A note on the behaviour of the Tate conjecture under finitely generated field extensions

Abstract

We show that the $\ell$-adic Tate conjecture for divisors on smooth proper varieties over finitely generated fields of positive characteristic follows from the $\ell$-adic Tate conjecture for divisors on smooth projective surfaces over finite fields. Similar results for cycles of higher co-dimension are given.

v1:6 pages. Comments are welcome. v2: Corrected a gap in the proof of Theorem 1.1.2. Changed Proposition 3.1.2 accordingly. v3: final version

References in corpus (3)

• The Tate conjecture for K3 surfaces over finite fields
• On the Tate and Mumford-Tate conjectures in codimension one for varieties with h^{2,0}=1
• 2-adic integral canonical models and the Tate conjecture in characteristic 2

Cited by in corpus (2)

• Sur la conjecture de Tate pour les diviseurs
• On the Brauer groups of fibrations