A regulator map for 1-cycles with modulus

Abstract

Let $k$ be a field of characteristic 0. We define a map from the additive higher Chow group of 1-cycles with strong sup $m$-modulus $CH_1(A_k(m), n)_{ssup}$ to the module of absolute Kähler differentials of $k$ with twisted $k^*$-action $\Omega^{n-2}_k\langle \omega \rangle$ of weight $\omega$. We will call the map a regulator map, and we show that the regulator map is surjective if $k$ is an algebraically closed field. In case $\omega = m+1$, this map specializes to Park's regulator map. We study a relationship between the cyclic homology and the additive higher Chow group with strong sup modulus by using our regulator map.