The motive of a classifying space

Abstract

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 Q. As a tool, we give several characterizations of the varieties which satisfy Kunneth 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 Kunneth 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.

44 pages; v2: references added, some arguments simplified

References in corpus (2)

• A geometric invariant of a finite group
• Adjoint functors on the derived category of motives

Cited by in corpus (27)

• Exceptional collections, and the Neron-Severi lattice for surfaces
• The intersection motive of the moduli stack of shtukas
• Algebraic cycles and Todorov surfaces
• Exotic Motivic Periodicities
• A formula for the Voevodsky motive of the moduli stack of vector bundles on a curve
• Hypersurfaces that are not stably rational
• Adjoint functors on the derived category of motives
• A characterization of finite étale morphisms in tensor triangular geometry
• Motive of the moduli stack of rational curves on a weighted projective stack
• Theta-stratifications, Theta-reductive stacks, and applications
• Finite Groups Generated in Low Real Codimension
• The eleventh cohomology group of ${\bar{\mathcal{M}}}_{g,n}$
• Extensions of tautological rings and motivic structures in the cohomology of ${\overline{\mathcal{M}}}_{g,n}$
• Enumerating Non-Stable Vector Bundles
• The integral cohomology of the Hilbert scheme of points on a surface
• Surjectivity of cycle maps for singular varieties
• Motivic splitting principle
• On the motive of intersections of two Grassmannians in ${\mathbb{P}}^9$
• Stably A^1-connected varieties and universal triviality of CH_0
• On the Mixed Tate property and the motivic class of the classifying stack of a finite group
• $T$-equivariant motives of flag varieties
• ${\mathbb{A}}^1$-homotopy equivalences and a theorem of Whitehead
• Torsion birational motives of surfaces and unramified cohomology
• Mixed Hodge modules on stacks
• Feynman quadrics-motive of the massive sunset graph
• Lannes's $T$-functor and equivariant Chow rings
• Motivic classes of classifying stacks of finite groups and unramified cohomology