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: $ρ_1,$ from the infinitesimal part of the motivic cohomology ${\rm H}^2 _{M}(X,\mathbb{Q}(2))$ of $X$ to ${\rm ker}({\rm H}^{0}(X,Ω^{1} _{X}/d\mathcal{O}_{X}) \to {\rm H}^{0}(\underline{X},Ω^{1} _{\underline{X}}/d\mathcal{O}_{\underline{X}});$ and $ρ_2,$ from ${\rm ker}(ρ_1)$ to ${\rm H}^1(X,D_{1}(\mathcal{O}_{X})),$ where $D_{1}(\mathcal{O}_X)$ is the Zariski sheaf associated to the first André-Quillen homology. The main tool is a generalization of our additive dilogarithm construction. Using Goodwillie's theorem, we deduce that $ρ_2$ 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 $k.$