An additive variant of Somekawa's $K$-groups and K\"ahler differentials

Abstract

We introduce a Milnor type $K$-group associated to commutative algebraic groups over a perfect field. It is an additive variant of Somekawa's $K$-group. We show that the $K$-group associated to the additive group and $q$ multiplicative groups of a field is isomorphic to the space of absolute K\"ahler differentials of degree $q$ of the field, thus giving us a geometric interpretation of the space of absolute K\"ahler differentials. We also show that the $K$-group associated to the additive group and Jacobian variety of a curve is isomorphic to the homology group of a certain complex.