On the K-theory of algebraic tori

Given an algebraic torus T over a field F, its lattice of characters $Λ$ gives rise to a topological torus T(T)=ΛR/Λ with a continuous action of the absolute Galois group G. We construct a natural equivalence between the algebraic K-theory K_(T) and the equivariant homology HG_(T(T);KG(F)) of the topological torus T(T) with coefficients in the G-equivariant K-theory of F. This generalizes a computation of K₀(T) due to Merkurjev and Panin. We obtain this equivalence by analyzing the motive KFT in the stable motivic category SH(F) of Voevodsky and Morel, where KF is the motivic spectrum representing homotopy K-theory. We construct a natural comparison map F KF[BΛ] → KFT from the KF-homology of the étale delooping of $Λ$ to KFT as a special case of a motivic Fourier transform and prove that it is an equivalence by using a motivic Eilenberg--Moore formula for classifying spaces of tori.