We characterize the finite codimension sub-${\mathbf {k}}$-algebras of ${\mathbf {k}}[\![t]\!]$ as the solutions of a computable finite family of higher differential operators. For this end, we establish a duality between such a sub-algebras and the finite codimension ${\mathbf {k}}$-vector spaces of ${\mathbf {k}}[u]$, this ring acts on ${\mathbf {k}}[\![t]\!]$ by differentiation.

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 ${\mathcal{D}}={\mathcal{D}}(R,k)$ 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 $M$.