On additive higher Chow groups of affine schemes

Abstract

We show that the multivariate additive higher Chow groups of a smooth affine $k$-scheme $\mathrm{Spec}(R)$ essentially of finite type over a perfect field $k$ of characteristic $\ne 2$ form a differential graded module over the big de Rham–Witt complex ${\mathbb{W}}_m\Omega$^bullet_R. In the univariate case, we show that additive higher Chow groups of $\mathrm{Spec}(R)$ form a Witt-complex over $R$. We use these structures to prove an étale descent for multivariate additive higher Chow groups.