Degree one Milnor K-invariants of groups of multiplicative type

Abstract

Let G be a commutative affine algebraic group over a field F, and let H:FieldsF→AbGrps be a functor. A (homomorphic) H-invariant of G is a natural transformation Tors(−,G)→H, where Tors(−,G) is the functor FieldsF→AbGrps taking a field extension L/F to the group of isomorphism classes of GL-torsors over Spec(L). The goal of this paper is to compute the group Invhom1(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×⊗ZQ/Z.