A regulator map for 1-cycles with modulus (Q725647)

In [Am. J. Math. 131, No. 1, 257--276 (2009; Zbl 1176.14001)] \textit{J. Park} defined cubical additive higher Chow groups \(\mathrm{ACH}_p(X,n;m)\) with modulus \(m\) for any irreducible scheme \(X\) over a field \(k\) of characteristic zero. For the case \(p=1\) and \(X = \mathrm{Spec} \;k \;\) he constructed a regulator map \(R_{2,m} : \mathrm{ACH}_1(k,n;m) \rightarrow \Omega^{n-2}_{k/\mathbb{Z}}\). Park asked the question whether \(R_{2,2}\) is an isomorphism and he stated his belief that the additive Chow groups are related to the Hochschild cohomology. The paper under review contributes to this line of investigation. The author remarks that there are two different definitions for the additive Chow groups of a field, he writes them \(\mathrm{CH}_d(A_k(m),n)_{\mathrm{ssup}}\) and \(\mathrm{CH}_d(A_k(m),n)_{\sup}\), a natural map goes from the first to the second. Park's regulator's domain was \(\mathrm{CH}_1(A_k(m),n)_{\sup}\), but it appears that \(R_{2,m}\) is not well defined for this choice. Let \( \Omega^{n-2}_{k / \mathbb{Z} } \langle w \rangle \) be the vector space of absolute Kähler differentials with the twisted \(k^*\) action of weight \(w\): Oneda's first main result says that for \( m \leq c < 2m \) there are regulator like weight preserving maps \(L^n_c : \mathrm{CH}_1(A_k(m),n)_{\mathrm{ssup}} \to \Omega_k^{n-2} \langle c \rangle \) such that \( L^n_{m+1}\) is equal to the composition of the projection to \(\mathrm{CH}_1(A_k(m),n)_{\sup}\) with Park's \(R_{2,m}\). Moreover if \(k\) is algebraically closed then the map \(L^n \;: \;\mathrm{CH}_1(A_k(m),n)_{\mathrm{ssup}} \to \bigoplus_{m \leq c < 2m } \;\Omega_k^{n-2} \langle c \rangle \) is surjective. Elements in \(\mathrm{CH}_1\) are given by curves, when \(n=2\) parametric curves are used to provide a map \(\Phi ^{\prime} : x^m \;k[ x ] \to \;\mathrm{CH}_1(A_k(m),2)_{\mathrm{ssup}} \). The author then shows that the composition of \(\Phi ^{\prime} \) with the arrow \(\mathrm{CH}_1(A_k(m),2)_{\mathrm{ssup}} \to \mathrm{CH}_1(A_k(m),2)_{\sup}\) factors through \(x^m \;k[ x ] / x^{2m}\). Next he proves that \(L^2 \;\Phi ^{\prime}\) amounts to the natural projection \( x^m \;k[ x ] \to x^m \;k[ x ] / x^{2m}\). The connection of this computations with Hochschild cohomology lies in the result that for a number field \(k\) the space \(x^{m+1} \;k[ x ] / x^{2m}\) is in fact isomorphic to \(HC_2 (k[ x ] / x^{m},(x))\), i.e. to the relative cyclic homology of the truncated polynomial ring.