Math 513C: Noncommutative and Noetherian Rings
Paul Smith
Spring 1999, MWF 3:30-4:20
Course Organization
One of the fundamental questions motivating the study of
noncommutative algebras is the problem of classifying/understanding the
solutions to equations in matrices. For example, finding matrices
x,y,z satisfying xy-yx=z, yz-zy=x,
zx-xz=y is equivalent to the problem of understanding the finite
dimensional representations of the Lie algebra so(3). Associated
to such a system of equations is a noncommutative ring (for this example
the ring is the enveloping algebra of so(3)). The structure of
this ring yields information about the solutions to the system. If one
only wants solutions in 1x1 matrices, or equivalently, if one only wants
solutions in the base field over which one is working, then one can take
the ring to be commutative, and then the solutions are in bijection with
the points on the algebraic variety corresponding to the ring. This is
the starting point of algebraic geometry.
In the past decade there has been an effort to obtain a similar
geometric point of view when one seeks solutions in
n x n matrices (for all n). This
two-quarter course (beginning Spring 1999, continuing in Fall 1999) will
examine some of these developments which go under the general heading of
``noncommutative algebraic geometry.'' I will focus particularly on
noncommutative (projective) surfaces. The classification of
commutative algebraic surfaces is one of the high points of
classical algebraic geometry (the work of Castelnuovo, Enriques, Severi,
et alia, between 1895 and 1910). We will review the relevant parts of
that and examine noncommutative surfaces from a similar viewpoint.
A large part of the course will be devoted to the
theory of noncommutative Noetherian rings; these algebraic preliminaries
provide the infrastructure needed for the geometric part of the course.
Among the basic examples we will study are the enveloping algebras of
finite dimensional Lie algebras, and rings of differential operators on
curves.
Course Organization
I will construct a Web site for the course, and a chat room. All
the course materials, lectures, reading assignments, homework etc., will
be posted on the Web. I will advertise the course at other universities
where there are people studying noncommutative rings and noncommutative
algebraic geometry. I hope that the course will be followed by a variety
of people from various places. I hope that the chat room will provide a
forum for students here and elsewhere to discuss the course, ask
questions, get their questions answered et cetera. The URL is
http://www.math.washington.edu/~smith/Teaching/513nag/nag99.html.