ag.algebraic geometry - Which is the correct generalization of Euler sequence to the projectivization of a vector bundle?

Hi,

let $E$ be a vector bundle over a smooth projective variety $X$ and $pi:mathbb{P}(E)rightarrow X$ its projectivization, $T_{pi}:=ker(pi_{ * })$ where $pi_{*}:T_{mathbb{P}(E)}rightarrow pi^{*}T_X$, let $mathcal{O}_E(-1)hookrightarrowpi^{*}E$ be the "tautological" bundle over $mathbb{P}(E)$. The following it is not a proof but a first reasoning to understand things, if i restrict to a point $xin X$ i have the usual Euler sequence

$0 rightarrow mathcal{O}_{E_x}(-1)rightarrow ({pi^{*}}E)_x rightarrow (T_{pi})_{x}otimes mathcal{O}_{E_x}(-1)rightarrow 0 $

so my thought is that the generalized euler sequence becomes

$0 rightarrow mathcal{O}_{E}(-1)rightarrow pi^{*}Erightarrow T_{pi}otimes mathcal{O}_{E}(-1)rightarrow 0 $

is this the right path or i'm wrong?

thank you in advance.