pullback is injective on picard groups? Let $E \rightarrow X$ be a rank two holomorphic vector bundle over a complex manifold $X$. I was recently asked on exam to prove an assertion that I believe boils down to showing that the pullback map $Pic(X) \rightarrow Pic(E)$ is injective, i.e. the pullback of a nontrivial line bundle on $X$ is a nontrivial line bundle on $E$.  
This was a bit surprising to me, and I am curious how to prove it. If there were a holomorphic section $X \rightarrow E$, then we would be done, but I don't believe there should be one in general? 
EDIT: For reference, the full question was to show that $Pic(X) \times \mathbb{Z} \rightarrow Pic(E)$ is injective, where $(0,n) \rightarrow L^{\otimes n}$ and $L$ Is the tautological bundle on $E$. Obviously my claim is a special case, but I think the full statement is easily reduced to it after considering the restriction to a fiber. 
 A: Let $f\colon E\to X$ be any schematically dominant morphism of schemes, e.g., a surjective morphism with $X$ reduced. Let $U_{E/X}$ be the cokernel of the canonical morphism $f^{\flat}\colon\mathbb G_{m,X}\to f_{*}\mathbb G_{m,E}$ of etale sheaves on $X$ (this morphism is injective by the schematically dominant hypothesis. The sheaf just defined is, or should be, called the sheaf of relative units of $E$ over $X$). Then there exists a canonical exact sequence of abelian groups
$$
0\to\varGamma(X, \mathbb G_{m,X})\to\varGamma(E, \mathbb G_{m,E})\to
U_{E/X}(X)\to\textrm{Pic}(X)\overset{f^{*}}{\to}\textrm{Pic}(E).
$$
So, you must prove that in your particular case the map $\varGamma(E, \mathbb G_{m,E})\to U_{E/X}(X)$ is surjective. Well, is it?
Regarding the proof: exactness of the above exact sequence boils down to checking that the kernel of $f^{*}\colon \textrm{Pic}(X)\to\textrm{Pic}(E)$ is the same as the kernel of $H^{1}(f^{\flat})$. This follows from the fact that $f^{*}$ factors as
$$
H^{1}(X,\mathbb G_{m,X})\to H^{1}(X,f_{*}\mathbb G_{m,E})\hookrightarrow H^{1}(E,\mathbb G_{m,E}),
$$
where the first map is $H^{1}(f^{\flat})$ and the second map is the first edge morphism in the Cartan-Leray spectral sequence $H^{ r}(X, R^{ s} f_{*}\mathbb G_{m,E})\Rightarrow H^{ r+s}(E, \mathbb G_{m,E})$. 
Observe now the following helpful fact: If $f$ has a section $\sigma\colon X\to E$, then $\mathbb G_{m,E}\to \sigma_{*}\mathbb G_{m,X}$ induces a morphism $f_{*}\mathbb G_{m,E}\to f_{*}\sigma_{*}\mathbb G_{m,X}=\mathbb G_{m,X}$ which splits the sequence
$$
0\to \mathbb G_{m,X}\to f_{*}\mathbb G_{m,E}\to U_{E/X}\to 0.
$$
In this case the surjectivity of $\varGamma(E, \mathbb G_{m,E})\to U_{E/X}(X)$ is evident. Does your map have a section?
