Noncommutative Geometry and Cayley-smooth Orders explains the
theory of Cayley-smooth orders in central simple algebras over
function fields of varieties. In particular, the book describes the
etale local structure of such orders as well as their central
singularities and finite dimensional representations. After an
introduction to partial desingularizations of commutative
singularities from noncommutative algebras, the book presents the
invariant theoretic description of orders and their centers. It
proceeds to introduce etale topology and its use in noncommutative
algebra as well as to collect the necessary material on
representations of quivers. The subsequent chapters explain the
etale local structure of a Cayley-smooth order in a semisimple
representation, classify the associated central singularity to
smooth equivalence, describe the nullcone of these marked quiver
representations, and relate them to the study of all isomorphism
classes of n-dimensional representations of a Cayley-smooth order.
The final chapters study Quillen-smooth algebras via their finite
dimensional representations. Noncommutative Geometry and
Cayley-smooth Orders provides a gentle introduction to one of
mathematics' and physics' hottest topics.
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