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 \(\mathrm {Aut}(X)\). 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.
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Notes
The results of [20] hold in the much broader setting of sheaves of Azumaya algebras \(\mathscr {A}\) on quasi-compact quasi-separated schemes X with a finite number of connected components. The characteristic restriction is as follows: For a K-linear additive invariant E, with K a commutative ring, the product r of the ranks of \(\mathscr {A}\) on the components of X must be a unit in K.
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Acknowledgments
Thanks to Amnon Yekutieli for pointing out a number of helpful references. Thanks also to Ben Antieau for a number of thoughtful comments and to James Zhang, whose inquiries are responsible for the materials of Sect. 6. Thanks to the referee for many helpful comments.
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This work was supported by NSF Postdoctoral Research Fellowship DMS-1503147.
