Subsection 4.3. (0N06) · ScalingStacks

Theorem 4.9. Assume that (PT) and (FO) hold. Then there is an equivalence of categories

Moreover, for all primes $\ell\neq p$ the $\ell$ -adic realisation functor $\operatorname{Real}_{\ell}$ gives an isomorphism $E\otimes_{\mathbb{Q}}\mathbb{Q}_{\ell}\cong E^{\acute{e}t}_{\ell}$ and acts as a degrading functor with respect to the Tate-twist $(1)$ in the sense of [BGS96]

0MYZ

Proposition 4.10. Assume that the conditions (PT) and (FO) are fulfilled. Then the collection of objects $\widehat{\mathcal{T}}^{Spr}$ in $\operatorname{DM}_{G}(\mathcal{N})$ is tilting.

0MZ0

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. ∎

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. ∎

4.3. Main results