Subsection 3.4. (0N00) · ScalingStacks

3.4. Equivariant mixed Tate motives

There is also an equivariant version of the category of mixed Tate motives defined in Section 3.2. For the remainder of the section let $k=\overline{\mathbb{F}}_{p}.$

0MY4

Definition 3.9. The category of equivariant mixed Tate motives on a point

\operatorname{DTM}_{G}(k,\mathbb{Q})\subset\operatorname{DM}_{G}(k,\mathbb{Q})

is the full subcategory of objects $M$ such that $\operatorname{For}(M)\in\operatorname{DTM}(k,\mathbb{Q}),$

We now explain how the explicit description of this category in [SVW18, Theorem II.3.1] can be obtained using the weight complex functor.

0MY5

Proposition 3.10. There is a weight structure $w$ on $\operatorname{DTM}_{G}(k,\mathbb{Q})$ such that

	$\displaystyle M\in\operatorname{DTM}_{G}(k,\mathbb{Q})^{w\geq 0}$	$\displaystyle\iff\operatorname{For}(M)\in\operatorname{DTM}(k,\mathbb{Q})^{w\geq 0}\text{ and }$
	$\displaystyle M\in\operatorname{DTM}_{G}(k,\mathbb{Q})^{w\leq 0}$	$\displaystyle\iff\operatorname{For}(M)\in\operatorname{DTM}(k,\mathbb{Q})^{w\leq 0}.$

0MY6

Proof. This is [SVW18, Proposition II.4.10]. ∎

Denote by $G^{0}\subset G$ the connected component of the identity. Then one can consider the object $\operatorname{Ind}^{G}_{G^{0}}(\mathbb{Q})\in\operatorname{DM}_{G}(k)$ which plays the role of the local system on $BG$ with free action of the component group $\pi_{0}(G)=G/G^{0}.$ The heart of $w$ admits the following explicit description by [SVW18, Proposition II.4.5].

0MY7

Proposition 3.11. The heart of the weight structure on $\operatorname{DTM}_{G}(k,\mathbb{Q})$ is generated by the objects $\operatorname{Ind}^{G}_{G^{0}}(\mathbb{Q})(n)[2n]$ with respect to isomorphism, direct summands and finite direct sums

\operatorname{DTM}_{G}(k,\mathbb{Q})^{w=0}=\langle\operatorname{Ind}^{G}_{G^{0}}(\mathbb{Q})(n)[2n]\mid n\in\mathbb{Z}\rangle_{\cong,\inplus,\oplus}\subset\operatorname{DTM}_{G}(k,\mathbb{Q}).

For the moment, assume that $G=G_{0}$ is connected. By [SVW18, Proposition I.7.6] the categories $\operatorname{DM}_{G}$ just depend on the quotient $G/R_{u}(G)$ of $G$ by its unipotent radical. Hence, we can assume that $G$ is reductive. Denote by $T$ a maximal torus in $G$ and by $W=N_{G}(T)/T$ the Weyl group. Let $S=\operatorname{Sym}(X(T)_{\mathbb{Q}})$ be the symmetric algebra of the rationalized character lattice of $T.$ The algebra $S$ is isomorphic to a polynomial ring in $\operatorname{rank}(T)$ many variables and graded where we put $X(T)$ in degree one.

Since we are only considering rational coefficients, the $T$ - and $G$ -equivariant Chow rings agree with equivariant cohomology rings. In particular, the Chern class map induces isomorphisms of graded algebras

(3.7)

\displaystyle S\cong\operatorname{CH}^{\bullet}_{T}(k)_{\mathbb{Q}}\text{ and }S^{W}\cong\operatorname{CH}^{\bullet}_{T}(k)_{\mathbb{Q}}^{W}\cong\operatorname{CH}^{\bullet}_{G}(k)_{\mathbb{Q}},

see [Tot99] or [EG98, Section 3.2]. Similarly, for higher Chow groups

(3.8)

\displaystyle\operatorname{CH}^{\bullet}_{G}(k,i)_{\mathbb{Q}}\cong(\operatorname{CH}^{\bullet}_{T}(k,i)_{\mathbb{Q}})^{W}\cong(S\otimes\operatorname{CH}^{\bullet}(k,i)_{\mathbb{Q}})^{W}

using [Kri17, Theorem 1.5] and [Kri13, Theorem 5.7].

If $G$ is not connected, we consider the extension algebra

\displaystyle E

\displaystyle=\bigoplus_{n\in\mathbb{Z}}\operatorname{Hom}_{\operatorname{DM}_{G}(k,\mathbb{Q})}(\operatorname{Ind}^{G}_{G^{0}}(\mathbb{Q}),\operatorname{Ind}^{G}_{G^{0}}(\mathbb{Q})(n)[2n])

which can be thought of as the Chow ring of $BG$ with coefficients in the local system given by the regular representation of $\pi_{0}(G).$ For an explicit description denote by $S^{W}\ltimes\mathbb{Q}[\pi_{0}(G)]$ the twisted group algebra, see [SVW18, A.2.3], which as a graded vector space is just the tensor product $S^{W}\otimes\mathbb{Q}[\pi_{0}(G)]$ with $\mathbb{Q}[\pi_{0}(G)]$ concentrated in degree zero.

0MY8

Proposition 3.12. There is a natural isomorphism $E\cong S^{W}\ltimes\mathbb{Q}[\pi_{0}(G)].$

0MY9

Proof. If $G$ is connected this is (3.7). Otherwise the statements follow from a transfer argument for finite group torsors, see [SVW18, Section A.2.2]. ∎

0MYA

Proposition 3.13. The collection of objects $\{\operatorname{Ind}^{G}_{G^{0}}(\mathbb{Q})(n)[2n]\mid n\in\mathbb{Z}\}$ is tilting.

0MYB

Proof. Assume first that $G$ is connected. By (3.7) it suffices to show that the higher Chow groups in (3.7) vanish for $i\neq 0.$ This holds by Lemma 3.1 using $k=\overline{\mathbb{F}}_{p}.$ Again, the statement for $G$ not connected follows from [SVW18, Section A.2.2]. ∎

With Corollary 2.16 we obtain the following explicit description of $\operatorname{DTM}_{G}(k,\mathbb{Q}).$

0MYC

Theorem 3.14. The weight complex functor induces an equivalence

t:\operatorname{DTM}_{G}(k,\mathbb{Q})\to\operatorname{D}^{\mathbb{Z}}_{\operatorname{perf}}(S^{W}\ltimes\mathbb{Q}[\pi_{0}(G)]).

Since $S^{W}\ltimes\mathbb{Q}[\pi_{0}(G)]$ has finite cohomological dimension, $\operatorname{D}^{\mathbb{Z}}_{\operatorname{perf}}(S^{W}\ltimes\mathbb{Q}[\pi_{0}(G)])$ is just the bounded derived category of finitely generated graded modules. A similar result was shown in [SVW18, Theorem II.3.1] using slightly different arguments.

Jens Niklas Eberhardt, Catharina Stroppel

Original source: arXiv:2109.00305v2