i$$

hence the $\mathcal{O}_{\mathbb{P}^n_k}$-module $\pi^*\pi_*(G)$ is free on the set $I$. If you require $G$ to have an isomorphism

$$\pi^*\pi_*(G) \cong G$$

it follows $G$ is a free $\mathcal{O}_{\mathbb{P}^n_k}$-module on $I$.

Question: "I'm trying to see that this implies Rif∗G=0 for all i≥1, for which flatness of G over X is needed."

Answer: In Hartshorne, Ex.III.8.4 they compute the higher direct images

$$R^n\pi_*(\mathcal{O}(l))$$

for any projective bundle $\mathbb{P}(E^*)$ and any locally trivial sheaf $E$. In particular they calculate it for the example above with $E:=V$, $l=0$ and $\mathcal{O}_{\mathbb{P}^n_k}$. Note that if $f:X\rightarrow Y$ is a morphism of schemes and if $E,F$ are $\mathcal{O}_X$-modules it follows

$$f_*(E\oplus F)\cong f_*E \oplus f_*F.$$

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