Modules over regular algebras of dimension 3

Existence of critical modules of GK-dimension 2 over elliptic algebras

“…As in [ATV1], [ATV2], let (E, σ, L) be the triplet associated to the algebra A. We assume throughout that σ has infinite order and no fixed point.…”

“…Interpreted geometrically in the language of non-commutative projective geometry, this means that in a quantum plane there exist "irreducible curve" modules of all possible degrees. It is well-known from [ATV2] that critical A-modules of GKdimension 2 exist in multiplicity 1: up to a shift, these are precisely the line modules. But very little seems to be known about the critical modules of GK-dimension 2 in higher multiplicities; indeed, it is natural to question their existence.…”

Generalizing the notion of rank to noncommutative quadratic forms

“…Remark 1.2. As was shown in [ATV1], if the point modules of S are parametrized by P n−1 , then S is a twist (in the sense of [ATV2,§8]) of the polynomial ring by a graded degreezero automorphism τ ∈ Aut(R) (see Definition 2.1 below). This case occurs if and only if µ ik = µ ij µ jk for all i, j, k. This is the situation throughout Section 2, since there the assumption that n = 2 causes the point modules of S to be parametrized by P 1 .…”

Quantum sections and gauge algebras

“…the point-and line-modules, cf. [ATV2], and the fat point modules introduced in [Art] . If we restrict attention to the geometry of innocent quantum spaces we can use the gauge algebra and its quantum sections to put a "scheme" structure on Proj(~) which reduces the study of (fat) point-modules to that of finite dimensional representations of algebas, cf.…”