Subsection 3.3. (0MZZ) · ScalingStacks

3.3. Equivariant motivic sheaves

Soergel–Virk–Wendt [SVW18] introduced an equivariant version of the above formalism. For a variety $X$ with an action of a linear algebraic group $G$ they define the category $\operatorname{DM}_{G}(X,\mathbb{Q})$ of $G$ -equivariant motivic sheaves on $X.$ This system of categories still carries a six-functor-formalism³³ 3 For technical reasons, Soergel–Virk–Wendt construct a six-functor-formalism for the full subcategory $\operatorname{DM}^{+}_{G}(X,\mathbb{Q})\subset\operatorname{DM}_{G}(X,\mathbb{Q})$ of objects that are bounded below with respect to the homotopy $t$ -structure, see [SVW18, Section I.6]. All objects that we consider here are automatically in $\operatorname{DM}^{+}_{G}(X,\mathbb{Q})$ and we simply ignore this technicality. and induction/restriction functors changing the group.

Similarly to the non-equivariant case, there is a realization functor to the equivariant derived category of $\ell$ -adic sheaves $\operatorname{D}_{G}(X,\mathbb{Q}_{\ell})$ of Bernstein–Lunts [BL94]

\operatorname{Real}_{\ell}:\operatorname{DM}_{G}(X,\mathbb{Q})\to\operatorname{D}_{G}(X,\mathbb{Q}_{\ell}).

Moreover, for $X$ smooth there is a natural isomorphism

(3.5)

\displaystyle\operatorname{Hom}_{\operatorname{DM}_{G}(X,\mathbb{Q})}(\mathbb{Q}_{X},\mathbb{Q}_{X}(m)[n])

\displaystyle\cong\operatorname{CH}^{m}_{G}(X,2m-n)_{\mathbb{Q}}

to the equivariant higher Chow groups as defined by Totaro [Tot99] and Edidin–Graham [EG98].

There is a forgetful functor from equivariant to non-equivariant motivic sheaves

\operatorname{For}:\operatorname{DM}_{G}(X,\mathbb{Q})\to\operatorname{DM}(X,\mathbb{Q})

commuting with the six functors.

An important property of equivariant (motivic) sheaves is the induction equivalence, which allows to describe $G$ -equivariant motivic sheaves on a $G$ -orbit $G/H$ in terms of $H$ -equivariant motives on a point:

0MY1

Proposition 3.7. Let $i:H\hookrightarrow G$ be a closed subgroup. Denote by $s:X\to G\times_{H}X,x\mapsto[e,x].$ If the anti-diagonal action of $H$ on $G\times X$ is free then there is an equivalence of categories

(3.6)

\displaystyle(i,s)^{*}:\operatorname{DM}_{G}(G\times_{H}X)\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{DM}_{H}(X).

0MY2

Proof. See [SVW18, Proposition I.7.4]. ∎

0MY3

Remark 3.8. In [SVW18] the language of derivators is used and it is shown that $\operatorname{DM}_{G}(X,\mathbb{Q})$ admits an enhancements as stable derivator. In [RS20], [RS21a] and [RS21b] Richarz–Scholbach provide a similar construction in the language of $\infty$ -categories which shows that $\operatorname{DM}_{G}(X,\mathbb{Q})$ admits an enhancement as stable $\infty$ -category.

Jens Niklas Eberhardt, Catharina Stroppel

Original source: arXiv:2109.00305v2