ScalingStacks

Proof. We first show that $\mu_{!}(\mathbb{Q}_{M})$ is $*$-pointwise pure Tate. Let $i_{x}:\operatorname{pt}\to N$ be the inclusion of a point $x\in X.$ We have to show that

$$i_{x}^{*}\operatorname{For}\mu_{!}(\mathbb{Q}_{M})\in\operatorname{DTM}(k,\mathbb{Q})^{w=0}.$$

By applying base change with respect to the Cartesian diagram

and the fact that $\operatorname{For}$ commutes with the six operations, we have

$$i_{x}^{*}\operatorname{For}\mu_{!}(\mathbb{Q}_{M})=\mu^{\prime}_{!}l^{*}\mathbb{Q}_{M}=\mu^{\prime}_{!}\mathbb{Q}_{\mu^{-1}(x)}\in\operatorname{DM}(k,\mathbb{Q}).$$

Now $\mu^{\prime}_{!}\mathbb{Q}_{\mu^{-1}(x)}$ is pure Tate since it is Verdier dual to the motive $\operatorname{M}^{c}(\mu^{-1}(\{x\})=\operatorname{M}(\mu^{-1}(\{x\})$ and Verdier duality preserves pure Tate motives.

That $\mu_{!}(\mathbb{Q}_{M})=\mu_{*}(\mathbb{Q}_{M})$ is $!$-pointwise pure Tate follows by using Verdier dual arguments. ∎