# Which sheaves on a projective bundle are flat over the base scheme?

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.

**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

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