A Criterion for Perfectoid Purity and the Rationality of Thresholds
Shou Yoshikawa
TL;DR
The paper develops a splitting-order sequence as a numerical invariant to detect perfectoid purity for hypersurfaces in unramified regular local rings, enabling an explicit computation of the perfectoid-pure threshold via $\mathrm{ppt}(A/f,p)=\sum_{i\ge1} \frac{p-1-s_i}{p^i}$ when all $s_i\le p-1$. It shows that $\mathrm{ppt}(R,p)$ is always rational for regular local rings and extends the framework to graded and Calabi–Yau cone settings, yielding new examples of perfectoid-pure singularities for large primes. A computational method akin to Fedder’s criterion is developed to determine splitting-order sequences, linking the $p$-adic threshold to the classical $F$-pure threshold in the regular case. These results provide a practical, algebraic toolkit for testing perfectoid purity, with broad geometric applications and a clear pathway to explicit threshold calculations.
Abstract
We introduce a new criterion providing a sufficient condition for a hypersurface in an unramified regular local ring to be perfectoid pure. The criterion is formulated in terms of an explicitly computable sequence of integers, called the splitting-order sequence. Our main theorem shows that if all entries of the sequence are at most $p-1$, then the hypersurface is perfectoid pure, and the perfectoid-pure threshold can be computed explicitly from it. As a consequence, we prove that for any regular local ring $R$, the perfectoid pure threshold $\mathrm{ppt}(R,p)$ with respect to $p$ is always a rational number. Moreover, we show that for sufficiently large primes $p$, the cone over a Fermat type Calabi-Yau hypersurface is perfectoid pure, revealing new and unexpected examples of perfectoid pure singularities. Moreover, we show that for sufficiently large primes $p$, the cone over a Fermat type Calabi-Yau hypersurface is perfectoid pure, revealing new and unexpected examples of perfectoid pure singularities.