Ideal classes of three-dimensional Sklyanin algebras

Abstract

In this paper we classify graded reflexive ideals, up to isomorphism and shift, in certain threedimensional Artin-Schelter regular algebras. This classification is similar to the classification of right ideals in the first Weyl algebra, a problem that was completely settled recently. The situation we consider is substantially more complicated however.  2004 Elsevier Inc. All rights reserved.

Sklyanin algebras and Hilbert schemes of points

“…When S = Skl(E, σ ) for an automorphism σ of infinite order, de Naeghel and Van den Bergh [19] have independently used quiver varieties to obtain a geometric description of the set of isomorphism classes of line bundles in qgr-S. One advantage of their approach is that it shows that, in stark contrast to the commutative case, (P S \ E) [n] is affine.…”

Ideal classes of three dimensional Artin–Schelter regular algebras

“…The Hilbert scheme Hilb n (P 2 q ) was constructed in [27] (see also [17] for a somewhat less general result and [10] in the case where A is the homogenization of the first Weyl algebra). The definition of Hilb n (P 2 q ) is not entirely straightforward since in general P 2 q will have very few zero dimensional non-commutative subschemes (see [31]), so a different approach is needed.…”

“…By applying to it the right exact functor Ext 3 In case M ∈ grmod(A) is torsion free of rank one and normalized it turns out that the invariant of M is non-negative [17,27].…”

“…The main result of [27] is that Hilb n (P 2 q ) is represented by a smooth scheme Hilb n (P 2 q ) of dimension 2n (see also [17] for a special case, treated with a different method which yields some extra information).…”

On incidence between strata of the Hilbert scheme of points on $$\mathbb{P}^{2}$$

“…The authors became interested in the incidence problem while they were studying the deformations of the Hilbert schemes of P 2 which come from non-commutative geometry, see [16,6,7].…”