Picard group action on the category of twisted sheaves

In this paper, we study the category of twisted sheaves over a scheme X . Let \mathcal{M} be a quasi-coherent sheaf on X and \alpha in \mathrm{Br}(X) . We show that the functor -\otimes_{\mathcal{O}_X}\mathcal{M}: \operatorname{QCoh}(X, \alpha) \to \operatorname{QCoh}(X, \alpha) is naturally isomorphic to the identity functor if and only if \mathcal{M}\cong \mathcal{O}_{X} . As a corollary, the action of \operatorname{Pic}(X) on D^{b}(X, \alpha) is faithful for any Noetherian scheme X . We also show that taking Brauer twists of varieties does not yield new Calabi–Yau categories.