Local-global type conjectures on the image of Chow groups in étale cohomology. (Conjectures de type local-global sur l’image des groupes de Chow dans la cohomologie étale.) (French) Zbl 0981.14003

Raskind, Wayne (ed.) et al., Algebraic \(K\)-theory. Proceedings of an AMS-IMS-SIAM summer research conference, Seattle, WA, USA, July 13-24, 1997. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 67, 1-12 (1999).
Let \(X\) be a smooth projective variety of dimension \(d\) over a global field \(k\). Assume that \(X\) is geometrically integral. For such a variety, the author formulates the following conjecture:
Let \(i\) be a positive integer and \(l\) be a prime number different from the characteristic of \(k\). Assume that for each place \(v\) of \(k\) a cycle \(z_v\) is given, belonging to \(\text{CH}^i(X_v)\), the Chow group of codimension \(i\) cycles on \(X_v\), modulo rational equivalence (here \(X_v\) denotes \(X\times_k k_v\)). Assume also that every class in \(H^{2j}(X,\mathbb Q_l/\mathbb Z_l(j))\) is orthogonal to the whole family \(\{z_v\}\). Then, for all positive \(n\), there exists a global cycle \(z_n\in \text{CH}^i(X)\), such that the class of \(z_n\) in \(\tilde H^{2i}(X_v, \mu^{\otimes i}_{l^n})\) coincides with the class of \(z_v\) for all \(v\).
The author then proves the following two facts, as a consequence of a general proposition:
Proposition 3.1: (i) the proposed conjecture is true if \(k\) is a number field, the codimension \(i\) is one and the Tate-Shafarevich group of the Picard variety of \(X\) is a finite group;
(ii) if \(k\) is the function field of a smooth projective curve \(C\) over a finite field and \(X/k\) admits a regular model \(\mathcal X /C\), then the conjecture for \(\text{CH}^i(X)\) follows from a suitable form of the Tate conjecture for the Chow groups on \(\mathcal X\).
For the entire collection see [Zbl 0931.00031].


14C15 (Equivariant) Chow groups and rings; motives
14F20 Étale and other Grothendieck topologies and (co)homologies