Modules Over Elliptic Algebras and Quantum Planes

This is a study of modules over elliptic algebras, especially modules of Gelfand‐Kirillov dimension 2. An elliptic algebra A is associated with a certain automorphism of a one‐dimensional scheme E , generally an elliptic curve, and every elliptic algebra defines a ‘non‐commutative projective plane’ proj‐ A , sometimes called a quantum plane. Therefore, the study of modules translates into an interplay between the geometries of E and of quantum planes. The relation to geometry is studied by looking at ‘the points of a given module M ’ and corresponding ‘incidence relations’ (a point p of E is said to be a point of M if there is a non‐zero map from M to the corresponding point module N p ).