Geodesic
On the Chow ring of the classifying stack of algebraic tori
Documenta mathematica, Tome 27 (2022), pp. 917-932.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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 equivariant Chow group $A^2_T$ in the general case. Building on this work, Scavia exhibited an example where $(A^2_T)_{\text{tors}}\neq 0$. \par 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 $A^*_T$ 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.
Classification : 14C15, 14C17, 14A20
Keywords: Chow ring, torsion classes, classifying stack, torsors 
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     author = {Sala, Francesco},
     title = {On the {Chow} ring of the classifying stack of algebraic tori},
     journal = {Documenta mathematica},
     pages = {917--932},
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     volume = {27},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2022__27__a43/}
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Sala, Francesco. On the Chow ring of the classifying stack of algebraic tori. Documenta mathematica, Tome 27 (2022), pp. 917-932. http://geodesic.mathdoc.fr/item/DOCMA_2022__27__a43/