In the past decade a similar geometric point of view has been developed to understand the solutions in n-by-n matrices (for all n). This two quarter course will examine some of these developments which go under the general heading of ``non-commutative algebraic geometry''. The course will focus particularly on non-commutative (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 1895-1910). We will review the relevant parts of that and examine non-commutative surfaces from a similar viewpoint.
We will begin with some standard examples of non-commutative rings: n-by-n matrices; the Weyl algebras (a typical ring of differential operators); the enveloping algebras of the 2-dimensional non-abelian Lie algebra, the 3-dimensional Heisenberg Lie algebra, and sl(2); some group algebras; the 4-dimensional Kronecker algebra; some graded algebras arising in quantum groups. We will then introduce some of the language of category theory since our basic geometric object will be a Grothendieck category; an important example is the module category over a ring. The examples of non-commutative rings given earlier will then be re-visited and examined in a more geometric fashion using the language just introduced. They provide examples of non-commutative affine planes and quadric surfaces. We will study the points and lines on them, and maps between them.
We will then cover some basic results about non-commutative noetherian rings: Goldie's theorem, Gelfand-Kirillov dimension, and a little homological algebra. Graded rings will be introduced in order to provide us with non-commutative projective varieties. We will review the results of Artin-Tate-van den Bergh that provide us with the quantum projective planes. We will examine how the affine examples studied earlier embed in these, and other, quantum projective spaces. The cohomology of non-commutative projective schemes will be developed and studied. We will use a little K-theory in order to prove a Bezout's Theorem for the quantum projective planes, and to develop an intersection theory for non-commutative surfaces.
You can find out a little more about this subject by reading the short introduction on my research webpage.