ℚℓ-versus Fℓ -coefficients in the Grothendieck–Serre/Tate conjectures

Abstract

We investigate the relation between the Grothendieck–Serre/Tate (G-S/T for short) conjectures with ℚℓ- and Fℓ -coefficients for ℓ ≫ 0 going through their ultraproduct formulations. Our main result roughly asserts that the G-S/T conjecture with Fℓ -coefficients for ℓ ≫ 0 always implies the G-S/T conjecture with ℚℓ-coefficients for ℓ ≫ 0 and that the converse implication holds at least in characteristic p > 0. In characteristic p > 0, this completes partly the motivic picture predicting that the G-S/T conjecture should be independent of the field of coefficients. As a concrete application of our result, we obtain that over an arbitrary finitely generated fields of characteristic p > 0, the Tate conjecture with ℚℓ-coefficients for divisors and some ℓ ≠ p is equivalent to the finiteness of the Galois-fixed part of the prime-to-p torsion subgroup of the geometric Brauer group. This generalizes a well-known theorem of Tate over finite fields.