This question is mainly a reference request about the order of a Brauer class on a smooth projective variety over $\mathbb{C}$. Namely, let $X$ be a smooth complex projective variety and $\alpha$ be a Brauer class on $X$.

I have read in a few papers on the ArXiv that the order of $\alpha$ divides the rank of any $\alpha$-twisted sheaf. In particular, the existence of a $\alpha$-twisted line bundle would imply that $\alpha$ is trivial.

On the other hand, let $ p : Y \longrightarrow X$ be a non-trivial Brauer-Severi variety associated to the class $\alpha \in Br(X)$. It seems that one can define a $p^*\alpha$-twisted line bundle $\mathcal{O}_{Y/X}(1)$ and it seems this twisted line bundle is not a line bundle (so that the class $p^* \alpha$ is non trivial).

I am not sure how to reconcile these two claims. Is there something obvious I am missing?

Thanks a lot!