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

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

arXiv comment19 pages