ScalingStacks

Proof. Let $M,N\in\operatorname{DTM}^{Spr}_{G}(\mathcal{N}).$ All objects in $\widehat{\mathcal{T}}^{Spr}$ are supported on a closed subset of $\mathcal{N}$ consisting of finitely many $G$-orbits by condition (FO) and are pointwise pure Tate using condition (PT) and Proposition 3.21. Now $M$ and $N$ are constructed from the objects in $\widehat{\mathcal{T}}^{Spr}$ by a finite combination of taking direct summands, finite direct sums and triangles. Hence $M$ and $N$ are also supported on a closed subset of $\mathcal{N}$ consisting finitely many $G$-orbits and are pointwise mixed Tate. Now the statement follows from Proposition 3.20. ∎