Infinitesimal Bloch regulator

Abstract

In this paper, we continue our project of defining and studying the infinitesimal versions of the classical, real analytic, invariants of motives. Here, we construct an infinitesimal analog of Bloch's regulator. Let $X/k$ be a scheme of finite type over a field $k$ of characteristic 0. Suppose that $\underline{X} \hookrightarrow X$ is a closed subscheme, smooth over $k,$ and defined by a square-zero sheaf of ideals, which is locally free on $\underline{X}.$ We define two regulators: $\rho_1,$ from the infinitesimal part of the motivic cohomology ${\rm H}^2 _{M}(X,\mathbb{Q}(2))$ of $X$ to ${\rm ker}({\rm H}^{0}(X,\Omega^{1} _{X}/d\mathcal{O}_{X}) \to {\rm H}^{0}(\underline{X},\Omega^{1} _{\underline{X}}/d\mathcal{O}_{\underline{X}});$ and $\rho_2,$ from ${\rm ker}(\rho_1)$ to ${\rm H}^1(X,D_{1}(\mathcal{O}_{X})),$ where $D_{1}(\mathcal{O}_X)$ is the Zariski sheaf associated to the first Andr\'{e}-Quillen homology. The main tool is a generalization of our additive dilogarithm construction. Using Goodwillie's theorem, we deduce that $\rho_2$ is an isomorphism. We also reinterpret the above results in terms of the infinitesimal Deligne-Vologodsky crystalline complex $\mathcal{D}_{X}^{\circ}(2),$ when $X$ is smooth over the dual numbers of $k.$