Student AG Seminar: Picard group action on the category of twisted sheaves

In this talk, we study the category of twisted sheaves over a scheme \$X\$. Let \$M\$ be a quasi-coherent sheaf on \$X\$, and \$\alpha\$ in \$\mathrm{Br}(X)\$. We show that the functor \$- \otimes_{\mathcal{O}_X} M:\mathrm{QCoh}(X,\alpha) \to \mathrm{QCoh}(X,\alpha)\$ is naturally isomorphic to the identity functor if and only if \$M \cong \mathcal{O}_X\$. As a corollary, the action of \$\mathrm{Pic}(X)\$ on \$\mathrm{D}^b(X,\alpha)\$ is faithful for any Noetherian scheme \$X\$.