The motive of a classifying space

Totaro, Burt¹

¹ Department of Mathematics, University of California, Los Angeles, Box 951555, Los Angeles, CA 90095-1555, United States

Geometry & topology, Tome 20 (2016) no. 4, pp. 2079-2133.

Résumé

We give the first examples of finite groups $G$ such that the Chow ring of the classifying space $BG$ depends on the base field, even for fields containing the algebraic closure of $\mathbb{Q}$. As a tool, we give several characterizations of the varieties that satisfy Künneth properties for Chow groups or motivic homology.

We define the (compactly supported) motive of a quotient stack in Voevodsky’s derived category of motives. This makes it possible to ask when the motive of $BG$ is mixed Tate, which is equivalent to the motivic Künneth property. We prove that $BG$ is mixed Tate for various “well-behaved” finite groups $G$, such as the finite general linear groups in cross-characteristic and the symmetric groups.

Détail

DOI : 10.2140/gt.2016.20.2079

Classification : 14C15, 14F42, 14M20, 14A20
Keywords: Chow ring, mixed Tate motive, classifying space

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