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

TL;DR

This paper introduces a new additive variant of Somekawa's K-groups linked to algebraic groups over perfect fields, revealing their connection to K"ahler differentials and homology groups, thus providing geometric insights.

Contribution

It defines a Milnor type K-group for algebraic groups and establishes its isomorphism with K"ahler differentials and homology groups, offering new geometric interpretations.

Findings

K-group for additive and multiplicative groups is isomorphic to K"ahler differentials.

K-group for additive group and Jacobian is isomorphic to a homology group.

Provides a geometric interpretation of 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.