Yuri Berest, George Wilson; Ideal classes of the Weyl algebra and noncommutative projective geometry (with an appendix by Michel Van den Bergh). Int Math Res Notices 2002; 2002 (26): 1347-1396. doi: 10.1155/S1073792802108051
Let R be the set of isomorphism classes of right ideals in the Weyl algebra A = A1(ℂ), and let C be the set of isomorphism classes of triples , where V is a finite-dimensional (complex) vector space, and are endomorphisms of V such that has rank 1. Following a suggestion of L. Le Bruyn, we define a map θR → C by appropriately extending an ideal of A to a sheaf over a quantum projective plane, and then using standard methods of homological algebra. We prove that θ is inverse to a bijection ω:C → R, constructed, by the authors in 2000, by a completely different method. The main step in the proof is to show that θ is equivariant with respect to natural actions of the group G = Aut(A) on R and C: for that we have to study also the extensions of an ideal to certain weighted quantum projective planes. Along the way, we find an elementary description of θ.