For a natural class of cohomology theories with support (including étale or pro-étale cohomology with suitable coefficients), we prove a moving lemma for cohomology classes with support on smooth quasi-projective k-varieties that admit a smooth projective compactification (e.g. if char(k)=0). This has the following consequences for such k-varieties and cohomology theories: a local and global generalization of the effacement theorem of Quillen, Bloch--Ogus, and Gabber, a finite level version of the Gersten conjecture in characteristic zero, and a generalization of the injectivity property and the codimension 1 purity theorem for étale cohomology. Our results imply that the refined unramified cohomology groups from [Sch23] are motivic.

Comment: 50 pages, to appear in EPIGA (special volume in honour of Claire Voisin)

• 14C15 - (Equivariant) Chow groups and rings; motives
• 14C25 - Algebraic cycles
• 14F20 - Étale and other Grothendieck topologies and (co)homologies