Iterating the algebraic étale-Brauer set

In this paper, we iterate the algebraic etale-Brauer set for any nice variety X over a number field k with π 1 et ( X ‾ ) finite and we show that the iterated set coincides with the original algebraic etale-Brauer set. This provides some evidence towards the conjectures by Colliot-Thelene on the arithmetic of rational points on nice geometrically rationally connected varieties over k and by Skorobogatov on the arithmetic of rational points on K3 surfaces over k; moreover, it gives a partial answer to an “algebraic” analogue of a question by Poonen about iterating the descent set.