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!



Twisted line bundles Brauer class
OK so I think I understand what I missed from the beginning. If I take a particular $\alpha$-twisted sheaf, say $F$, then the object $F$ itself, as a twisted sheaf, **depends on** the Cech cocycle I use to represent $\alpha$. In particular, the Cech cocycle used to define $\mathcal{O}\_{Y/X}(1)$ on a Severi Brauer variety is non-trivial.


On the other hand, the **category** of $\alpha$-twisted sheaves only depends (**up to equivalence**) on the cohomology class $\alpha \in H^2(X,\mathcal{O}^{\star}\_X)\_{et}$. 


Hence, though $\mathcal{O}\_{Y/X}(1)$ is not a line bundle, one still has that the category of $p^\* \alpha$-twisted sheaves on $Y$ is equivalent to the category of untwisted sheaves on $Y$!


