On additive higher Chow groups of affine schemes

TL;DR

This paper establishes that multivariate additive higher Chow groups of smooth affine schemes over a perfect field form a differential graded module over the de Rham-Witt complex, and proves étale descent for these groups.

Contribution

It introduces a new structure of differential graded modules over the de Rham-Witt complex for additive higher Chow groups and proves their étale descent.

Findings

Additive higher Chow groups form a differential graded module over the de Rham-Witt complex.

In the univariate case, they form a Witt-complex over the scheme.

Étale descent holds for multivariate additive higher Chow groups.

Abstract

We show that the multivariate additive higher Chow groups of a smooth affine $k$-scheme $\Spec (R)$ essentially of finite type over a perfect field $k$ of characteristic $\not = 2$ form a differential graded module over the big de Rham-Witt complex $\W_m\Omega^{\bullet}_{R}$. In the univariate case, we show that additive higher Chow groups of $\Spec (R)$ form a Witt-complex over $R$. We use these structures to prove an \'etale descent for multivariate additive higher Chow groups.