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After a brief reminder of classical finiteness results for étale cohomology, we plan to explain some extensions in two directions: uniformity and calculability.
Uniformity. We will prove that the direct image of a "uniformly tame-constructible" family of complexes by a proper morphism is uniformly tame-constructible. This is a generalization of the following fact: if X is a scheme separated of finite type over an algebraically closed field, the mod ? Betti numbers of X are bounded independently of ?. (If time permits, the non-proper case and other 6-operations will be considered.)
Calculability. We will prove the algorithmic computability of the mod ? cohomology of a scheme X (as above), as well as the maps induced by functoriality (e.g. Galois action). An extension to higher direct images of constructible sheaves, with respect to proper morphisms, will also be proven.
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