Independence of perfectoid pure thresholds from the chosen pure perfectoid algebra

Determine whether, for a mixed-characteristic Noetherian local perfectoid-pure ring R and an ideal a, the perfectoid pure threshold computed with any pure perfectoid R-algebra B satisfies $\ppt(a)=\ppt(a;B)$.

Background

The paper defines the global perfectoid pure threshold $\ppt(a)$ as a supremum over suitable perfectoid R-algebras, while $\ppt(a;B)$ is the threshold computed relative to one chosen perfectoid algebra B.

The authors show that different choices of B can produce different relative thresholds when the maps from R are not both pure. They explicitly leave open whether this dependence persists when both R-to-B maps are pure. The paper later proves equality in positive characteristic and establishes a partial mixed-characteristic result for complete regular local rings, but does not resolve the general mixed-characteristic question.

References

On the other hand, we do not know whether it is possible that $\ppt(a; B_1) \neq \ppt(a; B_2)$ when $R\to B_1$ and $R\to B_2$ are both pure.

— Perfectoid Pure and Signature Pairs  (2610.22059 - Cai et al., 15 Oct 2026) in Section 13, subsection “Perfectoid pure thresholds,” Remark following the definition of $\ppt(a;\Delta)$

As we mentioned, we do not know a good analog of \autoref{prop: recover char p} in mixed characteristic.

\begin{question} Let $(R, m)$ be a Noetherian local ring of mixed characteristic $(0,p>0)$ that is perfectoid pure. Then is $\ppt(a) = \ppt (a; B)$ whenever $R \to B$ is pure and $B$ is perfectoid? \end{question}

— Perfectoid Pure and Signature Pairs  (2610.22059 - Cai et al., 15 Oct 2026) in Section 13, subsection “Localization and regular extensions,” immediately before the proposition on complete regular local rings