On the de Rham homology of affine varieties in characteristic 0

Abstract

Let $Y$ be an affine variety over a field $k$ of characteristic 0, embedded in a smooth scheme $X$. We show that the Hodge-de Rham spectral sequence abutting to algebraic de Rham homology of $Y$ associated to this embedding is independent of the choice of $X$ and the embedding (up to a bidegree shift, and starting on the $E_2$ page), and that the terms in the spectral sequence are finite-dimensional $k$-spaces (again, starting on the $E_2$ page).