On the de Rham homology of affine varieties in characteristic 0

Abstract

Let K be a field of characteristic 0, let S be a complete local ring with coefficient field K, let K[[x1,…,xn]] be the ring of formal power series in variables x1,…,xn with coefficients from K, let K[[x1,…,xn]]→S be a K-algebra surjection and let E••,• be the associated Hodge-de Rham spectral sequence for the computation of the de Rham homology of S. Nicholas Switala [12] proved that this spectral sequence is independent of the surjection beginning with the E2 page, and the groups E2p,q are all finite-dimensional over K.In this paper we extend this result to affine varieties. Namely, let Y be an affine variety over K, let X be a non-singular affine variety over K, let Y⊂X be an embedding over K and let E••,• be the associated Hodge-de Rham spectral sequence for the computation of the de Rham homology of Y. Then this spectral sequence is independent of the embedding beginning with the E2 page, and the groups E2p,q are all finite-dimensional over K.