On the Chow ring of the classifying stack of algebraic tori

TL;DR

This paper studies the structure of the Chow ring of classifying stacks of algebraic tori, extending previous results and identifying cases where torsion appears, using elementary methods to determine the ring explicitly.

Contribution

It extends Karpenko's results to special tori and fully determines the Chow ring for tori with certain resolutions, revealing torsion phenomena.

Findings

Chow ring structure is determined for tori with specific resolutions.

Torsion can occur in the Chow ring of certain algebraic tori.

Elementary approach simplifies understanding of the Chow ring in these cases.

Abstract

We investigate the structure of the Chow ring of the classifying stacks $BT$ of algebraic tori, as it has been defined by B. Totaro. Some previous work of N. Karpenko, A. Merkurjev, S. Blinstein and F. Scavia has shed some light on the structure of such rings. In particular Karpenko showed the absence of torsion classes in the case of permutation tori, while Merkurjev and Blinstein described in a very effective way the second Chow group $A^2(BT)$ in the general case. Building on this work, Scavia exhibited an example where $A^2(BT)_\text{tors}\neq 0$. Here, by making use of a very elementary approach, we extend the result of Karpenko to special tori and we completely determine the Chow ring $A^*(BT)$ when $T$ is an algebraic torus admitting a resolution with special tori $0\rightarrow T\rightarrow Q\rightarrow P$. In particular we show that there can be torsion in the Chow ring of such tori.