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Abstract

We construct and describe compactified moduli stacks of Azumaya algebras on a smooth projective morphism XS. These stacks are the algebro-geometric version of the (suitably compactified) stacks of principal PGLn-bundles and they also have strong connections to arithmetic. A geometric approach to the problem leads one to study stacks of (semistable) twisted sheaves. We show that these stacks are very similar to the stacks of semistable sheaves. This gives a way of understanding the structure of the stack of principal PGL n-bundles and its coarse moduli space in terms of fairly well-understood spaces. In particular, when XS is a smooth projective curve or surface over an algebraically closed field, our method yields concrete theorems about the structure of these stacks (at least as certain natural invariants are allowed to increase without bound). On the arithmetic side, we use the geometry and rationality properties of these moduli spaces to study a classical question about the Brauer group of a function field K, known as the “period-index problem”: for which classes α in Br(K) of order n does there exist a division algebra D of rank n 2 with [D] = α? We give an answer to this question when K is the function field of a curve or surface over an algebraically closed, finite, or local field and when α is an unramified Brauer class of order prime to the characteristic of K . In the general case, we relate the unramified period-index problem to rationality questions on Galois twists of moduli spaces of semistable sheaves. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)

Details

Title
Moduli of twisted sheaves and generalized Azumaya algebras
Author
Lieblich, Max
Year
2004
Publisher
ProQuest Dissertations & Theses
Source type
Dissertation or Thesis
Language of publication
English
ProQuest document ID
305088000
Copyright
Database copyright ProQuest LLC; ProQuest does not claim copyright in the individual underlying works.