An extended version of additive K-theory

摘要

There are two infinitesimal (i.e., additive) versions of the K-theory of a fieldF: one introduced by Cathelineau, which is an F-module, and the otherintroduced by Bloch-Esnault, which is an F~*-module. Both versions areequipped with a regulator map, when F is the field of complex numbers. 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 comparisonmap between our extended version and Cathelineau's group. Our results were motivated by two unrelated sources: Neumann's work onthe extended Bloch group (which is isomorphic to indecomposable K_3ofthe complex numbers), and the study of singularities of generating series ofhypergeometric multisums.