ScalingStacks

Definition 3.16. Let $?\in\{*,!\}.$ An object $M\in\operatorname{DM}_{G}(X)$ is called $?$-pointwise mixed Tate or $?$-pointwise pure Tate, respectively, if for each point $i_{x}:\operatorname{pt}\to X$

$\displaystyle i_{x}^{?}\operatorname{For}M\in\operatorname{DTM}(k,\mathbb{Q})\text{ or }i_{x}^{?}\operatorname{For}M\in\operatorname{DTM}(k,\mathbb{Q})^{w=0}\text{, respectively.}$

Here the functor $i_{x}^{?}\operatorname{For}$ is the composition

$$\operatorname{DM}_{G}(X,\mathbb{Q})\stackrel{{\scriptstyle\operatorname{For}}}{{\to}}\operatorname{DM}(X,\mathbb{Q})\stackrel{{\scriptstyle i_{x}^{?}}}{{\to}}\operatorname{DM}(k,\mathbb{Q}).$$

The object $M$ is called pointwise mixed (pure) Tate if it is $*$- and $!$-pointwise mixed (pure) Tate.