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

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 $M$.