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 Np).

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We describe the residue complex for three-dimensional Sklyanin algebras, which are the interesting special cases of quantum polynomial rings in three variables. In particular, we show that the multiplicities of the point modules in the residue complex are all one, just as in the classical case of commutative polynomial rings in three variables. We explain why the residue complex in the quantum case has the same multiplicities of point modules as that in the commutative case (even if there are fewer point modules in the quantum case) by pointing out two quantum anomalies. Academic PressContents.

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Abstract. We show that over an elliptic algebra, critical modules of GelfandKirillov dimension 2 exist in all multiplicities (assuming the ground field is uncountable, algebraically closed). Geometrically, this shows that in a quantum plane there exist "irreducible curve" modules of all possible degrees.