Moduli of twisted sheaves and Azumaya algebras
-- Max Lieblich, January 23, 2004

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We construct and describe moduli spaces of Azumaya algebras on
 a smooth projective surface.  These spaces are the
 algebro-geometric version of the spaces of principal
 PGL(n)-bundles and they also have strong connections to
 arithmetic.  A geometric approach to the problem leads one to
 study moduli spaces of twisted sheaves.  We show that these
 spaces are very similar to the moduli spaces of semi-stable
 sheaves.  On the arithmetic side, we use the geometry of these
 moduli spaces to answer a classical question about the Brauer
 group of a function field K in two variables over a finite
field, known as the "period-index problem": for which
 classes alpha in Br(K) of order n does there
 exist a division algebra D of rank n^2 with |D| = alpha?