Completion of motivic sheaves

We study the process of $\mathrm{\ell }$-adic completion of motivic sheaves. We observe that, in equal characteristic, when restricted to constructible objets, it is compatible with the six operations. This implies that one can reconstruct $\mathrm{\ell }$-adic sheaves of geometric origin over a scheme of finite type over a field from $\mathrm{\ell }$-adic cohomology of smooth schemes. In the case of finite fields, this includes perverse $\mathrm{\ell }$-adic sheaves of geometric orgin. However, the analogous behaviour fails systematically in mixed characteristic: the reason is that it would imply strong independence of $\mathrm{\ell }$ results that can be proven to be too optimistic.