The Rings of Noncommutative Projective Geometry

In the past 15 years a study of “noncommutative projective geometry” has flourished. By using and generalizing techniques of commutative projective geometry, one can study certain noncommutative graded rings and obtain results for which no purely algebraic proof is known. For instance, noncommutative graded domains of quadratic growth, or “noncommutative curves,” have now been classified by geometric data and these rings must be noetherian. Rings of cubic growth, or “noncommutative surfaces,” are not yet classified, but a rich theory is currently forming. In this survey, we describe some of these results and examine the question of which rings should be included in noncommutative projective geometry.