Picard group action on the category of twisted sheaves

TL;DR

This paper investigates the action of the Picard group on the derived category of twisted sheaves, establishing conditions for functorial identities and demonstrating faithfulness of the Picard action on these categories.

Contribution

It proves that tensoring with a quasi-coherent sheaf is identity only when the sheaf is trivial, and shows the Picard group's faithful action on the derived category of twisted sheaves.

Findings

Tensoring with a sheaf is identity iff the sheaf is trivial.

The Picard group acts faithfully on the derived category of twisted sheaves.

The results hold for any Noetherian scheme.

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