On completion of graded D-modules

TL;DR

This paper proves that for finitely generated graded D-modules over polynomial rings, the de Rham cohomology remains unchanged when extended to formal power series rings, confirming a conjecture for a broad class of modules.

Contribution

It establishes that the natural maps on de Rham cohomology are isomorphisms for finitely generated graded D-modules, answering a question posed by Hartshorne and Polini.

Findings

De Rham cohomology is preserved under extension to formal power series rings.

The result applies to all finitely generated graded D-modules with finite-dimensional cohomology.

Confirms a conjecture for a broad class of holonomic D-modules.

Abstract

Let $R = k[x_1, \ldots, x_n]$ be a polynomial ring over a field $k$ of characteristic zero and $\cR$ be the formal power series ring $k[[x_1, \ldots, x_n]]$. If $M$ is a $\D$-module over $R$, then $\cR \otimes_R M$ is naturally a $\D$-module over $\cR$. Hartshorne and Polini asked whether the natural maps $H^i_{\dR}(M)\to H^i_{\dR}(\cR \otimes_R M)$ (induced by $M\to \cR \otimes_R M$) are isomorphisms whenever $M$ is graded and holonomic. We give a positive answer to their question, as a corollary of the following stronger result. Let $M$ be a finitely generated graded $\D$-module: for each integer $i$ such that $\dim_kH^i_{\dR}(M)<\infty$, the natural map $H^i_{\dR}(M)\to H^i_{\dR}(\cR \otimes_R M)$ (induced by $M\to \cR \otimes_R M$) is an isomorphism.