Completion of motivic sheaves
Denis-Charles Cisinski
TL;DR
The paper investigates the $oldsymbol{ ext{ell}}$-adic completion of motivic sheaves and shows that in equal characteristic, for $X$ of finite type over a field and constructible $M$, the natural map $oldsymbol{Z}_{oldsymbol{ ext{ell}},X}\,\otimes^{ ext{L}} M \rightarrow M_{oldsymbol{ ext{ell}}}$ is an isomorphism and pullbacks commute with completion, enabling reconstruction of the six-functor formalism from $oldsymbol{ ext{ell}}$-adic cohomology of smooth schemes. It defines and analyzes the category $ ext{D}^{b}_{ ext{gm}}(X,ar{oldsymbol{Q}}_oldsymbol{ ext{ell}})$ of geometric-origin $oldsymbol{ ext{ell}}$-adic objects, showing it is stable under the six operations and whose perverse-heart consists of perverse sheaves of geometric origin, linked to pushforwards from smooth projective varieties. In mixed characteristic, however, these favorable properties fail, as demonstrated by a counterexample that would force unrealistically strong equivalences between $oldsymbol{ ext{ell}}$-adic and rigid cohomologies; this highlights fundamental obstructions to a universal independence-of-$oldsymbol{ ext{ell}}$ principle. Overall, the work clarifies the extent to which motivic and $oldsymbol{ ext{ell}}$-adic realizations can be reconciled and delineates the limitations imposed by mixed characteristic.
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.