ScalingStacks

Remark 1.1. The analogous statement replacing motivic sheaves by mixed $\ell$-adic sheaves (or mixed Hodge modules) fails because there are non-trivial extensions between the Tate objects $\mathbb{Q}_{\ell}(n)$ in the category of mixed $\ell$-adic sheaves. These unwanted extensions were first addressed in the context of perverse sheaves on flag varieties and category $\mathcal{O}$ in [BGS96, Section 4]. There, a workaround is proposed using the derived category of mixed $\ell$-adic perverse sheaves which have a semisimple Frobenius action on their associated graded with respect to the weight filtration. This construction has several drawbacks when compared to motivic sheaves. First, it is not clear how to extend it to the equivariant case and to settings where no perverse $t$-structure exists. Secondly, it is difficult to determine if the six functors preserve the semisimplicity of the Frobenius. This and the independence of the prime $\ell$ are the main technical advantage of motivic sheaves.