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

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 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 E₂ -terms, are independent of the chosen surjection (up to a degree shift in the homology case) and consist of finite-dimensional k -spaces. These E₂ -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 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 D=D (R, k) is the ring of k -linear differential operators on R, then the Matlis dual D (M) of any left D -module M can again be given a structure of left <jats: inline