Duality and de Rham cohomology for graded $D$-modules
We consider the (graded) Matlis dual $\DD(M)$ of a graded $\D$-module $M$ over the polynomial ring $R = k[x_1, \ldots, x_n]$ ($k$ is a field of characteristic zero), and show that it can be given a structure of $\D$-module in such a way that, whenever $\dim_kH^i_{dR}(M)$ is finite, then $H^i_{dR}(M)$ is $k$-dual to $H^{n-i}_{dR}(\DD(M))$. As a consequence, we show that if $M$ is a graded $\D$-module such that $H^n_{dR}(M)$ is a finite-dimensional $k$-space, then $\dim_k(H^n_{dR}(M))$ is the maximal integer $s$ for which there exists a surjective $\D$-linear homomorphism $M \rightarrow E^s$, where $E$ is the top local cohomology module $H^n_{(x_1, \ldots, x_n)}(R)$. 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 $\dim_kH^i_{dR}(M)$ is finite for some $i$, we generalize the above result further, showing that $H^{i}_{dR}(M)$ is $k$-dual to $\Ext_{\D}^{n-i}(M, \E)$.