Completion of motivic sheaves

Abstract

We study the process of $\ell$-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 $\ell$-adic sheaves of geometric origin over a scheme of finite type over a field from $\ell$-adic cohomology of smooth schemes. In the case of finite fields, this includes perverse $\ell$-adic sheaves of geometric orgin. However, the analogous behaviour fails systematically in mixed characteristic: the reason is that it would imply strong independence of $\ell$ results that can be proven to be too optimistic.