Let the group G of m-th roots of unity act on the complex line by multiplication, inducing an action on the algebra, Diff, of polynomial differential operators on the line. Following Crawley-Boevey and Holland, we introduce a multiparameter deformation, D_c, of the smash-product (Diff # G). Our main result provides natural bijections between (roughly speaking) the following spaces:
(1) G-equivariant version of Wilson's adelic Grassmannian of rank r;
(2) Rank r projective D_c-modules (equipped with generic trivialization);
(3) Rank r torsion-free sheaves on a `noncommutative quadric';
(4) Disjoint union of Nakajima quiver varieties for the cyclic quiver with m vertices. The bijection between (1) and (2) is provided by a version of Riemann-Hilbert correspondence between D-modules and sheaves. The bijections between (2), (3) and (4) were motivated by our previous work math.AG/0103068. The resulting bijection between (1) and (4) reduces, in the very special case: r=1 and G=1, to the partition of (rank 1) adelic Grassmannian into a union of Calogero-Moser spaces, discovered by Wilson. This gives, in particular, a natural and purely algebraic approach to Wilson's result.
Noncommutative Grassmannian of codimension two has coherent coordinate ring
Efimov，Luntz，and Orlov在“Deformation theory of objects in homotopy and导出范畴III：阿贝尔范畴”中引入了一个非对易格拉斯曼NGr（m，n），作为一个与P^n上n-m+1个凝聚层的例外集合相关联的非对易代数。它是一个n-m +1维的分次Calabi-Yau Z-代数。我们证明了这个代数是相干的，只要格拉斯曼的余维数d = n-m是2。根据前引书，这给出了非对易格拉斯曼凝聚层的导出范畴上的t-结构。该证明与最近对某些分次3维Calabi-Yau代数的相干性的证明有很大不同，并且基于该代数的PBW-基的性质。
