On completion of graded $D$-modules

Let $R = k[x_1, \dotsc , x_n]$ be a polynomial ring over a field $k$ of characteristic zero and $\widehat{R}$ be the formal power series ring $k[[x_1, \dotsc , x_n]]$. If $M$ is a $\mathcal{D}$-module over $R$, then $\widehat{R} \otimes_R M$ is naturally a $\mathcal{D}$-module over $\widehat{R}$. Hartshorne and Polini asked whether the natural maps $H^i_{\mathrm{dR}} (M) \to H^i_{\mathrm{dR}} (\widehat{R} \otimes_R M)$ (induced by $M \to \widehat{R} \otimes_R M$) 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 $\mathcal{D}$-module: for each integer $i$ such that $\dim_k H^i_{\mathrm{dR}} (M) \lt \infty$, the natural map $H^i_{\mathrm{dR}} (M) \to H^i_{\mathrm{dR}} (\widehat{R} \otimes_R M)$ (induced by $M \to \widehat{R} \otimes_R M$) is an isomorphism.

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The first author gratefully acknowledges NSF support through grant DMS-1604503. The second author is partially supported by the NSF through DMS-1606414 and CAREER grant DMS-1752081.

Received 13 September 2018

Accepted 14 January 2019

Published 20 August 2020