The derived Picard group of an affine Azumaya algebra

TL;DR

This paper characterizes the derived Picard group of affine Azumaya algebras, revealing its dependence on the scheme's Picard group, automorphisms, and Brauer class, and refines existing descriptions for special cases.

Contribution

It provides a new description of the derived Picard group of affine Azumaya algebras, extending previous work and offering an alternative proof of a known relation to Brauer classes.

Findings

Derived Picard group depends on global sections, Picard group, and automorphisms.

Refines Yekutieli's description for trivial Azumaya algebras.

Provides an alternative proof relating derived and Brauer equivalences.

Abstract

We describe the derived Picard group of an Azumaya algebra A on an affine scheme X in terms of global sections of the constant sheaf of integers on X, the Picard group of X, and the stabilizer of the Brauer class of A under the action of Aut(X). In particular, we find that the derived Picard group of an Azumaya algebra is generally not isomorphic to that of the underlying scheme. In the case of the trivial Azumaya algebra, our result refines Yekutieli's description of the derived Picard group of a commutative algebra. We also get, as a corollary, an alternate proof of a result of Antieau which relates derived equivalences to Brauer equivalences for affine Azumaya algebras. The example of a Weyl algebra in finite characteristic is examined in some detail.