Msri Graduate Workshop: Lecture Notes for Course on Noncommutative Projective Geometry

These notes contain the material about noncommutative projective algebraic geometry that the author lectured on at the graduate workshop in June 2012 at MSRI. The notes generally contain everything covered in the lectures but may contain more than we are able to say in the lectures. Still, there are many facts we assume without proof in the lectures and for length reasons we generally do not put in the proofs here either. The main topic of our focus is the theory of Artin-Schelter regular algebras. We chose this for several reasons. First, the study of these by Artin and Schelter was some of the earliest work in the subject of noncommutative projective geometry, and motivated most of the later developments. Regular algebras are sufficiently structured to admit classification in dimensions at most 3, but this classification is non-trivial and uses many interesting techniques. There are still many open questions about them, most prominently whether or not the dimension 4 regular algebras can also be classified. Intuitively, the geometry of a regular algebra (at least one with quadratic relations) should be thought of as a noncommutative projective space. Thus these geometric spaces should play the same fundamental role in a theory of noncommutative projective geometry that projective spaces do in commutative projective geometry. This is another reason to focus on them initially. Also, regular algebras provide some down-to-earth examples of a more general construction defined by Ginzburg called a Calabi-Yau algebra, which is of especially current and wide interest and is connected to some of the other lecture courses at the workshop. Using regular algebras as examples and motivation, we discuss two important aspects of the geometry of noncommutative graded rings: the parameter space of point modules (a special case of the theory of noncommutative Hilbert schemes), and the noncommutative projective scheme associated to a noncommutative graded ring (which is a certain abelian category). Finally, we discuss some aspects of the problem of classification of noncommutative curves and surfaces.