Duality and de Rham cohomology for graded [formula omitted]-modules

Abstract

We consider the (graded) Matlis dual D(M) of a graded D-module M over the polynomial ring R=k[x1,…,xn] (k is a field of characteristic zero), and show that it can be given a structure of D-module in such a way that, whenever dimk⁡HdRi(M) is finite, then HdRi(M) is k-dual to HdRn−i(D(M)). As a consequence, we show that if M is a graded D-module such that HdRn(M) is a finite-dimensional k-space, then dimk⁡(HdRn(M)) is the maximal integer s for which there exists a surjective D-linear homomorphism M→Es, where E is the top local cohomology module H(x1,…,xn)n(R). This extends a recent result of Hartshorne and Polini on formal power series rings to the case of polynomial rings; we also apply the same circle of ideas to provide an alternate proof of their result. When M is a finitely generated graded D-module such that dimk⁡HdRi(M) is finite, we generalize the above result further, showing that HdRn−i(M) is k-dual to ExtDi(M,E).