Degree One Milnor $K$-invariants of Groups of Multiplicative Type

TL;DR

This paper computes the first degree Milnor $K$-invariants of groups of multiplicative type, providing explicit descriptions of these invariants in the context of algebraic groups over fields.

Contribution

It explicitly determines the group of homomorphic invariants for commutative affine algebraic groups of multiplicative type with respect to a specific functor involving Milnor $K$-theory.

Findings

Computed the group of $H$-invariants for groups of multiplicative type.

Established a connection between torsors and Milnor $K$-theory invariants.

Provided explicit descriptions of invariants in terms of field extensions.

Abstract

Let $G$ be a commutative affine algebraic group over a field $F$, and let $H \colon \mathrm{Fields}_{F} \to \mathrm{AbGrps}$ be a functor. A (homomorphic) $H$-invariant of $G$ is a natural transformation $\mathrm{Tors}(-, G) \to H$, where $\mathrm{Tors}(-, G)$ is the functor $\mathrm{Fields}_{F} \to \mathrm{AbGrps}$ taking a field extension $L/F$ to the group of isomorphism classes of $G_{L}$-torsors over $\mathrm{Spec}(L)$. The goal of this paper is to compute the group $\mathrm{Inv}_{\mathrm{hom}}^{1}(G, H)$ of $H$-invariants of $G$ when $G$ is a group of multiplicative type, and $H$ is the functor taking a field extension $L/F$ to $L^{\times} \otimes_{\mathbb{Z}} \mathbb{Q}/\mathbb{Z}$.