ScalingStacks

Proposition 3.17. Let $?\in\{*,!\}$ and $M\in\operatorname{DM}_{G}(X).$ Then $M$ is $?$-pointwise mixed or pure Tate if and only if for each orbit $\mathcal{O}\hookrightarrow X$

$\displaystyle(i,s)^{*}j^{?}M\in\operatorname{DTM}_{H}(k,\mathbb{Q})\text{ or }(i,s)^{*}j^{?}M\in\operatorname{DTM}_{H}(k,\mathbb{Q})^{w=0}\text{, respectively.}$

Here $j:G/H\cong\mathcal{O}\hookrightarrow X$ and the functor $(i,s)^{*}j^{?}$ is the composition

$$\operatorname{DM}_{G}(X,\mathbb{Q})\stackrel{{\scriptstyle j^{?}}}{{\to}}\operatorname{DM}_{G}(G/H,\mathbb{Q})\stackrel{{\scriptstyle(i,s)^{*}}}{{\to}}\operatorname{DM}_{H}(\operatorname{pt},\mathbb{Q})$$

of pullback to the orbit and the induction equivalence (3.6).