Abstract: In algebraic geometry, purity refers to a diverse range of phenomena in which certain invariants or categories associated to geometric objects are insensitive to the removal of closed subsets of large codimensions. In this talk, we present Zariski—Nagata purity concerning finite étale covers on smooth schemes over Prüfer rings by proving Auslander’s flatness criterion in this non-Noetherian context. By Gabber—Ramero’s upper bound of projective dimensions, we present an Auslander—Buchsbaum formula. Finally, we take advantage of these cohomological results to establish the parafactoriality and the purity of torsors over Prüfer bases. This is a joint work with Fei Liu.