Proposition 4.11 (0MZ1) · ScalingStacks

0MZ1

Proposition 4.11. Assume that the conditions (PT) and (FO) are fulfilled. Let $M,N\in\operatorname{DTM}^{Spr}_{G}(\mathcal{N})$ and $\ell\neq p$ be a prime. The natural isomorphisms $\operatorname{Real}_{\ell}\circ(1)\to\operatorname{Real}_{\ell}$ from (3.1) induce isomorphisms

\bigoplus_{n\in\mathbb{Z}}\operatorname{Hom}_{\operatorname{DM}_{G}(N,\mathbb{Q}_{\ell})}(M,N(n))\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{Hom}_{\operatorname{D}_{G}(\mathcal{N},\mathbb{Q}_{\ell})}(\operatorname{Real}_{\ell}(M),\operatorname{Real}_{\ell}(N))

and for $M,N\in\operatorname{DTM}^{Spr}_{G}(\mathcal{N})^{w=0}$ isomorphisms

\operatorname{Hom}_{\operatorname{DM}_{G}(N,\mathbb{Q}_{\ell})}(M,N)\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{Hom}_{\operatorname{D}_{G}(\mathcal{N},\mathbb{Q}_{\ell})}(\operatorname{Real}_{\ell}(M),\operatorname{Real}_{\ell}(N)).

0MZ2

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. ∎