Given an algebraic torus T over a field F, its lattice of characters Λ gives rise to a topological torus T (T) =Λ ₑ/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 H^G_ (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 K₅^T in the stable motivic category SH (F) of Voevodsky and Morel, where K₅ is the motivic spectrum representing homotopy K-theory. We construct a natural comparison map F K₅BΛ K₅^T from the K₅-homology of the étale delooping of Λ to K₅^T 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.
Bai et al. (Thu,) studied this question.