On the K-theory of algebraic tori

TL;DR

This paper establishes a natural equivalence between the algebraic K-theory of algebraic tori and the equivariant homology of associated topological tori, generalizing previous computations and using motivic homotopy theory techniques.

Contribution

It constructs a new equivalence linking algebraic K-theory of tori with equivariant topological homology, extending prior results and employing motivic Fourier transforms.

Findings

Established a natural equivalence between $K_*(T)$ and equivariant homology $H^G_*(\mathfrak{T}(T);K_G(F))$.

Generalized Merkurjev and Panin's computation of $K_0(T)$ to higher K-theory.

Used motivic Eilenberg--Moore formula to prove the equivalence.

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 $H^{G}_{\ast}(\mathfrak{T}(T);K_G(F))$ 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 $\mathfrak{F}\colon \mathbb{K}_{F}[B\Lambda] \to \mathbb{K}_{F}^{T}$ from the $\mathbb{K}_{F}$-homology of the \'etale 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.