The Beilinson Equivalence for Differential Operators and Lie Algebroids

Details The Beilinson Equivalence for Differential Operators and Lie Algebroids The Beilinson Equivalence for Differential Operators and Lie Algebroids

2009-08-25

arxiv.org/abs/0908.3662v1

J. Pure Appl. Algebra 12 (2010) 2124--2143

Mathematics

Quantum Algebra

Scientific paper

10.1016/j.jpaa.2010.02.016

Let D be the ring of differential operators on a smooth irreducible affine variety X over the complex numbers; or, more generally, the enveloping algebra of any locally free Lie algebroid on X. The category of finitely-generated graded modules of the Rees algebra D has a natural quotient category qgr(D) which imitates the category of modules on Proj of a graded commutative ring. We show that the derived category D^b(qgr(D)) is equivalent to the derived category of finitely-generated modules of a sheaf of algebras E on X which is coherent over X. This generalizes the usual Beilinson equivalence for projective space, and also the Beilinson equivalence for differential operators on a smooth curve used by Ben-Zvi and Nevins to describe the moduli space of left ideals in D.

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