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 CH₁(A_k(m), n)_ssup to the module of absolute Kähler differentials of k with twisted k*-action Ω_k^n-2 <ω> of weight ω. 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 ω = 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.