I want to prove that "Given a bundle $E$, for any line bundle $L$ the projectivizations of $E$ and $E$ tensor $L$ are isomorphic i.e $P(E)≅P(E⊗L)$".


The statement can also be seen on the Wikipedia page of the projective bundle. The reference they give is for Hartshorne, Algebraic Geometry. But unfortunately, I am not aware of that.


By bundle, you can as well assume it to be a smooth vector bundle and their projectivization to be a fibre bundle (in the topological sense) with fibre 1-dim subspaces of the vector space fibres.


What I tried:-
At the fibre level, we can define it $[v]→[u⊗v]$ for an arbitrary non-zero element $u$ in the fibre of $L$ and $v$ in the fibre of $E$. This is clearly a well-defined map. Now, how can I get a continuous map at the total space level, at least locally we can do it as we have an explicit description of the projective bundles locally, will this suffice?


Any suggestions will be of great help.


Thanks and regards in advance



Projective bundle is stable under twisting by a line bundle
$\DeclareMathOperator{\Hom}{Hom}$There's a nice explicit map $P\Hom(L,E)\to P(E)$ that takes a homomorphism to its image. Then use that $E\otimes L \cong \Hom(L^\vee,E)$ where $L^\vee$ is the dual.


But if you work through what this does, it's simply the map $P(E\otimes L)\to P(E)$ that takes $[u\otimes v] \mapsto [u]$ for nonzero $u$ and $v$.


