2019/04/14 by Sinan Unver, Unver, Sinan
Mathematics · #14C25 #19E15 #Algebraic Geometry (math.AG) #FOS: Mathematics #K-Theory and Homology (math.KT) #math.AG #math.KT #msc:14C25 #msc:19E15
paper · pdf · doi:10.48550/arxiv.1904.06694
arxiv created 2019/04/14 · arxiv updated 2019/04/16
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 \underlineX \hookrightarrow X is a closed subscheme, smooth over k, and defined by a square-zero sheaf of ideals, which is locally free on \underlineX. We define two regulators: ρ1, from the infinitesimal part of the motivic cohomology \rm H2 M(X,ℚ(2)) of X to \rm ker(\rm H0(X,Ω1 X/dOX) → \rm H0(\underlineX,Ω1 _\underlineX/dO_\underlineX); and ρ2, from \rm ker(ρ1) to \rm H1(X,D1(OX)), where D1(OX) 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 ρ2 is an isomorphism. We also reinterpret the above results in terms of the infinitesimal Deligne-Vologodsky crystalline complex DX∘(2), when X is smooth over the dual numbers of k.