Refined unramified cohomology and algebraic cycles

We introduce refined unramified cohomology groups, explain their relation to classical unramified cohomology, and prove some general comparison theorems to certain cycle groups. This generalizes and simplifies work of Bloch–Ogus, Colliot-Thelene–Voisin, Voisin, and Ma, who dealt with cycles of low (co-)dimension. Our approach has several applications. For instance, it allows to construct the first example of a variety whose Griffiths group has infinite torsion subgroup.