Moduli schemes of generically simple Azumaya modules

Abstract

Let A be an Azumaya algebra over a smooth projective variety X or more generally, a torsion free coherent sheaf of algebras over X whose generic fiber is a central simple algebra. We show that generically simple torsion free A-module sheaves have a projective coarse moduli scheme; it is smooth and even symplectic if X is an abelian or K3 surface and A is Azumaya. We explain a relation to the classical theory of the Brandt groupoid.

20 pages. v2: generic setup slightly generalized (from rank one modules over a central division algebra to simple modules over a central simple algebra), a few references added