I - Noncommutative projective geometry

Summary

These notes are a significantly expanded version of the author’s lectures at the graduate workshop “Noncommutative algebraic geometry” held at the Mathematical Sciences Research Institute in June 2012. The main point of entry to the subject we chose was the idea of an Artin–Schelter regular algebra. The introduction of such algebras by Artin and Schelter motivated many of the later developments in the subject. Regular algebras are sufficiently rigid to admit classification in dimension at most 3, yet this classification is nontrivial and uses many interesting techniques. There are also many open questions about regular algebras, including the classification in dimension 4. Intuitively, regular algebras with quadratic relations can be thought of as the coordinate rings of noncommutative projective spaces; thus, they provide examples of the simplest, most fundamental noncommutative projective varieties. In addition, regular algebras provide some down-to-earth examples of Calabi–Yau algebras. This is a class of algebras defined by Ginzburg more recently, which is related to several of the other lecture courses given at the workshop.

Section 1 reviews some important background and introduces noncommutative Gr¨obner bases. We also include as part of Exercise Set 1 a few exercises using the computer algebra system GAP. Section 2 presents some of the main ideas of the theory of Artin–Schelter regular algebras. Then, using regular algebras as examples and motivation, in Sections 3 and 4 we discuss two important aspects of the geometry of noncommutative graded rings: the parameter space of point modules for a graded algebra, and the noncommutative projective scheme associated to a noetherian graded ring. Finally, in Section 5 we discuss some aspects of the classification of noncommutative curves and surfaces, including a review of some more recent results.

We have tried to keep these notes as accessible as possible to readers of varying backgrounds. In particular, Sections 1 and 2 assume only some basic familiarity with noncommutative rings and homological algebra, as found for example in [117] and [193]. Only knowledge of the concept of a projective space is needed to understand the main ideas about point modules in the first half of Section 3. In the final two sections, however, we will of necessity assume that the reader has a more thorough background in algebraic geometry, including the theory of schemes and sheaves as in Hartshorne’s textbook [128].