On the de Rham homology of affine varieties in characteristic 0

Let K be a field of characteristic 0, let S be a complete local ring with coefficient field K, let K[[ x 1 , ... , x n ]] be the ring of formal power series in variables x 1 , ... , xn n with coefficients from K, let K[[ x 1 , ... , x n ]] -> S be a K-algebra surjection and let E center dot , center dot center dot 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 E 2 page, and the groups E p,q 2 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 subset of X be an embedding over K and let E center dot , center dot center dot 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 E 2 page, and the groups E 2 p,q are all finite- dimensional over K. (c) 2023 Published by Elsevier Inc.