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

Abstract

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

Proofs in Section 5 are replaced with more conceptual ones and the two D-module structures on Matlis dual are reconciled. Comments welcome

References in corpus (4)

• Local Cohomology and Matlis duality
• On the de Rham homology and cohomology of a complete local ring in equicharacteristic zero
• Simple D-module components of local cohomology modules
• On Switala's Matlis Duality

Cited by in corpus (2)

• On derived functors of graded local cohomology modules-II
• On derived functors of Graded local cohomology modules