On the de Rham homology of affine varieties in characteristic 0

TL;DR

This paper extends Switala's results on the independence and finiteness of the Hodge-de Rham spectral sequence from complete local rings to affine varieties over characteristic zero fields.

Contribution

It generalizes the known properties of the spectral sequence to affine varieties, showing independence from embeddings and finite-dimensionality of groups.

Findings

Spectral sequence is independent of the embedding from the E2 page onward.

E2 groups are finite-dimensional over the base field.

Results hold for affine varieties over characteristic zero fields.

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 $\mathbb K$.