0MYY Theorem 4.9. Assume that (PT) and (FO) hold. Then there is an equivalence of categoriesDMGSโpโrโก(๐ฉ,โ){\lx@inpgf@ignorespaces\operatorname{DM}^{Spr}_{G}(\mathcal{N},\mathbb{Q})}Dperfโคโก(E).{\lx@inpgf@ignorespaces{\operatorname{D}^{\mathbb{Z}}_{\operatorname{perf}}(E)}.}โผ\scriptstyle{\lx@inpgf@ignorespaces\sim} Moreover, for all primes โโ p\ell\neq p the โ\ell-adic realisation functor Realโ\operatorname{Real}_{\ell} gives an isomorphism Eโโโโโ EโeยดโtE\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)(1) in the sense of [BGS96]DMGSโpโrโก(๐ฉ,โโ){\lx@inpgf@ignorespaces\operatorname{DM}^{Spr}_{G}(\mathcal{N},\mathbb{Q}_{\ell})}DGSโpโrโก(๐ฉ,โโ).{\lx@inpgf@ignorespaces{\operatorname{D}^{Spr}_{G}(\mathcal{N},\mathbb{Q}_{\ell}).}}Realโ\scriptstyle{\lx@inpgf@ignorespaces\operatorname{Real}_{\ell}}