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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.
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.
This proceedings is composed of the papers resulting from the NATO
work-shop "Perspectives in Ring Theory" and the work-shop "Geometry
and Invariant The ory of Representations of Quivers" . Three
reports on problem sessions have been induced in the part
corresponding to the work-shop where they belonged. One more report
on a problem session, the "lost" problem session, will be published
elsewhere eventually. vii Acknowledgement The meeting became
possible by the financial support of the Scientific Affairs
Division of NATO. The people at this division have been very
helpful in the orga nization of the meeting, in particular we
commemorate Dr. Mario di Lullo, who died unexpectedly last year,
but who has been very helpful with the organization of earlier
meetings in Ring Theory. For additional financial support we thank
the national foundation for scientific research (NFWO), the rector
of the University of Antwerp, UIA, and the Belgian Ministry of
Education. We also gladly acknowledge support from the Belgian
Friends of the Hebrew University and the chairman Prof. P. Van
Remoortere who honored Prof. S. Amitsur for his continuous
contributions to the mathematical activities at the University of
Antwerp. I thank the authors who contributed their paper(s) to this
proceedings and the lecturers for their undisposable contributions
towards the success of the work-shop. Finally I thank Danielle for
allowing me to spoil another holiday period in favor of a
congress."
This proceedings is composed of the papers resulting from the NATO
work-shop "Perspectives in Ring Theory" and the work-shop "Geometry
and Invariant The ory of Representations of Quivers" . Three
reports on problem sessions have been induced in the part
corresponding to the work-shop where they belonged. One more report
on a problem session, the "lost" problem session, will be published
elsewhere eventually. vii Acknowledgement The meeting became
possible by the financial support of the Scientific Affairs
Division of NATO. The people at this division have been very
helpful in the orga nization of the meeting, in particular we
commemorate Dr. Mario di Lullo, who died unexpectedly last year,
but who has been very helpful with the organization of earlier
meetings in Ring Theory. For additional financial support we thank
the national foundation for scientific research (NFWO), the rector
of the University of Antwerp, UIA, and the Belgian Ministry of
Education. We also gladly acknowledge support from the Belgian
Friends of the Hebrew University and the chairman Prof. P. Van
Remoortere who honored Prof. S. Amitsur for his continuous
contributions to the mathematical activities at the University of
Antwerp. I thank the authors who contributed their paper(s) to this
proceedings and the lecturers for their undisposable contributions
towards the success of the work-shop. Finally I thank Danielle for
allowing me to spoil another holiday period in favor of a
congress."
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