On the de Rham homology of affine varieties in characteristic 0

Let $\mathbb{K}$ be a field of characteristic 0, let $S$ be a complete local ring with coefficient field $\mathbb{K}$, let $\mathbb{K}[[x_1,\dots ,x_n]]$ be the ring of formal power series in variables $x_1,\dots ,x_n$ with coefficients from $\mathbb{K}$, let $\mathbb{K}[[x_1,\dots ,x_n]]\to S$ be a $\mathbb K$-algebrasurjectionandlet$E_\bullet^{\bullet,\bullet} $ be the associated Hodge-de Rham spectral sequence for the computation of the de Rham homology of $S$. Nicholas Switala proved that this spectral sequence is independent of the surjection beginning with the $E_2$ page, and the groups $E_2^{p,q}$ are all finite-dimensional over $\mathbb{K}$. In this paper we extend this result to affine varieties. Namely, let $Y$ be an affine variety over $\mathbb{K}$, let $X$ be a non-singular affine variety over $\mathbb{K}$, let $Y\subset X$ be an embedding over $\mathbb{K}$ and let $E_{\bullet }^{\bullet ,\bullet }$ 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 $\mathbb{K}$.