On the K-theory of algebraic tori

Abstract

Given an algebraic torus $T$ over a field $F$, its lattice of characters $\Lambda$ gives rise to a topological torus $\mathfrak{T}(T)={\Lambda }_{\mathbb{R}}/\Lambda$ with a continuous action of the absolute Galois group $G$. We construct a natural equivalence between the algebraic $K$-theory $K_{\ast }(T)$ and the equivariant homology $H_{\ast }^G(\mathfrak{T}(T);K_G(F))$ of the topological torus $\mathfrak{T}(T)$ with coefficients in the $G$-equivariant $K$-theory of $F$. This generalizes a computation of $K_0(T)$ due to Merkurjev and Panin. We obtain this equivalence by analyzing the motive ${\mathbb{K}}_F^T$ in the stable motivic category $\mathrm{SH}(F)$ of Voevodsky and Morel, where ${\mathbb{K}}_F$ is the motivic spectrum representing homotopy $K$-theory. We construct a natural comparison map $\mathfrak{F}:{\mathbb{K}}_F[B\Lambda ]\to {\mathbb{K}}_F^T$ from the ${\mathbb{K}}_F$-homology of the étale delooping of $\Lambda$ to ${\mathbb{K}}_F^T$ 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.

45 pages, comments are welcome