Finite generation conjectures for cohomology over finite fields

We construct an intermediate cohmology between motivic cohomology and Weil-etale cohomology. Using this, the Bass conjecture on finite generation of motivic cohomology, and the Beilinson-Tate on the finite generation of Weil-etale cohomology are related.

💡 Research Summary

The paper introduces a new “Frobenius cohomology” theory that sits between motivic cohomology and Weil‑étale cohomology for schemes of finite type over a finite field. Using Bloch’s higher cycle complex (Z(n)), three cohomology groups are defined:

1. Motivic cohomology (H_i^M(X,\mathbb Z(n))), the usual higher Chow groups.
2. Frobenius cohomology (H_i^F(X,\mathbb Z(n))), obtained as the homology of the double complex (\operatorname{Cone}(1-\varphi)) where (\varphi) is the Frobenius action of the Weil group (G) on the cycle complex of the geometric fibre (\bar X).
3. Kato cohomology (H_i^K(X,\mathbb Z(n))), the homology of the (G)-invariants of the same cycle complex; it generalises the integral Kato homology introduced by Kato.

Natural maps (\alpha: H_i^M\to H_i^F) and (\beta: H_i^F\to H_i^K) fit into a long exact sequence
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