This talk presents recent results on purity theorems over valuation rings. We begin by investigating the Hartogs extension phenomenon for smooth algebras over valuation rings. We then discuss Zariski-Nagata purity, focusing on finite étale covers.
To support these geometric results, we review the homological characterization of normal coherent rings, such as Serre's $(R_1)$ and $(S_2)$ conditions, can be formulated using homological invariants like depth and weak dimension in the coherent, non-Noetherian setting.
Finally, we analyze cohomological purity, specifically establishing the purity of Brauer groups for syntomic algebras over valuation rings by using perfectoid techniques of Cesnavicius-Scholze. | 
