Computation method for perfectoid purity and perfectoid BCM-regularity

Abstract

In this paper, we introduce the notion of quasi-$F$-splitting for rings in mixed characteristic. By comparing quasi-$F$-splitting with perfectoid purity, we obtain a new inversion of adjunction-type result. Furthermore, we study the possible values of the perfectoid pure threshold of $\mathrm{div}(p)$ and construct new examples of perfectoid pure rings.

43 pages, v7:Revided the proof of Proposition 6.8