Let $X = \mathbb{P}(V)$, where $V$ is a vector bundle on $S$, a noetherian scheme. $\pi : X \longrightarrow S$. Let $T$ be an $S$-scheme.
I am wondering what the projective bundle of the pullback of $V$ to $T$ is; in particular, whether $\mathbb{P}(V\otimes O_T) = X\times T$?
Thanks!