Graded duality for filtered D-modules

TL;DR

This paper establishes a duality relationship for filtered D-modules, linking graded pieces to local cohomology complexes, and provides methods to compute these sheaves via higher direct images, extending Grothendieck's local duality.

Contribution

It proves a graded duality theorem for filtered D-modules and introduces a new approach to compute local cohomology sheaves using higher direct images.

Findings

Dual of graded pieces is isomorphic to graded pieces of local cohomology complex.

Local cohomology sheaves can be computed via higher direct images.

Results extend classical local duality to filtered D-modules.

Abstract

For a coherent filtered D-module we show that the dual of each graded piece over the structure sheaf is isomorphic to a certain graded piece of the ring-theoretic local cohomology complex of the graded quotient of the dual of the filtered D-module along the zero-section of the cotangent bundle. This follows from a similar assertion for coherent graded modules over a polynomial algebra over the structure sheaf. We also prove that the local cohomology sheaves can be calculated by using the higher direct images of the twists of the associated sheaf complex on the projective cotangent bundle. These are closely related to local duality, essentially due to Grothendieck.