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Infinitesimal Bloch regulator. (English) Zbl 1453.19005

Summary: The aim of the paper is to define an infinitesimal analog of the Bloch regulator, which attaches to a pair of meromorphic functions on a Riemann surface, a line bundle with connection on the punctured surface. In the infinitesimal context, we consider a pair \((X, \underline{X})\) of schemes over a field of characteristic 0, such that the regular scheme \(\underline{X}\) is defined in \(X\) by a square-zero sheaf of ideals which is locally free on \(\underline{X}\). We propose a definition of the weight two motivic cohomology of \(X\) based on the Bloch group, which is defined in terms of the functional equation of the dilogarithm. The analog of the Bloch regulator is a map from a subspace of the infinitesimal part of \(\operatorname{H}_{\mathcal{M}}^2(X, \mathbb{Q}(2))\) to the first cohomology group of the Zariski sheaf associated to an André-Quillen homology group. Using Goodwillie’s theorem, we deduce that this map is an isomorphism, which is an infinitesimal analog of the injectivity conjecture for the Bloch regulator.

MSC:

19E15 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
14C25 Algebraic cycles
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References:

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