Picard group action on the category of twisted sheaves

Abstract

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 $-{\otimes }_{{\mathcal{O}}_X}\mathcal{M}:\mathrm{QCoh}(X,\alpha )\to \mathrm{QCoh}(X,\alpha )$ is naturally isomorphic to the identity functor if and only if $\mathcal{M}\cong {\mathcal{O}}_X$. As a corollary, the action of $\mathrm{Pic}(X)$ 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.