ScalingStacks

Proof. The condition (PT) implies that all objects in $\widehat{\mathcal{T}}^{Spr}$ are pointwise pure Tate by Proposition 3.21. Now, let $M=\mu_{i,!}(\mathbb{Q}_{\widetilde{\mathcal{N}}_{i}})(k)[2k]$ and $N=\mu_{j,!}(\mathbb{Q}_{\widetilde{\mathcal{N}}_{j}})(l)[2l].$ The subvariety $k_{i,j}:\mathcal{N}_{i,j}=\mu_{i}(\widetilde{\mathcal{N}}_{i})\cup\mu_{j}(\widetilde{\mathcal{N}}_{j})\hookrightarrow\mathcal{N}$ is closed since $\mu_{i}$ and $\mu_{j}$ are proper. Since $M$ and $N$ are supported on $\mathcal{N}_{i,j}$ there is an equality

(4.4)

$$\operatorname{Hom}_{\operatorname{DM}_{G}(N,\mathbb{Q})}(M,N[n])=\operatorname{Hom}_{\operatorname{DM}_{G}(\mathcal{N}_{i,j},\mathbb{Q})}(k_{i,j}^{*}M,k_{i,j}^{!}N[n]).$$

The condition (FO) ensures that there are only finitely many $G$-orbits in $\mathcal{N}_{i,j}.$ Moreover $k_{i,j}^{*}M$ and $k_{i,j}^{!}N$ are $*$- and $!$-pointwise pure Tate. Hence by Proposition 3.19 the right hand side of (4.4) vanishes for $n\neq 0$ and the statement follows. ∎