On the functoriality of refined unramified cohomology

In this paper, we generalise the construction of the functorial pullback of refined unramified cohomology between smooth schemes, by following the ideas of Fulton's intersection theory and Rost's cycle modules. We also define standard actions of algebraic cycles on the refined unramified cohomology groups of smooth proper schemes avoiding Chow's moving lemma, which coincide with Schreieder's constructions for smooth projective schemes. As applications, we prove the projective bundle and blow-up formulas for refined unramified cohomology groups and we reduce the Rost nilpotence principle in characteristic zero to a statement concerning certain refined unramified cohomology groups. Moreover, we compute the refined unramified cohomology for smooth proper linear varieties and show that Rost's nilpotence principle holds for these varieties in characteristic zero.