On the de Rham homology and cohomology of a complete local ring in equicharacteristic zero

TL;DR

This paper investigates the invariance of de Rham homology and cohomology for complete local rings in characteristic zero, establishing their independence from certain choices and introducing related invariants akin to Lyubeznik numbers.

Contribution

It proves the invariance of Hodge-de Rham spectral sequences for these rings and develops a duality theory for $ ext{D}$-modules that links de Rham cohomology spaces via Matlis duality.

Findings

Hodge-de Rham spectral sequences are independent of surjection choices.

The $E_2$-terms are finite-dimensional $k$-spaces and serve as invariants.

Matlis duality for $ ext{D}$-modules relates de Rham cohomology of dual modules.

Abstract

Let $A$ be a complete local ring with a coefficient field $k$ of characteristic zero, and let $Y$ be its spectrum. The de Rham homology and cohomology of $Y$ have been defined by R. Hartshorne using a choice of surjection $R \rightarrow A$ where $R$ is a complete regular local $k$-algebra: the resulting objects are independent of the chosen surjection. We prove that the Hodge-de Rham spectral sequences abutting to the de Rham homology and cohomology of $Y$, beginning with their $E_2$-terms, are independent of the chosen surjection (up to a degree shift in the homology case) and consist of finite-dimensional $k$-spaces. These $E_2$-terms therefore provide invariants of $A$ analogous to the Lyubeznik numbers. As part of our proofs we develop a theory of Matlis duality in relation to $\mathcal{D}$-modules that is of independent interest. Some of the highlights of this theory are that if $R$ is a complete regular local ring containing $k$ and $\mathcal{D}$ is the ring of $k$-linear differential operators on $R$, then the Matlis dual $D(M)$ of any left $\mathcal{D}$-module $M$ can again be given a structure of left $\mathcal{D}$-module, and if $M$ is a holonomic $\mathcal{D}$-module, then the de Rham cohomology spaces of $D(M)$ are $k$-dual to those of $M$.