Purity and quasi-split torsors over Pr\"ufer bases

We establish an analogue of the Zariski--Nagata purity theorem for finite \'etale covers on smooth schemes over Pr\"ufer rings by demonstrating Auslander's flatness criterion in this non-Noetherian context. We derive an Auslander--Buchsbaum formula for general local rings, which provides a useful tool for studying the algebraic structures involved in our work. Through analysis of reflexive sheaves, we prove various purity theorems for torsors under certain group algebraic spaces, such as the reductive ones. Specifically, using results from EGAIV4 on parafactoriality on smooth schemes over normal bases, we prove the purity for cohomology groups of multiplicative type groups at this level of generality. Subsequently, we leverage the aforementioned purity results to resolve the Grothendieck--Serre conjecture for torsors under a quasi-split reductive group scheme over schemes smooth over Pr\"ufer rings. Along the way, we also prove a version of the Nisnevich purity conjecture for quasi-split reductive group schemes in our Pr\"uferian context, inspired by the recent work of Cesnavicius.