MATH 739 — Fall 2000
Topics in Algebra:
Examples in Non-commutative Geometry

The idea of 'algebraicizing geometry' is perhaps one of the oldest in modern mathematics. It traces back to at least Descartes (introduction of Cartesian coordinates and analytic geometry), and culminates in a perfect union of Algebraic geometry and Commutative algebra. In many situations geometric and (commutative) algebraic points of view are not merely related but are complementary and essentially equivalent to each other. The well-known examples of equivalences between geometric and algebraic categories are:

• {affine algebraic varieties over a field k} <—><-> {commutative algebras finitely generated over k};
• {locally compact Hausdorff topological spaces} <—><-> {commutative Banach algebras with involution};
• {quasi-coherent sheaves on a projective variety, X say} <—><-> {graded O(X)-modules modulo torsion}, etc.

The key idea that usually underlies 'non-commutative geometry' is to take geometric (or topological) concepts, rephrase them in algebraic terms (using one of the above equivalences) and then extend in a meaningful way to a wider category of non-commutative algebras. There are at least two reasons for developing such an extension. First, this often puts classical theorems of (commutative) geometry into proper perspective, and on the other hand, brings new tools and geometric intuition into the study of non-commutative rings. Second, in many instances a natural object to study is a highly singular space which cannot be defined in purely geometric and/or topological terms. Whereas this singular space does not exist topologically, there is often a non-commutative algebra associated to it which plays a role of the ring of functions and thus reflects the basic properties of our space.

Though the ideas outlined above are certainly not new there is no yet a systematic account of Non-commutative algebraic geometry. Its basic language, notions and tools are still under development. The purpose of this course is to provide an introduction and motivations to the subject mostly through the problems and examples arising in other fields. The selection of such examples (of course, strongly influenced by instructor's tastes and limitations) will include rings of differential operators on algebraic varieties, low-dimensional enveloping algebras, path algebras of quivers as well as some interesting examples arising in Mathematical physics (e.g., Sklyanin algebras). As a central part of the course, we shall develop Non-commutative Projective Geometry following mostly recent publications by Artin, Zhang, Smith, Stafford and others. A detailed syllabus will be presented at the first lecture.

Some prerequisites would be desirable for this course (for example, familiarity with categorical language) but I shall try to make it accessible for first/second graduate students by giving extensive reviews and, when necessary, indicating precise references to the available literature.

Last modified: April 7, 2003