Duality and de Rham cohomology for graded $D$-modules

TL;DR

This paper explores the duality between de Rham cohomology groups of graded D-modules over polynomial rings and their Matlis duals, extending known results and providing new duality formulas.

Contribution

It establishes a duality between de Rham cohomology of graded D-modules and their Matlis duals, extending results from power series to polynomial rings and generalizing to finitely generated modules.

Findings

De Rham cohomology groups are dual to those of the Matlis dual in finite-dimensional cases.

The dimension of top de Rham cohomology equals the maximal number of surjective maps to the top local cohomology module.

Provides an alternative proof of Hartshorne and Polini's result for polynomial rings.

Abstract

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 $\Ext_{\D}^{n-i}(M, \E)$.