Zariski realization and torsion-freeness conjecture for etale motivic complexes

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Let α\alpha be the morphism from the etale site to the Zariski site, and let Zl(n)c{\Bbb Z}_l(n)^c be the continuous etale Tate-twist complex. On the big Zariski site of Spec⁡Fp\operatorname{Spec}{\Bbb F}_p, consider the natural morphism

Z(n)⊗Zl→τ≤nRα∗Zl(n)c.{\Bbb Z}(n)\otimes {\Bbb Z}_l\to \tau_{\le n}R\alpha_*{\Bbb Z}_l(n)^c.

Zariski realization conjecture. (a) This morphism is an isomorphism. (b) The Zariski sheaf Rn+1α∗Zl(n)cR^{n+1}\alpha_*{\Bbb Z}_l(n)^c is torsion-free. Under resolution of singularities and the Kato conjecture, this is related to finite generation statements for motivic cohomology.

References

Primary source

Bruno Kahn, “A sheaf-theoretic reformulation of the Tate conjecture”, arXiv:math/9801017 (1998).

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