Quasi-$F^{\infty}$-split height versus quasi-$F$-regular height for rational double points and graded rings

TL;DR

This paper investigates the relationship between quasi-$F$-singularities, showing that under certain conditions, the finiteness of the quasi-$F^{ abla}$-split height implies quasi-$F$-regularity, and these heights coincide for specific Gorenstein singularities.

Contribution

It establishes the equality of quasi-$F^{ abla}$-split height and quasi-$F$-regular height for rational double points and certain graded rings, advancing understanding of $F$-singularity invariants.

Findings

For rational double points, $ ext{ht}^ abla = ext{ht}^{ ext{reg}}$ for non-$F$-pure cases.

In graded Gorenstein rings with $F$-rational punctured spectrum, the heights coincide.

Finiteness of $ ext{ht}^ abla$ implies quasi-$F$-regularity under suitable conditions.

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 $\mathrm{ht}^\infty = \mathrm{ht}^{\mathrm{reg}}$.