On the Chow ring of the classifying stack of algebraic tori

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.