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 $\operatorname{Br}(X)$. 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 $X$.