On the de Rham homology of affine varieties in characteristic 0

Abstract

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 $\mathbb K$-algebra surjection and let $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 $E^{p,q}_2$ 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 $E^{p,q}_2$ are all finite-dimensional over $\mathbb K$.