Completion of motivic sheaves

Denis-Charles Cisinski

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arXiv · 2503.24033

Completion of motivic sheaves

Denis-Charles Cisinski

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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.

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Denis-Charles Cisinski. 2025-03-31. Completion of motivic sheaves. https://arxiv.org/abs/2503.24033

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