Quasi-$F^{\infty}$-split height versus quasi-$F$-regular height for rational double points and graded rings
Teppei Takamatsu, Shou Yoshikawa
TL;DR
The paper investigates when finiteness of the quasi-F^∞-split height ht^∞ forces quasi-F-regularity, and whether ht^∞ coincides with ht^{reg}. It provides a complete computation of ht^e, ht^∞, and ht^{reg} for rational double points, showing ht^∞ = ht^{reg} for all non-F-pure RDPs, and establishes a graded-case result: for certain graded non-F-pure Gorenstein rings with F-rational punctured spectrum, ht^∞(R) = ht^{reg}(R). The work relies on Fedder-type criteria for quasi-F^e-splitting and quasi-F-regularity, including explicit D1–D3 conditions and test-ideal considerations, to derive precise height formulas across RDPs and to confirm the graded-case equality. These results advance understanding of quasi-F-singularities by linking finiteness of ht^∞ to quasi-F-regularity in key singularity classes and by providing explicit height data for RDPs and graded localizations.
Abstract
In this paper, we study a phenomenon concerning quasi-$F$-singularities: under suitable hypotheses, the finiteness of the quasi-$F^{\infty}$-split height ($\mathrm{ht}^{\infty}$) implies quasi-$F$-regularity, and moreover, $\mathrm{ht}^{\infty}$ coincides with the quasi-$F$-regular height ($\mathrm{ht}^{\mathrm{reg}}$). We establish this coincidence for two important classes of isolated Gorenstein singularities. First, we explicitly compute $\mathrm{ht}^{\infty}$ and $\mathrm{ht}^{\mathrm{reg}}$ for all rational double points, showing that every non-$F$-pure rational double point satisfies $\mathrm{ht}^\infty = \mathrm{ht}^{\mathrm{reg}}$. Second, for localizations of graded non-$F$-pure normal Gorenstein rings with $F$-rational punctured spectrum, we again obtain the equality $\mathrm{ht}^\infty = \mathrm{ht}^{\mathrm{reg}}$.