An Extended Version of Additive K-theory

There are two infinitesimal (i.e., additive) versions of the K-theory of a field F : one introduced by Cathelineau, which is an F -module, and another one introduced by Bloch-Esnault, which is an F ∗module. Both versions are equipped with a regulator map, when F is the field of complex numbers. In our short paper we will introduce an extended version of Cathelineau’s group, and a complex-valued regulator map given by the entropy. We will also give a comparison map between our extended version and Cathelineau’s group. Our results were motivated by two unrelated sources: Neumann’s work on the extended Bloch group (which is isomorphic to indecomposable K3 of the complex numbers), and the study of singularities of generating series of hypergeometric multisums.