Ideal Classes of the Weyl Algebra and Noncommutative Projective Geometry (with an Appendix by M. Van den Bergh) TeX Export

@misc{citeulike:486109, abstract = {Let R be the set of isomorphism classes of ideals in the Weyl algebra \$A=A\_{1}\$, and let C be the set of isomorphism classes of triples (V; X, Y), where V is a finite-dimensional (complex) vector space, and X, Y are endomorphisms of V such that [X,Y]+I has rank 1. Following a suggestion of L. Le Bruyn, we define a map \$\theta: R \to 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 \$\theta\$ is inverse to a bijection \$\omega: C \to R\$ constructed in \cite{BW} by a completely different method. The main step in the proof is to show that \$\theta\$ 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 \theta.}, archivePrefix = {arXiv}, author = {Berest, Yuri and Wilson, George}, citeulike-article-id = {486109}, citeulike-linkout-0 = {http://arxiv.org/abs/math.AG/0104248}, citeulike-linkout-1 = {http://arxiv.org/pdf/math.AG/0104248}, eprint = {math.AG/0104248}, keywords = {moduli, noncommutative, weylalgebra}, month = {Aug}, posted-at = {2006-01-30 19:58:32}, priority = {0}, title = {Ideal Classes of the Weyl Algebra and Noncommutative Projective Geometry (with an Appendix by M. Van den Bergh)}, url = {http://arxiv.org/abs/math.AG/0104248}, year = {2001} }

TY - ELEC ID - citeulike:486109 L3 - citeulike-article-id:486109 N2 - Let R be the set of isomorphism classes of ideals in the Weyl algebra $A=A_{1}$, and let C be the set of isomorphism classes of triples (V; X, Y), where V is a finite-dimensional (complex) vector space, and X, Y are endomorphisms of V such that [X,Y]+I has rank 1. Following a suggestion of L. Le Bruyn, we define a map $\theta: R \to C$ 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 $\theta$ is inverse to a bijection $\omega: C \to R$ constructed in \cite{BW} by a completely different method. The main step in the proof is to show that $\theta$ 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 \theta. TI - Ideal Classes of the Weyl Algebra and Noncommutative Projective Geometry (with an Appendix by M. Van den Bergh) KW - moduli KW - noncommutative KW - weylalgebra AU - Berest, Yuri AU - Wilson, George PY - 2001/Aug/08/ UR - http://arxiv.org/abs/math.AG/0104248 ER -