On the Chow ring of the classifying stack of algebraic tori

  • Francesco Sala

    Scuola Normale Superiore, Piazza dei cavalieri 7, 56126 Pisa, Italy
On the Chow ring of the classifying stack of algebraic tori cover
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Abstract

We investigate the structure of the Chow ring of the classifying stacks BTBT 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 equivariant Chow group AT2A^2_T in the general case. Building on this work, Scavia exhibited an example where (AT2)tors0(A^2_T)_{\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 ATA^*_T when TT is an algebraic torus admitting a resolution with special tori 0TQP0\rightarrow T\rightarrow Q\rightarrow P. In particular we show that there can be torsion in the Chow ring of such tori.

Cite this article

Francesco Sala, On the Chow ring of the classifying stack of algebraic tori. Doc. Math. 27 (2022), pp. 917–932

DOI 10.4171/DM/888